diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzliyj" "b/data_all_eng_slimpj/shuffled/split2/finalzzliyj" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzliyj" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec:introduction}\n\nPhase transitions in two-dimensional (2D) fermionic\nsystems are a central topic of theoretical and experimental condensed matter\nphysics. Correlated quasi-2D materials with rich phase diagrams include\nhigh-temperature superconductors \\cite{RevModPhys.78.17} and transition-metal\ndichalcogenides \\cite{manzeli20172d}. Dirac fermions in two dimensions can be\ninvestigated in graphene \\cite{Neto_rev}. \nStrongly correlated 2D fermions exhibit exotic phases\n\\cite{RevModPhys.89.025003} and phase transitions\n\\cite{Senthil04_2}, and can support long-range order\nat $T>0$ \\cite{PhysRevLett.17.1133}. While magnetism originates\nfrom short-range Coulomb repulsion, the main mechanism behind the numerous\ncharge-density-wave (CDW) phases found experimentally is electron-phonon\ncoupling. In addition to polaron effects, the latter leads to\na phonon-mediated, retarded electron-electron interaction and an intricate\ninterplay of spin, charge, and lattice fluctuations. \n\nQuantum Monte Carlo (QMC) simulations are a key tool to investigate \ncorrelated 2D quantum systems. Although simulations\nare significantly harder for fermions than for spins or bosons, QMC methods have been very\nsuccessfully applied to fermionic models. However, whereas the phase diagram\nand critical behavior of, \\eg, the 2D honeycomb Hubbard model is known in detail\n\\cite{Sorella12,Assaad13,Toldin14,Otsuka16}, the same is not true even for\nthe simplest Holstein molecular-crystal model of electron-phonon\ninteraction. Most notably, simulations with phonons are often severely restricted\nby long autocorrelation times also away from critical points \\cite{Hohenadler2008}. \nCurrently, reliable critical temperatures, convincing analysis of critical\nbehavior, and the ground-state phase diagram remain key open problems. In fact, even the\nsimpler 1D case had until recently been discussed controversially\n\\cite{MHHF2017}, with earlier claims of dominant pairing correlations refuted\nby direct calculations of the correlation functions and traced back to spin\ngap formation \\cite{PhysRevB.92.245132}.\n\nHere, we use large-scale continuous-time QMC simulations to investigate\nthe CDW transition in the 2D Holstein-Hubbard model. Although the latter has\nbeen extensively studied in the past, important open questions remain.\nAt strong coupling and half-filling, the ground state is either a CDW\ninsulator or an antiferromagnetic Mott insulator. Recent variational QMC studies\n\\cite{ohgoe2017competitions,1709.00278} argue in\nfavor of a third phase (metallic or superconducting), seemingly in\ncontradiction with theoretical arguments based on weak-coupling\ninstabilities of the Fermi liquid \\cite{PhysRevLett.56.2732,PhysRevB.42.2416}. We use finite-size\nscaling to determine $T_c$ of the CDW transition, show that the latter can\nalso be detected by the fidelity susceptibility, and provide evidence for its\nIsing critical behavior. Moreover, we present arguments and data for the\nexistence of a metallic bipolaron phase at $T>T_c$ and address the possibility of a\nmetallic or a superconducting ground state.\n\nThe paper is organized as follows. Section~\\ref{sec:model} introduces the\nrelevant models, Sec.~\\ref{sec:methods} gives a brief review of the numerical\nmethods, Sec.~\\ref{sec:results} discusses the results, and Sec.~\\ref{sec:conclusions} \ncontains our conclusions.\n\n\\section{Models}\\label{sec:model}\n\nThe Holstein-Hubbard Hamiltonian \\cite{HOLSTEIN1959325} reads\n\\begin{align}\\label{eq:model}\\nonumber\n \\hat{H}\n =\n &-t \\sum_{\\las i,j\\ras \\sigma} \\hat{c}^\\dag_{i\\sigma} \\hat{c}^{\\phantom{\\dag}}_{j\\sigma} \n +\n \\sum_{i}\n \\left[\n \\mbox{$\\frac{1}{2M}$} \\hat{P}^2_{i}\n +\n \\mbox{$\\frac{K}{2}$} \\hat{Q}_{i}^2\n \\right]\n \\\\\n &-\n g\n \\sum_{i} \\hat{Q}_{i} \n \\hat{\\rho}_i\n + U \n \\sum_i (\\hat{n}_{i\\UP}-\\mbox{$\\frac{1}{2}$}) (\\hat{n}_{i\\DO}-\\mbox{$\\frac{1}{2}$})\n \\,.\n\\end{align}\nThe first two terms describe free electrons and free phonons, respectively.\nHere, $\\hat{c}^\\dag_{i\\sigma}$ creates an electron with spin $\\sigma$ at lattice\nsite $i$ and electrons hop with amplitude $t$ between nearest-neighbor\nsites on a square lattice. The phonons are of the Einstein type with frequency\n$\\omega_0=\\sqrt{K\/M}$; their displacements $\\hat{Q}_i$ couple to local\nfluctuations $\\hat{\\rho}_i=\\hat{n}_i-1$ of the electron occupation $\\hat{n}_{i} =\n\\sum_\\sigma \\hat{n}_{i\\sigma}$ where $\\hat{n}_{i\\sigma} = \\hat{c}^\\dag_{i\\sigma}\\hat{c}^{\\phantom{\\dag}}_{i\\sigma}$. The\nlast term describes a Hubbard onsite repulsion of strength $U$.\nWe simulated $L\\times L$ lattices with periodic boundary conditions at\nhalf-filling ($\\las\\hat{n}_{i}\\ras=1$, chemical potential $\\mu=0$). A\nuseful dimensionless coupling parameter is $\\lambda=g^2\/(W K)$ with the free\nbandwidth $W=8t$. We set $\\hbar$, $k_\\text{B}$, and the lattice constant to one and\nuse $t$ as the energy unit.\n\nFor $U=0$, Eq.~(\\ref{eq:model}) reduces to the Holstein model. Its\nrelative simplicity has motivated numerous QMC investigations of CDW\nformation and superconductivity \\cite{%\nPhysRevB.40.197,PhysRevB.42.2416,PhysRevB.42.4143,PhysRevLett.66.778,PhysRevB.43.10413,PhysRevB.46.271,PhysRevB.48.7643,PhysRevB.48.16011,PhysRevB.55.3803}. Equation~(\\ref{eq:model})\nwith $g=0$ corresponds to the repulsive Hubbard model on the square lattice. At half-filling, the ground state of the latter is an\nantiferromagnetic Mott insulator for any $U>0$ \\cite{Hirsch89}. However, in contrast to \nCDW order, antiferromagnetism is restricted to $T=0$ in two dimensions by the\nMermin-Wagner theorem \\cite{PhysRevLett.17.1133}.\nThe full Holstein-Hubbard Hamiltonian~(\\ref{eq:model}) captures the\ncompetition between Mott and CDW ground states\n\\cite{PhysRevB.52.4806,PhysRevLett.75.2570,PhysRevB.75.014503,PhysRevB.92.195102,PhysRevLett.109.246404,PhysRevB.87.235133,ohgoe2017competitions}.\nWhereas early work unanimously agreed on the absence of a disordered\nor a superconducting ground state at half-filling, such a phase has recently\nbeen advocated by numerical results \\cite{ohgoe2017competitions,1709.00278}. \n\nBecause it is sufficient to address many of the open questions of interest,\nwe will mainly consider the case $U=0$. However, selected results for\nthe impact of the Hubbard repulsion will also be reported. For\nEq.~(\\ref{eq:model}) with $U=0$, mean-field theory (exact for $\\omega_0=0$ and\n$T=0$) predicts a CDW ground state with a checkerboard pattern for\nthe lattice displacements and the charge density [ordering vector ${\\bm\n Q}=(\\pi,\\pi)$, see inset of Fig.~\\ref{fig:phasediagram}] \nat half-filling \\cite{PhysRevB.40.197,PhysRevB.42.2416,PhysRevLett.66.778}.\nHere, we systematically explore the impact of quantum and thermal fluctuations.\n\nAn important limiting case is the antiadiabatic limit $\\omega_0\\to\\infty$, in\nwhich the Holstein-Hubbard model maps to a Hubbard model with Hamiltonian\n\\begin{align}\\label{eq:model2\n \\hat{H}\n &=\n -t \\sum_{\\las i,j\\ras \\sigma} \\hat{c}^\\dag_{i\\sigma} \\hat{c}^{\\phantom{\\dag}}_{j\\sigma} \n +\n U_\\infty \n \\sum_{i} (\\hat{n}_{i\\UP}-\\mbox{$\\frac{1}{2}$}) (\\hat{n}_{i\\DO}-\\mbox{$\\frac{1}{2}$})\n\\end{align}\nand effective interaction $U_\\infty=U-\\lambda W$. For $U=0$, interactions are\npurely attractive and give rise to coexisting CDW and superconducting order\nfor any $\\lambda>0$ at $T=0$. However, at half-filling, this order is minimal\nin the sense that $T_c=0$ \\cite{Hirsch85}, which is related to a perfect\ndegeneracy of CDW and pairing correlations and an associated continuous SO(3)\norder parameter for which the Mermin-Wagner theorem applies \\cite{PhysRevLett.17.1133}.\n\n\\section{Methods}\\label{sec:methods}\n\nExtending previous applications to 1D electron-phonon models \\cite{Ho.As.Fe.12,PhysRevB87.075149,PhysRevLett.117.206404}, we use the\ncontinuous-time interaction expansion (CT-INT) method \\cite{Rubtsov05}. To\nthis end, we express the partition function as a functional integral\n\\begin{align} \\label{partitionfunction}\n Z = \\int \\mathcal{D}(\\bar{c},c) \\ e^{-S_0\\left[\\bar{c},c\\right]-S_1\\left[\\bar{c},c\\right]} \\int \\mathcal{D}(\\bar{b},b) \\ e^{-S_\\text{ep}\\left[\\bar{c},c,\\bar{b},b\\right]}\n\\end{align}\nusing coherent states.\nSplitting the action into the free-fermion part $S_0$, the Hubbard interaction $S_1$, \nand the remainder $S_\\text{ep}$ that contains the free-phonon contribution\nand the electron-phonon coupling, the phonons are integrated out analytically\nto arrive at a fermionic model with both an instantaneous Hubbard interaction ($S_1$)\nand a retarded, phonon-mediated interaction ($S_2$) \\cite{Assaad07}. This model can\nbe simulated by the CT-INT method by sampling both types of vertices\n\\cite{Assaad07} to stochastically sum the weak-coupling Dyson expansion\n\\cite{Rubtsov05} around $S_0$. Because the latter converges for fermionic systems in\na finite spacetime volume, CT-INT is exact apart from statistical errors.\nTechnical reviews can be found in Refs.~\\cite{Gull_rev,Assaad14_rev}.\n\nIn contrast to the determinant QMC (DetQMC) method \\cite{Blankenbecler81}\nused in almost all previous works on Holstein-Hubbard-type models, CT-INT\nhas significantly smaller autocorrelation times \\cite{Hohenadler2008}.\nCT-INT simulation times scale as ${O}(n^3)$, where $n$\n[$\\approx {O}(\\lambda\\beta L^2)$ for $U=0$] is the average expansion\norder and $\\beta=1\/T$. Although DetQMC formally has a better\n${O}(\\beta L^6)$ scaling, CT-INT benefits from\nreduced expansion orders at weak coupling and seems to outperform DetQMC for\nmost parameters considered despite being limited for $\\omega_0\\gtrsim t$ by a\nsign problem. Whereas even the noninteracting case is challenging for DetQMC, CT-INT\ntrivially gives exact results for $\\lambda=0$ and can in principle\nsimulate the entire range of phonon frequencies, including the\nexperimentally important adiabatic regime $\\omega_00$ (the focus of this work) remain the same. \n\n\\begin{figure}[b]\n \n \\includegraphics[width=0.45\\textwidth]{fig2.pdf} \n \\caption{\\label{fig:criticalpoint} (a) Determination of the critical value $\\lambda_c$ \n from (a) the crossing of the correlation ratios $R_\\text{c}$ for\n different system sizes $L$ and (b) the maximum in the fidelity\n susceptibility. Here, $\\omega_0\/t=0.1$, $U=0$, and (a) $T\/t=0.05$, (b)\n $T\/t=0.2$. \n }\n\\end{figure}\n\n\\subsection{Critical values}\\label{sec:results:Tc}\n\nTo obtain the critical values shown in\nFig.~\\ref{fig:phasediagram}, we calculated the correlation ratio\n\\cite{Binder1981}\n\\begin{equation}\\label{eq:rcdw}\n R_\\text{c} =\n 1-\\frac{S_\\text{c}(\\bm{Q}-\\delta{\\bm q})}{S_\\text{c}(\\bm{Q})}\n\\end{equation}\n(with $|\\delta{\\bm q}|=2\\pi\/L$) from the charge structure factor\n\\begin{equation}\\label{eq:scdw}\nS_\\text{c}({\\bm q}) = \\frac{1}{L^2}\\sum_{ij}\ne^{\\text{i}(\\bm{r}_i-\\bm{r}_j)\\cdot\\bm{q}}\\,\\las \\hat{n}_i \\hat{n}_j\\ras\n\\end{equation}\neither at fixed $\\lambda$ or at fixed $T$. Here, ${\\bm Q}=(\\pi,\\pi)$. \nBy definition, a divergence of $S_\\text{c}(\\bm{Q})$ with $L$ in the CDW phase\nimplies $R_\\text{c}\\to 1$ for $L\\to\\infty$, whereas $R_\\text{c}\\to 0$ in\nthe absence of long-range CDW order. Moreover, because $R_\\text{c}$ is\na renormalization group invariant \\cite{Binder1981}, the critical point can be\nestimated from the crossing of curves for different $L$, as illustrated in\nFig.~\\ref{fig:criticalpoint}(a) for $\\omega_0\/t=0.1$ and $T\/t=0.05$. \nWhile the correlation ratio~(\\ref{eq:rcdw}) is expected to exhibit smaller\nfinite-size corrections than the structure factor~(\\ref{eq:scdw}), a shift\nof consecutive crossing points is observed on the accessible system sizes,\nmaking it necessary to extrapolate to\n$L=\\infty$. To this end, we used a fit function\n\\begin{equation}\\label{eq:fitfunction}\nf(L) = a + b L^c\\,.\n\\end{equation}\nExamples for such extrapolations are shown for $\\om_0\/t=0$ in Fig.~\\ref{fig:extrapolation}(a)\nand for $\\om_0\/t=0.1$ in Fig.~\\ref{fig:extrapolation}(b). For classical\nphonons, we can access significantly larger system sizes up to $L=28$. The\npoints in Fig.~\\ref{fig:extrapolation}(a) correspond to crossing points of\n$R_\\text{c}$ for $L$, $L-2$ (\\ie, $\\Delta L=2$) and $L$, $L-4$ ($\\Delta\nL=4$), respectively. Fitting to Eq.~(\\ref{eq:fitfunction}), these two choices\nyield identical results for $T_c$ within error bars. The errors take into\naccount the statistical errors of the QMC results as well as the errors in\ndetermining the crossing points using parabolic fits (obtained from a\nbootstrap analysis) and extrapolating to $L=\\infty$. They are smaller than\nthe symbol size in Fig.~\\ref{fig:phasediagram} but naturally do not capture\npossible variations due to the choice of fit function or observable. \nFor quantum phonons, we systematically used $L=4,6,8,10,12$ and hence $\\Delta L=2$, as\nillustrated in Fig.~\\ref{fig:extrapolation}(b). A similar extrapolation gives\n$\\lambda_c=0.101(1)$ for the parameters of Fig.~\\ref{fig:criticalpoint}(a).\n\n\\begin{figure}[t]\n \n \\includegraphics[width=0.45\\textwidth]{fig3.pdf}\n \\caption{\\label{fig:extrapolation} \n Finite-size extrapolation of the crossing points of $R_\\text{c}(L)$,\n $R_\\text{c}(L-\\Delta L)$ using the fit function~(\\ref{eq:fitfunction}). \n Here, (a) $\\omega_0=0$, $\\lambda=0.1$, $T_c=0.0506(1)$ and (b)\n $\\omega_0\/t=0.1$, $T\/t = 0.2$, $\\lambda_c=0.244(1)$.\n }\n\\end{figure}\n\nThe phase transition can also be detected using the fidelity susceptibility\n$\\chi_{F}$ \\cite{2008arXiv0811.3127G}, an unbiased diagnostic to\ndetect critical points without any knowledge about the order parameter.\nIt essentially relies on calculating the overlap of the ground states of (in\nthe present case) Holstein \nHamiltonians with couplings $\\lambda$ and $\\lambda+\\delta\\lambda$. A\nfinite-temperature generalization has been given in Refs.~\\cite{PhysRevE.76.022101,PhysRevLett.103.170501, PhysRevB.81.064418}\nand CT-INT estimators in Refs.~\\cite{PhysRevX.5.031007,PhysRevB.94.245138}.\nAlthough these estimators have rather large statistical errors at low\ntemperatures, $\\chi_{F}\/L^2$ for $T\/t=0.20$ in\nFig.~\\ref{fig:criticalpoint}(b) shows the expected peak at a position that is\nconsistent with Fig.~\\ref{fig:phasediagram} and\n$\\lambda_c=0.244(1)$ from Fig.~\\ref{fig:extrapolation}(b).\n\nFigure~\\ref{fig:phasediagram} shows $T_c(\\lambda)$ for\ndifferent $\\omega_0$, covering the entire adiabatic regime $0\\leq\n\\omega_0 \\leq t$. The mean-field result $T_c\\sim e^{-1\/\\sqrt{\\lambda}}$ for\nthe 2D Holstein model---compared to $T_c\\sim e^{-1\/\\lambda}$ in dynamical mean-field\ntheory (DMFT) \\cite{PhysRevB.63.115114}---is expected to overestimate\n$T_c$ even at $\\omega_0=0$ and does not capture the expected maximum at\n$\\lambda<\\infty$ \\cite{PhysRevB.63.115114}. The latter is outside\nthe range of couplings considered here. Quantum lattice fluctuations\nsuppress $T_c$ at a given $\\lambda$. For $\\omega_0\/t=0.1$, $T_c$ shows only\nminor deviations from the result for classical phonons, whereas for larger $\\omega_0$ \nquantum fluctuation effects are clearly visible over the entire parameter range shown.\n The systematic suppression of $T_c$ with\nincreasing $\\omega_0$ is perfectly consistent with the fact that $T_c=0$ for the\nattractive Hubbard model \\cite{Hirsch85}, to which the Holstein model maps in the limit\n$\\omega_0\\to\\infty$ \\cite{Hirsch83a}. This connection and a possible metallic phase at low temperatures\nas a result of quantum fluctuations will be discussed below. At $T>0$, a\nmetallic region is naturally expected in the phase diagram of the 2D\nHolstein-Hubbard model because the antiferromagnetic Mott state arising from\nthe Hubbard interaction is confined to $T=0$. In contrast to previous DMFT results\n\\cite{PhysRevB.63.115114}, the critical temperatures in\nFig.~\\ref{fig:phasediagram} were obtained by taking into account all (spatial\nand temporal) fluctuations on the square lattice. \n\n\\begin{figure}[t]\n \n \\includegraphics[width=0.45\\textwidth]{fig4.pdf} \n \\caption{\\label{fig:TcvsU} Critical temperature of the CDW transition in\n the Holstein-Hubbard model. (a) Suppression of $T_c$ with increasing $U$\n at $\\lambda=0.25$ from finite-size scaling, (b) comparison of Holstein\n and Holstein-Hubbard results in terms of the effective coupling\n $\\lambda_\\text{eff}=\\lambda-U\/W$. The points labeled `Holstein'\n correspond to $\\lambda_c$ at different temperatures from Fig.~\\ref{fig:phasediagram}. The points\n labeled `Hol-Hub' (Holstein-Hubbard) are for $T_c$ at $\\lambda=0.25$ and\n $U\/t=0,0.25,0.50$ from (a). Here, $\\omega_0\/t=0.1$.}\n\\end{figure}\n\nThe Hubbard repulsion suppresses CDW order\n\\cite{PhysRevB.52.4806,PhysRevLett.75.2570,PhysRevB.75.014503,PhysRevB.92.195102,PhysRevLett.109.246404,PhysRevB.87.235133,ohgoe2017competitions}.\nThis is already apparent from the effective Hubbard model~(\\ref{eq:model2}) in the limit\n$\\omega_0\\to\\infty$ where a nonzero $U$ reduces the effective, attractive\ninteraction and thereby the CDW gap at $T=0$. Whereas CDW order is restricted\nto $T=0$ in this limit, here we consider the Holstein-Hubbard model in the\nopposite, adiabatic regime. Specifically, we take $\\omega_0\/t=0.1$ and $\\lambda=0.25$.\n\nTo quantify the effect of $U$, we show in Fig.~\\ref{fig:TcvsU}(a) the\nsuppression of $T_c$ as a function of $U$. Starting from\n$T_c\/t=0.204(1)$ at $U=0$, $T_c$ decreases by about 15 percent in the range\n$U\\in[0,0.5t]$. In principle, in the spirit of an effective Holstein model,\nwe can try to capture this effect by a coupling $\\lambda_\\text{eff}=\\lambda-U\/W$.\nHowever, Fig.~\\ref{fig:TcvsU}(b) reveals that for the parameters considered\nthis overestimates the effect of the Hubbard repulsion because $T_c$ at a\ngiven $\\lambda_\\text{eff}$ in the Holstein model ($U=0$) is significantly\nlower than in the Holstein-Hubbard model ($U>0$). We attribute this finding\nto (i) the stronger suppression of the antiferromagnetic\ncorrelations (long-range magnetic order only exists at $T=0$) compared to the\nCDW correlations (CDW order exists also at $T>0$) at the temperatures considered,\nand (ii) retardation effects. A DMFT analysis of the Holstein-Hubbard model revealed that $T_c$ is suppressed with increasing $U$\nat weak electron-phonon coupling but initially enhanced at strong\ncoupling. This behavior was explained in terms of a reduction of the\nbipolaron mass due to the onsite repulsion \\cite{PhysRevLett.75.2570}.\n\n\\begin{figure}[b]\n \n \\includegraphics[width=0.45\\textwidth]{fig5.pdf} \n \\caption{\\label{fig:quising} Scaling collapse of (a) the structure\n factor and (b) the correlation ratio for $\\omega_0\/t=0.1$,\n $\\lambda=0.25$, and $U=0$ using the critical exponents of the 2D Ising\n model. The critical temperatures $T_c$ were determined from the best\n scaling collapse and are given in the text.\n}\n\\end{figure}\n\n\\subsection{Critical behavior}\\label{sec:results:critical}\n\nIn the thermodynamic limit, the long-range CDW\norder at $T0$. \n\nThere are two well-understood limits. The {\\it classical} Holstein\nmodel ($\\omega_0=0$) has a CDW ground state for any $\\lambda>0$ and\n$T_c>0$ (see Sec.~\\ref{sec:results:Tc}). This follows from mean-field theory,\nwhich becomes exact at $T=0$. In the opposite, antiadiabatic limit\n$\\omega_0\\to\\infty$, the Holstein model maps to the attractive Hubbard model,\nwhose ground state has coexisting CDW and superconducting order but $T_c=0$. \nHence, as a function of $\\omega_0$, the Holstein model interpolates between\ntwo limits that both exhibit long-range CDW order at $T=0$. \n\nBetween these limiting cases (\\ie, for $0<\\omega_0<\\infty$), there appear to be \ntwo distinct scenarios for the shape of the phase boundary $T_c(\\lambda)$, as\nillustrated in Fig.~\\ref{fig:tcschematic}. In\nscenario (I), $T_c>0$ for any $\\lambda>0$, so that the ground state is always\na CDW insulator. By contrast, in scenario (II), $T_c=0$ for\n$\\lambda<\\lambda_c(\\omega_0)$ and $T_c>0$ for $\\lambda>\\lambda_c(\\omega_0)$. \nCase (II) can further be divided into (IIa) where CDW order exists at $T=0$\nfor any $\\lambda$, and (IIb) with a disordered phase at $T=0$ below\n$\\lambda_c(\\omega_0)$. In scenario (I), the adiabatic (classical) fixed\npoint determines the behavior for any finite $\\omega_0$. On the other hand,\nin scenario (IIa), the physics is determined by the antiadiabatic fixed point\nfor $\\lambda<\\lambda_c(\\omega_0)$ and by the adiabatic fixed point for\n$\\lambda>\\lambda_c(\\omega_0)$. Note that CDW order with $T_c=0$ requires an\nemergent continuous order parameter, as realized for the attractive Hubbard\nmodel ($\\omega_0=\\infty$). However, the corresponding symmetry is broken for\n$\\omega_0<\\infty$ by retardation effects in the Holstein model \\cite{Hirsch83a}.\n\n\\begin{figure}[t]\n \\includegraphics[width=0.25\\textwidth]{fig6.pdf}\n \\caption{\\label{fig:tcschematic} The two possible scenarios for the phase\n diagram of the Holstein model. In scenario (I), we have CDW order with\n $T_c>0$ for any $\\lambda>0$. In scenario (II), $T_c=0$ for $\\lambda<\\lambda_c(\\omega_0)$.}\n\\end{figure}\n\nA CDW ground state for any $\\lambda>0$ may be expected based on the\ninstability of the Fermi liquid. For the half-filled square lattice with\nnearest-neighbor hopping, the noninteracting charge susceptibility\n$\\chi^{(0)}_\\text{c}({\\bm Q})\\sim\\ln^2\\beta t$ due to the\ncombined effect of nesting and Van Hove singularities\n\\cite{PhysRevLett.56.2732,PhysRevB.42.2416}. In the Hubbard model, such\ndivergences underlie the existence of an antiferromagnetic Mott insulator\nfor any $U>0$, and coexisting CDW and superconducting order for any $U<0$ \\cite{Hirsch85}.\nFor the Holstein model that does not have a symmetry-imposed degeneracy of CDW and pairing\ncorrelations, superconducting correlations were found \nto be weaker than CDW correlations at half-filling \\cite{PhysRevB.42.2416},\nconsistent with the weaker divergence of the $\\bm{Q}=0$ pairing susceptibility\n$\\chi^{(0)}_\\text{p}({\\bm Q})\\sim\\ln\\beta t$.\n\nDespite these theoretical arguments, metallic and superconducting ground\nstates were recently suggested for the half-filled Holstein and\nHolstein-Hubbard models based on variational QMC simulations\n\\cite{ohgoe2017competitions,1709.00278}. A metallic phase is also found\nwithin DMFT \\cite{Koller04,PhysRevB.70.125114,PhysRevB.88.125126}, where a Van\nHove singularity is absent. For $\\omega_0\\ll t$, the results of\nFig.~\\ref{fig:phasediagram} appear consistent with CDW\norder even at $T=0$ for any $\\lambda>0$. On the other hand, the phase\nboundary $T_c(\\lambda)$ in Fig.~\\ref{fig:phasediagram} undergoes an\nincreasingly strong shift to larger $\\lambda$ with increasing\n$\\omega_0$, in principle compatible with $T_c=0$ at sufficiently weak\ncoupling [scenario (II)]. In the significantly better understood 1D case,\nnumerical results show that for $\\omega_0>0$ the ground state remains metallic\nfor $\\lambda<\\lambda_c$ despite a $\\ln \\beta t$ nesting-related\ndivergence of the charge susceptibility \\cite{MHHF2017}. Since $T_c=0$ in the\n1D case, this corresponds to scenario (IIb) above and is consistent with the\n$\\omega_0=\\infty$ limit, the 1D attractive Hubbard model. The latter has a\nmetallic but spin-gapped Luther-Emery liquid \\cite{Lu.Em.74} ground state and\nno long-range order. Functional renormalization group calculations for the\n2D Holstein-Hubbard model exclude metallic or superconducting behavior at\nhalf-filling except for an extremely small region where $T_c$ is essentially\nzero \\cite{PhysRevB.92.195102}.\n\nTo address the ground-state phase diagram directly, we calculated the\ncorrelation ratios\n\\begin{align}\\label{eq:Rchic}\n R^\\chi_\\text{c} \n &= 1-\\frac{\\chi_\\text{c}(\\bm{Q}-\\delta{\\bm\n q})}{\\chi_\\text{c}(\\bm{Q})}\\,,\\quad\n \\bm{Q}=(\\pi,\\pi)\\,,\n \\\\\n R^\\chi_\\text{p} \n &= 1-\\frac{\\chi_\\text{c}(\\bm{Q}-\\delta{\\bm q})}{\\chi_\\text{c}(\\bm{Q})}\\,,\n \\quad \\bm{Q} = (0,0)\\,.\n\\end{align}\nfor CDW and s-wave pairing based on the susceptibilities\n\\begin{align}\\label{eq:chic}\n \\chi_\\text{c}(\\bm Q) \n &= \\frac{1}{L^2} \\sum_{ij}\n e^{\\text{i}(\\bm{r}_i-\\bm{r}_j)\\cdot\\bm{Q}} \\int_0^\\beta \\text{d} \\tau\n \\las \\hat{n}_{i}(\\tau) \\hat{n}_{j}\\ras\\,,\n \\\\\\label{eq:chip}\n \\chi_\\text{p}(\\bm Q) \n &= \\frac{1}{L^2} \\sum_{ij}\n e^{\\text{i}(\\bm{r}_i-\\bm{r}_j)\\cdot\\bm{Q}} \\int_0^\\beta \\text{d} \\tau\n \\las \\hat{\\Delta}^\\dag_{i}(\\tau) \\hat{\\Delta}^{\\phantom{\\dag}}_{j}\\ras\\,,\n\\end{align}\nwhere $\\hat{\\Delta}_i=c_{i\\UP}c_{i\\DO}$. The susceptibilities generally\nexhibit better finite-size scaling behavior than the corresponding static\nstructure factors [cf. Eq.~(\\ref{eq:scdw})]. We take a coupling\n$\\lambda=0.075$, for which Refs.~\\cite{ohgoe2017competitions,1709.00278}\nsuggest the absence of CDW order at $U=0$ over a large range of phonon\nfrequencies. The inverse temperature was scaled as $\\beta t=2L$ (with $4\\leq\nL\\leq 16$), which is at the current limit of the CT-INT method due to the sign problem. \n\n\\begin{figure}[t]\n \n \\includegraphics[width=0.45\\textwidth]{fig7.pdf}\n \\caption{\\label{fig:beta2L} (a) Charge and (b) pairing correlation ratios\n for different phonon frequencies. Here, $\\beta t=2L$, $\\lambda=0.075$,\n $U=0$.}\n\\end{figure}\n\n\\begin{figure}[t]\n \n \\includegraphics[width=0.45\\textwidth]{fig8.pdf} \n \\caption{\\label{fig:beta2L_U} (a) Charge and (b) pairing correlation ratios\n for different Hubbard repulsions. Here, $\\beta t=2L$, $\\lambda=0.075$,\n $\\omega_0\/t=1$.}\n\\end{figure}\n\nThe correlation ratios shown in Figs.~\\ref{fig:beta2L} and~\\ref{fig:beta2L_U} \nhave the same properties as discussed in Sec.~\\ref{sec:results:Tc};\nlong-range order is revealed by $R^\\chi_\\alpha\\to 1$ for $L\\to\\infty$, and\na larger correlation ratio indicates stronger correlations in the\ncorresponding channel. For $\\om_0=0.1$, the results in Fig.~\\ref{fig:beta2L}(a)\nsuggest long-range CDW order, consistent with Fig.~\\ref{fig:phasediagram}.\nAt the same time, the pairing correlation ratio in Fig.~\\ref{fig:beta2L}(b)\nis strongly suppressed. Upon increasing $\\omega_0$, CDW correlations are\nsuppressed and pairing correlations enhanced, but\n$R^\\chi_\\text{c}>R^\\chi_\\text{p}$ for any $\\omega_0<\\infty$. Degenerate CDW\nand pairing correlations are only observed for the attractive Hubbard model\n($\\omega_0=\\infty$). \nThe fact that CDW correlations at $\\omega_0<\\infty$ are stronger\nthan for $\\omega_0=\\infty$ suggests a CDW ground state also for the Holstein\nmodel and likely no superconducting order since $T_c$ is already minimal for\n$\\omega_0=\\infty$. As demonstrated in Fig.~\\ref{fig:beta2L_U}, a nonzero\nHubbard repulsion suppresses both CDW and pairing correlations while\nenhancing antiferromagnetic correlations (not shown).\n\nFigure~\\ref{fig:beta2L} also reveals that in the weak-coupling regime where\nan absence of CDW order was predicted \\cite{ohgoe2017competitions,1709.00278},\nit is challenging to unequivocally detect the known $T=0$ long-range\norder of the attractive Hubbard model in terms of $R^\\chi_\\text{c},R^\\chi_\\text{p}\\to 1$ for\n$L\\to\\infty$. The same should be true for the Holstein and Holstein-Hubbard\nmodel in the regime where $T_c$ is small. Therefore, leaving aside the approximations inherent to\nvariational QMC methods, the reported absence of CDW order \n\\cite{ohgoe2017competitions,1709.00278} should also be taken with care.\n\nWhile we are unable to provide a definitive $T=0$ phase diagram, the\nresults of Fig.~\\ref{fig:beta2L} together with the\nobservation that long-range CDW order is known to exist at $T=0$ for both\n$\\om_0=0$ and $\\om_0=\\infty$ are consistent with CDW order but no\nsuperconductivity in the half-filled Holstein model at $T=0$. \nFurthermore, in the absence of a higher symmetry relating CDW and\nsuperconducting order as in the attractive Hubbard model, we expect $T_c>0$\n(although potentially exponentially small) and hence\nscenario (I) depicted in Fig.~\\ref{fig:tcschematic}.\n\n\\subsection{Bipolaron liquid}\\label{sec:results:bipolarons}\n\n\\begin{figure}[t]\n \n \\includegraphics[width=0.45\\textwidth]{fig9.pdf}\n \\caption{\\label{fig:atthubbard2} Local spin and charge susceptibilities\n [Eq.~(\\ref{eq:localsusc})] for $\\lambda=0.1$, $U=0$, and $L=8$. Open\n symbols in (a) are for $\\lambda=0$,\n arrows indicate maxima.}\n\\end{figure}\n\nA final interesting point is the\nnature of the metallic phase at $T>T_c$. In the CDW phase, spin, charge, and\nhence also single-particle excitations are gapped. For 1D electron-phonon models, the\nspin gap persists in the metallic phase \\cite{MHHF2017} and the $T=0$ CDW\ntransition occurs at the two-particle level via the ordering of preformed\npairs (singlet bipolarons) and the opening of a charge gap. The same is true for\nthe 2D attractive Hubbard model for which the spin gap can be made\narbitrarily large by increasing $U$ while keeping $T_c=0$. Hence, the\ndisordered phase at low but finite temperatures is not a Fermi liquid but a metal with gapped\nsingle-particle and spin excitations \\cite{PhysRevB.50.635,PhysRevLett.69.2001}, the 2D analog of a Luther-Emery liquid \\cite{Lu.Em.74}. Singlet bipolarons\nin principle also form for any $\\lambda>0$ in the 2D Holstein model, \nalthough their binding energy ($\\sim \\lambda$) can be small\n\\cite{PhysRevB.69.245111}. Nevertheless, we expect a spin-gapped metallic\nphase for suitable parameters. At\nsufficiently high temperatures, bipolarons undergo thermal\ndissociation \\cite{PhysRevB.71.184309}.\n\nTo detect signatures of a spin-gapped metal, we consider the static charge and spin susceptibilities\n\\begin{equation}\\label{eq:localsusc}\n\\chi_\\text{c} =\\beta (\\las \\hat{N}^2\\ras - \\las\\hat{N}\\ras^2), \\quad \n\\chi_\\text{s} =\\beta (\\las \\hat{M}^2\\ras - \\las\\hat{M}\\ras^2)\\quad \n\\end{equation}\nwith $\\hat{N} = \\sum_i \\hat{n}_i$, $\\hat{M} = \\sum_i \\hat{S}^x_i$.\nFigure~\\ref{fig:atthubbard2}(a) shows results for $\\lambda=0.1$ and\n$\\omega_0\/t=\\infty$. Whereas $\\chi_\\text{s}\/L^2$ diverges with decreasing\ntemperature in a Fermi liquid (open symbols), it is strongly suppressed \nas $T\\to0$ by the spin gap. The charge susceptibility is also suppressed at\nvery low $T$, but $\\chi_\\text{c}\/L^2$ approaches a finite value determined by\nthe density of $T=0$ charge fluctuations. The distinct temperature scales reflected by the maxima of $\\chi_\\text{s}\/L^2$ and\n$\\chi_\\text{c}\/L^2$ reveal the spin-gapped metallic phase at $T>0$ in the\nattractive Hubbard model. For the\nHolstein model, $\\chi_\\text{s}\/L^2$ is cut off by the spin\ngap, whereas $\\chi_\\text{c}\/L^2$ is cut off by the charge gap that\nappears at the CDW transition at $T=T_c$. The distinct maxima visible even in the\nadiabatic regime [Figs.~\\ref{fig:atthubbard2}(b)\nand~\\ref{fig:atthubbard2}(c)] are consistent with a spin-gapped phase at\n$T>T_c$. The extent of the latter appears to decrease with\ndecreasing $\\omega_0\/t$ and the phase is expected to be absent\nin the classical or mean-field limit ($\\omega_0=0$) where charge and spin gaps become\nequal. An immediate and important corollary of the existence of a spin-gapped\nmetal of bipolarons above $T_c$ would be that, contrary to expectations in previous work\n\\cite{PhysRevB.48.7643,PhysRevB.48.16011}, the appearance\nof a gap in the density of states does in general not imply CDW order.\nThe additional spin-gap component is also compatible with\nexperimentally observed large gap to $T_c$ ratios \\cite{PhysRevB.63.115114}.\n\nIn principle, a spin-gapped phase without long-range order (CDW or\nsuperconductivity) could also exist at $T=0$, but the discussion in\nSec.~\\ref{sec:results:phasediagram} provided arguments against a disordered phase.\nWhile well established in 1D electron-phonon models in terms of a Luther-Emery\nliquid \\cite{MHHF2017}, it would correspond to a so-called Bose metal\n\\cite{PhysRevB.60.1261} in higher dimensions. An interesting question\nregarding the recent findings of\nRefs.~\\cite{ohgoe2017competitions,1709.00278} is whether the variational wave\nfunctions used can distinguish between spin-gap formation and\nsuperconductivity. To this end, it would be useful to test this method for\nthe intricate but well understood 1D Holstein model.\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nWe applied exact, continuous-time QMC simulations to the half-filled\nHolstein-Hubbard model on the square lattice. The critical temperature for\nthe CDW transition was determined as a function of phonon frequency,\nelectron-phonon coupling, and Hubbard repulsion from finite-size scaling. We\nalso demonstrated the expected 2D Ising universality of this transition and\naddressed the ground-state phase diagram, providing data and theoretical\narguments for the likely absence of a metallic or superconducting phase at\nweak coupling. Finally, we discussed the possibility of a spin-gapped metallic\nphase of bipolarons above $T_c$. The quantitative ground-state phase diagram\nremains an important open problem.\n\n\\vspace*{1em}\n\\begin{acknowledgments}\nWe thank F. Assaad, N. Costa, P. Br\\\"ocker, T. Lang, and R. Scalettar for helpful discussions and the DFG for\nsupport via SFB 1170 and FOR~1807. We gratefully acknowledge the\ncomputing time granted by the John von Neumann Institute for Computing (NIC)\nand provided on the supercomputer JURECA \\cite{jureca} at the J\\\"{u}lich\nSupercomputing Centre.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nSince the first unambiguous discovery of an exoplanet in 1995 \\citep[][]{Mayor1995} over 4,000 more have been confirmed. Studies of their characteristics have unveiled an extremely wide range of planetary properties in terms of planetary mass, size, system architecture and orbital periods, greatly revolutionising our understanding of how these bodies form and evolve.\n\nThe transit method, whereby we observe a temporary decrease in the brightness of a star due to a planet passing in front of its host star, is to date the most successful method for planet detection, having discovered over 75\\% of the planets listed on the NASA Exoplanet Archive\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}. It yields a wealth of information including planet radius, orbital period, system orientation and potentially even atmospheric composition. Furthermore, when combined with Radial Velocity \\citep[RV; e.g.,][]{Mayor1995, Marcy1997} observations, which yield the planetary mass, we can infer planet densities, and thus their internal bulk compositions. Other indirect detection methods include radio pulsar timing \\citep[e.g.,][]{Wolszczan1992} and microlensing \\citep[e.g.,][]{Gaudi2012}.\n\n\nThe \\textit{Transiting Exoplanet Survey Satellite} mission \\citep[\\protect\\emph{TESS};][]{ricker15} is currently in its extended mission, searching for transiting planets orbiting bright ($V < 11$\\,mag) nearby stars. Over the course of the two year nominal mission, \\emph{TESS}\\ monitored around 85 per cent of the sky, split up into 26 rectangular sectors of 96 $\\times$ 24 deg each (13 per hemisphere). Each sector is monitored for $\\approx$ 27.4 continuous days, measuring the brightness of $\\approx$ 20,000 pre-selected stars every two minutes. In addition to these short cadence (SC) observations, the \\emph{TESS}\\ mission provides Full Frame Images (FFI) that span across all pixels of all CCDs and are taken at a cadence of 30 minutes. While most of the targets ($\\sim$ 63 per cent) will be observed for $\\approx$ 27.4 continuous days, around $\\sim$ 2 per cent of the targets at the ecliptic poles are located in the `continuous viewing zones' and will be continuously monitored for $\\sim$ 356 days.\n\nStars themselves are extremely complex, with phenomena ranging from outbursts to long and short term variability and oscillations, which manifest themselves in the light curves. These signals, as well as systematic effects and artifacts introduced by the telescope and instruments, mean that standard periodic search methods, such as the Box-Least-Squared method \\citep{bls2002} can struggle to identify certain transit events, especially if the observed signal is dominated by natural stellar variability. Standard detection pipelines also tend to bias the detection of short period planets, as they typically require a minimum of two transit events in order to gain the signal-to-noise ratio (SNR) required for detection.\n\nOne of the prime science goals of the \\emph{TESS}\\ mission is to further our understanding of the overall planet population, an active area of research that is strongly affected by observational and detection biases. In order for exoplanet population studies to be able to draw meaningful conclusions, they require a certain level of completeness in the sample of known exoplanets as well as a robust sample of validated planets spanning a wide range of parameter space. \\textcolor{black}{Due to this, we independently search the \\emph{TESS}\\ light curves for transiting planets via visual vetting in order to detect candidates that were either intentionally ignored by the main \\emph{TESS}\\ pipelines, which require at least two transits for a detection, missed because of stellar variability or instrumental artefacts, or were identified but subsequently erroneously discounted at the vetting stage, usually because the period found by the pipeline was incorrect. These candidates can help populate under-explored regions of parameter space and will, for example, benefit the study of planet occurrence rates around different stellar types as well as inform theories of physical processes involved with the formation and evolution of different types of exoplanets.}\n\nHuman brains excel in activities related to pattern recognition, making the task of identifying transiting events in light curves, even when the pattern is in the midst of a strong varying signal, ideally suited for visual vetting. Early citizen science projects, such as Planet Hunters \\citep[PH;][]{fischer12} and Exoplanet Explorers \\citep{Christiansen2018}, successfully harnessed the analytic power of a large number of volunteers and made substantial contributions to the field of exoplanet discoveries. The PH project, for example, showed that human vetting has a higher detection efficiency than automated detection algorithms for certain types of transits. In particular, they showed that citizen science can outperform on the detection of single (long-period) transits \\citep[e.g.,][]{wang13, schmitt14a}, aperiodic transits \\citep[e.g. circumbinary planets;][]{schwamb13} and planets around variable stars \\citep[e.g., young systems,][]{fischer12}. Both PH and Exoplanet Explorers, which are hosted by the world's largest citizen science platform Zooniverse \\citep{lintott08}, ensured easy access to \\textit{Kepler} and \\textit{K2} data by making them publicly available online in an immediately accessible graphical format that is easy to understand for non-specialists. The popularity of these projects is reflected in the number of participants, with PH attracting 144,466 volunteers from 137 different countries over 9 years of the project being active.\n\nFollowing the end of the \\textit{Kepler} mission and the launch of the \\emph{TESS}\\ satellite in 2018, PH was relaunched as the new citizen science project \\textit{Planet Hunters TESS} (PHT) \\footnote{\\url{www.planethunters.org}}, with the aim of identifying transit events in the \\emph{TESS}\\ data that were \\textcolor{black}{intentionally ignored or missed} by the main \\emph{TESS}\\ pipelines. \\textcolor{black}{Such a search complements other methods methods via its sensitivity to single-transit, and, therefore, longer period planets. Additionally, other dedicated non-citizen science based methods are also employed to look for single transit candidates \\citep[see e.g., the Bayesian transit fitting method by ][]{Gill2020, Osborn2016}}.\n\nCitizen science transit searches specialise in finding the rare events that the standard detection pipelines miss, however, these results are of limited use without an indication of the completeness of the search. Addressing the problem of completeness was therefore one of our highest priorities while designing PHT as discussed throughout this paper. \n\nThe layout of the remainder of the paper will be as follows. An overview of the Planet Hunters TESS project is found in Section~\\ref{sec:PHT}, followed by an in depth description of how the project identifies planet candidates in Section~\\ref{sec:method}. The recovery efficiency of the citizen science approach is assessed in Section~\\ref{sec:recovery_efficiency}, followed by a description of the in-depth vetting of candidates and ground based-follow up efforts in Section~\\ref{sec:vetting} and \\ref{sec:follow_up}, respectively. Planet Candidates and noteworthy systems identified by Planet Hunters TESS are outlined in Section~\\ref{sec:PHT_canidates}, followed by a discussion of the results in Section~\\ref{sec:condlusion}.\n\n\\section{Planet Hunters TESS}\n\\label{sec:PHT}\n\nThe PHT project works by displaying \\emph{TESS}\\ light curves (Figure~\\ref{fig:interface}), and asking volunteers to identify transit-like signals. Only the two-minute cadence targets, which are produced by the \\emph{TESS}\\ pipeline at the Science Processing Operations Center \\citep[SPOC,][]{Jenkins2018} and made publicly available by the Mikulski Archive for Space Telescopes (MAST)\\footnote{\\url{http:\/\/archive.stsci.edu\/tess\/}}, are searched by PHT. First-time visitors to the PHT site, or returning visitors who have not logged in are prompted to look through a short tutorial, which briefly explains the main aim of the project and shows examples of transit events and other stellar phenomena. Scientific explanation of the project can be found elsewhere on the site in the `field guide' and on the project's `About' page. \n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=\\textwidth]{Figures\/PHT_new_interface.png}\n \\caption{\n PHT user interface showing a simulated light curve. The transit events are highlighted with white partially-transparent columns that are drawn on using the mouse. Stellar information on the target star is available by clicking on `subject info' below the light curve.} \n \\label{fig:interface}\n\\end{figure*}\n\nAfter viewing the tutorial, volunteers are ready to participate in the project and are presented with \\emph{TESS}\\ light curves (known as `subjects') that need to be classified. The project was designed to be as simple as possible and therefore only asks one question: \\textit{`Do you see a transit?}'. Users identify transit-like events, and the time of their occurrence, by drawing a column over the event using the mouse button, as shown in Figure~\\ref{fig:interface}. There is no limit on the number of transit-like events that can be marked in a light curve. No markings indicate that there are no transit-like events present in the light curve. Once the subject has been analysed, users submit their classification and continue to view the next light curve by clicking `Done'. \n\nAlongside each light curve, users are offered information on the stellar properties of the target, such as the radius, effective temperature and magnitude (subject to availability, see \\cite{Stassun18}). However, in order to reduce biases in the classifications, the TESS Input Catalog (TIC) ID of the target star is not provided until after the subject classification has been submitted.\n\nIn addition to classifying the data, users are given the option to comment on light curves via the `Talk' discussion forum. Each light curve has its own discussion page to allow volunteers to discuss and comment, as well as to `tag' light curves using searchable hashtags, and to bring promising candidates to the attention of other users and the research team. The talk discussion forums complement the main PHT analysis and have been shown to yield interesting objects which may be challenging to detect using automated algorithms \\citep[e.g.,][]{eisner2019RN}. Unlike in the initial PH project, there are no questions in the main interface regarding stellar variability, however, volunteers are encouraged to mention astrophysical phenomenon or \\textit{unusual} features, such as eclipsing binaries or stellar flares, using the `Talk' discussion forum. \n\nThe subject TIC IDs are revealed on the subject discussion pages, allowing volunteers to carry out further analysis on specific targets of interest and to report and discuss their findings. This is extremely valuable for both other volunteers and the PHT science team, as it can speed up the process of identifying candidates as well as rule out false positives in a fast and effective manner. \n\nSince the launch of PHT on 6 December 2018, there has been one significant makeover to the user interface. The initial PHT user interface (UI1), which was used for sectors 1 through 9, split the \\emph{TESS}\\ light curves up into either three or four chunks (depending on the data gaps in each sector) which lasted around seven days each. This allowed for a more `zoomed' in view of the data, making it easier to identify transit-like events than when the full $\\sim$ 30 day light curves were shown. The results from a PHT beta project, which displayed only simulated data, showed that a more zoomed in view of the light curve was likely to yield a higher transit recovery rate.\n\nThe updated, and current, user interface (UI2) allows users to manually zoom in on the x-axis (time) of the data. Due to this additional feature, each target has been displayed as a single light curve as of Sector 10. In order to verify that the changes in interface did not affect our findings, all of the Sector 9 subjects were classified using both UI1 and UI2. We saw no significant change in the number of candidates recovered (see Section~\\ref{sec:recovery_efficiency} for a description of how we quantified detection efficiency).\n\n\n\\subsection{Simulated Data}\n\\label{subsec:sims} \n\nIn addition to the real data, volunteers are shown simulated light curves, which are generated by randomly injecting simulated transit signals, provided by the SPOC pipeline \\citep[][]{Jenkins2018}, into real \\emph{TESS}\\ light curves. The simulated data play an important role in assessing the sensitivity of the project, training the users and providing immediate feedback, and to gauge the relative abilities of individual users (see Sec~\\ref{subsec:weighting}). \n\nWe calculate a signal to noise ratio (SNR) of the injected signal by dividing the injected transit depth by the Root Mean Square Combined Differential Photometric Precision (RMS CDPP) of the light curve on 0.5-, 1- or 2-hr time scales (whichever is closest to the duration of the injected transit signal). Only simulations with a SNR greater than 7 in UI1 and greater than 4 for UI2 are shown to volunteers.\n\nSimulated light curves are randomly shown to the volunteers and classified in the exact same manner as the real data. The user is always notified after a simulated light curve has been classified and given feedback as to whether the injected signal was correctly identified or not. For each sector, we generate between one and two thousand simulated light curves, using the real data from that sector in order to ensure that the sector specific systematic effects and data gaps of the simulated data do not differ from the real data. The rate at which a volunteer is shown simulated light curves decreases from an initial rate of 30 per cent for the first 10 classifications, down to a rate of 1 per cent by the time that the user has classified 100 light curves. \n\n\n\\section{Identifying Candidates}\n\\label{sec:method}\n\nEach subject is seen by multiple volunteers, before it is `retired' from the site, and the classifications are combined (see Section~\\ref{subsec:DBscan}) in order to assess the likelihood of a transit event. For sectors 1 through 9, the subjects were retired after 8 classifications if the first 8 volunteers who saw the light curves did not mark any transit events, after 10 classifications if the first 10 volunteers all marked a transit event and after 15 classifications if there was not complete consensus amongst the users. As of Sector 9 with UI2, all subjects were classified by 15 volunteers, regardless of whether or not any transit-like events were marked. Sector 9, which was classified with both UI1 and UI2, was also classified with both retirement rules.\n\nThere were a total of 12,617,038 individual classifications completed across the project on the nominal mission data. 95.4 per cent of these classifications were made by 22,341 registered volunteers, with the rest made by unregistered volunteers. Around 25 per cent of the registered volunteers complete more than 100 classifications, 11.8 per cent more than 300, 8.4 per cent more than 500, 5.4 per cent more than 1000 and 1.1 per cent more than 10,000. The registered volunteers completed a mean and median of 647 and 33 classifications, respectively. Figure~\\ref{fig:user_count} shows the distribution in user effort for logged in users who made between 0 and 300 classifications. \n\nThe distribution in the number of classifications made by the registered volunteers is assessed using the Gini coefficient, which ranges from 0 (equal contributions from all users) to 1 (large disparity in the contributions). The Gini coefficients for individual sectors ranges from 0.84 to 0.91 with a mean of 0.87, while the Gini coefficient for the overall project (all of the sectors combined) is 0.94. The mean Gini coefficient among other astronomy Zooniverse projects lies at 0.82 \\citep{spiers2019}. We note that the only other Zooniverse project with an equally high Gini coefficient as PHT is \\textit{Supernova Hunters}, a project which, similarly to PHT and unlike most other Zooniverse projects, has periodic data releases that are accompanied by an e-newsletter sent to all project volunteers. Periodic e-newsletters have the effect of promoting the project to both regularly and irregularly participating volunteers, who may only complete a couple of classifications as they explore the task, as well as to returning users who complete a large number of classifications following every data release, increasing the disparity in user contributions (the Gini coefficient).\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.5\\textwidth]{Figures\/user_count.png}\n \\caption{\n The distribution of the number of classifications by the registered volunteers, using a bin size of 5 from 0 to 300 classifications. A total of 11.8 per cent of the registered volunteers completed more than 300 classifications.} \n \\label{fig:user_count}\n\\end{figure}\n\n\n\\subsection{User Weighting}\n\\label{subsec:weighting} \nUser weights are calculated for each individual volunteer in order to identify users who are more sensitive to detecting transit-like signals and those who are more likely to mark false positives. The weighting scheme is based on the weighting scheme described by \\cite{schwamb12}.\n\nUser weights are calculated independently for each observation sector, using the simulated light curves shown alongside the data from that sector. All users start off with a weighting of one, which is then increased or decreased when a simulated transit event is correctly or incorrectly identified, respectively. \n\nSimulated transits are deemed correctly identified, or `True', if the mid-point of a user's marking falls within the width of the simulated transit events. If none of the user's markings fall within this range, the simulated transit is deemed not identified, or `False'. If more than one of a user's markings coincide with the same simulated signal, it is only counted as being correct once, such that the total number of `True' markings cannot exceed the number of injected signals. For each classification, we record the number of `Extra' markings, which is the total number of markings made by the user minus the number of correctly identified simulated transits. \n\nEach simulated light curve, identified by superscript $i$ (where $i=1$, \\ldots, $N$) was seen by $K^{(i)}$ users (the mean value of $K^{(i)}$\nwas 10), and contained $T^{(i)}$ simulated transits (where $T^{(i)}$ depends on the period of the simulated transit signal and the duration of the light curve). For a specific light curve $i$, each user who saw the light curve is identified by a subscript $k$ (where $k=1$, \\ldots, $K^{(i)}$) and each injected transit by a subscript $t$ (where $t=1$, \\ldots, $T^{(i)}$). \n\nIn order to distinguish between users who are able to identify obvious transits and those who are also able to find those that are more difficult to see, we start by defining a `recoverability' $r^{(i)}_t$ for each injected transit $t$ in each light curve. This is defined empirically, as the number of users who identified the transit correctly divided by $K^{(i)}$ (the total number of users who saw the light curve in question).\n\nNext, we quantify the performance of each user on each light curve as follows (this performance is analogous to the `seed' defined in \\citealt{schwamb12}, but we define it slightly differently):\n\\begin{equation}\n p^{(i)}_{k} = C_{\\rm E} ~ \\frac{E^{(i)}_{k}}{\\langle E^{(i)} \\rangle} + \\sum_{t=1}^{T^{(i)}} \\begin{cases}\n C_{\\rm T} ~ \\left[ r^{(i)}_t \\right]^{-1}, & \\text{if $m^{(i)}_{t,k} = $`True'}\\\\\n C_{\\rm F} ~ r^{(i)}_t, & \\text{if $m^{(i)}_{t,k} = $`False'},\n \\end{cases}\n\\end{equation}\nwhere $m^{(i)}_{t,k}$ is the identification of transit $t$ by user $k$ in light curve $i$, which is either `True' or `False'; $E^{(i)}_{k}$ is the number of `Extra' markings made by user $k$ for light curve $i$, and $\\langle E^{(i)} \\rangle$ is the mean number of `Extra' markings made by all users who saw subject $i$. The parameters $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ control the impact of the `Extra', `True' and `False' markings on the overall user weightings, and are optimized empirically as discussed below in Section~\\ref{subsec:optimizesearch}. \n\nFollowing \\citealt{schwamb12}, we then assign a global `weight' $w_k$ to each user $k$, which is defined as:\n\\begin{equation}\n\\begin{split}\n\tw_k = I \\times (1 + \\log_{10} N_k)^{\\nicefrac{\\sum_i p^{(i)}_k}{N_k}}\n\\label{equ:weight}\n\\end{split}\n\\end{equation}\nwhere $I$ is an empirical normalization factor, such that the distribution of user weights remains centred on one, $N_k$ is the total number of simulated transit events that user $k$ assessed, and the sum over $i$ concerns only the light curves that user $k$ saw. \nWe limit the user weights to the range 0.05--3 \\emph{a posteriori}.\n\n\nWe experimented with a number of alternative ways to define the user weights, including the simpler $w_k=\\nicefrac{\\sum_i p^{(i)}_k}{N_k}$, but Eqn.~\\ref{equ:weight} was found to give the best results (see Section~\\ref{sec:recovery_efficiency} for how this was evaluated).\n\n\\subsection{Systematic Removal}\n\\label{subsec:sysrem} \nSystematic effects, for example caused by the spacecraft or background events, can result in spurious signals that affect a large subset of the data, resulting in an excess in markings of transit-like events at certain times within an observation sector. As the four \\emph{TESS}\\ cameras can yield unique systematic effects, the times of systematics were identified uniquely for each camera. The times were identified using a Kernel Density Estimation \\citep[KDE;][]{rosenblatt1956} with a cosine kernel and a bandwidth of 0.1 days, applied across all of the markings from that sector for each camera. Fig.~\\ref{fig:sys_rem} shows the KDE of all marked transit-events made during Sector 17 for TESS's cameras 1 (top panel) to 4 (bottom panel). The isolated spikes, or prominences, in the number of marked events, such as at T = 21-22 days in the bottom panel, are assumed to be caused by systematic effects that affect multiple light curves. Prominences are considered significant if they exceed a factor four times the standard deviation of the kernel output, which was empirically determined to be the highest cut-off to not miss clearly visible systematics. All user-markings within the full width at half maximum of these peaks are omitted from all further analysis. \\textcolor{black}{The KDE profiles for each Sector are provided as electronic supplementary material.}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.46\\textwidth]{Figures\/systematics_sec17.png}\n \\caption{\n Kernel density estimation of the user-markings made for Sector 17, for targets observed with TESS's observational Cameras 1 (top panel) to 4 (bottom panel). The orange vertical lines the indicate prominences that are at least four times greater than the standard deviation of the distribution. The black points underneath the figures show the mid-points of all of the volunteer-markings, where darker regions represent a higher density of markings.}\n \\label{fig:sys_rem}\n\\end{figure}\n\n\\subsection{Density Based Clustering}\n\\label{subsec:DBscan} \n\nThe times and likelihoods of transit-like events are determined by combining all of the classifications made for each subject and identifying times where multiple volunteers identified a signal. We do this using an unsupervised machine learning method, known as DBSCAN \\citep[][Density-Based Spatial Clustering of Applications with Noise]{ester1996DB}. DBSCAN is a non-parametric density based clustering algorithm that helps to distinguish between dense clusters of data and sparse noise. For a data point to belong to a cluster it must be closer than a given distance ($\\epsilon$) to at least a set minimum number of other points (minPoints). \n\nIn our case, the data points are one-dimensional arrays of times of transits events, as identified by the volunteers, and clusters are times where multiple volunteers identified the same event. For each cluster a `transit score' ($s_i$) is determined, which is the sum of the user weights of the volunteers who contribute to the given cluster divided by the sum of the user weights of volunteers who saw that light curve. These transit scores are used to rank subjects from most to least likely to contain a transit-like event. Subjects which contain multiple successful clusters with different scores are ranked by the highest transit score. \n\n\\subsection{Optimizing the search}\n\\label{subsec:optimizesearch}\n\nThe methodology described in Sections~\\ref{subsec:weighting} to \\ref{subsec:DBscan} has five free parameters: the number of markings required to constitute a cluster ($minPoints$), the maximum separation of markings required for members of a cluster ($\\epsilon$), and $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ used in the weighting scheme. The values of these parameters were optimized via a grid search, where $C_{\\rm E}$ and $C_{\\rm F}$ ranged from -5 to 0, $C_{\\rm T}$ ranged from 0 to 20, and $minPoints$ ranged from 1 to 8, all in steps of 1. ($\\epsilon$) ranged from 0.5 to 1.5 in steps of 0.5. This grid search was carried out on 4 sectors, two from UI1 and two from UI2, for various variations of Equation~\\ref{equ:weight}. \n\nThe success of each combination of parameters was assessed by the fractions of TOIs and TCEs that were recovered within the top highest ranked 500 candidates, as discussed in more detail Section~\\ref{sec:recovery_efficiency}. We found the most successful combination of parameters to be $minPoints$ = 4 markings, $\\epsilon$, = 1 day, $C_{\\rm T}$ = 3, $C_{\\rm F}$= -2 and $C_{\\rm E}$ = -2.\n\n\\subsection{MAST deliverables}\n\\label{subsec:deliverables}\n\nThe analysis described above is carried out both in real-time as classifications are made, as well as offline after all of the light curves of a given sector have been classified. When the real-time analysis identifies a successful DB cluster (i.e. when at least four citizen scientists identified a transit within a day of the \\emph{TESS}\\ data of one another), the potential candidate is automatically uploaded to the open access Planet Hunters Analysis Database (PHAD) \\footnote{\\url{https:\/\/mast.stsci.edu\/phad\/}} hosted by the Mikulski Archive for Space Telescopes (MAST) \\footnote{\\url{https:\/\/archive.stsci.edu\/}}. While PHAD does not list every single classification made on PHT, it does display all transit candidates which had significant consensus amongst the volunteers who saw that light curve, along with the user-weight-weighted transit scores. This analysis does not apply the systematics removal described in Section~\\ref{subsec:sysrem}. The aim of PHAD is to provide an open source database of potential planet candidates identified by PHT, and to credit the volunteers who identified said targets. \n\nThe offline analysis is carried out following the complete classifications of all of the data from a given \\emph{TESS}\\ sector. The combination of all of the classifications allows us to identify and remove times of systematics and calculate better calibrated and more representative user weights. The remainder of this paper will only discuss the results from the offline analysis.\n\n\\section{Recovery Efficiency}\n\\label{sec:recovery_efficiency}\n\\subsection{Recovery of simulated transits}\n\nThe recovery efficiency is, in part, assessed by analysing the recovery rate of the injected transit-like signals (see Section~\\ref{subsec:sims}). Figure~\\ref{fig:SIM_recovery} shows the median and mean transit scores (fraction of volunteers who correctly identified a given transit scaled by user weights) of the simulated transits within SNR bins ranging from 4 to 20 in steps of 0.5. Simulations with a SNR less than 4 were not shown on PHT. The figure highlights that transit signals with a SNR of 7.5 or greater are correctly identified by the vast majority of volunteers. \n\n\\textcolor{black}{As the simulated data solely consist of real light curves with synthetically injected transit signals, we do not have any light curves, simulated or otherwise, which we can guarantee do not contain any planetary transits (real or injected). As such, this prohibits us from using simulated data to infer an analogous false-positive rate.}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.47\\textwidth]{Figures\/SIMS_recovery.png}\n \\caption{The median (blue) and mean (orange) transit scores for injected transits with SNR ranges between 4 and 20. The mean and median are calculated in SNR bins with a width of 0.5, as indicated by the horizontal lines around each data point. \n }\n \\label{fig:SIM_recovery}\n\\end{figure}\n\n\\subsection{Recovery of TCEs and TOIs}\n\\label{subsec:TCE_TOI}\nThe recovery efficiency of PHT is assessed further using the planet candidates identified by the SPOC pipeline \\citep{Jenkins2018}. The SPOC pipeline extracts and processes all of the 2-minute cadence \\emph{TESS}\\ light curves prior to performing a large scale transit search. Data Validation (DV) reports, which include a range of transit diagnostic tests, are generated by the pipeline for around 1250 Threshold Crossing Events (TCEs), which were flagged as containing two or more transit-like features. Visual vetting is then performed by the \\emph{TESS}\\ science team on these targets, and promising candidates are added to the catalog of \\emph{TESS}\\ Objects of Interest (TOIs). Each sector yields around 80 TOIs \\textcolor{black}{and a mean of 1025 TCEs.}\n\nFig~\\ref{fig:TCE_TOI_recovery} shows the fraction of TOIs and TCEs (top and bottom panel respectively) that we recover with PHT as a function of the rank, where a higher rank corresponds to a lower transit score, for Sectors 1 through 26. TOIs and TCEs with R < 2 $R_{\\oplus}$ are not included in this analysis, as the initial PH showed that human vetting alone is unable to reliably recover planets smaller than 2 $R_{\\oplus}$ \\citep{schwamb12}. Planets smaller than 2 $R_{\\oplus}$ are, therefore, not the main focus of our search.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI-recovery_radlim2.png}\n \\caption{The fraction of recovered TOIs and TCEs (top and bottom panel respectively) with R > 2$R_{\\oplus}$ as a function of the rank, for sectors 1 to 26. The lines represent the results from different observation sectors.}\n \\label{fig:TCE_TOI_recovery}\n\\end{figure}\n\n\nFig~\\ref{fig:TCE_TOI_recovery} shows a steep increase in the fractional TOI recovery rate up to a rank of $\\sim$ 500. Within the 500 highest ranked PHT candidates for a given sector, we are able to recover between 46 and 62 \\% (mean of 53 \\%) of all of the TOIs (R > 2 $R_{\\oplus}$), a median 90 \\% of the TOIs where the SNR of the transit events are greater than 7.5 and median 88 \\% of TOIs where the SNR of the transit events are greater than 5.\n\nThe relation between planet recovery rate and the SNR of the transit events is further highlighted in Figure~\\ref{fig:TOI_properties}, which shows the SNR vs the orbital period of the recovered TOIs. The colour of the markers indicate the TOI's rank within a given sector, with the lighter colours representing a lower rank. The circles and crosses represent candidates at a rank lower and higher than 500, respectively. The figure shows that transit events with a SNR less than 3.5 are missed by the majority of volunteers, whereas events with a SNR greater than 5 are mostly recovered within the top 500 highest ranked candidates. \n\nThe steep increase in the fractional TOI recovery rate at lower ranks, as shown in figure~\\ref{fig:TCE_TOI_recovery}, is therefore due to the detection of the high SNR candidates that are identified by most, if not all, of the PHT volunteers who classified those targets. At a rank of around 500, the SNR of the TOIs tends towards the limit of what human vetting can detect and thus the identification of TOIs beyond a rank of 500 is more sporadic.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.49\\textwidth]{Figures\/TOI_recovery_properties.png}\n \\caption{The SNR vs orbital period of TOIs with R > 2$R_{\\oplus}$. The colour represents their rank within the sector, as determined by the weighted DB clustering algorithm. Circles indicate that they were identified at a rank < 500, while crosses indicate that they were not within the top 500 highest ranked candidates of a given sector.\n }\n \\label{fig:TOI_properties}\n\\end{figure}\n\nThe fractional TCE recovery rate (bottom panel of Figure~\\ref{fig:TCE_TOI_recovery}) is systematically lower than that of the TOIs. There are qualitative reasons as to why humans might not identify a TCE as opposed to a TOI, including that TCEs may be caused by artefacts or periodic stellar signals that the SPOC pipeline identified as a potential transit but that the human eye would either miss or be able to rule out as systematic effect. This leads to a lower recovery fraction of TCEs comparatively, an effect that is further amplified by the much larger number of TCEs.\n\nThe detection efficiency of PHT is estimated using the fractional recovery rate of TOIs for a range of radius and period bins, as shown in Figure~\\ref{fig:recovery_rank500_radius_period}. A TOI is considered to be recovered if its detection rank is less than 500 within the given sector. Out of the total 1913 TOIs, to date, \\textcolor{black}{PHT recovered 715 TOIs among the highest ranked candidates across the 26 sectors. This corresponds to a mean of 12.7~\\% of the top 500 ranked candidates per sector being TOIs. In comparison, the primary \\emph{TESS}\\ team on average visually vets 1025 TCEs per sector, out of which a mean of 17.3~\\% are promoted to TOI status.} We find that, independent of the orbital period, PHT is over 85~\\% complete in the recovery of TOIs with radii equal to or greater than 4 $R_{\\oplus}$. This agrees with the findings from the initial Planet Hunters project \\citep{schwamb12}. The detection efficiency decreases to 51~\\% for 3 - 4 $R_{\\oplus}$ TOIs, 49~\\% for 2 - 3 $R_{\\oplus}$ TOIs and to less than 40~\\% for TOIs with radii less than 2 $R_{\\oplus}$. Fig~\\ref{fig:recovery_rank500_radius_period} shows that the orbital period does not have a strong effect on the detection efficiency for periods greater than $\\sim$~1~day, which highlights that human vetting efficiency is independent of the number of transits present within a light curve. For periods shorter than around 1~day, the detection efficiency decreases even for larger planets, due to the high frequency of events seen in the light curve. For these light curves, many volunteers will only mark a subset of the transits, which may not overlap with the subset marked by other volunteers. Due to the methodology used to identify and rank the candidates, as described in Section~\\ref{sec:method}, this will actively disfavour the recovery of very short period planets. Although this obviously introduces biases in the detectability of very short period signals, the major detection pipelines are specifically designed to identify these types of planets and thus this does not present a serious detriment to our main science goal of finding planets that were \\textcolor{black}{intentionally ignored or missed} by the main automated pipelines.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{Figures\/TOI_recovery_grid.png}\n \\caption{TOI recovery rate as a function of planet radius and orbital period. A TOI is considered recovered if it is amongst the top 500 highest ranked candidates within a given sector. The logarithmically spaced grid ranges from 0.2 to 225 d and 0.6 to 55 $R{_\\oplus}$ for the orbital period and planet radius, respectively. The fraction of TOIs recovered using PHT is computed for each cell and represented by the colour the grid. Cells with less than 10 TOIs are considered incomplete for statistical analysis and are shown by the hatched lines. White cells contain no TOIs. The annotations for each cell indicate the number of recovered TOIs followed by the Poisson uncertainty in brackets. The filled in and empty grey circles indicated the recovered and not-recovered TOIs, respectively.}\n \\label{fig:recovery_rank500_radius_period}\n\\end{figure*}\n\n\nFinally, we assessed whether the detection efficiency varies across different sectors by assessing the fraction of recovered TOIs and TCEs within the highest ranked 500 candidates. We found the recovery of TOIs within the top 500 highest ranked candidates to remain relatively constant across all sectors, while the fraction of recovered TCEs in the top 500 highest ranked candidates increases in later sectors, as shown in Figure~\\ref{fig:recovery_rank500}). After applying a Spearman's rank test we find a positive correlation of 0.86 (pvalue = 5.9 $\\times$ $10^{-8}$) and 0.57 (pvalue = 0.003) between the observation sector and TCE and TOI recovery rates, respectively. These correlations suggest that the ability of users to detect transit-like events improves as they classify more subjects. The improvement of volunteers over time can also be seen in Fig~\\ref{fig:user_weights}, which shows the mean (unnormalized) user weight per sector for volunteers who completed one or more classifications in at least one sector (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors 26 sectors from the nominal \\emph{TESS}\\ mission (pink). The figure highlights an overall improvement in the mean user weight in later sectors, as well as a positive correlation between the overall increase in user weight and the number of sectors that volunteers have participated in.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI_rank500.png}\n \\caption{The fractional recovery rate of the TOIs (blue circles) and TCEs (teal squares) at a rank of 500 for each sector. Sector 1-9 (white background) represent southern hemisphere sectors classified with UI1, sectors 9-14 (light grey background) show the southern hemisphere sectors classified with UI2, and sectors 14-24 (dark grey background) show the northern hemisphere sectors classified with US2.}\n \\label{fig:recovery_rank500}\n\\end{figure}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.50\\textwidth]{Figures\/user_weights_sectors.png}\n \\caption{Mean user weights per sector. The solid lines show the user weights for the old user interface and the dashed line for the new interface, separated by the black line (Sector 9). The different coloured lines show the mean user weights calculated considering user who participated in any number of sectors (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors observed during the nominal \\emph{TESS}\\ mission (pink).}\n \\label{fig:user_weights}\n\\end{figure}\n\n\n\\section{Candidate vetting}\n\\label{sec:vetting}\n\nFor each observation sector the subjects are ranked according to their transit scores, and the 500 highest ranked targets (excluding TOIs) visually vetted by the PHT science team in order to identify potential candidates and rule out false positives. A vetting cut-off rank of 500 was chosen as we found this to maximise the number of found candidates while minimising the number of likely false positives. In the initial round of vetting, which is completed via a separate Zooniverse classification interface that is only accessible to the core science team, a minimum of three members of the team sort the highest ranked targets into either `keep for further analysis', `eclipsing binary' or `discard'. The sorting is based on the inspection of the full \\emph{TESS}\\ light curve of the target, with the times of the satellite momentum dumps indicated. Additionally, around the time of each likely transit event (i.e. time of successful DB clusters) we inspect the background flux and the x and y centroid positions. Stellar parameters are provided for each candidate, subject to availability, alongside links to the SPOC Data Validation (DV) reports for candidates that had been flagged as TCEs but were never promoted to TOIs status.\n\nCandidates where at least two of the reviewers indicated that the signal is consistent with a planetary transit are kept for further analysis. \\textcolor{black}{This constitute a $\\sim$~5~\\% retention rate of the 500 highest ranked candidates per sector between the initial citizen science classification stage and the PHT science team vetting stage. Considering that the known planets and TOIs are not included at this stage of vetting, it is not surprising that our retention rate is lower that the true-positive rates of TCEs (see Section~\\ref{subsec:TCE_TOI}). Furthermore, this false-positive rate is consistent with the the findings of the initial Planet Hunters project \\citep{schwamb12}.}\n\nThe rest of the 500 candidates were grouped into $\\sim$~37~\\% `eclipsing binary' and $\\sim$~58~\\% `discard'. The most common reasons for discarding light curves are due to events caused by momentum dumps and due to background events, such as background eclipsing binaries, that mimic transit-like signals in the light curve. The targets identified as eclipsing binaries are analysed further by the \\emph{TESS}\\ Eclipsing Binaries Working Group (Prsa et al, in prep).\n\n\n\n\nFor the second round of candidate vetting we generate our own data validation reports for all candidates classified as `keep for further analysis'. The reports are generated using the open source software {\\sc latte} \\citep[Lightcurve Analysis Tool for Transiting Exoplanets;][]{LATTE2020}, which includes a range of standard diagnostic plots that are specifically designed to help identify transit-like signals and weed out astrophysical false positives in \\emph{TESS}\\ data. In brief the diagnostics consist of:\n\n\\textbf{Momentum Dumps}. The times of the \\emph{TESS}\\ reaction wheel momentum dumps that can result in instrumental effects that mimic astrophysical signals.\n\n\\textbf{Background Flux}. The background flux to help identify trends caused by background events such as asteroids or fireflies \\citep{vanderspek2018tess} passing through the field of view.\n\n\\textbf{x and y centroid positions}. The CCD column and row local position of the target's flux-weighted centroid, and the CCD column and row motion which considers differential velocity aberration (DVA), pointing drift, and thermal effects. This can help identify signals caused by systematics due to the satellite. \n\\textbf{Aperture size test}. The target light curve around the time of the transit-like event extracted using two apertures of different sizes. This can help identify signals resulting from background eclipsing binaries.\n \n\\textbf{Pixel-level centroid analysis}. A comparison between the average in-transit and average out-of-transit flux, as well as the difference between them. This can help identify signals resulting from background eclipsing binaries.\n\n\\textbf{Nearby companion stars}. The location of nearby stars brighter than V-band magnitude 15 as queried from the Gaia Data Release 2 catalog \\citep{gaia2018gaia} and the DSS2 red field of view around the target star in order to identify nearby contaminating sources. \n\n\\textbf{Nearest neighbour light curves}. Normalized flux light curves of the five short-cadence \\emph{TESS}\\ stars with the smallest projected distances to the target star, used to identify alternative sources of the signal or systematic effects that affect multiple target stars. \n\n\\textbf{Pixel level light curves}. Individual light curves extracted for each pixel around the target. Used to identify signals resulting from background eclipsing binaries, background events and systematics.\n\n\\textbf{Box-Least-Squares fit}. Results from two consecutive BLS searches, where the identified signals from the initial search are removed prior to the second BLS search.\n\nThe {\\sc latte} validation reports are assessed by the PHT science team in order to identify planetary candidates that warrant further investigation. Around 10~\\% of the targets assessed at this stage of vetting are kept for further investigation, resulting in $\\sim$~3 promising planet candidates per observation sector. The discarded candidates can be loosely categorized into (background) eclipsing binaries ($\\sim$~40~\\%), systematic effects ($\\sim$~25~\\%), background events ($\\sim$~15~\\%) and other (stellar signals such as spots; $\\sim$~10~\\%).\n\n\nWe use \\texttt{pyaneti}\\ \\citep{pyaneti} to infer the planetary and orbital parameters of our most promising candidates. For multi-transit candidates we fit for seven parameters per planet, time of mid-transit $T_0$, orbital period $P$, impact parameter $b$, scaled semi-major axis $a\/R_\\star$, scaled planet radius $r_{\\rm p}\/R_\\star$, and two limb darkening coefficients following a \\citet{Mandel2002} quadratic limb darkening model, implemented with the $q_1$ and $q_2$ parametrization suggested by \\citet{Kipping2013}. Orbits were assumed to be circular.\nFor the mono-transit candidates, we fit the same parameters as for the multi-transit case, except for the orbital period and scaled semi-major axis which cannot be known for single transits. We follow \\citet{Osborn2016} to estimate the orbital period of the mono-transit candidates assuming circular orbits.\n\nWe note that some of our candidates are V-shaped, consistent with a grazing transit configuration. For these cases, we set uniform priors between 0 and 0.15 for $r_{\\rm p}\/R_\\star$ and between 0 and 1.15 for the impact parameter in order to avoid large radii caused by the $r_{\\rm p}\/R_\\star - b$ degeneracy. Thus, the $r_{\\rm p}\/R_\\star$ for these candidates should not be trusted. A full characterisation of these grazing transits is out of the scope of this manuscript.\n\nFigure~\\ref{fig:PHT_pyaneti} shows the \\emph{TESS}\\ transits together with the inferred model for each candidate. Table~\\ref{tab:PHT-caniddates} shows the inferred main parameters, the values and their uncertainties are given by the median and 68.3\\% credible interval of the posterior distributions.\n\n\n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_one.png}\n \\caption{All of the PHT candidates modelled using \\texttt{pyaneti}. The parameters of the best fits are summarised in Table~\\protect\\ref{tab:PHT-caniddates}. The blue and magenta fits show the multi and single transit event candidates, respectively.} \n \\label{fig:PHT_pyaneti}\n\\end{figure*}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_two.png}\n \\addtocounter{figure}{-1}\n \\caption{\\textbf{PHT candidates (continued)}} \n\\end{figure*}\n\n\nCandidates that pass all of our rounds of vetting are uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS) website\\footnote{\\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}} as community TOIs (cTOIs).\n\n\\section{Follow-up observations}\n\\label{sec:follow_up}\n\nMany astrophysical false positive scenarios can be ruled out from the detailed examination of the \\emph{TESS}\\ data, both from the light curves themselves and from the target pixel files. However, not all of the false positive scenarios can be ruled out from these data alone, due in part to the large \\emph{TESS}\\ pixels (20 arcsconds). Our third stage of vetting, therefore, consists of following up the candidates with ground based observations including photometry, reconnaissance spectroscopy and speckle imaging. The results from these observations will be discussed in detail in a dedicated follow-up paper. \n\n\\subsection{Photometry}\n\nWe make use of the LCO global network of fully robotic 0.4-m\/SBIG and 1.0-m\/Sinistro facilities \\citep{LCO2013} to observe additional transits, where the orbital period is known, in order to refine the ephemeris and confirm that the transit events are not due to a blended eclipsing binary in the vicinity of the main target. Snapshot images are taken of single transit event candidates in order to identify nearby contaminating sources. \n\n\n\\subsection{Spectroscopy}\n\nWe perform high-resolution optical spectroscopy using telescopes from across the globe in order to cover a wide range of RA and Dec:\n\\begin{itemize}\n\\item The Las Cumbres Observatory (LCO) telescopes with the Network of Robotic Echelle Spectrographs \\citep[NRES,][]{LCO2013}. These fibre-fed spectrographs, mounted on 1.0-m telescopes around the globe, have a resolution of R = 53,000 and a wavelength coverage of 380 to 860 nm. \n\n\\item The MINERVA Australis Telescope facility, located at Mount Kent Observatory in Queensland, Australia \\citep{addison2019}. This facility is made up of four 0.7m CDK700 telescopes, which individually feed light via optic fibre into a KiwiSpec high-resolution (R = 80,000) stabilised spectrograph \\citep{barnes2012} that covers wavelengths from 480 nm to 620 nm. \n\n\\item The CHIRON spectrograph mounted on the SMARTS 1.5-m telescope \\citep{Tokovinin2018}, located at the Cerro Tololo\nInter-American Observatory (CTIO) in Chile. The high resolution cross-dispersed echelle spectrometer is fiber-fed followed by an image slicer. It has a resolution of R = 80,000 and covers wavelengths ranging from 410 to 870 nm.\n\n\\item The SOPHIE echelle spectrograph mounted on the 1.93-m Haute-Provence Observatory (OHP), France\n\\citep{2008Perruchot,2009Bouchy}. The high resolution cross-dispersed stabilized echelle spectrometer is fed by two optical fibers. Observations were taken in high-resolution mode (R = 75,000) with a wavelength range of 387 to 694 nm.\n\n\\end{itemize}\n\nReconnaissance spectroscopy with these instruments allow us to extract stellar parameters, identify spectroscopic binaries, and place upper limits on the companion masses. Spectroscopic binaries and targets whose spectral type is incompatible with the initial planet hypothesis and\/or precludes precision RV observations (giant or early type stars) are not followed up further. Promising targets, however, are monitored in order to constrain their period and place limits on their mass. \n\n\\subsection{Speckle Imaging}\n\nFor our most promising candidates we perform high resolution speckle imaging using the `Alopeke instrument on the 8.1-m Frederick C. Gillett Gemini North telescope in Maunakea, Hawaii, USA, and its twin, Zorro, on the 8.1-m Gemini South telescope on Cerro Pach\\'{o}n, Chile \\citep{Matson2019, Howell2011}. Speckle interferometric observations provide extremely high resolution images reaching the diffraction limit of the telescope. We obtain simultaneous 562 nm and 832 nm rapid exposure (60 msec) images in succession that effectively `freeze out' atmospheric turbulence and through Fourier analysis are used to search for close companion stars at 5-8 magnitude contrast levels. This analysis, along with the reconstructed images, allow us to identify nearby companions and to quantify their light contribution to the TESS aperture and thus the transit signal.\n\n\n\\section{Planet candidates and Noteworthy Systems}\n\\label{sec:PHT_canidates}\n\\subsection{Planet candidate properties}\n\nIn this final part of the paper we discuss the 90 PHT candidates around 88 host stars that passed the initial two stages of vetting and that were uploaded to ExoFOP as cTOIs. At the time of discovery none of these candidates were TOIs. The properties of all of the PHT candidates are summarised in Table~\\ref{tab:PHT-caniddates}. Candidates that have been promoted to TOI status since their PHT discovery are highlighted with an asterisk following the TIC ID, and candidates that have been shown to be false positives, based on the ground-based follow-up observations, are marked with a dagger symbol ($\\dagger$). The majority (81\\%) of PHT candidates are single transit events, indicated by an `s' following the orbital period presented in the table. \\textcolor{black}{18 of the PHT candidates were flagged as TCEs by the \\emph{TESS}\\ pipeline, but not initially promoted to TOI status. The most common reasons for this was that the pipeline identified a single-transit event as well as times of systematics (often caused by momentum dumps), due to its two-transit minimum detection threshold. This resulted in the candidate being discarded on the basis of it not passing the `odd-even' transit depth test. Out of the 18 TCEs, 14 have become TOI's since the PHT discovery. More detail on the TCE candidates can be found in Appendix~\\ref{appendixA}.} \n\nAll planet parameters (columns 2 to 8) are derived from the \\texttt{pyaneti}\\ modelling as described in Section~\\ref{sec:vetting}. Finally, the table summarises the ground-based follow-up observations (see Sec~\\ref{sec:follow_up}) that have been obtained to date, where the bracketed numbers following the observing instruments indicate the number of epochs. Unless otherwise noted, the follow-up observations are consistent with a planetary scenario. More in depth descriptions of individual targets for which we have additional information to complement the results in Table~\\ref{tab:PHT-caniddates} can be found in Appendix~\\ref{appendixA}.\n\n\\subsection{Planet candidate analysis}\n\n\nThe majority of the TOIs (87.7\\%) have orbital periods shorter than 15 days due to the requirement of observing at least two transits included in all major pipelines \\textcolor{black}{combined with the observing strategy of \\emph{TESS}}. As visual vetting does not impose these limits, the candidates outlined in this paper are helping to populate the relatively under-explored long-period region of parameter space. This is highlighted in Figure~\\ref{fig:PHT_candidates}, which shows the transit depths vs the orbital periods of the PHT single transit candidates (orange circles) and the multi-transit candidates (magenta squares) compared to the TOIs (blue circles). Values of the orbital periods and transit depths were obtained via transit modelling using \\texttt{pyaneti} (see Section~\\ref{sec:vetting}). The orbital period of single transit events are poorly constrained, which is reflected by the large errorbars in Figure~\\ref{fig:PHT_candidates}. Figure~\\ref{fig:PHT_candidates} also highlights that with PHT we are able to recover a similar range of transit depths as the pipeline found TOIs, as was previously shown in Figure~\\ref{fig:recovery_rank500_radius_period}.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_candidate_period_depth_plot_errobrars.png}\n \\caption{The properties of the PHT single transit (orange circles) and multi transit (magenta squares) candidates compared to the properties of the TOIs (blue circles). All parameters (listed in Table~\\ref{fig:PHT_candidates}) were extracted using \\texttt{pyaneti}\\ modelling.}\n \\label{fig:PHT_candidates}\n\\end{figure}\n\nThe PHT candidates were further compared to the TOIs in terms of the properties of their host stars. Figure~\\ref{fig:eep} shows the effective temperature and stellar radii as taken from the TIC \\citep{Stassun18}, for TOIs (blue dots) and the PHT candidates (magenta circles). The solid and dashed lines indicate the main sequence and post-main sequence MIST stellar evolutionary tracks \\citep{choi2016}, respectively, for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. This shows that around 10\\% of the host stars are in the process of, or have recently evolved off the main sequence. The models assume solar metalicity, no stellar rotation and no additional internal mixing.\n\n\\textcolor{black}{Ground based follow-up spectroscopy has revealed that six of the PHT candidates listed in Table~\\ref{tab:PHT-caniddates} are astrophysical false positives. As the follow-up campaign of the targets is still underway, the true false-positive rate of the candidates to have made it through all stages of the vetting process, as outlined in the methodology, will be be assessed in future PHT papers once the true nature of more of the candidates has been independently verified.}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_eep.png}\n \\caption{Stellar evolution tracks showing main sequence (solid black lines) and post-main sequence (dashed grey lines) MIST stellar evolution for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. The blue dots show the TOIs and the magenta circles show the PHT candidates.} \n \\label{fig:eep}\n\\end{figure}\n\n\n\\subsection{Stellar systems}\n\\label{subsec:PHT_stars}\n\nIn addition to the planetary candidates, citizen science allows for the identification of interesting stellar systems and astrophysical phenomena, in particular where the signals are aperiodic or small compared to the dominant stellar signal. These include light curves that exhibit multiple transit-like signals, possibly as a result of a multiple stellar system or a blend of eclipsing binaries. We have investigated all light curves that were flagged as possible multi-stellar systems via the PHT discussion boards. Similar to the planet vetting, as described in Section~\\ref{sec:vetting}, we generated {\\sc latte} data validation reports in order to assess the nature of the signal. Additionally, we subjected these systems to an iterative signal removal process, whereby we phase-folded the light curve on the dominant orbital period, binned the light curve into between 200-500 phase bins, created an interpolation model, and then subtracted said signal in order to evaluate the individual transit signals. The period of each signal, as listed in Table~\\ref{tab:PHT-multis}, was determined by phase folding the light curve at a number of trial periods and assessing by eye the best fit period and corresponding uncertainty.\n\nDue to the large \\emph{TESS}\\ pixels, blends are expected to be common. We searched for blends by generating phase folded light curves for each pixel around the source of the target in order to better locate the source of each signal. Shifts in the \\emph{TESS}\\ x and y centroid positions were also found to be good indicators of visually separated sources. Nearby sources with a magnitude difference greater than 5 mags were ruled out as possible contaminators. We consider a candidate to be a confirmed blend when the centroids are separated by more than 1 \\emph{TESS}\\ pixel, as this corresponds to an angular separation > 21 arcseconds meaning that the systems are highly unlikely to be gravitationally bound. Systems where the signal appears to be coming from the same \\emph{TESS}\\ pixel and that show no clear centroid shifts are considered to be candidate multiple systems. We note that blends are still possible, however, without further investigation we cannot conclusively rule these out as possible multi stellar systems. \n\nAll of the systems are summarised in Table~\\ref{tab:PHT-multis}. Out of the 26 systems, 6 are confirmed multiple systems which have either been published or are being prepared for publication; 7 are visually separated eclipsing binaries (confirmed blends); and 13 are candidate multiple system. Additional observations will be required to determine whether or not these candidate multiple systems are in fact gravitationally bound or photometric blends as a results of the large \\emph{TESS}\\ pixels or due to a line of sight happenstance. \n\n\\begin{landscape}\n\\begin{table}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period} & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n & \\textbf{Name} & \\textbf{(\\textcolor{black}{BJD - 2457000})} & \\textbf{(days)} & & $(R_{\\oplus})$ & \\textbf{Parameter} & \\textbf{(hours)} & & & & & \\\\\n\\hline\n101641905 & TWOMASS 11412617+3441004 & $1917.26335 _{ - 0.00072 } ^ { + 0.00071 }$ & $14.52 _{ - 5.25 } ^ { + 6.21 }(s)$ & $0.1135 _{ - 0.0064 } ^ { + 0.0032 }$ & $9.76 _{ - 0.69 } ^ { + 0.65 }$ & $0.691 _{ - 0.183 } ^ { + 0.077 }$ & $3.163 _{ - 0.088 } ^ { + 0.093 }$ & 12.196 & & & & \\\\\n103633672* & TYC 4387-00923-1 & $1850.3211 _{ - 0.00077 } ^ { + 0.00135 }$ & $90.9 _{ - 23.7 } ^ { + 46.4 }(s)$ & $0.0395 _{ - 0.0013 } ^ { + 0.0013 }$ & $3.45 _{ - 0.24 } ^ { + 0.26 }$ & $0.3 _{ - 0.21 } ^ { + 0.26 }$ & $6.7 _{ - 0.11 } ^ { + 0.12 }$ & 10.586 & & NRES (1) & & \\\\\n110996418 & TWOMASS 12344723-1019107 & $1580.6406 _{ - 0.0038 } ^ { + 0.0037 }$ & $5.18 _{ - 2.93 } ^ { + 6.86 }(s)$ & $0.1044 _{ - 0.0067 } ^ { + 0.008 }$ & $12.7 _{ - 0.99 } ^ { + 1.15 }$ & $0.44 _{ - 0.3 } ^ { + 0.3 }$ & $3.53 _{ - 0.27 } ^ { + 0.36 }$ & 13.945 & & & & \\\\\n128703021 & HIP 71639 & $1601.8442 _{ - 0.00108 } ^ { + 0.00093 }$ & $26.0 _{ - 8.22 } ^ { + 22.35 }(s)$ & $0.0254 _{ - 0.00049 } ^ { + 0.00072 }$ & $4.44 _{ - 0.2 } ^ { + 0.23 }$ & $0.47 _{ - 0.3 } ^ { + 0.22 }$ & $7.283 _{ - 0.091 } ^ { + 0.141 }$ & 6.06 & & NRES (2);MINERVA (34) & Gemini & \\\\\n138126035 & TYC 1450-00833-1 & $1954.3229 _{ - 0.0041 } ^ { + 0.0067 }$ & $28.8 _{ - 14.0 } ^ { + 203.2 }(s)$ & $0.0375 _{ - 0.0026 } ^ { + 0.0069 }$ & $4.01 _{ - 0.35 } ^ { + 0.74 }$ & $0.58 _{ - 0.38 } ^ { + 0.35 }$ & $4.65 _{ - 0.32 } ^ { + 0.85 }$ & 10.349 & & & & \\\\\n142087638 & TYC 9189-00274-1 & $1512.1673 _{ - 0.0043 } ^ { + 0.0034 }$ & $3.14 _{ - 1.41 } ^ { + 12.04 }(s)$ & $0.0469 _{ - 0.0035 } ^ { + 0.0063 }$ & $6.05 _{ - 0.54 } ^ { + 0.89 }$ & $0.5 _{ - 0.35 } ^ { + 0.36 }$ & $2.72 _{ - 0.23 } ^ { + 0.5 }$ & 11.526 & & & & \\\\\n159159904 & HIP 64812 & $1918.6109 _{ - 0.0067 } ^ { + 0.0091 }$ & $584.0 _{ - 215.0 } ^ { + 1724.0 }(s)$ & $0.0237 _{ - 0.0011 } ^ { + 0.0026 }$ & $3.12 _{ - 0.22 } ^ { + 0.36 }$ & $0.49 _{ - 0.34 } ^ { + 0.35 }$ & $15.11 _{ - 0.54 } ^ { + 0.7 }$ & 9.2 & & NRES (2) & & \\\\\n160039081* & HIP 78892 & $1752.9261 _{ - 0.0045 } ^ { + 0.005 }$ & $30.19918 _{ - 0.00099 } ^ { + 0.00094 }$ & $0.0211 _{ - 0.0013 } ^ { + 0.0035 }$ & $2.67 _{ - 0.21 } ^ { + 0.43 }$ & $0.52 _{ - 0.34 } ^ { + 0.36 }$ & $4.93 _{ - 0.27 } ^ { + 0.37 }$ & 8.35 & SBIG (1) & NRES (1);SOPHIE (4) & Gemini & \\\\\n162631539 & HIP 80264 & $1978.2794 _{ - 0.0044 } ^ { + 0.0051 }$ & $17.32 _{ - 6.66 } ^ { + 52.35 }(s)$ & $0.0195 _{ - 0.0011 } ^ { + 0.0024 }$ & $2.94 _{ - 0.24 } ^ { + 0.38 }$ & $0.48 _{ - 0.33 } ^ { + 0.36 }$ & $5.54 _{ - 0.33 } ^ { + 0.41 }$ & 7.42 & & & & \\\\\n166184426* & TWOMASS 13442500-4020122 & $1600.4409 _{ - 0.003 } ^ { + 0.0036 }$ & $16.3325 _{ - 0.0066 } ^ { + 0.0052 }$ & $0.0545 _{ - 0.0031 } ^ { + 0.0039 }$ & $1.85 _{ - 0.12 } ^ { + 0.15 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.98 _{ - 0.22 } ^ { + 0.17 }$ & 12.911 & & & & \\\\\n167661160$\\dagger$ & TYC 7054-01577-1 & $1442.0703 _{ - 0.0028 } ^ { + 0.004 }$ & $36.802 _{ - 0.07 } ^ { + 0.069 }$ & $0.0307 _{ - 0.0014 } ^ { + 0.0024 }$ & $4.07 _{ - 0.32 } ^ { + 0.43 }$ & $0.37 _{ - 0.26 } ^ { + 0.33 }$ & $5.09 _{ - 0.23 } ^ { + 0.21 }$ & 9.927 & & NRES (9);MINERVA (4) & & EB from MINERVA observations \\\\\n172370679* & TWOMASS 19574239+4008357 & $1711.95923 _{ - 0.00099 } ^ { + 0.001 }$ & $32.84 _{ - 4.17 } ^ { + 5.59 }(s)$ & $0.1968 _{ - 0.0032 } ^ { + 0.0022 }$ & $13.24 _{ - 0.43 } ^ { + 0.43 }$ & $0.22 _{ - 0.15 } ^ { + 0.14 }$ & $4.999 _{ - 0.097 } ^ { + 0.111 }$ & 14.88 & & & & Confirmed planet \\citep{canas2020}. \\\\\n174302697* & TYC 3641-01789-1 & $1743.7267 _{ - 0.00092 } ^ { + 0.00093 }$ & $498.2 _{ - 80.0 } ^ { + 95.3 }(s)$ & $0.07622 _{ - 0.00068 } ^ { + 0.00063 }$ & $13.34 _{ - 0.57 } ^ { + 0.58 }$ & $0.642 _{ - 0.029 } ^ { + 0.024 }$ & $17.71 _{ - 0.12 } ^ { + 0.13 }$ & 9.309 & SBIG (1) & & & \\\\\n179582003 & TYC 9166-00745-1 & $1518.4688 _{ - 0.0016 } ^ { + 0.0016 }$ & $104.6137 _{ - 0.0022 } ^ { + 0.0022 }$ & $0.06324 _{ - 0.0008 } ^ { + 0.0008 }$ & $7.51 _{ - 0.35 } ^ { + 0.35 }$ & $0.21 _{ - 0.15 } ^ { + 0.19 }$ & $9.073 _{ - 0.084 } ^ { + 0.097 }$ & 10.806 & & & & \\\\\n192415680 & TYC 2859-00682-1 & $1796.0265 _{ - 0.0012 } ^ { + 0.0013 }$ & $18.47 _{ - 6.34 } ^ { + 21.73 }(s)$ & $0.0478 _{ - 0.0017 } ^ { + 0.0027 }$ & $4.43 _{ - 0.33 } ^ { + 0.38 }$ & $0.45 _{ - 0.31 } ^ { + 0.31 }$ & $3.94 _{ - 0.1 } ^ { + 0.12 }$ & 9.838 & SBIG (1) & SOPHIE (2) & & \\\\\n192790476 & TYC 7595-00649-1 & $1452.3341 _{ - 0.0014 } ^ { + 0.002 }$ & $16.09 _{ - 5.73 } ^ { + 15.49 }(s)$ & $0.0438 _{ - 0.0018 } ^ { + 0.0026 }$ & $3.24 _{ - 0.34 } ^ { + 0.37 }$ & $0.37 _{ - 0.25 } ^ { + 0.3 }$ & $3.395 _{ - 0.099 } ^ { + 0.192 }$ & 10.772 & & & & \\\\\n206361691$\\dagger$ & HIP 117250 & $1363.2224 _{ - 0.0082 } ^ { + 0.009 }$ & $237.7 _{ - 81.0 } ^ { + 314.4 }(s)$ & $0.01762 _{ - 0.00088 } ^ { + 0.00125 }$ & $2.69 _{ - 0.19 } ^ { + 0.25 }$ & $0.43 _{ - 0.28 } ^ { + 0.32 }$ & $13.91 _{ - 0.53 } ^ { + 0.52 }$ & 8.88 & & CHIRON (2) & & SB2 from CHIRON \\\\\n207501148 & TYC 3881-00527-1 & $2007.7273 _{ - 0.0011 } ^ { + 0.0011 }$ & $39.9 _{ - 10.3 } ^ { + 14.3 }(s)$ & $0.0981 _{ - 0.0047 } ^ { + 0.011 }$ & $13.31 _{ - 0.95 } ^ { + 1.56 }$ & $0.9 _{ - 0.03 } ^ { + 0.039 }$ & $4.73 _{ - 0.14 } ^ { + 0.14 }$ & 10.385 & & & & \\\\\n219466784* & TYC 4409-00437-1 & $1872.6879 _{ - 0.0097 } ^ { + 0.0108 }$ & $318.0 _{ - 147.0 } ^ { + 1448.0 }(s)$ & $0.0332 _{ - 0.0024 } ^ { + 0.0048 }$ & $3.26 _{ - 0.31 } ^ { + 0.49 }$ & $0.55 _{ - 0.39 } ^ { + 0.34 }$ & $10.06 _{ - 0.81 } ^ { + 1.12 }$ & 11.099 & & & & \\\\\n219501568 & HIP 79876 & $1961.7879 _{ - 0.0018 } ^ { + 0.002 }$ & $16.5931 _{ - 0.0017 } ^ { + 0.0015 }$ & $0.0221 _{ - 0.0012 } ^ { + 0.0015 }$ & $4.22 _{ - 0.3 } ^ { + 0.35 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.615 _{ - 0.077 } ^ { + 0.093 }$ & 8.38 & & & & \\\\\n229055790 & TYC 7492-01197-1 & $1337.866 _{ - 0.0022 } ^ { + 0.0019 }$ & $48.0 _{ - 12.8 } ^ { + 48.4 }(s)$ & $0.0304 _{ - 0.00097 } ^ { + 0.00115 }$ & $3.52 _{ - 0.2 } ^ { + 0.24 }$ & $0.37 _{ - 0.26 } ^ { + 0.32 }$ & $6.53 _{ - 0.11 } ^ { + 0.14 }$ & 9.642 & & NRES (2) & & \\\\\n229608594 & TWOMASS 18180283+7428005 & $1960.0319 _{ - 0.0037 } ^ { + 0.0045 }$ & $152.4 _{ - 54.1 } ^ { + 152.6 }(s)$ & $0.0474 _{ - 0.0023 } ^ { + 0.0024 }$ & $3.42 _{ - 0.34 } ^ { + 0.36 }$ & $0.38 _{ - 0.26 } ^ { + 0.3 }$ & $6.98 _{ - 0.23 } ^ { + 0.37 }$ & 12.302 & & & & \\\\\n229742722* & TYC 4434-00596-1 & $1689.688 _{ - 0.025 } ^ { + 0.02 }$ & $29.0 _{ - 16.4 } ^ { + 66.3 }(s)$ & $0.019 _{ - 0.0028 } ^ { + 0.0029 }$ & $2.9 _{ - 0.44 } ^ { + 0.48 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $4.27 _{ - 0.09 } ^ { + 0.11 }$ & 10.33 & & NRES (8);SOPHIE (4) & Gemini & \\\\\n233194447 & TYC 4211-00650-1 & $1770.4924 _{ - 0.0065 } ^ { + 0.0107 }$ & $373.0 _{ - 101.0 } ^ { + 284.0 }(s)$ & $0.02121 _{ - 0.00073 } ^ { + 0.001 }$ & $5.08 _{ - 0.28 } ^ { + 0.33 }$ & $0.34 _{ - 0.24 } ^ { + 0.29 }$ & $24.45 _{ - 0.47 } ^ { + 0.5 }$ & 9.178 & & NRES (2) & Gemini & \\\\\n235943205 & TYC 4588-00127-1 & $1827.0267 _{ - 0.004 } ^ { + 0.0034 }$ & $121.3394 _{ - 0.0063 } ^ { + 0.0065 }$ & $0.0402 _{ - 0.0016 } ^ { + 0.0019 }$ & $4.2 _{ - 0.25 } ^ { + 0.29 }$ & $0.4 _{ - 0.27 } ^ { + 0.28 }$ & $6.37 _{ - 0.2 } ^ { + 0.3 }$ & 11.076 & & NRES (1);SOPHIE (2) & & \\\\\n237201858 & TYC 4452-00759-1 & $1811.5032 _{ - 0.0069 } ^ { + 0.0067 }$ & $129.7 _{ - 41.5 } ^ { + 146.8 }(s)$ & $0.0258 _{ - 0.0013 } ^ { + 0.0015 }$ & $4.12 _{ - 0.27 } ^ { + 0.3 }$ & $0.4 _{ - 0.28 } ^ { + 0.31 }$ & $10.94 _{ - 0.37 } ^ { + 0.53 }$ & 10.344 & & NRES (1) & & \\\\\n243187830* & HIP 5286 & $1783.7671 _{ - 0.0017 } ^ { + 0.0019 }$ & $4.05 _{ - 1.53 } ^ { + 9.21 }(s)$ & $0.0268 _{ - 0.0015 } ^ { + 0.0027 }$ & $2.06 _{ - 0.17 } ^ { + 0.23 }$ & $0.47 _{ - 0.32 } ^ { + 0.34 }$ & $2.02 _{ - 0.12 } ^ { + 0.15 }$ & 8.407 & SBIG (1) & & & \\\\\n243417115 & TYC 8262-02120-1 & $1614.4796 _{ - 0.0028 } ^ { + 0.0022 }$ & $1.81 _{ - 0.73 } ^ { + 3.45 }(s)$ & $0.0523 _{ - 0.0035 } ^ { + 0.005 }$ & $5.39 _{ - 0.47 } ^ { + 0.64 }$ & $0.47 _{ - 0.33 } ^ { + 0.34 }$ & $2.03 _{ - 0.16 } ^ { + 0.23 }$ & 11.553 & & & & \\\\\n256429408 & TYC 4462-01942-1 & $1962.16 _{ - 0.0022 } ^ { + 0.0023 }$ & $382.0 _{ - 132.0 } ^ { + 265.0 }(s)$ & $0.03582 _{ - 0.00086 } ^ { + 0.00094 }$ & $6.12 _{ - 0.29 } ^ { + 0.3 }$ & $0.51 _{ - 0.36 } ^ { + 0.18 }$ & $16.96 _{ - 0.2 } ^ { + 0.24 }$ & 8.898 & & & & \\\\\n264544388* & TYC 4607-01275-1 & $1824.8438 _{ - 0.0076 } ^ { + 0.0078 }$ & $7030.0 _{ - 6260.0 } ^ { + 3330.0 }(s)$ & $0.0288 _{ - 0.0029 } ^ { + 0.0018 }$ & $4.58 _{ - 0.43 } ^ { + 0.35 }$ & $0.936 _{ - 0.363 } ^ { + 0.011 }$ & $19.13 _{ - 1.35 } ^ { + 0.84 }$ & 8.758 & & NRES (1) & & \\\\\n264766922 & TYC 8565-01780-1 & $1538.69518 _{ - 0.00091 } ^ { + 0.00091 }$ & $3.28 _{ - 0.94 } ^ { + 1.25 }(s)$ & $0.0933 _{ - 0.0063 } ^ { + 0.0176 }$ & $16.95 _{ - 1.33 } ^ { + 3.19 }$ & $0.908 _{ - 0.039 } ^ { + 0.048 }$ & $2.73 _{ - 0.11 } ^ { + 0.11 }$ & 10.747 & & & & \\\\\n26547036* & TYC 3921-01563-1 & $1712.30464 _{ - 0.00041 } ^ { + 0.0004 }$ & $73.0 _{ - 13.6 } ^ { + 16.5 }(s)$ & $0.10034 _{ - 0.0007 } ^ { + 0.00078 }$ & $11.75 _{ - 0.59 } ^ { + 0.58 }$ & $0.17 _{ - 0.12 } ^ { + 0.11 }$ & $8.681 _{ - 0.049 } ^ { + 0.052 }$ & 9.849 & & NRES (4) & Gemini & \\\\\n267542728$\\dagger$ & TYC 4583-01499-1 & $1708.4956 _{ - 0.0073 } ^ { + 0.0085 }$ & $39.7382 _{ - 0.0023 } ^ { + 0.0023 }$ & $0.03267 _{ - 0.00089 } ^ { + 0.00175 }$ & $18.46 _{ - 0.94 } ^ { + 1.14 }$ & $0.38 _{ - 0.26 } ^ { + 0.27 }$ & $24.16 _{ - 0.39 } ^ { + 0.45 }$ & 11.474 & & & & EB from HIRES RVs. \\\\\n270371513$\\dagger$ & HIP 10047 & $1426.2967 _{ - 0.0023 } ^ { + 0.002 }$ & $0.39 _{ - 0.17 } ^ { + 1.79 }(s)$ & $0.024 _{ - 0.0015 } ^ { + 0.0032 }$ & $4.8 _{ - 0.38 } ^ { + 0.64 }$ & $0.5 _{ - 0.34 } ^ { + 0.39 }$ & $1.93 _{ - 0.16 } ^ { + 0.19 }$ & 6.98515 & & MINERVA (20) & & SB 2 from MINERVA observations. \\\\\n274599700 & TWOMASS 17011885+5131455 & $2002.1202 _{ - 0.0024 } ^ { + 0.0024 }$ & $32.9754 _{ - 0.005 } ^ { + 0.005 }$ & $0.0847 _{ - 0.0021 } ^ { + 0.0018 }$ & $13.25 _{ - 0.83 } ^ { + 0.83 }$ & $0.37 _{ - 0.24 } ^ { + 0.19 }$ & $8.2 _{ - 0.18 } ^ { + 0.21 }$ & 12.411 & & & & \\\\\n278990954 & TYC 8548-00717-1 & $1650.0191 _{ - 0.0086 } ^ { + 0.0105 }$ & $18.45 _{ - 8.66 } ^ { + 230.7 }(s)$ & $0.034 _{ - 0.0024 } ^ { + 0.0115 }$ & $9.65 _{ - 0.92 } ^ { + 3.13 }$ & $0.58 _{ - 0.4 } ^ { + 0.36 }$ & $10.62 _{ - 0.66 } ^ { + 2.46 }$ & 10.749 & & & & \\\\\n280865159* & TYC 9384-01533-1 & $1387.0749 _{ - 0.0045 } ^ { + 0.0044 }$ & $1045.0 _{ - 249.0 } ^ { + 536.0 }(s)$ & $0.0406 _{ - 0.0011 } ^ { + 0.0014 }$ & $4.75 _{ - 0.26 } ^ { + 0.28 }$ & $0.35 _{ - 0.24 } ^ { + 0.23 }$ & $19.08 _{ - 0.32 } ^ { + 0.36 }$ & 11.517 & & & Gemini & \\\\\n284361752 & TYC 3924-01678-1 & $2032.093 _{ - 0.0078 } ^ { + 0.008 }$ & $140.6 _{ - 46.6 } ^ { + 159.1 }(s)$ & $0.0259 _{ - 0.0014 } ^ { + 0.0017 }$ & $3.62 _{ - 0.26 } ^ { + 0.31 }$ & $0.4 _{ - 0.27 } ^ { + 0.34 }$ & $8.98 _{ - 0.66 } ^ { + 0.86 }$ & 10.221 & & & & \\\\\n288240183 & TYC 4634-01225-1 & $1896.941 _{ - 0.0051 } ^ { + 0.0047 }$ & $119.0502 _{ - 0.0091 } ^ { + 0.0089 }$ & $0.02826 _{ - 0.00089 } ^ { + 0.00119 }$ & $4.28 _{ - 0.35 } ^ { + 0.36 }$ & $0.55 _{ - 0.37 } ^ { + 0.25 }$ & $17.49 _{ - 0.36 } ^ { + 0.6 }$ & 9.546 & & & & \\\\\n29169215 & TWOMASS 09011787+4727085 & $1872.5047 _{ - 0.0032 } ^ { + 0.0036 }$ & $14.89 _{ - 6.12 } ^ { + 24.84 }(s)$ & $0.0403 _{ - 0.0025 } ^ { + 0.0033 }$ & $3.28 _{ - 0.37 } ^ { + 0.45 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $3.56 _{ - 0.21 } ^ { + 0.32 }$ & 11.828 & & & & \\\\\n293649602 & TYC 8103-00266-1 & $1511.2109 _{ - 0.004 } ^ { + 0.0037 }$ & $12.85 _{ - 5.34 } ^ { + 42.21 }(s)$ & $0.04 _{ - 0.0024 } ^ { + 0.0039 }$ & $4.66 _{ - 0.36 } ^ { + 0.5 }$ & $0.5 _{ - 0.35 } ^ { + 0.34 }$ & $4.1 _{ - 0.31 } ^ { + 0.56 }$ & 10.925 & & & & \\\\\n296737508 & TYC 5472-01060-1 & $1538.0036 _{ - 0.0015 } ^ { + 0.0016 }$ & $18.27 _{ - 5.06 } ^ { + 17.45 }(s)$ & $0.0425 _{ - 0.0014 } ^ { + 0.0019 }$ & $5.33 _{ - 0.22 } ^ { + 0.27 }$ & $0.44 _{ - 0.3 } ^ { + 0.26 }$ & $5.13 _{ - 0.13 } ^ { + 0.15 }$ & 9.772 & Sinistro (1) & NRES (1);MINERVA (1) & Gemini & \\\\\n298663873 & TYC 3913-01781-1 & $1830.76819 _{ - 0.00099 } ^ { + 0.00099 }$ & $479.9 _{ - 89.4 } ^ { + 109.4 }(s)$ & $0.06231 _{ - 0.00034 } ^ { + 0.00045 }$ & $11.07 _{ - 0.57 } ^ { + 0.57 }$ & $0.16 _{ - 0.11 } ^ { + 0.13 }$ & $23.99 _{ - 0.093 } ^ { + 0.1 }$ & 9.162 & & NRES (2) & Gemini & Dalba et al. (in prep) \\\\\n303050301 & TYC 6979-01108-1 & $1366.1301 _{ - 0.0022 } ^ { + 0.0023 }$ & $281.0 _{ - 170.0 } ^ { + 264.0 }(s)$ & $0.0514 _{ - 0.0027 } ^ { + 0.0018 }$ & $4.85 _{ - 0.32 } ^ { + 0.32 }$ & $0.73 _{ - 0.48 } ^ { + 0.1 }$ & $7.91 _{ - 0.31 } ^ { + 0.36 }$ & 10.048 & & NRES (1) & Gemini & \\\\\n303317324 & TYC 6983-00438-1 & $1365.1845 _{ - 0.0023 } ^ { + 0.0028 }$ & $69.0 _{ - 25.5 } ^ { + 78.1 }(s)$ & $0.0365 _{ - 0.0013 } ^ { + 0.0016 }$ & $2.88 _{ - 0.3 } ^ { + 0.31 }$ & $0.39 _{ - 0.26 } ^ { + 0.32 }$ & $5.78 _{ - 0.18 } ^ { + 0.24 }$ & 10.799 & & & & \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\emph{Note} -- Candidates that have become TOIs following the PHT discovery are marked with an asterisk (*). The `s' following the orbital period indicates that the candidates is a single transit event. The ground-based follow-up observations are summarized in columns 10-12, where the bracketed numbers correspond the number of epochs obtained with each instrument. See Section~\\ref{sec:follow_up} for description of each instrument. The $\\dagger$ symbol indicates candidates that have been shown to be astrophysical false positives based on the ground based follow-up observations.}\n\\label{tab:PHT-caniddates}\n\\end{table}\n\\end{landscape}\n\n\\begin{landscape}\n\\begin{table}\n\\addtocounter{table}{-1}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period} & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n & \\textbf{Name} & \\textbf{(\\textcolor{black}{BJD - 2457000})} & \\textbf{(days)} & & $(R_{\\oplus})$ & \\textbf{Parameter} & \\textbf{(hours)} & & & & & \\\\\n\\hline\n303586471$\\dagger$ & HIP 115828 & $1363.7692 _{ - 0.0033 } ^ { + 0.0027 }$ & $13.85 _{ - 4.19 } ^ { + 18.2 }(s)$ & $0.0214 _{ - 0.001 } ^ { + 0.0014 }$ & $2.52 _{ - 0.16 } ^ { + 0.2 }$ & $0.4 _{ - 0.27 } ^ { + 0.33 }$ & $4.23 _{ - 0.19 } ^ { + 0.16 }$ & 8.27 & & MINERVA (11) & & SB 2 from MINERVA observations. \\\\\n304142124* & HIP 53719 & $1585.28023 _{ - 0.0008 } ^ { + 0.0008 }$ & $42.8 _{ - 10.0 } ^ { + 18.2 }(s)$ & $0.04311 _{ - 0.00093 } ^ { + 0.00153 }$ & $4.1 _{ - 0.23 } ^ { + 0.24 }$ & $0.33 _{ - 0.21 } ^ { + 0.21 }$ & $5.66 _{ - 0.067 } ^ { + 0.09 }$ & 8.62 & & NRES (1);MINERVA (4) & & Confirmed planet \\citep{diaz2020} \\\\\n304339227 & TYC 9290-01087-1 & $1673.3242 _{ - 0.009 } ^ { + 0.0128 }$ & $111.9 _{ - 72.2 } ^ { + 4844.1 }(s)$ & $0.0253 _{ - 0.0024 } ^ { + 0.0481 }$ & $3.27 _{ - 0.61 } ^ { + 5.72 }$ & $0.67 _{ - 0.47 } ^ { + 0.36 }$ & $7.44 _{ - 0.86 } ^ { + 2.84 }$ & 9.169 & & & & \\\\\n307958020 & TYC 4191-00309-1 & $1864.82 _{ - 0.014 } ^ { + 0.013 }$ & $169.0 _{ - 107.0 } ^ { + 10194.0 }(s)$ & $0.0223 _{ - 0.0022 } ^ { + 0.0543 }$ & $3.92 _{ - 0.52 } ^ { + 9.27 }$ & $0.71 _{ - 0.53 } ^ { + 0.33 }$ & $12.48 _{ - 1.1 } ^ { + 5.41 }$ & 9.017 & & & & \\\\\n308301091 & TYC 2081-01273-1 & $2030.3691 _{ - 0.0024 } ^ { + 0.0026 }$ & $29.24 _{ - 8.49 } ^ { + 22.46 }(s)$ & $0.0362 _{ - 0.0013 } ^ { + 0.0014 }$ & $5.41 _{ - 0.34 } ^ { + 0.35 }$ & $0.35 _{ - 0.25 } ^ { + 0.29 }$ & $6.57 _{ - 0.14 } ^ { + 0.19 }$ & 10.273 & & & & \\\\\n313006381 & HIP 45012 & $1705.687 _{ - 0.0081 } ^ { + 0.0045 }$ & $21.56 _{ - 8.9 } ^ { + 54.15 }(s)$ & $0.0261 _{ - 0.0017 } ^ { + 0.0027 }$ & $2.34 _{ - 0.2 } ^ { + 0.27 }$ & $0.45 _{ - 0.3 } ^ { + 0.38 }$ & $3.85 _{ - 0.51 } ^ { + 0.31 }$ & 9.39 & & & & \\\\\n323295479* & TYC 9506-01881-1 & $1622.9258 _{ - 0.00083 } ^ { + 0.00087 }$ & $117.8 _{ - 25.8 } ^ { + 30.9 }(s)$ & $0.0981 _{ - 0.0021 } ^ { + 0.0023 }$ & $11.35 _{ - 0.67 } ^ { + 0.66 }$ & $0.839 _{ - 0.024 } ^ { + 0.019 }$ & $6.7 _{ - 0.14 } ^ { + 0.15 }$ & 10.595 & & & & \\\\\n328933398.01* & TYC 4634-01435-1 & $1880.9878 _{ - 0.0039 } ^ { + 0.0042 }$ & $24.9335 _{ - 0.0046 } ^ { + 0.005 }$ & $0.0437 _{ - 0.0022 } ^ { + 0.0023 }$ & $4.62 _{ - 0.32 } ^ { + 0.33 }$ & $0.38 _{ - 0.25 } ^ { + 0.27 }$ & $5.02 _{ - 0.22 } ^ { + 0.27 }$ & 11.215 & & & & Potential multi-planet system. \\\\\n328933398.02* & TYC 4634-01435-1 & $1848.6557 _{ - 0.0053 } ^ { + 0.0072 }$ & $50.5 _{ - 22.4 } ^ { + 77.1 }(s)$ & $0.0296 _{ - 0.0028 } ^ { + 0.0033 }$ & $3.14 _{ - 0.33 } ^ { + 0.39 }$ & $0.41 _{ - 0.28 } ^ { + 0.35 }$ & $5.99 _{ - 0.8 } ^ { + 0.77 }$ & 11.215 & & & & \\\\\n331644554 & TYC 3609-00469-1 & $1757.0354 _{ - 0.0031 } ^ { + 0.0033 }$ & $947.0 _{ - 215.0 } ^ { + 274.0 }(s)$ & $0.12 _{ - 0.025 } ^ { + 0.021 }$ & $21.84 _{ - 4.57 } ^ { + 3.86 }$ & $1.018 _{ - 0.036 } ^ { + 0.028 }$ & $10.93 _{ - 0.34 } ^ { + 0.35 }$ & 9.752 & & & & \\\\\n332657786 & TWOMASS 09595797-1609323 & $1536.7659 _{ - 0.0015 } ^ { + 0.0015 }$ & $63.76 _{ - 9.52 } ^ { + 11.13 }(s)$ & $0.14961 _{ - 0.00064 } ^ { + 0.00029 }$ & $3.83 _{ - 0.12 } ^ { + 0.12 }$ & $0.059 _{ - 0.041 } ^ { + 0.064 }$ & $3.333 _{ - 0.095 } ^ { + 0.096 }$ & 15.99 & & & & \\\\\n336075472 & TYC 3526-00332-1 & $2028.1762 _{ - 0.0043 } ^ { + 0.0037 }$ & $61.9 _{ - 24.0 } ^ { + 95.6 }(s)$ & $0.0402 _{ - 0.0022 } ^ { + 0.0033 }$ & $3.09 _{ - 0.34 } ^ { + 0.4 }$ & $0.43 _{ - 0.29 } ^ { + 0.32 }$ & $5.39 _{ - 0.23 } ^ { + 0.37 }$ & 11.842 & & & & \\\\\n349488688.01 & TYC 1529-00224-1 & $1994.283 _{ - 0.0038 } ^ { + 0.0033 }$ & $11.6254 _{ - 0.005 } ^ { + 0.0052 }$ & $0.02195 _{ - 0.00096 } ^ { + 0.00122 }$ & $3.44 _{ - 0.18 } ^ { + 0.21 }$ & $0.39 _{ - 0.27 } ^ { + 0.3 }$ & $5.58 _{ - 0.15 } ^ { + 0.18 }$ & 8.855 & & NRES (2);SOPHIE (2) & & Potential multi-planet system. \\\\\n349488688.02 & TYC 1529-00224-1 & $2002.77063 _{ - 0.00097 } ^ { + 0.00103 }$ & $15.35 _{ - 1.94 } ^ { + 4.15 }(s)$ & $0.03688 _{ - 0.00067 } ^ { + 0.00069 }$ & $5.78 _{ - 0.18 } ^ { + 0.18 }$ & $0.24 _{ - 0.16 } ^ { + 0.21 }$ & $6.291 _{ - 0.058 } ^ { + 0.074 }$ & 8.855 & & NRES (2);SOPHIE (2) & & \\\\\n356700488* & TYC 4420-01295-1 & $1756.638 _{ - 0.013 } ^ { + 0.011 }$ & $184.5 _{ - 64.7 } ^ { + 333.1 }(s)$ & $0.0173 _{ - 0.0011 } ^ { + 0.0015 }$ & $2.92 _{ - 0.2 } ^ { + 0.28 }$ & $0.44 _{ - 0.3 } ^ { + 0.34 }$ & $11.76 _{ - 0.65 } ^ { + 1.03 }$ & 8.413 & & & & \\\\\n356710041* & TYC 1993-00419-1 & $1932.2939 _{ - 0.0019 } ^ { + 0.0019 }$ & $29.6 _{ - 14.0 } ^ { + 19.0 }(s)$ & $0.0496 _{ - 0.0021 } ^ { + 0.0011 }$ & $14.82 _{ - 0.85 } ^ { + 0.84 }$ & $0.66 _{ - 0.42 } ^ { + 0.11 }$ & $12.76 _{ - 0.24 } ^ { + 0.24 }$ & 9.646 & & & & \\\\\n369532319 & TYC 2743-01716-1 & $1755.8158 _{ - 0.006 } ^ { + 0.0051 }$ & $35.4 _{ - 12.0 } ^ { + 51.6 }(s)$ & $0.0316 _{ - 0.0023 } ^ { + 0.0028 }$ & $3.43 _{ - 0.3 } ^ { + 0.37 }$ & $0.41 _{ - 0.29 } ^ { + 0.34 }$ & $5.5 _{ - 0.32 } ^ { + 0.32 }$ & 10.594 & & & Gemini & \\\\\n369779127 & TYC 9510-00090-1 & $1643.9403 _{ - 0.0046 } ^ { + 0.0058 }$ & $9.93 _{ - 3.38 } ^ { + 19.74 }(s)$ & $0.0288 _{ - 0.0015 } ^ { + 0.0033 }$ & $4.89 _{ - 0.31 } ^ { + 0.56 }$ & $0.46 _{ - 0.31 } ^ { + 0.33 }$ & $5.64 _{ - 0.38 } ^ { + 0.33 }$ & 9.279 & & & & \\\\\n384159646* & TYC 9454-00957-1 & $1630.39405 _{ - 0.00079 } ^ { + 0.00079 }$ & $11.68 _{ - 2.75 } ^ { + 4.21 }(s)$ & $0.0658 _{ - 0.0012 } ^ { + 0.0011 }$ & $9.87 _{ - 0.45 } ^ { + 0.44 }$ & $0.27 _{ - 0.18 } ^ { + 0.21 }$ & $5.152 _{ - 0.069 } ^ { + 0.087 }$ & 10.158 & SBIG (1) & NRES (8);MINERVA (6) & Gemini & \\\\\n385557214 & TYC 1807-00046-1 & $1791.58399 _{ - 0.00068 } ^ { + 0.0007 }$ & $5.62451 _{ - 0.0004 } ^ { + 0.00043 }$ & $0.096 _{ - 0.019 } ^ { + 0.032 }$ & $8.32 _{ - 2.06 } ^ { + 2.77 }$ & $0.95 _{ - 0.075 } ^ { + 0.053 }$ & $1.221 _{ - 0.094 } ^ { + 0.058 }$ & 10.856 & & & & \\\\\n388134787 & TYC 4260-00427-1 & $1811.034 _{ - 0.015 } ^ { + 0.017 }$ & $246.0 _{ - 127.0 } ^ { + 6209.0 }(s)$ & $0.0265 _{ - 0.0024 } ^ { + 0.023 }$ & $2.57 _{ - 0.28 } ^ { + 2.19 }$ & $0.55 _{ - 0.39 } ^ { + 0.44 }$ & $8.85 _{ - 1.13 } ^ { + 1.84 }$ & 10.95 & & NRES (1) & Gemini & \\\\\n404518509 & HIP 16038 & $1431.2696 _{ - 0.0037 } ^ { + 0.0035 }$ & $26.83 _{ - 9.46 } ^ { + 56.14 }(s)$ & $0.0259 _{ - 0.0013 } ^ { + 0.0022 }$ & $2.94 _{ - 0.21 } ^ { + 0.29 }$ & $0.47 _{ - 0.31 } ^ { + 0.34 }$ & $5.02 _{ - 0.23 } ^ { + 0.28 }$ & 9.17 & & & & \\\\\n408636441* & TYC 4266-00736-1 & $1745.4668 _{ - 0.0016 } ^ { + 0.0015 }$ & $37.695 _{ - 0.0034 } ^ { + 0.0033 }$ & $0.0485 _{ - 0.0019 } ^ { + 0.0023 }$ & $3.32 _{ - 0.16 } ^ { + 0.19 }$ & $0.39 _{ - 0.27 } ^ { + 0.29 }$ & $3.63 _{ - 0.1 } ^ { + 0.14 }$ & 11.93 & SBIG (1) & & Gemini & Half of the period likely. \\\\\n418255064 & TWOMASS 13063680-8037015 & $1629.3304 _{ - 0.0018 } ^ { + 0.0018 }$ & $25.37 _{ - 7.06 } ^ { + 15.41 }(s)$ & $0.0732 _{ - 0.0029 } ^ { + 0.0031 }$ & $5.57 _{ - 0.36 } ^ { + 0.38 }$ & $0.37 _{ - 0.25 } ^ { + 0.25 }$ & $3.83 _{ - 0.13 } ^ { + 0.14 }$ & 12.478 & SBIG (1) & & Gemini & \\\\\n420645189$\\dagger$ & TYC 4508-00478-1 & $1837.4767 _{ - 0.0018 } ^ { + 0.0017 }$ & $250.2 _{ - 66.6 } ^ { + 99.4 }(s)$ & $0.0784 _{ - 0.0033 } ^ { + 0.0046 }$ & $8.82 _{ - 0.55 } ^ { + 0.7 }$ & $0.892 _{ - 0.026 } ^ { + 0.028 }$ & $6.95 _{ - 0.27 } ^ { + 0.3 }$ & 10.595 & & MINERVA (1) & & SB 2 from MINERVA observations. \\\\\n422914082 & TYC 0046-00133-1 & $1431.5538 _{ - 0.0014 } ^ { + 0.0017 }$ & $12.91 _{ - 3.91 } ^ { + 8.97 }(s)$ & $0.0418 _{ - 0.0015 } ^ { + 0.0016 }$ & $3.96 _{ - 0.32 } ^ { + 0.35 }$ & $0.36 _{ - 0.25 } ^ { + 0.28 }$ & $4.07 _{ - 0.09 } ^ { + 0.126 }$ & 11.026 & Sinistro (1) & NRES (1) & & \\\\\n427344083 & TWOMASS 22563609+7040518 & $1961.8967 _{ - 0.0031 } ^ { + 0.0036 }$ & $7.77 _{ - 5.6 } ^ { + 9.65 }(s)$ & $0.107 _{ - 0.016 } ^ { + 0.025 }$ & $12.27 _{ - 1.87 } ^ { + 2.9 }$ & $0.834 _{ - 0.484 } ^ { + 0.094 }$ & $2.88 _{ - 0.3 } ^ { + 0.42 }$ & 13.404 & & & & \\\\\n436873727 & HIP 13224 & $1803.83679 _{ - 0.00058 } ^ { + 0.00056 }$ & $19.26 _{ - 5.95 } ^ { + 6.73 }(s)$ & $0.05246 _{ - 0.00061 } ^ { + 0.00059 }$ & $10.02 _{ - 0.43 } ^ { + 0.41 }$ & $0.767 _{ - 0.057 } ^ { + 0.038 }$ & $5.462 _{ - 0.081 } ^ { + 0.074 }$ & 7.51 & & & & \\\\ \n441642457* & TYC 3858-00452-1 & $1745.5102 _{ - 0.0108 } ^ { + 0.0097 }$ & $79.8072 _{ - 0.0071 } ^ { + 0.0076 }$ & $0.0281 _{ - 0.0024 } ^ { + 0.0033 }$ & $3.55 _{ - 0.34 } ^ { + 0.46 }$ & $0.934 _{ - 0.023 } ^ { + 0.026 }$ & $6.9 _{ - 0.39 } ^ { + 0.6 }$ & 9.996 & & & & \\\\\n441765914* & TWOMASS 17253007+7552562 & $1769.6154 _{ - 0.0058 } ^ { + 0.0093 }$ & $161.6 _{ - 58.2 } ^ { + 1460.1 }(s)$ & $0.0411 _{ - 0.0024 } ^ { + 0.0119 }$ & $3.6 _{ - 0.3 } ^ { + 1.01 }$ & $0.45 _{ - 0.32 } ^ { + 0.48 }$ & $7.44 _{ - 0.36 } ^ { + 1.08 }$ & 11.638 & & & & \\\\\n452920657 & TWOMASS 00332018+5906355 & $1810.5765 _{ - 0.0031 } ^ { + 0.003 }$ & $53.2 _{ - 29.0 } ^ { + 34.3 }(s)$ & $0.135 _{ - 0.016 } ^ { + 0.012 }$ & $9.71 _{ - 1.16 } ^ { + 0.9 }$ & $0.73 _{ - 0.48 } ^ { + 0.11 }$ & $4.6 _{ - 0.26 } ^ { + 0.29 }$ & 14.167 & SBIG (1) & & & \\\\\n455737331 & TYC 2779-00785-1 & $1780.7084 _{ - 0.008 } ^ { + 0.0073 }$ & $50.4 _{ - 17.6 } ^ { + 75.0 }(s)$ & $0.0257 _{ - 0.0016 } ^ { + 0.002 }$ & $3.05 _{ - 0.24 } ^ { + 0.29 }$ & $0.43 _{ - 0.29 } ^ { + 0.33 }$ & $6.6 _{ - 0.43 } ^ { + 0.5 }$ & 10.189 & SBIG (1) & & Gemini & \\\\\n456909420 & TYC 1208-01094-1 & $1779.4109 _{ - 0.0026 } ^ { + 0.0022 }$ & $5.78 _{ - 5.29 } ^ { + 5.95 }(s)$ & $0.078 _{ - 0.031 } ^ { + 0.045 }$ & $9.15 _{ - 3.61 } ^ { + 5.27 }$ & $0.973 _{ - 0.495 } ^ { + 0.063 }$ & $1.73 _{ - 0.27 } ^ { + 0.28 }$ & 10.941 & & & & \\\\\n458451774 & TWOMASS 12551793+4431260 & $1917.1875 _{ - 0.0019 } ^ { + 0.0019 }$ & $12.39 _{ - 6.34 } ^ { + 83.97 }(s)$ & $0.0752 _{ - 0.0054 } ^ { + 0.0211 }$ & $3.33 _{ - 0.26 } ^ { + 0.92 }$ & $0.61 _{ - 0.43 } ^ { + 0.32 }$ & $2.08 _{ - 0.19 } ^ { + 0.59 }$ & 13.713 & & & & \\\\\n48018596 & TYC 3548-00800-1 & $1713.4514 _{ - 0.0063 } ^ { + 0.0046 }$ & $100.1145 _{ - 0.0018 } ^ { + 0.0021 }$ & $0.049 _{ - 0.0081 } ^ { + 0.018 }$ & $7.88 _{ - 1.33 } ^ { + 2.9 }$ & $0.984 _{ - 0.028 } ^ { + 0.027 }$ & $2.83 _{ - 0.26 } ^ { + 0.29 }$ & 9.595 & & NRES (1) & Gemini & \\\\\n53309262 & TWOMASS 07475406+5741549 & $1863.1133 _{ - 0.0064 } ^ { + 0.0061 }$ & $294.8 _{ - 96.0 } ^ { + 327.0 }(s)$ & $0.1239 _{ - 0.0075 } ^ { + 0.0098 }$ & $5.38 _{ - 0.36 } ^ { + 0.46 }$ & $0.46 _{ - 0.31 } ^ { + 0.28 }$ & $6.74 _{ - 0.45 } ^ { + 0.62 }$ & 15.51 & & & & \\\\\n53843023 & TYC 6956-00758-1 & $1328.0335 _{ - 0.0054 } ^ { + 0.0057 }$ & $202.0 _{ - 189.0 } ^ { + 272.0 }(s)$ & $0.058 _{ - 0.02 } ^ { + 0.056 }$ & $5.14 _{ - 1.77 } ^ { + 4.99 }$ & $0.962 _{ - 0.597 } ^ { + 0.083 }$ & $4.25 _{ - 0.72 } ^ { + 0.66 }$ & 11.571 & & & & \\\\\n55525572* & TYC 8876-01059-1 & $1454.6713 _{ - 0.0066 } ^ { + 0.0065 }$ & $83.8951 _{ - 0.004 } ^ { + 0.004 }$ & $0.0343 _{ - 0.001 } ^ { + 0.0021 }$ & $7.31 _{ - 0.46 } ^ { + 0.56 }$ & $0.43 _{ - 0.29 } ^ { + 0.31 }$ & $13.54 _{ - 0.3 } ^ { + 0.51 }$ & 10.358 & & CHIRON (5) & Gemini & Confirmed planet \\citep{2020eisner} \\\\\n63698669* & TYC 6993-00729-1 & $1364.6226 _{ - 0.0074 } ^ { + 0.0067 }$ & $73.6 _{ - 26.8 } ^ { + 133.6 }(s)$ & $0.0248 _{ - 0.0019 } ^ { + 0.0023 }$ & $2.15 _{ - 0.2 } ^ { + 0.25 }$ & $0.42 _{ - 0.29 } ^ { + 0.35 }$ & $5.63 _{ - 0.32 } ^ { + 0.57 }$ & 10.701 & SBIG (1) & & & \\\\\n70887357* & TYC 5883-01412-1 & $1454.3341 _{ - 0.0016 } ^ { + 0.0015 }$ & $56.1 _{ - 15.3 } ^ { + 18.8 }(s)$ & $0.0605 _{ - 0.0027 } ^ { + 0.0027 }$ & $12.84 _{ - 0.86 } ^ { + 0.9 }$ & $0.917 _{ - 0.028 } ^ { + 0.016 }$ & $7.29 _{ - 0.18 } ^ { + 0.19 }$ & 9.293 & & & & \\\\\n7422496$\\dagger$ & HIP 25359 & $1470.3625 _{ - 0.0031 } ^ { + 0.0023 }$ & $61.4 _{ - 16.7 } ^ { + 49.0 }(s)$ & $0.0255 _{ - 0.001 } ^ { + 0.0011 }$ & $2.44 _{ - 0.15 } ^ { + 0.16 }$ & $0.37 _{ - 0.25 } ^ { + 0.29 }$ & $5.89 _{ - 0.15 } ^ { + 0.15 }$ & 9.36 & & MINERVA (4) & & SB 2 from MINERVA observations. \\\\\n82452140 & TYC 3076-00921-1 & $1964.292 _{ - 0.011 } ^ { + 0.011 }$ & $21.1338 _{ - 0.0052 } ^ { + 0.0066 }$ & $0.0266 _{ - 0.0019 } ^ { + 0.0027 }$ & $2.95 _{ - 0.25 } ^ { + 0.34 }$ & $0.42 _{ - 0.29 } ^ { + 0.36 }$ & $5.87 _{ - 0.62 } ^ { + 0.94 }$ & 10.616 & & & & \\\\\n88840705 & TYC 3091-00808-1 & $2026.6489 _{ - 0.001 } ^ { + 0.001 }$ & $260.6 _{ - 87.6 } ^ { + 142.2 }(s)$ & $0.109 _{ - 0.023 } ^ { + 0.027 }$ & $9.98 _{ - 2.28 } ^ { + 2.75 }$ & $1.001 _{ - 0.042 } ^ { + 0.037 }$ & $4.72 _{ - 0.13 } ^ { + 0.15 }$ & 9.443 & & & & \\\\\n91987762* & HIP 47288 & $1894.25381 _{ - 0.00051 } ^ { + 0.00047 }$ & $10.51 _{ - 3.48 } ^ { + 3.67 }(s)$ & $0.05459 _{ - 0.00106 } ^ { + 0.00097 }$ & $9.56 _{ - 0.56 } ^ { + 0.52 }$ & $0.771 _{ - 0.062 } ^ { + 0.033 }$ & $4.342 _{ - 0.073 } ^ { + 0.063 }$ & 7.87 & & NRES (4) & Gemini & \\\\\n95768667 & TYC 1434-00331-1 & $1918.3318 _{ - 0.0093 } ^ { + 0.0079 }$ & $26.9 _{ - 12.4 } ^ { + 72.3 }(s)$ & $0.0282 _{ - 0.0022 } ^ { + 0.0031 }$ & $3.54 _{ - 0.32 } ^ { + 0.43 }$ & $0.48 _{ - 0.33 } ^ { + 0.35 }$ & $5.4 _{ - 0.64 } ^ { + 0.76 }$ & 10.318 & & & & \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\textbf{Properties of PHT candidates (continued)}}\n\\label{tab:PHT-caniddates2}\n\\end{table}\n\\end{landscape}\n\n\n\\section{Conclusion}\n\\label{sec:condlusion}\n\nWe present the results from the analysis of the first 26 \\emph{TESS}\\ sectors. The outlined citizen science approach engages over 22 thousand registered citizen scientists who completed 12,617,038 classifications from December 2018 through August 2020 for the sectors observed during the first two years of the \\emph{TESS}\\ mission. We applied a systematic search for planetary candidates using visual vetting by multiple volunteers to identify \\emph{TESS}\\ targets that are most likely to host a planet. Between 8 and 15 volunteers have inspected each \\emph{TESS}\\ light curve and marked times of transit-like events using the PHT online interface. For each light curve, the markings from all the volunteers who saw that target were combined using an unsupervised machine learning method, known as DBSCAN, in order to identify likely transit-like events. Each of these identified events was given a transit score based on the number of volunteers who identified a given event and on the user weighting of each of those volunteers. Individual user weights were calculated based on the user's ability to identify simulated transit events, injected into real \\emph{TESS}\\ light curves, that are displayed on the PHT site alongside of the real data. The transit scores were then used to generate a ranked list of candidates that range from most likely to least likely to host a planet candidate. The top 500 highest ranked candidates were further vetted by the PHT science team. This stage of vetting primarily made use of the open source {\\sc latte} \\citep{LATTE2020} tool which generates a number of standard diagnostic plots that help identify promising candidates and weed out false positive signals. \n\nOn average we found around three high priority candidates per sector which were followed up using ground based telescopes, where possible. To date, PHT has statistically confirmed one planet, TOI-813 \\citep{2020eisner}: a Saturn-sized planet on an 84 day orbit around a subgiant host star. Other PHT identified planets listed in this paper are being followed up by other teams of astronomers, such as TOI-1899 (TIC 172370679) which was recently confirmed to be a warm Jupiter transiting an M-dwarf \\citep{canas2020}. The remaining candidates outlined in this paper require further follow-up observations to confirm their planetary nature.\n\nThe sensitivity of our transit search effort was assessed using synthetic data, as well as the known TOI and TCE candidates flagged by the SPOC pipeline. For simulated planets (where simulated signals are injected into real \\emph{TESS}\\ light curves) we have shown that the recovery efficiency of human vetting starts to decrease for transit-signals that have a SNR less than 7.5. The detection efficiency was further evaluated by the fractional recovery of the TOI and TCEs. We have shown that PHT is over 85 \\% complete in the recovery of planets that have a radius greater than 4 $R_{\\oplus}$, 51 \\% complete for radii between 3 and 4 $R_{\\oplus}$ and 49 \\% complete for radii between 2 and 3 $R_{\\oplus}$. Furthermore, we have shown that human vetting is not sensitive to the number of transits present in the light curve, meaning that they are equally likely to identify candidates on longer orbital periods as they are those with shorter orbital periods for periods greater than $\\sim$ 1 day. Planets with periods shorter than around 1 day exhibit over 20 transits within one \\emph{TESS}\\ sectors resulting in a decrease in identification by the volunteers. This is due to many volunteers only marking a random subset of these events, resulting in a lack of consensus on any given transit event and thus decreasing the overall transit score of these light curves. \n\nIn addition to searching for signals due to transiting exoplanets, PHT provides a platform that can be used to identify other stellar phenomena that may otherwise be difficult to identify with automated pipelines. Such phenomena, including eclipsing binaries, multiple stellar systems, dwarf novae, and stellar flares are often mentioned on the PHT discussion forums where volunteers can use searchable hashtags and comments to bring these systems to the attention of other citizen scientists as well as the PHT science team. All of the eclipsing binaries identified on the site, for example, are being used and vetted by the \\emph{TESS}\\ Eclipsing Binary Working Group (Prsa et al. in prep). Furthermore, we have investigated the nature of all of the targets that were identified as possible multiple stellar systems, as summarised in Table~\\ref{tab:PHT-multis}.\n\nOverall we have shown that large scale visual vetting can complement the findings \\textcolor{black}{from the major \\emph{TESS}\\ pipeline} by identifying longer period planets that may only exhibit a single transit event in their light curve, as well as in finding signals that are aperiodic or embedded in a strong varying stellar signal. The identification of planets around stars with variable signals allow us to potentially characterise the host-star (e.g., with asteroseismology or spot modulation). Additionally, the longer period planets are integral to our understanding of how planet systems form and evolve, as they allow us to investigate the evolution of planets that are farther away from their host star and therefore less dependent on stellar radiation. \\textcolor{black}{While automated pipelines specifically designed to identify single transit events in the \\emph{TESS}\\ data exist \\citep[e.g., ][]{Gill2020}, neither their methodology nor the full list of their findings are yet publicly available and thus we are unable to compare results.} \n\nThe planets that PHT finds have longer periods ($\\gtrsim$ 27 d) than those found in \\emph{TESS}\\ data using automated pipelines, and are more typical of the Kepler sample (25\\% of Kepler confirmed planets have periods greater than 27 days\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}). However, the Kepler planets are considerably fainter, and thus less amenable to ground-based follow-up or atmospheric characterisation from space (CHEOPS and JWST). Thus PHT helps to bridge the parameter spaces covered by these two missions, by identifying longer period planet candidates around bright, nearby stars, for which we can ultimately obtain precise planetary mass estimates. Although statistical characterisation of exo-planetary systems is no doubt important, precise mass measurements are key to developing our understanding of exoplanets and the physics which dictate their evolution. In particular, identification of this PHT sample provides follow-up targets to investigate the dependence of photo-evaporation on the mass of planets as well as on the planet radius, and will help our understanding of the photo-evaporation valley at longer orbital periods \\citep{Owen2013}. \n\nPHT will continue to operate throughout the \\emph{TESS}\\ extended mission, hopefully allowing us to identify even longer period planets as well as help verify some of the existing candidates with additional transits. \n\n\n\n\\begin{table*}\n\\resizebox{0.95\\textwidth}{!}{\n\\begin{tabular}{cccccccccc}\n\\textbf{TIC} & \\textbf{Period (days)} & \\textbf{Epoch (\\textcolor{black}{BJD - 2457000})} & \\textbf{Depth (ppm)} & \\textbf{Comment} \\\\\n\\hline\n13968858 & $3.4850 \\pm 0.001$ & $ 1684.780 \\pm 0.005$ & 410000 & Candidate multiple system \\\\\n & $1.4380 \\pm 0.001$ & $ 1684.335 \\pm 0.005$ & 50000 & \\\\\n35655828 & $ 8.073 \\pm 0.01$ & $ 1550.94 \\pm 0.01 $ & 23000 & Confirmed blend \\\\\n & $ 1.220 \\pm 0.001 $ & $ 1545.540 \\pm 0.005 $ & 2800 & \\\\\n63291675 & $ 8.099 \\pm 0.003 $ & $ 1685.1 \\pm 0.01 $ & 60000 & Confirmed blend \\\\\n & $ 1.4635 \\pm 0.0005 $ & $ 1683.8 \\pm 0.1 $ & 7000 & \\\\\n63459761 & $4.3630 \\pm 0.003 $ & $ 1714.350 \\pm 0.005 $ & 160000 & Candidate multiple system \\\\\n & $4.235 \\pm $ 0.005 & $ 1715.130 \\pm 0.03$ & 35000 & \\\\\n104909909 & $1.3060 \\pm 0.0001$ & $ 1684.470 \\pm 0.005$ & 32000 & Candidate multiple system \\\\\n & $2.5750 \\pm 0.003$ & $ 1684.400 \\pm 0.005$ & 65000 & \\\\\n115980439 & $ 4.615 \\pm 0.002 $ & $ 1818.05 \\pm 0.01 $ & 95000 & Confirmed blend \\\\\n & $ 0.742 \\pm 0.005 $ & $ 1816.23 \\pm 0.02 $ & 2000 & \\\\\n120362128 & $ 3.286 \\pm 0.002 $ & $ 1684.425 \\pm 0.01 $ & 33000 & Candidate multiple system \\\\\n & $ - $ & $ 1701.275 \\pm 0.02 $ & 12000 & \\\\\n & $ - $ & $ 1702.09 \\pm 0.02 $ & 36000 & \\\\\n121945407 & $ 0.9056768 \\pm 0.00000002$ & $-1948.76377 \\pm 0.0000001$ & 2500 & Confirmed multiple system $^{(\\mathrm{a})}$ \\\\\n & $ 45.4711 \\pm 0.00002$ & $-1500.0038 \\pm 0.0004 $ & 7500 & \\\\\n122275115 & $ - $ & $ 1821.779 \\pm 0.01 $ & 155000 & Candidate multiple system \\\\\n & $ - $ & $ 1830.628 \\pm 0.01 $ & 63000 & \\\\\n & $ - $ & $ 1838.505 \\pm 0.01 $ & 123000 & \\\\\n229804573 & $1.4641 \\pm 0.0005$ & $ 1326.135 \\pm 0.005$ & 180000 & Candidate multiple system \\\\\n & $0.5283 \\pm 0.0001$ & $ 1378.114 \\pm 0.005$ & 9000 & \\\\\n252403752 & $ - $ & $ 1817.73 \\pm 0.01 $ & 2800 & Candidate multiple system \\\\\n & $ - $ & $ 1829.76 \\pm 0.01 $ & 23000 & \\\\\n & $ - $ & $ 1833.63 \\pm 0.01 $ & 5500 & \\\\\n258837989 & $0.8870 \\pm 0.001$ & $ 1599.350 \\pm 0.005$ & 64000 & Candidate multiple system \\\\\n & $3.0730 \\pm 0.001$ & $ 1598.430 \\pm 0.005$ & 25000 & \\\\\n266958963 & $1.5753 \\pm 0.0002$ & $ 1816.425 \\pm 0.001$ & 265000 & Candidate multiple system \\\\\n & $2.3685 \\pm 0.0001$ & $ 1817.790 \\pm 0.001$ & 75000 & \\\\\n278956474 & $5.488068 \\pm 0.000016 $ & $ 1355.400 \\pm 0.005$ & 93900 & Confirmed multiple system $^{(\\mathrm{b})}$ \\\\\n & $5.674256 \\pm -0.000030$ & $ 1330.690 \\pm 0.005$ & 30000 & \\\\\n284925600 & $ 1.24571 \\pm 0.00001 $ & $ 1765.248 \\pm 0.005 $ & 490000 & Confirmed blend \\\\\n & $ 0.31828 \\pm 0.00001 $ & $ 1764.75 \\pm 0.005 $ & 35000 & \\\\\n293954660 & $2.814 \\pm 0.001 $ & $ 1739.177 \\pm 0.03 $ & 272000 & Confirmed blend \\\\\n & $4.904 \\pm 0.03 $ & $ 1739.73 \\pm 0.01 $ & 9500 & \\\\\n312353805 & $4.951 \\pm 0.003 $ & $ 1817.73 \\pm 0.01 $ & 66000 & Confirmed blend \\\\\n & $12.89 \\pm 0.01 $ & $ 1822.28 \\pm 0.01$ & 19000 & \\\\\n318210930 & $ 1.3055432 \\pm 0.000000033$ & $ -653.21602 \\pm 0.0000013$ & 570000 & Confirmed multiple system $^{(\\mathrm{c})}$ \\\\\n & $ 0.22771622 \\pm 0.0000000035$& $ -732.071119 \\pm 0.00000026 $ & 220000 & \\\\\n336434532 & $ 3.888 \\pm 0.002 $ & $ 1713.66 \\pm 0.01 $ & 22900 & Confirmed blend \\\\\n & $ 0.949 \\pm 0.003 $ & $ 1712.81 \\pm 0.01 $ & 2900 & \\\\\n350622185 & $1.1686 \\pm 0.0001$ & $ 1326.140 \\pm 0.005$ & 200000 & Candidate multiple system \\\\\n & $5.2410 \\pm 0.0005$ & $ 1326.885 \\pm 0.05$ & 4000 & \\\\\n375422201 & $9.9649 \\pm 0.001$ & $ 1711.937 \\pm 0.005$ & 245000 & Candidate multiple system \\\\\n & $4.0750 \\pm 0.001$ & $ 1713.210 \\pm 0.01 $ & 39000 & \\\\\n376606423 & $ 0.8547 \\pm 0.0002 $ & $ 1900.766 \\pm 0.005 $ & 9700 & Candidate multiple system \\\\\n & $ - $ & $ 1908.085 \\pm 0.01 $ & 33000 & \\\\\n394177355 & $ 94.22454 \\pm 0.00040 $ & $ - $ & - & Confirmed multiple system $^{(\\mathrm{d})}$ \\\\\n & $ 8.6530941 \\pm 0.0000016$ & $-2038.99492 \\pm 0.00017 $ & 140000 & \\\\\n & $ 1.5222468 \\pm 0.0000025$ & $ -2039.1201 \\pm 0.0014 $ & - & \\\\\n & $ 1.43420486 \\pm 0.00000012 $ & $-2039.23941 \\pm 0.00007 $ & - & \\\\\n424508303 & $ 2.0832649 \\pm 0.0000029 $ & $-3144.8661 \\pm 0.0034 $ & 430000 & Confirmed multiple system $^{(\\mathrm{e})}$ \\\\\n & $ 1.4200401 \\pm 0.0000042 $ & $-3142.5639 \\pm 0.0054 $ & 250000 & \\\\\n441794509 & $ 4.6687 \\pm 0.0002 $ & $ 1958.895 \\pm 0.005 $ & 34000 & Candidate multiple system \\\\\n & $ 14.785 \\pm 0.002 $ & $ 1960.845 \\pm 0.005 $ & 17000 & \\\\\n470710327 & $ 9.9733 \\pm 0.0001 $ & $ 1766.27 \\pm 0.005 $ & 51000 & Confirmed multiple system $^{(\\mathrm{f})}$ \\\\\n & $ 1.104686 \\pm 0.00001 $ & $ 1785.53266 \\pm 0.000005$ & 42000 & \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\nNote -- $^{(\\mathrm{a})}$ KOI-6139, \\citet{Borkovits2013}; \n$^{(\\mathrm{b})}$ \\citet{2020Rowden}\n$^{(\\mathrm{c})}$ \\citet{Koo2014}; \n$^{(\\mathrm{d})}$ KOI-3156, \\citet{2017Helminiak};\n$^{(\\mathrm{e})}$ V994 Her; \\citet{Zasche2016}; \n$^{(\\mathrm{f})}$ Eisner et al. {\\it in prep.}\n}\n\n\\label{tab:PHT-multis}\n\n\\end{table*}\n\n\\section*{Data Availability}\n\nAll of the \\emph{TESS}\\ data used within this article are hosted and made publicly available by the Mikulski Archive for Space Telescopes (MAST, \\url{http:\/\/archive.stsci.edu\/tess\/}). Similarly, the Planet Hunters TESS classifications made by the citizen scientists can be found on the Planet Hunters Analysis Database (PHAD, \\url{https:\/\/mast.stsci.edu\/phad\/}), which is also hosted by MAST. All planet candidates and their properties presented in this article have been uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS, \\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}) website as community TOIs (cTOIs), under their corresponding TIC IDs. The ground-based follow-up observations of individual targets will be shared on reasonable request to the corresponding author.\n\nThe models of individual transit events and the data validation reports used for the vetting of the targets were both generated using publicly available open software codes, \\texttt{pyaneti}\\ and {\\sc latte}.\n\n\\section*{Acknowledgements} \n\nThis project works under the in \\textit{populum veritas est} philosophy, and for that reason we would like to thank all of the citizen scientists who have taken part in the Planet Hunters TESS project and enable us to find many interesting astrophysical systems. \n\nSome of the observations in the paper made use of the High-Resolution Imaging instruments `Alopeke and Zorro. `Alopeke and Zorro were funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. `Alopeke and Zorro were mounted on the Gemini North and South telescope of the international Gemini Observatory, a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci\\'{o}n y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog\\'{i}a e Innovaci\\'{o}n (Argentina), Minist\\'{e}rio da Ci\\^{e}ncia, Tecnologia, Inova\\c{c}\\~{o}es e Comunica\\c{c}\\~{o}es (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). The authors also acknowledge the very significant cultural role and sacred nature of Maunakea. We are most fortunate to have the opportunity to conduct observations from this mountain.\n\nThis project has also received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement N$^\\circ$730890. This material reflects only the authors views and the Commission is not liable for any use that may be made of the information contained therein. This work makes use of observations from the Las Cumbres Observatory global telescope network, including the NRES spectrograph and the SBIG and Sinistro photometric instruments. \n\nFurthermore, NLE thanks the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant N$^\\circ$1829740, the Brinson Foundation, and the Moore Foundation; her participation in the program has benefited this work. Finally, CJ acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement N$^\\circ$670519: MAMSIE), and from the Research Foundation Flanders (FWO) under grant agreement G0A2917N (BlackGEM). \n\nThis research made use of Astropy, a community-developed core Python package for Astronomy \\citep{astropy2013}, matplotlib \\citep{matplotlib}, pandas \\citep{pandas}, NumPy \\citep{numpy}, astroquery \\citep{ginsburg2019astroquery} and sklearn \\citep{pedregosa2011scikit}. \n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Introduction}\n\nSince the first unambiguous discovery of an exoplanet in 1995 \\citep[][]{Mayor1995} over 4,000 more have been confirmed. Studies of their characteristics have unveiled an extremely wide range of planetary properties in terms of planetary mass, size, system architecture and orbital periods, greatly revolutionising our understanding of how these bodies form and evolve.\n\nThe transit method, whereby we observe a temporary decrease in the brightness of a star due to a planet passing in front of its host star, is to date the most successful method for planet detection, having discovered over 75\\% of the planets listed on the NASA Exoplanet Archive\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}. It yields a wealth of information including planet radius, orbital period, system orientation and potentially even atmospheric composition. Furthermore, when combined with Radial Velocity \\citep[RV; e.g.,][]{Mayor1995, Marcy1997} observations, which yield the planetary mass, we can infer planet densities, and thus their internal bulk compositions. Other indirect detection methods include radio pulsar timing \\citep[e.g.,][]{Wolszczan1992} and microlensing \\citep[e.g.,][]{Gaudi2012}.\n\n\nThe \\textit{Transiting Exoplanet Survey Satellite} mission \\citep[\\protect\\emph{TESS};][]{ricker15} is currently in its extended mission, searching for transiting planets orbiting bright ($V < 11$\\,mag) nearby stars. Over the course of the two year nominal mission, \\emph{TESS}\\ monitored around 85 per cent of the sky, split up into 26 rectangular sectors of 96 $\\times$ 24 deg each (13 per hemisphere). Each sector is monitored for $\\approx$ 27.4 continuous days, measuring the brightness of $\\approx$ 20,000 pre-selected stars every two minutes. In addition to these short cadence (SC) observations, the \\emph{TESS}\\ mission provides Full Frame Images (FFI) that span across all pixels of all CCDs and are taken at a cadence of 30 minutes. While most of the targets ($\\sim$ 63 per cent) will be observed for $\\approx$ 27.4 continuous days, around $\\sim$ 2 per cent of the targets at the ecliptic poles are located in the `continuous viewing zones' and will be continuously monitored for $\\sim$ 356 days.\n\nStars themselves are extremely complex, with phenomena ranging from outbursts to long and short term variability and oscillations, which manifest themselves in the light curves. These signals, as well as systematic effects and artifacts introduced by the telescope and instruments, mean that standard periodic search methods, such as the Box-Least-Squared method \\citep{bls2002} can struggle to identify certain transit events, especially if the observed signal is dominated by natural stellar variability. Standard detection pipelines also tend to bias the detection of short period planets, as they typically require a minimum of two transit events in order to gain the signal-to-noise ratio (SNR) required for detection.\n\nOne of the prime science goals of the \\emph{TESS}\\ mission is to further our understanding of the overall planet population, an active area of research that is strongly affected by observational and detection biases. In order for exoplanet population studies to be able to draw meaningful conclusions, they require a certain level of completeness in the sample of known exoplanets as well as a robust sample of validated planets spanning a wide range of parameter space. \\textcolor{red}{Due to this, we independently search the \\emph{TESS}\\ light curves for transiting planets via visual vetting in order to detect candidates that were either intentionally ignored by the main \\emph{TESS}\\ pipelines, which require at least two transits for a detection, missed because of stellar variability or instrumental artefacts, or were identified but subsequently erroneously discounted at the vetting stage, usually because the period found by the pipeline was incorrect. These candidates can help populate under-explored regions of parameter space and will, for example, benefit the study of planet occurrence rates around different stellar types as well as inform theories of physical processes involved with the formation and evolution of different types of exoplanets.}\n\nHuman brains excel in activities related to pattern recognition, making the task of identifying transiting events in light curves, even when the pattern is in the midst of a strong varying signal, ideally suited for visual vetting. Early citizen science projects, such as Planet Hunters \\citep[PH;][]{fischer12} and Exoplanet Explorers \\citep{Christiansen2018}, successfully harnessed the analytic power of a large number of volunteers and made substantial contributions to the field of exoplanet discoveries. The PH project, for example, showed that human vetting has a higher detection efficiency than automated detection algorithms for certain types of transits. In particular, they showed that citizen science can outperform on the detection of single (long-period) transits \\citep[e.g.,][]{wang13, schmitt14a}, aperiodic transits \\citep[e.g. circumbinary planets;][]{schwamb13} and planets around variable stars \\citep[e.g., young systems,][]{fischer12}. Both PH and Exoplanet Explorers, which are hosted by the world's largest citizen science platform Zooniverse \\citep{lintott08}, ensured easy access to \\textit{Kepler} and \\textit{K2} data by making them publicly available online in an immediately accessible graphical format that is easy to understand for non-specialists. The popularity of these projects is reflected in the number of participants, with PH attracting 144,466 volunteers from 137 different countries over 9 years of the project being active.\n\nFollowing the end of the \\textit{Kepler} mission and the launch of the \\emph{TESS}\\ satellite in 2018, PH was relaunched as the new citizen science project \\textit{Planet Hunters TESS} (PHT) \\footnote{\\url{www.planethunters.org}}, with the aim of identifying transit events in the \\emph{TESS}\\ data that were \\textcolor{red}{intentionally ignored or missed} by the main \\emph{TESS}\\ pipelines. \\textcolor{red}{Such a search complements other methods methods via its sensitivity to single-transit, and, therefore, longer period planets. Additionally, other dedicated non-citizen science based methods are also employed to look for single transit candidates \\citep[see e.g., the Bayesian transit fitting method by ][]{Gill2020, Osborn2016}}.\n\nCitizen science transit searches specialise in finding the rare events that the standard detection pipelines miss, however, these results are of limited use without an indication of the completeness of the search. Addressing the problem of completeness was therefore one of our highest priorities while designing PHT as discussed throughout this paper. \n\nThe layout of the remainder of the paper will be as follows. An overview of the Planet Hunters TESS project is found in Section~\\ref{sec:PHT}, followed by an in depth description of how the project identifies planet candidates in Section~\\ref{sec:method}. The recovery efficiency of the citizen science approach is assessed in Section~\\ref{sec:recovery_efficiency}, followed by a description of the in-depth vetting of candidates and ground based-follow up efforts in Section~\\ref{sec:vetting} and \\ref{sec:follow_up}, respectively. Planet Candidates and noteworthy systems identified by Planet Hunters TESS are outlined in Section~\\ref{sec:PHT_canidates}, followed by a discussion of the results in Section~\\ref{sec:condlusion}.\n\n\\section{Planet Hunters TESS}\n\\label{sec:PHT}\n\nThe PHT project works by displaying \\emph{TESS}\\ light curves (Figure~\\ref{fig:interface}), and asking volunteers to identify transit-like signals. Only the two-minute cadence targets, which are produced by the \\emph{TESS}\\ pipeline at the Science Processing Operations Center \\citep[SPOC,][]{Jenkins2018} and made publicly available by the Mikulski Archive for Space Telescopes (MAST)\\footnote{\\url{http:\/\/archive.stsci.edu\/tess\/}}, are searched by PHT. First-time visitors to the PHT site, or returning visitors who have not logged in are prompted to look through a short tutorial, which briefly explains the main aim of the project and shows examples of transit events and other stellar phenomena. Scientific explanation of the project can be found elsewhere on the site in the `field guide' and on the project's `About' page. \n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=\\textwidth]{Figures\/PHT_new_interface.png}\n \\caption{\n PHT user interface showing a simulated light curve. The transit events are highlighted with white partially-transparent columns that are drawn on using the mouse. Stellar information on the target star is available by clicking on `subject info' below the light curve.} \n \\label{fig:interface}\n\\end{figure*}\n\nAfter viewing the tutorial, volunteers are ready to participate in the project and are presented with \\emph{TESS}\\ light curves (known as `subjects') that need to be classified. The project was designed to be as simple as possible and therefore only asks one question: \\textit{`Do you see a transit?}'. Users identify transit-like events, and the time of their occurrence, by drawing a column over the event using the mouse button, as shown in Figure~\\ref{fig:interface}. There is no limit on the number of transit-like events that can be marked in a light curve. No markings indicate that there are no transit-like events present in the light curve. Once the subject has been analysed, users submit their classification and continue to view the next light curve by clicking `Done'. \n\nAlongside each light curve, users are offered information on the stellar properties of the target, such as the radius, effective temperature and magnitude (subject to availability, see \\cite{Stassun18}). However, in order to reduce biases in the classifications, the TESS Input Catalog (TIC) ID of the target star is not provided until after the subject classification has been submitted.\n\nIn addition to classifying the data, users are given the option to comment on light curves via the `Talk' discussion forum. Each light curve has its own discussion page to allow volunteers to discuss and comment, as well as to `tag' light curves using searchable hashtags, and to bring promising candidates to the attention of other users and the research team. The talk discussion forums complement the main PHT analysis and have been shown to yield interesting objects which may be challenging to detect using automated algorithms \\citep[e.g.,][]{eisner2019RN}. Unlike in the initial PH project, there are no questions in the main interface regarding stellar variability, however, volunteers are encouraged to mention astrophysical phenomenon or \\textit{unusual} features, such as eclipsing binaries or stellar flares, using the `Talk' discussion forum. \n\nThe subject TIC IDs are revealed on the subject discussion pages, allowing volunteers to carry out further analysis on specific targets of interest and to report and discuss their findings. This is extremely valuable for both other volunteers and the PHT science team, as it can speed up the process of identifying candidates as well as rule out false positives in a fast and effective manner. \n\nSince the launch of PHT on 6 December 2018, there has been one significant makeover to the user interface. The initial PHT user interface (UI1), which was used for sectors 1 through 9, split the \\emph{TESS}\\ light curves up into either three or four chunks (depending on the data gaps in each sector) which lasted around seven days each. This allowed for a more `zoomed' in view of the data, making it easier to identify transit-like events than when the full $\\sim$ 30 day light curves were shown. The results from a PHT beta project, which displayed only simulated data, showed that a more zoomed in view of the light curve was likely to yield a higher transit recovery rate.\n\nThe updated, and current, user interface (UI2) allows users to manually zoom in on the x-axis (time) of the data. Due to this additional feature, each target has been displayed as a single light curve as of Sector 10. In order to verify that the changes in interface did not affect our findings, all of the Sector 9 subjects were classified using both UI1 and UI2. We saw no significant change in the number of candidates recovered (see Section~\\ref{sec:recovery_efficiency} for a description of how we quantified detection efficiency).\n\n\n\\subsection{Simulated Data}\n\\label{subsec:sims} \n\nIn addition to the real data, volunteers are shown simulated light curves, which are generated by randomly injecting simulated transit signals, provided by the SPOC pipeline \\citep[][]{Jenkins2018}, into real \\emph{TESS}\\ light curves. The simulated data play an important role in assessing the sensitivity of the project, training the users and providing immediate feedback, and to gauge the relative abilities of individual users (see Sec~\\ref{subsec:weighting}). \n\nWe calculate a signal to noise ratio (SNR) of the injected signal by dividing the injected transit depth by the Root Mean Square Combined Differential Photometric Precision (RMS CDPP) of the light curve on 0.5-, 1- or 2-hr time scales (whichever is closest to the duration of the injected transit signal). Only simulations with a SNR greater than 7 in UI1 and greater than 4 for UI2 are shown to volunteers.\n\nSimulated light curves are randomly shown to the volunteers and classified in the exact same manner as the real data. The user is always notified after a simulated light curve has been classified and given feedback as to whether the injected signal was correctly identified or not. For each sector, we generate between one and two thousand simulated light curves, using the real data from that sector in order to ensure that the sector specific systematic effects and data gaps of the simulated data do not differ from the real data. The rate at which a volunteer is shown simulated light curves decreases from an initial rate of 30 per cent for the first 10 classifications, down to a rate of 1 per cent by the time that the user has classified 100 light curves. \n\n\n\\section{Identifying Candidates}\n\\label{sec:method}\n\nEach subject is seen by multiple volunteers, before it is `retired' from the site, and the classifications are combined (see Section~\\ref{subsec:DBscan}) in order to assess the likelihood of a transit event. For sectors 1 through 9, the subjects were retired after 8 classifications if the first 8 volunteers who saw the light curves did not mark any transit events, after 10 classifications if the first 10 volunteers all marked a transit event and after 15 classifications if there was not complete consensus amongst the users. As of Sector 9 with UI2, all subjects were classified by 15 volunteers, regardless of whether or not any transit-like events were marked. Sector 9, which was classified with both UI1 and UI2, was also classified with both retirement rules.\n\nThere were a total of 12,617,038 individual classifications completed across the project on the nominal mission data. 95.4 per cent of these classifications were made by 22,341 registered volunteers, with the rest made by unregistered volunteers. Around 25 per cent of the registered volunteers complete more than 100 classifications, 11.8 per cent more than 300, 8.4 per cent more than 500, 5.4 per cent more than 1000 and 1.1 per cent more than 10,000. The registered volunteers completed a mean and median of 647 and 33 classifications, respectively. Figure~\\ref{fig:user_count} shows the distribution in user effort for logged in users who made between 0 and 300 classifications. \n\nThe distribution in the number of classifications made by the registered volunteers is assessed using the Gini coefficient, which ranges from 0 (equal contributions from all users) to 1 (large disparity in the contributions). The Gini coefficients for individual sectors ranges from 0.84 to 0.91 with a mean of 0.87, while the Gini coefficient for the overall project (all of the sectors combined) is 0.94. The mean Gini coefficient among other astronomy Zooniverse projects lies at 0.82 \\citep{spiers2019}. We note that the only other Zooniverse project with an equally high Gini coefficient as PHT is \\textit{Supernova Hunters}, a project which, similarly to PHT and unlike most other Zooniverse projects, has periodic data releases that are accompanied by an e-newsletter sent to all project volunteers. Periodic e-newsletters have the effect of promoting the project to both regularly and irregularly participating volunteers, who may only complete a couple of classifications as they explore the task, as well as to returning users who complete a large number of classifications following every data release, increasing the disparity in user contributions (the Gini coefficient).\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.5\\textwidth]{Figures\/user_count.png}\n \\caption{\n The distribution of the number of classifications by the registered volunteers, using a bin size of 5 from 0 to 300 classifications. A total of 11.8 per cent of the registered volunteers completed more than 300 classifications.} \n \\label{fig:user_count}\n\\end{figure}\n\n\n\\subsection{User Weighting}\n\\label{subsec:weighting} \nUser weights are calculated for each individual volunteer in order to identify users who are more sensitive to detecting transit-like signals and those who are more likely to mark false positives. The weighting scheme is based on the weighting scheme described by \\cite{schwamb12}.\n\nUser weights are calculated independently for each observation sector, using the simulated light curves shown alongside the data from that sector. All users start off with a weighting of one, which is then increased or decreased when a simulated transit event is correctly or incorrectly identified, respectively. \n\nSimulated transits are deemed correctly identified, or `True', if the mid-point of a user's marking falls within the width of the simulated transit events. If none of the user's markings fall within this range, the simulated transit is deemed not identified, or `False'. If more than one of a user's markings coincide with the same simulated signal, it is only counted as being correct once, such that the total number of `True' markings cannot exceed the number of injected signals. For each classification, we record the number of `Extra' markings, which is the total number of markings made by the user minus the number of correctly identified simulated transits. \n\nEach simulated light curve, identified by superscript $i$ (where $i=1$, \\ldots, $N$) was seen by $K^{(i)}$ users (the mean value of $K^{(i)}$\nwas 10), and contained $T^{(i)}$ simulated transits (where $T^{(i)}$ depends on the period of the simulated transit signal and the duration of the light curve). For a specific light curve $i$, each user who saw the light curve is identified by a subscript $k$ (where $k=1$, \\ldots, $K^{(i)}$) and each injected transit by a subscript $t$ (where $t=1$, \\ldots, $T^{(i)}$). \n\nIn order to distinguish between users who are able to identify obvious transits and those who are also able to find those that are more difficult to see, we start by defining a `recoverability' $r^{(i)}_t$ for each injected transit $t$ in each light curve. This is defined empirically, as the number of users who identified the transit correctly divided by $K^{(i)}$ (the total number of users who saw the light curve in question).\n\nNext, we quantify the performance of each user on each light curve as follows (this performance is analogous to the `seed' defined in \\citealt{schwamb12}, but we define it slightly differently):\n\\begin{equation}\n p^{(i)}_{k} = C_{\\rm E} ~ \\frac{E^{(i)}_{k}}{\\langle E^{(i)} \\rangle} + \\sum_{t=1}^{T^{(i)}} \\begin{cases}\n C_{\\rm T} ~ \\left[ r^{(i)}_t \\right]^{-1}, & \\text{if $m^{(i)}_{t,k} = $`True'}\\\\\n C_{\\rm F} ~ r^{(i)}_t, & \\text{if $m^{(i)}_{t,k} = $`False'},\n \\end{cases}\n\\end{equation}\nwhere $m^{(i)}_{t,k}$ is the identification of transit $t$ by user $k$ in light curve $i$, which is either `True' or `False'; $E^{(i)}_{k}$ is the number of `Extra' markings made by user $k$ for light curve $i$, and $\\langle E^{(i)} \\rangle$ is the mean number of `Extra' markings made by all users who saw subject $i$. The parameters $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ control the impact of the `Extra', `True' and `False' markings on the overall user weightings, and are optimized empirically as discussed below in Section~\\ref{subsec:optimizesearch}. \n\nFollowing \\citealt{schwamb12}, we then assign a global `weight' $w_k$ to each user $k$, which is defined as:\n\\begin{equation}\n\\begin{split}\n\tw_k = I \\times (1 + \\log_{10} N_k)^{\\nicefrac{\\sum_i p^{(i)}_k}{N_k}}\n\\label{equ:weight}\n\\end{split}\n\\end{equation}\nwhere $I$ is an empirical normalization factor, such that the distribution of user weights remains centred on one, $N_k$ is the total number of simulated transit events that user $k$ assessed, and the sum over $i$ concerns only the light curves that user $k$ saw. \nWe limit the user weights to the range 0.05--3 \\emph{a posteriori}.\n\n\nWe experimented with a number of alternative ways to define the user weights, including the simpler $w_k=\\nicefrac{\\sum_i p^{(i)}_k}{N_k}$, but Eqn.~\\ref{equ:weight} was found to give the best results (see Section~\\ref{sec:recovery_efficiency} for how this was evaluated).\n\n\\subsection{Systematic Removal}\n\\label{subsec:sysrem} \nSystematic effects, for example caused by the spacecraft or background events, can result in spurious signals that affect a large subset of the data, resulting in an excess in markings of transit-like events at certain times within an observation sector. As the four \\emph{TESS}\\ cameras can yield unique systematic effects, the times of systematics were identified uniquely for each camera. The times were identified using a Kernel Density Estimation \\citep[KDE;][]{rosenblatt1956} with a cosine kernel and a bandwidth of 0.1 days, applied across all of the markings from that sector for each camera. Fig.~\\ref{fig:sys_rem} shows the KDE of all marked transit-events made during Sector 17 for TESS's cameras 1 (top panel) to 4 (bottom panel). The isolated spikes, or prominences, in the number of marked events, such as at T = 21-22 days in the bottom panel, are assumed to be caused by systematic effects that affect multiple light curves. Prominences are considered significant if they exceed a factor four times the standard deviation of the kernel output, which was empirically determined to be the highest cut-off to not miss clearly visible systematics. All user-markings within the full width at half maximum of these peaks are omitted from all further analysis. \\textcolor{red}{The KDE profiles for each Sector are provided as electronic supplementary material.}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.46\\textwidth]{Figures\/systematics_sec17.png}\n \\caption{\n Kernel density estimation of the user-markings made for Sector 17, for targets observed with TESS's observational Cameras 1 (top panel) to 4 (bottom panel). The orange vertical lines the indicate prominences that are at least four times greater than the standard deviation of the distribution. The black points underneath the figures show the mid-points of all of the volunteer-markings, where darker regions represent a higher density of markings.}\n \\label{fig:sys_rem}\n\\end{figure}\n\n\\subsection{Density Based Clustering}\n\\label{subsec:DBscan} \n\nThe times and likelihoods of transit-like events are determined by combining all of the classifications made for each subject and identifying times where multiple volunteers identified a signal. We do this using an unsupervised machine learning method, known as DBSCAN \\citep[][Density-Based Spatial Clustering of Applications with Noise]{ester1996DB}. DBSCAN is a non-parametric density based clustering algorithm that helps to distinguish between dense clusters of data and sparse noise. For a data point to belong to a cluster it must be closer than a given distance ($\\epsilon$) to at least a set minimum number of other points (minPoints). \n\nIn our case, the data points are one-dimensional arrays of times of transits events, as identified by the volunteers, and clusters are times where multiple volunteers identified the same event. For each cluster a `transit score' ($s_i$) is determined, which is the sum of the user weights of the volunteers who contribute to the given cluster divided by the sum of the user weights of volunteers who saw that light curve. These transit scores are used to rank subjects from most to least likely to contain a transit-like event. Subjects which contain multiple successful clusters with different scores are ranked by the highest transit score. \n\n\\subsection{Optimizing the search}\n\\label{subsec:optimizesearch}\n\nThe methodology described in Sections~\\ref{subsec:weighting} to \\ref{subsec:DBscan} has five free parameters: the number of markings required to constitute a cluster ($minPoints$), the maximum separation of markings required for members of a cluster ($\\epsilon$), and $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ used in the weighting scheme. The values of these parameters were optimized via a grid search, where $C_{\\rm E}$ and $C_{\\rm F}$ ranged from -5 to 0, $C_{\\rm T}$ ranged from 0 to 20, and $minPoints$ ranged from 1 to 8, all in steps of 1. ($\\epsilon$) ranged from 0.5 to 1.5 in steps of 0.5. This grid search was carried out on 4 sectors, two from UI1 and two from UI2, for various variations of Equation~\\ref{equ:weight}. \n\nThe success of each combination of parameters was assessed by the fractions of TOIs and TCEs that were recovered within the top highest ranked 500 candidates, as discussed in more detail Section~\\ref{sec:recovery_efficiency}. We found the most successful combination of parameters to be $minPoints$ = 4 markings, $\\epsilon$, = 1 day, $C_{\\rm T}$ = 3, $C_{\\rm F}$= -2 and $C_{\\rm E}$ = -2.\n\n\\subsection{MAST deliverables}\n\\label{subsec:deliverables}\n\nThe analysis described above is carried out both in real-time as classifications are made, as well as offline after all of the light curves of a given sector have been classified. When the real-time analysis identifies a successful DB cluster (i.e. when at least four citizen scientists identified a transit within a day of the \\emph{TESS}\\ data of one another), the potential candidate is automatically uploaded to the open access Planet Hunters Analysis Database (PHAD) \\footnote{\\url{https:\/\/mast.stsci.edu\/phad\/}} hosted by the Mikulski Archive for Space Telescopes (MAST) \\footnote{\\url{https:\/\/archive.stsci.edu\/}}. While PHAD does not list every single classification made on PHT, it does display all transit candidates which had significant consensus amongst the volunteers who saw that light curve, along with the user-weight-weighted transit scores. This analysis does not apply the systematics removal described in Section~\\ref{subsec:sysrem}. The aim of PHAD is to provide an open source database of potential planet candidates identified by PHT, and to credit the volunteers who identified said targets. \n\nThe offline analysis is carried out following the complete classifications of all of the data from a given \\emph{TESS}\\ sector. The combination of all of the classifications allows us to identify and remove times of systematics and calculate better calibrated and more representative user weights. The remainder of this paper will only discuss the results from the offline analysis.\n\n\\section{Recovery Efficiency}\n\\label{sec:recovery_efficiency}\n\\subsection{Recovery of simulated transits}\n\nThe recovery efficiency is, in part, assessed by analysing the recovery rate of the injected transit-like signals (see Section~\\ref{subsec:sims}). Figure~\\ref{fig:SIM_recovery} shows the median and mean transit scores (fraction of volunteers who correctly identified a given transit scaled by user weights) of the simulated transits within SNR bins ranging from 4 to 20 in steps of 0.5. Simulations with a SNR less than 4 were not shown on PHT. The figure highlights that transit signals with a SNR of 7.5 or greater are correctly identified by the vast majority of volunteers. \n\n\\textcolor{red}{As the simulated data solely consist of real light curves with synthetically injected transit signals, we do not have any light curves, simulated or otherwise, which we can guarantee do not contain any planetary transits (real or injected). As such, this prohibits us from using simulated data to infer an analogous false-positive rate.}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.47\\textwidth]{Figures\/SIMS_recovery.png}\n \\caption{The median (blue) and mean (orange) transit scores for injected transits with SNR ranges between 4 and 20. The mean and median are calculated in SNR bins with a width of 0.5, as indicated by the horizontal lines around each data point. \n }\n \\label{fig:SIM_recovery}\n\\end{figure}\n\n\\subsection{Recovery of TCEs and TOIs}\n\\label{subsec:TCE_TOI}\nThe recovery efficiency of PHT is assessed further using the planet candidates identified by the SPOC pipeline \\citep{Jenkins2018}. The SPOC pipeline extracts and processes all of the 2-minute cadence \\emph{TESS}\\ light curves prior to performing a large scale transit search. Data Validation (DV) reports, which include a range of transit diagnostic tests, are generated by the pipeline for around 1250 Threshold Crossing Events (TCEs), which were flagged as containing two or more transit-like features. Visual vetting is then performed by the \\emph{TESS}\\ science team on these targets, and promising candidates are added to the catalog of \\emph{TESS}\\ Objects of Interest (TOIs). Each sector yields around 80 TOIs \\textcolor{red}{and a mean of 1025 TCEs.}\n\nFig~\\ref{fig:TCE_TOI_recovery} shows the fraction of TOIs and TCEs (top and bottom panel respectively) that we recover with PHT as a function of the rank, where a higher rank corresponds to a lower transit score, for Sectors 1 through 26. TOIs and TCEs with R < 2 $R_{\\oplus}$ are not included in this analysis, as the initial PH showed that human vetting alone is unable to reliably recover planets smaller than 2 $R_{\\oplus}$ \\citep{schwamb12}. Planets smaller than 2 $R_{\\oplus}$ are, therefore, not the main focus of our search.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI-recovery_radlim2.png}\n \\caption{The fraction of recovered TOIs and TCEs (top and bottom panel respectively) with R > 2$R_{\\oplus}$ as a function of the rank, for sectors 1 to 26. The lines represent the results from different observation sectors.}\n \\label{fig:TCE_TOI_recovery}\n\\end{figure}\n\n\nFig~\\ref{fig:TCE_TOI_recovery} shows a steep increase in the fractional TOI recovery rate up to a rank of $\\sim$ 500. Within the 500 highest ranked PHT candidates for a given sector, we are able to recover between 46 and 62 \\% (mean of 53 \\%) of all of the TOIs (R > 2 $R_{\\oplus}$), a median 90 \\% of the TOIs where the SNR of the transit events are greater than 7.5 and median 88 \\% of TOIs where the SNR of the transit events are greater than 5.\n\nThe relation between planet recovery rate and the SNR of the transit events is further highlighted in Figure~\\ref{fig:TOI_properties}, which shows the SNR vs the orbital period of the recovered TOIs. The colour of the markers indicate the TOI's rank within a given sector, with the lighter colours representing a lower rank. The circles and crosses represent candidates at a rank lower and higher than 500, respectively. The figure shows that transit events with a SNR less than 3.5 are missed by the majority of volunteers, whereas events with a SNR greater than 5 are mostly recovered within the top 500 highest ranked candidates. \n\nThe steep increase in the fractional TOI recovery rate at lower ranks, as shown in figure~\\ref{fig:TCE_TOI_recovery}, is therefore due to the detection of the high SNR candidates that are identified by most, if not all, of the PHT volunteers who classified those targets. At a rank of around 500, the SNR of the TOIs tends towards the limit of what human vetting can detect and thus the identification of TOIs beyond a rank of 500 is more sporadic.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.49\\textwidth]{Figures\/TOI_recovery_properties.png}\n \\caption{The SNR vs orbital period of TOIs with R > 2$R_{\\oplus}$. The colour represents their rank within the sector, as determined by the weighted DB clustering algorithm. Circles indicate that they were identified at a rank < 500, while crosses indicate that they were not within the top 500 highest ranked candidates of a given sector.\n }\n \\label{fig:TOI_properties}\n\\end{figure}\n\nThe fractional TCE recovery rate (bottom panel of Figure~\\ref{fig:TCE_TOI_recovery}) is systematically lower than that of the TOIs. There are qualitative reasons as to why humans might not identify a TCE as opposed to a TOI, including that TCEs may be caused by artefacts or periodic stellar signals that the SPOC pipeline identified as a potential transit but that the human eye would either miss or be able to rule out as systematic effect. This leads to a lower recovery fraction of TCEs comparatively, an effect that is further amplified by the much larger number of TCEs.\n\nThe detection efficiency of PHT is estimated using the fractional recovery rate of TOIs for a range of radius and period bins, as shown in Figure~\\ref{fig:recovery_rank500_radius_period}. A TOI is considered to be recovered if its detection rank is less than 500 within the given sector. Out of the total 1913 TOIs, to date, \\textcolor{red}{PHT recovered 715 TOIs among the highest ranked candidates across the 26 sectors. This corresponds to a mean of 12.7~\\% of the top 500 ranked candidates per sector being TOIs. In comparison, the primary \\emph{TESS}\\ team on average visually vets 1025 TCEs per sector, out of which a mean of 17.3~\\% are promoted to TOI status.} We find that, independent of the orbital period, PHT is over 85~\\% complete in the recovery of TOIs with radii equal to or greater than 4 $R_{\\oplus}$. This agrees with the findings from the initial Planet Hunters project \\citep{schwamb12}. The detection efficiency decreases to 51~\\% for 3 - 4 $R_{\\oplus}$ TOIs, 49~\\% for 2 - 3 $R_{\\oplus}$ TOIs and to less than 40~\\% for TOIs with radii less than 2 $R_{\\oplus}$. Fig~\\ref{fig:recovery_rank500_radius_period} shows that the orbital period does not have a strong effect on the detection efficiency for periods greater than $\\sim$~1~day, which highlights that human vetting efficiency is independent of the number of transits present within a light curve. For periods shorter than around 1~day, the detection efficiency decreases even for larger planets, due to the high frequency of events seen in the light curve. For these light curves, many volunteers will only mark a subset of the transits, which may not overlap with the subset marked by other volunteers. Due to the methodology used to identify and rank the candidates, as described in Section~\\ref{sec:method}, this will actively disfavour the recovery of very short period planets. Although this obviously introduces biases in the detectability of very short period signals, the major detection pipelines are specifically designed to identify these types of planets and thus this does not present a serious detriment to our main science goal of finding planets that were \\textcolor{red}{intentionally ignored or missed} by the main automated pipelines.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{Figures\/TOI_recovery_grid.png}\n \\caption{TOI recovery rate as a function of planet radius and orbital period. A TOI is considered recovered if it is amongst the top 500 highest ranked candidates within a given sector. The logarithmically spaced grid ranges from 0.2 to 225 d and 0.6 to 55 $R{_\\oplus}$ for the orbital period and planet radius, respectively. The fraction of TOIs recovered using PHT is computed for each cell and represented by the colour the grid. Cells with less than 10 TOIs are considered incomplete for statistical analysis and are shown by the hatched lines. White cells contain no TOIs. The annotations for each cell indicate the number of recovered TOIs followed by the Poisson uncertainty in brackets. The filled in and empty grey circles indicated the recovered and not-recovered TOIs, respectively.}\n \\label{fig:recovery_rank500_radius_period}\n\\end{figure*}\n\n\nFinally, we assessed whether the detection efficiency varies across different sectors by assessing the fraction of recovered TOIs and TCEs within the highest ranked 500 candidates. We found the recovery of TOIs within the top 500 highest ranked candidates to remain relatively constant across all sectors, while the fraction of recovered TCEs in the top 500 highest ranked candidates increases in later sectors, as shown in Figure~\\ref{fig:recovery_rank500}). After applying a Spearman's rank test we find a positive correlation of 0.86 (pvalue = 5.9 $\\times$ $10^{-8}$) and 0.57 (pvalue = 0.003) between the observation sector and TCE and TOI recovery rates, respectively. These correlations suggest that the ability of users to detect transit-like events improves as they classify more subjects. The improvement of volunteers over time can also be seen in Fig~\\ref{fig:user_weights}, which shows the mean (unnormalized) user weight per sector for volunteers who completed one or more classifications in at least one sector (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors 26 sectors from the nominal \\emph{TESS}\\ mission (pink). The figure highlights an overall improvement in the mean user weight in later sectors, as well as a positive correlation between the overall increase in user weight and the number of sectors that volunteers have participated in.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI_rank500.png}\n \\caption{The fractional recovery rate of the TOIs (blue circles) and TCEs (teal squares) at a rank of 500 for each sector. Sector 1-9 (white background) represent southern hemisphere sectors classified with UI1, sectors 9-14 (light grey background) show the southern hemisphere sectors classified with UI2, and sectors 14-24 (dark grey background) show the northern hemisphere sectors classified with US2.}\n \\label{fig:recovery_rank500}\n\\end{figure}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.50\\textwidth]{Figures\/user_weights_sectors.png}\n \\caption{Mean user weights per sector. The solid lines show the user weights for the old user interface and the dashed line for the new interface, separated by the black line (Sector 9). The different coloured lines show the mean user weights calculated considering user who participated in any number of sectors (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors observed during the nominal \\emph{TESS}\\ mission (pink).}\n \\label{fig:user_weights}\n\\end{figure}\n\n\n\\section{Candidate vetting}\n\\label{sec:vetting}\n\nFor each observation sector the subjects are ranked according to their transit scores, and the 500 highest ranked targets (excluding TOIs) visually vetted by the PHT science team in order to identify potential candidates and rule out false positives. A vetting cut-off rank of 500 was chosen as we found this to maximise the number of found candidates while minimising the number of likely false positives. In the initial round of vetting, which is completed via a separate Zooniverse classification interface that is only accessible to the core science team, a minimum of three members of the team sort the highest ranked targets into either `keep for further analysis', `eclipsing binary' or `discard'. The sorting is based on the inspection of the full \\emph{TESS}\\ light curve of the target, with the times of the satellite momentum dumps indicated. Additionally, around the time of each likely transit event (i.e. time of successful DB clusters) we inspect the background flux and the x and y centroid positions. Stellar parameters are provided for each candidate, subject to availability, alongside links to the SPOC Data Validation (DV) reports for candidates that had been flagged as TCEs but were never promoted to TOIs status.\n\nCandidates where at least two of the reviewers indicated that the signal is consistent with a planetary transit are kept for further analysis. \\textcolor{red}{This constitute a $\\sim$~5~\\% retention rate of the 500 highest ranked candidates per sector between the initial citizen science classification stage and the PHT science team vetting stage. Considering that the known planets and TOIs are not included at this stage of vetting, it is not surprising that our retention rate is lower that the true-positive rates of TCEs (see Section~\\ref{subsec:TCE_TOI}). Furthermore, this false-positive rate is consistent with the the findings of the initial Planet Hunters project \\citep{schwamb12}.}\n\nThe rest of the 500 candidates were grouped into $\\sim$~37~\\% `eclipsing binary' and $\\sim$~58~\\% `discard'. The most common reasons for discarding light curves are due to events caused by momentum dumps and due to background events, such as background eclipsing binaries, that mimic transit-like signals in the light curve. The targets identified as eclipsing binaries are analysed further by the \\emph{TESS}\\ Eclipsing Binaries Working Group (Prsa et al, in prep).\n\n\n\n\nFor the second round of candidate vetting we generate our own data validation reports for all candidates classified as `keep for further analysis'. The reports are generated using the open source software {\\sc latte} \\citep[Lightcurve Analysis Tool for Transiting Exoplanets;][]{LATTE2020}, which includes a range of standard diagnostic plots that are specifically designed to help identify transit-like signals and weed out astrophysical false positives in \\emph{TESS}\\ data. In brief the diagnostics consist of:\n\n\\textbf{Momentum Dumps}. The times of the \\emph{TESS}\\ reaction wheel momentum dumps that can result in instrumental effects that mimic astrophysical signals.\n\n\\textbf{Background Flux}. The background flux to help identify trends caused by background events such as asteroids or fireflies \\citep{vanderspek2018tess} passing through the field of view.\n\n\\textbf{x and y centroid positions}. The CCD column and row local position of the target's flux-weighted centroid, and the CCD column and row motion which considers differential velocity aberration (DVA), pointing drift, and thermal effects. This can help identify signals caused by systematics due to the satellite. \n\\textbf{Aperture size test}. The target light curve around the time of the transit-like event extracted using two apertures of different sizes. This can help identify signals resulting from background eclipsing binaries.\n \n\\textbf{Pixel-level centroid analysis}. A comparison between the average in-transit and average out-of-transit flux, as well as the difference between them. This can help identify signals resulting from background eclipsing binaries.\n\n\\textbf{Nearby companion stars}. The location of nearby stars brighter than V-band magnitude 15 as queried from the Gaia Data Release 2 catalog \\citep{gaia2018gaia} and the DSS2 red field of view around the target star in order to identify nearby contaminating sources. \n\n\\textbf{Nearest neighbour light curves}. Normalized flux light curves of the five short-cadence \\emph{TESS}\\ stars with the smallest projected distances to the target star, used to identify alternative sources of the signal or systematic effects that affect multiple target stars. \n\n\\textbf{Pixel level light curves}. Individual light curves extracted for each pixel around the target. Used to identify signals resulting from background eclipsing binaries, background events and systematics.\n\n\\textbf{Box-Least-Squares fit}. Results from two consecutive BLS searches, where the identified signals from the initial search are removed prior to the second BLS search.\n\nThe {\\sc latte} validation reports are assessed by the PHT science team in order to identify planetary candidates that warrant further investigation. Around 10~\\% of the targets assessed at this stage of vetting are kept for further investigation, resulting in $\\sim$~3 promising planet candidates per observation sector. The discarded candidates can be loosely categorized into (background) eclipsing binaries ($\\sim$~40~\\%), systematic effects ($\\sim$~25~\\%), background events ($\\sim$~15~\\%) and other (stellar signals such as spots; $\\sim$~10~\\%).\n\n\nWe use \\texttt{pyaneti}\\ \\citep{pyaneti} to infer the planetary and orbital parameters of our most promising candidates. For multi-transit candidates we fit for seven parameters per planet, time of mid-transit $T_0$, orbital period $P$, impact parameter $b$, scaled semi-major axis $a\/R_\\star$, scaled planet radius $r_{\\rm p}\/R_\\star$, and two limb darkening coefficients following a \\citet{Mandel2002} quadratic limb darkening model, implemented with the $q_1$ and $q_2$ parametrization suggested by \\citet{Kipping2013}. Orbits were assumed to be circular.\nFor the mono-transit candidates, we fit the same parameters as for the multi-transit case, except for the orbital period and scaled semi-major axis which cannot be known for single transits. We follow \\citet{Osborn2016} to estimate the orbital period of the mono-transit candidates assuming circular orbits.\n\nWe note that some of our candidates are V-shaped, consistent with a grazing transit configuration. For these cases, we set uniform priors between 0 and 0.15 for $r_{\\rm p}\/R_\\star$ and between 0 and 1.15 for the impact parameter in order to avoid large radii caused by the $r_{\\rm p}\/R_\\star - b$ degeneracy. Thus, the $r_{\\rm p}\/R_\\star$ for these candidates should not be trusted. A full characterisation of these grazing transits is out of the scope of this manuscript.\n\nFigure~\\ref{fig:PHT_pyaneti} shows the \\emph{TESS}\\ transits together with the inferred model for each candidate. Table~\\ref{tab:PHT-caniddates} shows the inferred main parameters, the values and their uncertainties are given by the median and 68.3\\% credible interval of the posterior distributions.\n\n\n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_one.png}\n \\caption{All of the PHT candidates modelled using \\texttt{pyaneti}. The parameters of the best fits are summarised in Table~\\protect\\ref{tab:PHT-caniddates}. The blue and magenta fits show the multi and single transit event candidates, respectively.} \n \\label{fig:PHT_pyaneti}\n\\end{figure*}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_two.png}\n \\addtocounter{figure}{-1}\n \\caption{\\textbf{PHT candidates (continued)}} \n\\end{figure*}\n\n\nCandidates that pass all of our rounds of vetting are uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS) website\\footnote{\\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}} as community TOIs (cTOIs).\n\n\\section{Follow-up observations}\n\\label{sec:follow_up}\n\nMany astrophysical false positive scenarios can be ruled out from the detailed examination of the \\emph{TESS}\\ data, both from the light curves themselves and from the target pixel files. However, not all of the false positive scenarios can be ruled out from these data alone, due in part to the large \\emph{TESS}\\ pixels (20 arcsconds). Our third stage of vetting, therefore, consists of following up the candidates with ground based observations including photometry, reconnaissance spectroscopy and speckle imaging. The results from these observations will be discussed in detail in a dedicated follow-up paper. \n\n\\subsection{Photometry}\n\nWe make use of the LCO global network of fully robotic 0.4-m\/SBIG and 1.0-m\/Sinistro facilities \\citep{LCO2013} to observe additional transits, where the orbital period is known, in order to refine the ephemeris and confirm that the transit events are not due to a blended eclipsing binary in the vicinity of the main target. Snapshot images are taken of single transit event candidates in order to identify nearby contaminating sources. \n\n\n\\subsection{Spectroscopy}\n\nWe perform high-resolution optical spectroscopy using telescopes from across the globe in order to cover a wide range of RA and Dec:\n\\begin{itemize}\n\\item The Las Cumbres Observatory (LCO) telescopes with the Network of Robotic Echelle Spectrographs \\citep[NRES,][]{LCO2013}. These fibre-fed spectrographs, mounted on 1.0-m telescopes around the globe, have a resolution of R = 53,000 and a wavelength coverage of 380 to 860 nm. \n\n\\item The MINERVA Australis Telescope facility, located at Mount Kent Observatory in Queensland, Australia \\citep{addison2019}. This facility is made up of four 0.7m CDK700 telescopes, which individually feed light via optic fibre into a KiwiSpec high-resolution (R = 80,000) stabilised spectrograph \\citep{barnes2012} that covers wavelengths from 480 nm to 620 nm. \n\n\\item The CHIRON spectrograph mounted on the SMARTS 1.5-m telescope \\citep{Tokovinin2018}, located at the Cerro Tololo\nInter-American Observatory (CTIO) in Chile. The high resolution cross-dispersed echelle spectrometer is fiber-fed followed by an image slicer. It has a resolution of R = 80,000 and covers wavelengths ranging from 410 to 870 nm.\n\n\\item The SOPHIE echelle spectrograph mounted on the 1.93-m Haute-Provence Observatory (OHP), France\n\\citep{2008Perruchot,2009Bouchy}. The high resolution cross-dispersed stabilized echelle spectrometer is fed by two optical fibers. Observations were taken in high-resolution mode (R = 75,000) with a wavelength range of 387 to 694 nm.\n\n\\end{itemize}\n\nReconnaissance spectroscopy with these instruments allow us to extract stellar parameters, identify spectroscopic binaries, and place upper limits on the companion masses. Spectroscopic binaries and targets whose spectral type is incompatible with the initial planet hypothesis and\/or precludes precision RV observations (giant or early type stars) are not followed up further. Promising targets, however, are monitored in order to constrain their period and place limits on their mass. \n\n\\subsection{Speckle Imaging}\n\nFor our most promising candidates we perform high resolution speckle imaging using the `Alopeke instrument on the 8.1-m Frederick C. Gillett Gemini North telescope in Maunakea, Hawaii, USA, and its twin, Zorro, on the 8.1-m Gemini South telescope on Cerro Pach\\'{o}n, Chile \\citep{Matson2019, Howell2011}. Speckle interferometric observations provide extremely high resolution images reaching the diffraction limit of the telescope. We obtain simultaneous 562 nm and 832 nm rapid exposure (60 msec) images in succession that effectively `freeze out' atmospheric turbulence and through Fourier analysis are used to search for close companion stars at 5-8 magnitude contrast levels. This analysis, along with the reconstructed images, allow us to identify nearby companions and to quantify their light contribution to the TESS aperture and thus the transit signal.\n\n\n\\section{Planet candidates and Noteworthy Systems}\n\\label{sec:PHT_canidates}\n\\subsection{Planet candidate properties}\n\nIn this final part of the paper we discuss the 90 PHT candidates around 88 host stars that passed the initial two stages of vetting and that were uploaded to ExoFOP as cTOIs. At the time of discovery none of these candidates were TOIs. The properties of all of the PHT candidates are summarised in Table~\\ref{tab:PHT-caniddates}. Candidates that have been promoted to TOI status since their PHT discovery are highlighted with an asterisk following the TIC ID, and candidates that have been shown to be false positives, based on the ground-based follow-up observations, are marked with a dagger symbol ($\\dagger$). The majority (81\\%) of PHT candidates are single transit events, indicated by an `s' following the orbital period presented in the table. \\textcolor{red}{18 of the PHT candidates were flagged as TCEs by the \\emph{TESS}\\ pipeline, but not initially promoted to TOI status. The most common reasons for this was that the pipeline identified a single-transit event as well as times of systematics (often caused by momentum dumps), due to its two-transit minimum detection threshold. This resulted in the candidate being discarded on the basis of it not passing the `odd-even' transit depth test. Out of the 18 TCEs, 14 have become TOI's since the PHT discovery. More detail on the TCE candidates can be found in Appendix~\\ref{appendixA}.} \n\nAll planet parameters (columns 2 to 8) are derived from the \\texttt{pyaneti}\\ modelling as described in Section~\\ref{sec:vetting}. Finally, the table summarises the ground-based follow-up observations (see Sec~\\ref{sec:follow_up}) that have been obtained to date, where the bracketed numbers following the observing instruments indicate the number of epochs. Unless otherwise noted, the follow-up observations are consistent with a planetary scenario. More in depth descriptions of individual targets for which we have additional information to complement the results in Table~\\ref{tab:PHT-caniddates} can be found in Appendix~\\ref{appendixA}.\n\n\\subsection{Planet candidate analysis}\n\n\nThe majority of the TOIs (87.7\\%) have orbital periods shorter than 15 days due to the requirement of observing at least two transits included in all major pipelines \\textcolor{red}{combined with the observing strategy of \\emph{TESS}}. As visual vetting does not impose these limits, the candidates outlined in this paper are helping to populate the relatively under-explored long-period region of parameter space. This is highlighted in Figure~\\ref{fig:PHT_candidates}, which shows the transit depths vs the orbital periods of the PHT single transit candidates (orange circles) and the multi-transit candidates (magenta squares) compared to the TOIs (blue circles). Values of the orbital periods and transit depths were obtained via transit modelling using \\texttt{pyaneti} (see Section~\\ref{sec:vetting}). The orbital period of single transit events are poorly constrained, which is reflected by the large errorbars in Figure~\\ref{fig:PHT_candidates}. Figure~\\ref{fig:PHT_candidates} also highlights that with PHT we are able to recover a similar range of transit depths as the pipeline found TOIs, as was previously shown in Figure~\\ref{fig:recovery_rank500_radius_period}.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_candidate_period_depth_plot_errobrars.png}\n \\caption{The properties of the PHT single transit (orange circles) and multi transit (magenta squares) candidates compared to the properties TOIs (blue circles). All parameters (listed in Table~\\ref{fig:PHT_candidates}) were extracted using \\texttt{pyaneti}\\ modelling.}\n \\label{fig:PHT_candidates}\n\\end{figure}\n\nThe PHT candidates were further compared to the TOIs in terms of the properties of their host stars. Figure~\\ref{fig:eep} shows the effective temperature and stellar radii as taken from the TIC \\citep{Stassun18}, for TOIs (blue dots) and the PHT candidates (magenta circles). The solid and dashed lines indicate the main sequence and post-main sequence MIST stellar evolutionary tracks \\citep{choi2016}, respectively, for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. This shows that around 10\\% of the host stars are in the process of, or have recently evolved off the main sequence. The models assume solar metalicity, no stellar rotation and no additional internal mixing.\n\n\\textcolor{red}{Ground based follow-up spectroscopy has revealed that six of the PHT candidates listed in Table~\\ref{tab:PHT-caniddates} are astrophysical false positives. As the follow-up campaign of the targets is still underway, the true false-positive rate of the candidates to have made it through all stages of the vetting process, as outlined in the methodology, will be be assessed in future PHT papers once the true nature of more of the candidates has been independently verified.}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_eep.png}\n \\caption{Stellar evolution tracks showing main sequence (solid black lines) and post-main sequence (dashed grey lines) MIST stellar evolution for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. The blue dots show the TOIs and the magenta circles show the PHT candidates.} \n \\label{fig:eep}\n\\end{figure}\n\n\n\\subsection{Stellar systems}\n\\label{subsec:PHT_stars}\n\nIn addition to the planetary candidates, citizen science allows for the identification of interesting stellar systems and astrophysical phenomena, in particular where the signals are aperiodic or small compared to the dominant stellar signal. These include light curves that exhibit multiple transit-like signals, possibly as a result of a multiple stellar system or a blend of eclipsing binaries. We have investigated all light curves that were flagged as possible multi-stellar systems via the PHT discussion boards. Similar to the planet vetting, as described in Section~\\ref{sec:vetting}, we generated {\\sc latte} data validation reports in order to assess the nature of the signal. Additionally, we subjected these systems to an iterative signal removal process, whereby we phase-folded the light curve on the dominant orbital period, binned the light curve into between 200-500 phase bins, created an interpolation model, and then subtracted said signal in order to evaluate the individual transit signals. The period of each signal, as listed in Table~\\ref{tab:PHT-multis}, was determined by phase folding the light curve at a number of trial periods and assessing by eye the best fit period and corresponding uncertainty.\n\nDue to the large \\emph{TESS}\\ pixels, blends are expected to be common. We searched for blends by generating phase folded light curves for each pixel around the source of the target in order to better locate the source of each signal. Shifts in the \\emph{TESS}\\ x and y centroid positions were also found to be good indicators of visually separated sources. Nearby sources with a magnitude difference greater than 5 mags were ruled out as possible contaminators. We consider a candidate to be a confirmed blend when the centroids are separated by more than 1 \\emph{TESS}\\ pixel, as this corresponds to an angular separation > 21 arcseconds meaning that the systems are highly unlikely to be gravitationally bound. Systems where the signal appears to be coming from the same \\emph{TESS}\\ pixel and that show no clear centroid shifts are considered to be candidate multiple systems. We note that blends are still possible, however, without further investigation we cannot conclusively rule these out as possible multi stellar systems. \n\nAll of the systems are summarised in Table~\\ref{tab:PHT-multis}. Out of the 26 systems, 6 are confirmed multiple systems which have either been published or are being prepared for publication; 7 are visually separated eclipsing binaries (confirmed blends); and 13 are candidate multiple system. Additional observations will be required to determine whether or not these candidate multiple systems are in fact gravitationally bound or photometric blends as a results of the large \\emph{TESS}\\ pixels or due to a line of sight happenstance. \n\n\\begin{landscape}\n\\begin{table}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period} & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n & \\textbf{Name} & \\textbf{(\\textcolor{red}{BJD - 2457000})} & \\textbf{(days)} & & $(R_{\\oplus})$ & \\textbf{Parameter} & \\textbf{(hours)} & & & & & \\\\\n\\hline\n101641905 & TWOMASS 11412617+3441004 & $1917.26335 _{ - 0.00072 } ^ { + 0.00071 }$ & $14.52 _{ - 5.25 } ^ { + 6.21 }(s)$ & $0.1135 _{ - 0.0064 } ^ { + 0.0032 }$ & $9.76 _{ - 0.69 } ^ { + 0.65 }$ & $0.691 _{ - 0.183 } ^ { + 0.077 }$ & $3.163 _{ - 0.088 } ^ { + 0.093 }$ & 12.196 & & & & \\\\\n103633672* & TYC 4387-00923-1 & $1850.3211 _{ - 0.00077 } ^ { + 0.00135 }$ & $90.9 _{ - 23.7 } ^ { + 46.4 }(s)$ & $0.0395 _{ - 0.0013 } ^ { + 0.0013 }$ & $3.45 _{ - 0.24 } ^ { + 0.26 }$ & $0.3 _{ - 0.21 } ^ { + 0.26 }$ & $6.7 _{ - 0.11 } ^ { + 0.12 }$ & 10.586 & & NRES (1) & & \\\\\n110996418 & TWOMASS 12344723-1019107 & $1580.6406 _{ - 0.0038 } ^ { + 0.0037 }$ & $5.18 _{ - 2.93 } ^ { + 6.86 }(s)$ & $0.1044 _{ - 0.0067 } ^ { + 0.008 }$ & $12.7 _{ - 0.99 } ^ { + 1.15 }$ & $0.44 _{ - 0.3 } ^ { + 0.3 }$ & $3.53 _{ - 0.27 } ^ { + 0.36 }$ & 13.945 & & & & \\\\\n128703021 & HIP 71639 & $1601.8442 _{ - 0.00108 } ^ { + 0.00093 }$ & $26.0 _{ - 8.22 } ^ { + 22.35 }(s)$ & $0.0254 _{ - 0.00049 } ^ { + 0.00072 }$ & $4.44 _{ - 0.2 } ^ { + 0.23 }$ & $0.47 _{ - 0.3 } ^ { + 0.22 }$ & $7.283 _{ - 0.091 } ^ { + 0.141 }$ & 6.06 & & NRES (2);MINERVA (34) & Gemini & \\\\\n138126035 & TYC 1450-00833-1 & $1954.3229 _{ - 0.0041 } ^ { + 0.0067 }$ & $28.8 _{ - 14.0 } ^ { + 203.2 }(s)$ & $0.0375 _{ - 0.0026 } ^ { + 0.0069 }$ & $4.01 _{ - 0.35 } ^ { + 0.74 }$ & $0.58 _{ - 0.38 } ^ { + 0.35 }$ & $4.65 _{ - 0.32 } ^ { + 0.85 }$ & 10.349 & & & & \\\\\n142087638 & TYC 9189-00274-1 & $1512.1673 _{ - 0.0043 } ^ { + 0.0034 }$ & $3.14 _{ - 1.41 } ^ { + 12.04 }(s)$ & $0.0469 _{ - 0.0035 } ^ { + 0.0063 }$ & $6.05 _{ - 0.54 } ^ { + 0.89 }$ & $0.5 _{ - 0.35 } ^ { + 0.36 }$ & $2.72 _{ - 0.23 } ^ { + 0.5 }$ & 11.526 & & & & \\\\\n159159904 & HIP 64812 & $1918.6109 _{ - 0.0067 } ^ { + 0.0091 }$ & $584.0 _{ - 215.0 } ^ { + 1724.0 }(s)$ & $0.0237 _{ - 0.0011 } ^ { + 0.0026 }$ & $3.12 _{ - 0.22 } ^ { + 0.36 }$ & $0.49 _{ - 0.34 } ^ { + 0.35 }$ & $15.11 _{ - 0.54 } ^ { + 0.7 }$ & 9.2 & & NRES (2) & & \\\\\n160039081* & HIP 78892 & $1752.9261 _{ - 0.0045 } ^ { + 0.005 }$ & $30.19918 _{ - 0.00099 } ^ { + 0.00094 }$ & $0.0211 _{ - 0.0013 } ^ { + 0.0035 }$ & $2.67 _{ - 0.21 } ^ { + 0.43 }$ & $0.52 _{ - 0.34 } ^ { + 0.36 }$ & $4.93 _{ - 0.27 } ^ { + 0.37 }$ & 8.35 & SBIG (1) & NRES (1);SOPHIE (4) & Gemini & \\\\\n162631539 & HIP 80264 & $1978.2794 _{ - 0.0044 } ^ { + 0.0051 }$ & $17.32 _{ - 6.66 } ^ { + 52.35 }(s)$ & $0.0195 _{ - 0.0011 } ^ { + 0.0024 }$ & $2.94 _{ - 0.24 } ^ { + 0.38 }$ & $0.48 _{ - 0.33 } ^ { + 0.36 }$ & $5.54 _{ - 0.33 } ^ { + 0.41 }$ & 7.42 & & & & \\\\\n166184426* & TWOMASS 13442500-4020122 & $1600.4409 _{ - 0.003 } ^ { + 0.0036 }$ & $16.3325 _{ - 0.0066 } ^ { + 0.0052 }$ & $0.0545 _{ - 0.0031 } ^ { + 0.0039 }$ & $1.85 _{ - 0.12 } ^ { + 0.15 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.98 _{ - 0.22 } ^ { + 0.17 }$ & 12.911 & & & & \\\\\n167661160 & TYC 7054-01577-1 & $1442.0703 _{ - 0.0028 } ^ { + 0.004 }$ & $36.802 _{ - 0.07 } ^ { + 0.069 }$ & $0.0307 _{ - 0.0014 } ^ { + 0.0024 }$ & $4.07 _{ - 0.32 } ^ { + 0.43 }$ & $0.37 _{ - 0.26 } ^ { + 0.33 }$ & $5.09 _{ - 0.23 } ^ { + 0.21 }$ & 9.927 & & NRES (9);MINERVA (4) & & EB from MINERVA observations \\\\\n172370679* & TWOMASS 19574239+4008357 & $1711.95923 _{ - 0.00099 } ^ { + 0.001 }$ & $32.84 _{ - 4.17 } ^ { + 5.59 }(s)$ & $0.1968 _{ - 0.0032 } ^ { + 0.0022 }$ & $13.24 _{ - 0.43 } ^ { + 0.43 }$ & $0.22 _{ - 0.15 } ^ { + 0.14 }$ & $4.999 _{ - 0.097 } ^ { + 0.111 }$ & 14.88 & & & & Confirmed planet \\citep{canas2020}. \\\\\n174302697* & TYC 3641-01789-1 & $1743.7267 _{ - 0.00092 } ^ { + 0.00093 }$ & $498.2 _{ - 80.0 } ^ { + 95.3 }(s)$ & $0.07622 _{ - 0.00068 } ^ { + 0.00063 }$ & $13.34 _{ - 0.57 } ^ { + 0.58 }$ & $0.642 _{ - 0.029 } ^ { + 0.024 }$ & $17.71 _{ - 0.12 } ^ { + 0.13 }$ & 9.309 & SBIG (1) & & & \\\\\n179582003 & TYC 9166-00745-1 & $1518.4688 _{ - 0.0016 } ^ { + 0.0016 }$ & $104.6137 _{ - 0.0022 } ^ { + 0.0022 }$ & $0.06324 _{ - 0.0008 } ^ { + 0.0008 }$ & $7.51 _{ - 0.35 } ^ { + 0.35 }$ & $0.21 _{ - 0.15 } ^ { + 0.19 }$ & $9.073 _{ - 0.084 } ^ { + 0.097 }$ & 10.806 & & & & \\\\\n192415680 & TYC 2859-00682-1 & $1796.0265 _{ - 0.0012 } ^ { + 0.0013 }$ & $18.47 _{ - 6.34 } ^ { + 21.73 }(s)$ & $0.0478 _{ - 0.0017 } ^ { + 0.0027 }$ & $4.43 _{ - 0.33 } ^ { + 0.38 }$ & $0.45 _{ - 0.31 } ^ { + 0.31 }$ & $3.94 _{ - 0.1 } ^ { + 0.12 }$ & 9.838 & SBIG (1) & SOPHIE (2) & & \\\\\n192790476 & TYC 7595-00649-1 & $1452.3341 _{ - 0.0014 } ^ { + 0.002 }$ & $16.09 _{ - 5.73 } ^ { + 15.49 }(s)$ & $0.0438 _{ - 0.0018 } ^ { + 0.0026 }$ & $3.24 _{ - 0.34 } ^ { + 0.37 }$ & $0.37 _{ - 0.25 } ^ { + 0.3 }$ & $3.395 _{ - 0.099 } ^ { + 0.192 }$ & 10.772 & & & & \\\\\n206361691$\\dagger$ & HIP 117250 & $1363.2224 _{ - 0.0082 } ^ { + 0.009 }$ & $237.7 _{ - 81.0 } ^ { + 314.4 }(s)$ & $0.01762 _{ - 0.00088 } ^ { + 0.00125 }$ & $2.69 _{ - 0.19 } ^ { + 0.25 }$ & $0.43 _{ - 0.28 } ^ { + 0.32 }$ & $13.91 _{ - 0.53 } ^ { + 0.52 }$ & 8.88 & & CHIRON (2) & & SB2 from CHIRON \\\\\n207501148 & TYC 3881-00527-1 & $2007.7273 _{ - 0.0011 } ^ { + 0.0011 }$ & $39.9 _{ - 10.3 } ^ { + 14.3 }(s)$ & $0.0981 _{ - 0.0047 } ^ { + 0.011 }$ & $13.31 _{ - 0.95 } ^ { + 1.56 }$ & $0.9 _{ - 0.03 } ^ { + 0.039 }$ & $4.73 _{ - 0.14 } ^ { + 0.14 }$ & 10.385 & & & & \\\\\n219466784* & TYC 4409-00437-1 & $1872.6879 _{ - 0.0097 } ^ { + 0.0108 }$ & $318.0 _{ - 147.0 } ^ { + 1448.0 }(s)$ & $0.0332 _{ - 0.0024 } ^ { + 0.0048 }$ & $3.26 _{ - 0.31 } ^ { + 0.49 }$ & $0.55 _{ - 0.39 } ^ { + 0.34 }$ & $10.06 _{ - 0.81 } ^ { + 1.12 }$ & 11.099 & & & & \\\\\n219501568 & HIP 79876 & $1961.7879 _{ - 0.0018 } ^ { + 0.002 }$ & $16.5931 _{ - 0.0017 } ^ { + 0.0015 }$ & $0.0221 _{ - 0.0012 } ^ { + 0.0015 }$ & $4.22 _{ - 0.3 } ^ { + 0.35 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.615 _{ - 0.077 } ^ { + 0.093 }$ & 8.38 & & & & \\\\\n229055790 & TYC 7492-01197-1 & $1337.866 _{ - 0.0022 } ^ { + 0.0019 }$ & $48.0 _{ - 12.8 } ^ { + 48.4 }(s)$ & $0.0304 _{ - 0.00097 } ^ { + 0.00115 }$ & $3.52 _{ - 0.2 } ^ { + 0.24 }$ & $0.37 _{ - 0.26 } ^ { + 0.32 }$ & $6.53 _{ - 0.11 } ^ { + 0.14 }$ & 9.642 & & NRES (2) & & \\\\\n229608594 & TWOMASS 18180283+7428005 & $1960.0319 _{ - 0.0037 } ^ { + 0.0045 }$ & $152.4 _{ - 54.1 } ^ { + 152.6 }(s)$ & $0.0474 _{ - 0.0023 } ^ { + 0.0024 }$ & $3.42 _{ - 0.34 } ^ { + 0.36 }$ & $0.38 _{ - 0.26 } ^ { + 0.3 }$ & $6.98 _{ - 0.23 } ^ { + 0.37 }$ & 12.302 & & & & \\\\\n229742722* & TYC 4434-00596-1 & $1689.688 _{ - 0.025 } ^ { + 0.02 }$ & $29.0 _{ - 16.4 } ^ { + 66.3 }(s)$ & $0.019 _{ - 0.0028 } ^ { + 0.0029 }$ & $2.9 _{ - 0.44 } ^ { + 0.48 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $4.27 _{ - 0.09 } ^ { + 0.11 }$ & 10.33 & & NRES (8);SOPHIE (4) & Gemini & \\\\\n233194447 & TYC 4211-00650-1 & $1770.4924 _{ - 0.0065 } ^ { + 0.0107 }$ & $373.0 _{ - 101.0 } ^ { + 284.0 }(s)$ & $0.02121 _{ - 0.00073 } ^ { + 0.001 }$ & $5.08 _{ - 0.28 } ^ { + 0.33 }$ & $0.34 _{ - 0.24 } ^ { + 0.29 }$ & $24.45 _{ - 0.47 } ^ { + 0.5 }$ & 9.178 & & NRES (2) & Gemini & \\\\\n235943205 & TYC 4588-00127-1 & $1827.0267 _{ - 0.004 } ^ { + 0.0034 }$ & $121.3394 _{ - 0.0063 } ^ { + 0.0065 }$ & $0.0402 _{ - 0.0016 } ^ { + 0.0019 }$ & $4.2 _{ - 0.25 } ^ { + 0.29 }$ & $0.4 _{ - 0.27 } ^ { + 0.28 }$ & $6.37 _{ - 0.2 } ^ { + 0.3 }$ & 11.076 & & NRES (1);SOPHIE (2) & & \\\\\n237201858 & TYC 4452-00759-1 & $1811.5032 _{ - 0.0069 } ^ { + 0.0067 }$ & $129.7 _{ - 41.5 } ^ { + 146.8 }(s)$ & $0.0258 _{ - 0.0013 } ^ { + 0.0015 }$ & $4.12 _{ - 0.27 } ^ { + 0.3 }$ & $0.4 _{ - 0.28 } ^ { + 0.31 }$ & $10.94 _{ - 0.37 } ^ { + 0.53 }$ & 10.344 & & NRES (1) & & \\\\\n243187830* & HIP 5286 & $1783.7671 _{ - 0.0017 } ^ { + 0.0019 }$ & $4.05 _{ - 1.53 } ^ { + 9.21 }(s)$ & $0.0268 _{ - 0.0015 } ^ { + 0.0027 }$ & $2.06 _{ - 0.17 } ^ { + 0.23 }$ & $0.47 _{ - 0.32 } ^ { + 0.34 }$ & $2.02 _{ - 0.12 } ^ { + 0.15 }$ & 8.407 & SBIG (1) & & & \\\\\n243417115 & TYC 8262-02120-1 & $1614.4796 _{ - 0.0028 } ^ { + 0.0022 }$ & $1.81 _{ - 0.73 } ^ { + 3.45 }(s)$ & $0.0523 _{ - 0.0035 } ^ { + 0.005 }$ & $5.39 _{ - 0.47 } ^ { + 0.64 }$ & $0.47 _{ - 0.33 } ^ { + 0.34 }$ & $2.03 _{ - 0.16 } ^ { + 0.23 }$ & 11.553 & & & & \\\\\n256429408 & TYC 4462-01942-1 & $1962.16 _{ - 0.0022 } ^ { + 0.0023 }$ & $382.0 _{ - 132.0 } ^ { + 265.0 }(s)$ & $0.03582 _{ - 0.00086 } ^ { + 0.00094 }$ & $6.12 _{ - 0.29 } ^ { + 0.3 }$ & $0.51 _{ - 0.36 } ^ { + 0.18 }$ & $16.96 _{ - 0.2 } ^ { + 0.24 }$ & 8.898 & & & & \\\\\n264544388* & TYC 4607-01275-1 & $1824.8438 _{ - 0.0076 } ^ { + 0.0078 }$ & $7030.0 _{ - 6260.0 } ^ { + 3330.0 }(s)$ & $0.0288 _{ - 0.0029 } ^ { + 0.0018 }$ & $4.58 _{ - 0.43 } ^ { + 0.35 }$ & $0.936 _{ - 0.363 } ^ { + 0.011 }$ & $19.13 _{ - 1.35 } ^ { + 0.84 }$ & 8.758 & & NRES (1) & & \\\\\n264766922 & TYC 8565-01780-1 & $1538.69518 _{ - 0.00091 } ^ { + 0.00091 }$ & $3.28 _{ - 0.94 } ^ { + 1.25 }(s)$ & $0.0933 _{ - 0.0063 } ^ { + 0.0176 }$ & $16.95 _{ - 1.33 } ^ { + 3.19 }$ & $0.908 _{ - 0.039 } ^ { + 0.048 }$ & $2.73 _{ - 0.11 } ^ { + 0.11 }$ & 10.747 & & & & \\\\\n26547036* & TYC 3921-01563-1 & $1712.30464 _{ - 0.00041 } ^ { + 0.0004 }$ & $73.0 _{ - 13.6 } ^ { + 16.5 }(s)$ & $0.10034 _{ - 0.0007 } ^ { + 0.00078 }$ & $11.75 _{ - 0.59 } ^ { + 0.58 }$ & $0.17 _{ - 0.12 } ^ { + 0.11 }$ & $8.681 _{ - 0.049 } ^ { + 0.052 }$ & 9.849 & & NRES (4) & Gemini & \\\\\n267542728$\\dagger$ & TYC 4583-01499-1 & $1708.4956 _{ - 0.0073 } ^ { + 0.0085 }$ & $39.7382 _{ - 0.0023 } ^ { + 0.0023 }$ & $0.03267 _{ - 0.00089 } ^ { + 0.00175 }$ & $18.46 _{ - 0.94 } ^ { + 1.14 }$ & $0.38 _{ - 0.26 } ^ { + 0.27 }$ & $24.16 _{ - 0.39 } ^ { + 0.45 }$ & 11.474 & & & & EB from HIRES RVs. \\\\\n270371513$\\dagger$ & HIP 10047 & $1426.2967 _{ - 0.0023 } ^ { + 0.002 }$ & $0.39 _{ - 0.17 } ^ { + 1.79 }(s)$ & $0.024 _{ - 0.0015 } ^ { + 0.0032 }$ & $4.8 _{ - 0.38 } ^ { + 0.64 }$ & $0.5 _{ - 0.34 } ^ { + 0.39 }$ & $1.93 _{ - 0.16 } ^ { + 0.19 }$ & 6.98515 & & MINERVA (20) & & SB 2 from MINERVA observations. \\\\\n274599700 & TWOMASS 17011885+5131455 & $2002.1202 _{ - 0.0024 } ^ { + 0.0024 }$ & $32.9754 _{ - 0.005 } ^ { + 0.005 }$ & $0.0847 _{ - 0.0021 } ^ { + 0.0018 }$ & $13.25 _{ - 0.83 } ^ { + 0.83 }$ & $0.37 _{ - 0.24 } ^ { + 0.19 }$ & $8.2 _{ - 0.18 } ^ { + 0.21 }$ & 12.411 & & & & \\\\\n278990954 & TYC 8548-00717-1 & $1650.0191 _{ - 0.0086 } ^ { + 0.0105 }$ & $18.45 _{ - 8.66 } ^ { + 230.7 }(s)$ & $0.034 _{ - 0.0024 } ^ { + 0.0115 }$ & $9.65 _{ - 0.92 } ^ { + 3.13 }$ & $0.58 _{ - 0.4 } ^ { + 0.36 }$ & $10.62 _{ - 0.66 } ^ { + 2.46 }$ & 10.749 & & & & \\\\\n280865159* & TYC 9384-01533-1 & $1387.0749 _{ - 0.0045 } ^ { + 0.0044 }$ & $1045.0 _{ - 249.0 } ^ { + 536.0 }(s)$ & $0.0406 _{ - 0.0011 } ^ { + 0.0014 }$ & $4.75 _{ - 0.26 } ^ { + 0.28 }$ & $0.35 _{ - 0.24 } ^ { + 0.23 }$ & $19.08 _{ - 0.32 } ^ { + 0.36 }$ & 11.517 & & & Gemini & \\\\\n284361752 & TYC 3924-01678-1 & $2032.093 _{ - 0.0078 } ^ { + 0.008 }$ & $140.6 _{ - 46.6 } ^ { + 159.1 }(s)$ & $0.0259 _{ - 0.0014 } ^ { + 0.0017 }$ & $3.62 _{ - 0.26 } ^ { + 0.31 }$ & $0.4 _{ - 0.27 } ^ { + 0.34 }$ & $8.98 _{ - 0.66 } ^ { + 0.86 }$ & 10.221 & & & & \\\\\n288240183 & TYC 4634-01225-1 & $1896.941 _{ - 0.0051 } ^ { + 0.0047 }$ & $119.0502 _{ - 0.0091 } ^ { + 0.0089 }$ & $0.02826 _{ - 0.00089 } ^ { + 0.00119 }$ & $4.28 _{ - 0.35 } ^ { + 0.36 }$ & $0.55 _{ - 0.37 } ^ { + 0.25 }$ & $17.49 _{ - 0.36 } ^ { + 0.6 }$ & 9.546 & & & & \\\\\n29169215 & TWOMASS 09011787+4727085 & $1872.5047 _{ - 0.0032 } ^ { + 0.0036 }$ & $14.89 _{ - 6.12 } ^ { + 24.84 }(s)$ & $0.0403 _{ - 0.0025 } ^ { + 0.0033 }$ & $3.28 _{ - 0.37 } ^ { + 0.45 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $3.56 _{ - 0.21 } ^ { + 0.32 }$ & 11.828 & & & & \\\\\n293649602 & TYC 8103-00266-1 & $1511.2109 _{ - 0.004 } ^ { + 0.0037 }$ & $12.85 _{ - 5.34 } ^ { + 42.21 }(s)$ & $0.04 _{ - 0.0024 } ^ { + 0.0039 }$ & $4.66 _{ - 0.36 } ^ { + 0.5 }$ & $0.5 _{ - 0.35 } ^ { + 0.34 }$ & $4.1 _{ - 0.31 } ^ { + 0.56 }$ & 10.925 & & & & \\\\\n296737508 & TYC 5472-01060-1 & $1538.0036 _{ - 0.0015 } ^ { + 0.0016 }$ & $18.27 _{ - 5.06 } ^ { + 17.45 }(s)$ & $0.0425 _{ - 0.0014 } ^ { + 0.0019 }$ & $5.33 _{ - 0.22 } ^ { + 0.27 }$ & $0.44 _{ - 0.3 } ^ { + 0.26 }$ & $5.13 _{ - 0.13 } ^ { + 0.15 }$ & 9.772 & Sinistro (1) & NRES (1);MINERVA (1) & Gemini & \\\\\n298663873 & TYC 3913-01781-1 & $1830.76819 _{ - 0.00099 } ^ { + 0.00099 }$ & $479.9 _{ - 89.4 } ^ { + 109.4 }(s)$ & $0.06231 _{ - 0.00034 } ^ { + 0.00045 }$ & $11.07 _{ - 0.57 } ^ { + 0.57 }$ & $0.16 _{ - 0.11 } ^ { + 0.13 }$ & $23.99 _{ - 0.093 } ^ { + 0.1 }$ & 9.162 & & NRES (2) & Gemini & Dalba et al. (in prep) \\\\\n303050301 & TYC 6979-01108-1 & $1366.1301 _{ - 0.0022 } ^ { + 0.0023 }$ & $281.0 _{ - 170.0 } ^ { + 264.0 }(s)$ & $0.0514 _{ - 0.0027 } ^ { + 0.0018 }$ & $4.85 _{ - 0.32 } ^ { + 0.32 }$ & $0.73 _{ - 0.48 } ^ { + 0.1 }$ & $7.91 _{ - 0.31 } ^ { + 0.36 }$ & 10.048 & & NRES (1) & Gemini & \\\\\n303317324 & TYC 6983-00438-1 & $1365.1845 _{ - 0.0023 } ^ { + 0.0028 }$ & $69.0 _{ - 25.5 } ^ { + 78.1 }(s)$ & $0.0365 _{ - 0.0013 } ^ { + 0.0016 }$ & $2.88 _{ - 0.3 } ^ { + 0.31 }$ & $0.39 _{ - 0.26 } ^ { + 0.32 }$ & $5.78 _{ - 0.18 } ^ { + 0.24 }$ & 10.799 & & & & \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\emph{Note} -- Candidates that have become TOIs following the PHT discovery are marked with an asterisk (*). The `s' following the orbital period indicates that the candidates is a single transit event. The ground-based follow-up observations are summarized in columns 10-12, where the bracketed numbers correspond the number of epochs obtained with each instrument. See Section~\\ref{sec:follow_up} for description of each instrument. The $\\dagger$ symbol indicates candidates that have been shown to be astrophysical false positives based on the ground based follow-up observations.}\n\\label{tab:PHT-caniddates}\n\\end{table}\n\\end{landscape}\n\n\\begin{landscape}\n\\begin{table}\n\\addtocounter{table}{-1}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period} & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n & \\textbf{Name} & \\textbf{(\\textcolor{red}{BJD - 2457000})} & \\textbf{(days)} & & $(R_{\\oplus})$ & \\textbf{Parameter} & \\textbf{(hours)} & & & & & \\\\\n\\hline\n303586471$\\dagger$ & HIP 115828 & $1363.7692 _{ - 0.0033 } ^ { + 0.0027 }$ & $13.85 _{ - 4.19 } ^ { + 18.2 }(s)$ & $0.0214 _{ - 0.001 } ^ { + 0.0014 }$ & $2.52 _{ - 0.16 } ^ { + 0.2 }$ & $0.4 _{ - 0.27 } ^ { + 0.33 }$ & $4.23 _{ - 0.19 } ^ { + 0.16 }$ & 8.27 & & MINERVA (11) & & SB 2 from MINERVA observations. \\\\\n304142124* & HIP 53719 & $1585.28023 _{ - 0.0008 } ^ { + 0.0008 }$ & $42.8 _{ - 10.0 } ^ { + 18.2 }(s)$ & $0.04311 _{ - 0.00093 } ^ { + 0.00153 }$ & $4.1 _{ - 0.23 } ^ { + 0.24 }$ & $0.33 _{ - 0.21 } ^ { + 0.21 }$ & $5.66 _{ - 0.067 } ^ { + 0.09 }$ & 8.62 & & NRES (1);MINERVA (4) & & Confirmed planet \\citep{diaz2020} \\\\\n304339227 & TYC 9290-01087-1 & $1673.3242 _{ - 0.009 } ^ { + 0.0128 }$ & $111.9 _{ - 72.2 } ^ { + 4844.1 }(s)$ & $0.0253 _{ - 0.0024 } ^ { + 0.0481 }$ & $3.27 _{ - 0.61 } ^ { + 5.72 }$ & $0.67 _{ - 0.47 } ^ { + 0.36 }$ & $7.44 _{ - 0.86 } ^ { + 2.84 }$ & 9.169 & & & & \\\\\n307958020 & TYC 4191-00309-1 & $1864.82 _{ - 0.014 } ^ { + 0.013 }$ & $169.0 _{ - 107.0 } ^ { + 10194.0 }(s)$ & $0.0223 _{ - 0.0022 } ^ { + 0.0543 }$ & $3.92 _{ - 0.52 } ^ { + 9.27 }$ & $0.71 _{ - 0.53 } ^ { + 0.33 }$ & $12.48 _{ - 1.1 } ^ { + 5.41 }$ & 9.017 & & & & \\\\\n308301091 & TYC 2081-01273-1 & $2030.3691 _{ - 0.0024 } ^ { + 0.0026 }$ & $29.24 _{ - 8.49 } ^ { + 22.46 }(s)$ & $0.0362 _{ - 0.0013 } ^ { + 0.0014 }$ & $5.41 _{ - 0.34 } ^ { + 0.35 }$ & $0.35 _{ - 0.25 } ^ { + 0.29 }$ & $6.57 _{ - 0.14 } ^ { + 0.19 }$ & 10.273 & & & & \\\\\n313006381 & HIP 45012 & $1705.687 _{ - 0.0081 } ^ { + 0.0045 }$ & $21.56 _{ - 8.9 } ^ { + 54.15 }(s)$ & $0.0261 _{ - 0.0017 } ^ { + 0.0027 }$ & $2.34 _{ - 0.2 } ^ { + 0.27 }$ & $0.45 _{ - 0.3 } ^ { + 0.38 }$ & $3.85 _{ - 0.51 } ^ { + 0.31 }$ & 9.39 & & & & \\\\\n323295479* & TYC 9506-01881-1 & $1622.9258 _{ - 0.00083 } ^ { + 0.00087 }$ & $117.8 _{ - 25.8 } ^ { + 30.9 }(s)$ & $0.0981 _{ - 0.0021 } ^ { + 0.0023 }$ & $11.35 _{ - 0.67 } ^ { + 0.66 }$ & $0.839 _{ - 0.024 } ^ { + 0.019 }$ & $6.7 _{ - 0.14 } ^ { + 0.15 }$ & 10.595 & & & & \\\\\n328933398.01* & TYC 4634-01435-1 & $1880.9878 _{ - 0.0039 } ^ { + 0.0042 }$ & $24.9335 _{ - 0.0046 } ^ { + 0.005 }$ & $0.0437 _{ - 0.0022 } ^ { + 0.0023 }$ & $4.62 _{ - 0.32 } ^ { + 0.33 }$ & $0.38 _{ - 0.25 } ^ { + 0.27 }$ & $5.02 _{ - 0.22 } ^ { + 0.27 }$ & 11.215 & & & & Potential multi-planet system. \\\\\n328933398.02* & TYC 4634-01435-1 & $1848.6557 _{ - 0.0053 } ^ { + 0.0072 }$ & $50.5 _{ - 22.4 } ^ { + 77.1 }(s)$ & $0.0296 _{ - 0.0028 } ^ { + 0.0033 }$ & $3.14 _{ - 0.33 } ^ { + 0.39 }$ & $0.41 _{ - 0.28 } ^ { + 0.35 }$ & $5.99 _{ - 0.8 } ^ { + 0.77 }$ & 11.215 & & & & \\\\\n331644554 & TYC 3609-00469-1 & $1757.0354 _{ - 0.0031 } ^ { + 0.0033 }$ & $947.0 _{ - 215.0 } ^ { + 274.0 }(s)$ & $0.12 _{ - 0.025 } ^ { + 0.021 }$ & $21.84 _{ - 4.57 } ^ { + 3.86 }$ & $1.018 _{ - 0.036 } ^ { + 0.028 }$ & $10.93 _{ - 0.34 } ^ { + 0.35 }$ & 9.752 & & & & \\\\\n332657786 & TWOMASS 09595797-1609323 & $1536.7659 _{ - 0.0015 } ^ { + 0.0015 }$ & $63.76 _{ - 9.52 } ^ { + 11.13 }(s)$ & $0.14961 _{ - 0.00064 } ^ { + 0.00029 }$ & $3.83 _{ - 0.12 } ^ { + 0.12 }$ & $0.059 _{ - 0.041 } ^ { + 0.064 }$ & $3.333 _{ - 0.095 } ^ { + 0.096 }$ & 15.99 & & & & \\\\\n336075472 & TYC 3526-00332-1 & $2028.1762 _{ - 0.0043 } ^ { + 0.0037 }$ & $61.9 _{ - 24.0 } ^ { + 95.6 }(s)$ & $0.0402 _{ - 0.0022 } ^ { + 0.0033 }$ & $3.09 _{ - 0.34 } ^ { + 0.4 }$ & $0.43 _{ - 0.29 } ^ { + 0.32 }$ & $5.39 _{ - 0.23 } ^ { + 0.37 }$ & 11.842 & & & & \\\\\n349488688.01 & TYC 1529-00224-1 & $1994.283 _{ - 0.0038 } ^ { + 0.0033 }$ & $11.6254 _{ - 0.005 } ^ { + 0.0052 }$ & $0.02195 _{ - 0.00096 } ^ { + 0.00122 }$ & $3.44 _{ - 0.18 } ^ { + 0.21 }$ & $0.39 _{ - 0.27 } ^ { + 0.3 }$ & $5.58 _{ - 0.15 } ^ { + 0.18 }$ & 8.855 & & NRES (2);SOPHIE (2) & & Potential multi-planet system. \\\\\n349488688.02 & TYC 1529-00224-1 & $2002.77063 _{ - 0.00097 } ^ { + 0.00103 }$ & $15.35 _{ - 1.94 } ^ { + 4.15 }(s)$ & $0.03688 _{ - 0.00067 } ^ { + 0.00069 }$ & $5.78 _{ - 0.18 } ^ { + 0.18 }$ & $0.24 _{ - 0.16 } ^ { + 0.21 }$ & $6.291 _{ - 0.058 } ^ { + 0.074 }$ & 8.855 & & NRES (2);SOPHIE (2) & & \\\\\n356700488* & TYC 4420-01295-1 & $1756.638 _{ - 0.013 } ^ { + 0.011 }$ & $184.5 _{ - 64.7 } ^ { + 333.1 }(s)$ & $0.0173 _{ - 0.0011 } ^ { + 0.0015 }$ & $2.92 _{ - 0.2 } ^ { + 0.28 }$ & $0.44 _{ - 0.3 } ^ { + 0.34 }$ & $11.76 _{ - 0.65 } ^ { + 1.03 }$ & 8.413 & & & & \\\\\n356710041* & TYC 1993-00419-1 & $1932.2939 _{ - 0.0019 } ^ { + 0.0019 }$ & $29.6 _{ - 14.0 } ^ { + 19.0 }(s)$ & $0.0496 _{ - 0.0021 } ^ { + 0.0011 }$ & $14.82 _{ - 0.85 } ^ { + 0.84 }$ & $0.66 _{ - 0.42 } ^ { + 0.11 }$ & $12.76 _{ - 0.24 } ^ { + 0.24 }$ & 9.646 & & & & \\\\\n369532319 & TYC 2743-01716-1 & $1755.8158 _{ - 0.006 } ^ { + 0.0051 }$ & $35.4 _{ - 12.0 } ^ { + 51.6 }(s)$ & $0.0316 _{ - 0.0023 } ^ { + 0.0028 }$ & $3.43 _{ - 0.3 } ^ { + 0.37 }$ & $0.41 _{ - 0.29 } ^ { + 0.34 }$ & $5.5 _{ - 0.32 } ^ { + 0.32 }$ & 10.594 & & & Gemini & \\\\\n369779127 & TYC 9510-00090-1 & $1643.9403 _{ - 0.0046 } ^ { + 0.0058 }$ & $9.93 _{ - 3.38 } ^ { + 19.74 }(s)$ & $0.0288 _{ - 0.0015 } ^ { + 0.0033 }$ & $4.89 _{ - 0.31 } ^ { + 0.56 }$ & $0.46 _{ - 0.31 } ^ { + 0.33 }$ & $5.64 _{ - 0.38 } ^ { + 0.33 }$ & 9.279 & & & & \\\\\n384159646* & TYC 9454-00957-1 & $1630.39405 _{ - 0.00079 } ^ { + 0.00079 }$ & $11.68 _{ - 2.75 } ^ { + 4.21 }(s)$ & $0.0658 _{ - 0.0012 } ^ { + 0.0011 }$ & $9.87 _{ - 0.45 } ^ { + 0.44 }$ & $0.27 _{ - 0.18 } ^ { + 0.21 }$ & $5.152 _{ - 0.069 } ^ { + 0.087 }$ & 10.158 & SBIG (1) & NRES (8);MINERVA (6) & Gemini & \\\\\n385557214 & TYC 1807-00046-1 & $1791.58399 _{ - 0.00068 } ^ { + 0.0007 }$ & $5.62451 _{ - 0.0004 } ^ { + 0.00043 }$ & $0.096 _{ - 0.019 } ^ { + 0.032 }$ & $8.32 _{ - 2.06 } ^ { + 2.77 }$ & $0.95 _{ - 0.075 } ^ { + 0.053 }$ & $1.221 _{ - 0.094 } ^ { + 0.058 }$ & 10.856 & & & & \\\\\n388134787 & TYC 4260-00427-1 & $1811.034 _{ - 0.015 } ^ { + 0.017 }$ & $246.0 _{ - 127.0 } ^ { + 6209.0 }(s)$ & $0.0265 _{ - 0.0024 } ^ { + 0.023 }$ & $2.57 _{ - 0.28 } ^ { + 2.19 }$ & $0.55 _{ - 0.39 } ^ { + 0.44 }$ & $8.85 _{ - 1.13 } ^ { + 1.84 }$ & 10.95 & & NRES (1) & Gemini & \\\\\n404518509 & HIP 16038 & $1431.2696 _{ - 0.0037 } ^ { + 0.0035 }$ & $26.83 _{ - 9.46 } ^ { + 56.14 }(s)$ & $0.0259 _{ - 0.0013 } ^ { + 0.0022 }$ & $2.94 _{ - 0.21 } ^ { + 0.29 }$ & $0.47 _{ - 0.31 } ^ { + 0.34 }$ & $5.02 _{ - 0.23 } ^ { + 0.28 }$ & 9.17 & & & & \\\\\n408636441* & TYC 4266-00736-1 & $1745.4668 _{ - 0.0016 } ^ { + 0.0015 }$ & $37.695 _{ - 0.0034 } ^ { + 0.0033 }$ & $0.0485 _{ - 0.0019 } ^ { + 0.0023 }$ & $3.32 _{ - 0.16 } ^ { + 0.19 }$ & $0.39 _{ - 0.27 } ^ { + 0.29 }$ & $3.63 _{ - 0.1 } ^ { + 0.14 }$ & 11.93 & SBIG (1) & & Gemini & Half of the period likely. \\\\\n418255064 & TWOMASS 13063680-8037015 & $1629.3304 _{ - 0.0018 } ^ { + 0.0018 }$ & $25.37 _{ - 7.06 } ^ { + 15.41 }(s)$ & $0.0732 _{ - 0.0029 } ^ { + 0.0031 }$ & $5.57 _{ - 0.36 } ^ { + 0.38 }$ & $0.37 _{ - 0.25 } ^ { + 0.25 }$ & $3.83 _{ - 0.13 } ^ { + 0.14 }$ & 12.478 & SBIG (1) & & Gemini & \\\\\n420645189$\\dagger$ & TYC 4508-00478-1 & $1837.4767 _{ - 0.0018 } ^ { + 0.0017 }$ & $250.2 _{ - 66.6 } ^ { + 99.4 }(s)$ & $0.0784 _{ - 0.0033 } ^ { + 0.0046 }$ & $8.82 _{ - 0.55 } ^ { + 0.7 }$ & $0.892 _{ - 0.026 } ^ { + 0.028 }$ & $6.95 _{ - 0.27 } ^ { + 0.3 }$ & 10.595 & & MINERVA (1) & & SB 2 from MINERVA observations. \\\\\n422914082 & TYC 0046-00133-1 & $1431.5538 _{ - 0.0014 } ^ { + 0.0017 }$ & $12.91 _{ - 3.91 } ^ { + 8.97 }(s)$ & $0.0418 _{ - 0.0015 } ^ { + 0.0016 }$ & $3.96 _{ - 0.32 } ^ { + 0.35 }$ & $0.36 _{ - 0.25 } ^ { + 0.28 }$ & $4.07 _{ - 0.09 } ^ { + 0.126 }$ & 11.026 & Sinistro (1) & NRES (1) & & \\\\\n427344083 & TWOMASS 22563609+7040518 & $1961.8967 _{ - 0.0031 } ^ { + 0.0036 }$ & $7.77 _{ - 5.6 } ^ { + 9.65 }(s)$ & $0.107 _{ - 0.016 } ^ { + 0.025 }$ & $12.27 _{ - 1.87 } ^ { + 2.9 }$ & $0.834 _{ - 0.484 } ^ { + 0.094 }$ & $2.88 _{ - 0.3 } ^ { + 0.42 }$ & 13.404 & & & & \\\\\n436873727 & HIP 13224 & $1803.83679 _{ - 0.00058 } ^ { + 0.00056 }$ & $19.26 _{ - 5.95 } ^ { + 6.73 }(s)$ & $0.05246 _{ - 0.00061 } ^ { + 0.00059 }$ & $10.02 _{ - 0.43 } ^ { + 0.41 }$ & $0.767 _{ - 0.057 } ^ { + 0.038 }$ & $5.462 _{ - 0.081 } ^ { + 0.074 }$ & 7.51 & & & & \\\\ \n441642457* & TYC 3858-00452-1 & $1745.5102 _{ - 0.0108 } ^ { + 0.0097 }$ & $79.8072 _{ - 0.0071 } ^ { + 0.0076 }$ & $0.0281 _{ - 0.0024 } ^ { + 0.0033 }$ & $3.55 _{ - 0.34 } ^ { + 0.46 }$ & $0.934 _{ - 0.023 } ^ { + 0.026 }$ & $6.9 _{ - 0.39 } ^ { + 0.6 }$ & 9.996 & & & & \\\\\n441765914* & TWOMASS 17253007+7552562 & $1769.6154 _{ - 0.0058 } ^ { + 0.0093 }$ & $161.6 _{ - 58.2 } ^ { + 1460.1 }(s)$ & $0.0411 _{ - 0.0024 } ^ { + 0.0119 }$ & $3.6 _{ - 0.3 } ^ { + 1.01 }$ & $0.45 _{ - 0.32 } ^ { + 0.48 }$ & $7.44 _{ - 0.36 } ^ { + 1.08 }$ & 11.638 & & & & \\\\\n452920657 & TWOMASS 00332018+5906355 & $1810.5765 _{ - 0.0031 } ^ { + 0.003 }$ & $53.2 _{ - 29.0 } ^ { + 34.3 }(s)$ & $0.135 _{ - 0.016 } ^ { + 0.012 }$ & $9.71 _{ - 1.16 } ^ { + 0.9 }$ & $0.73 _{ - 0.48 } ^ { + 0.11 }$ & $4.6 _{ - 0.26 } ^ { + 0.29 }$ & 14.167 & SBIG (1) & & & \\\\\n455737331 & TYC 2779-00785-1 & $1780.7084 _{ - 0.008 } ^ { + 0.0073 }$ & $50.4 _{ - 17.6 } ^ { + 75.0 }(s)$ & $0.0257 _{ - 0.0016 } ^ { + 0.002 }$ & $3.05 _{ - 0.24 } ^ { + 0.29 }$ & $0.43 _{ - 0.29 } ^ { + 0.33 }$ & $6.6 _{ - 0.43 } ^ { + 0.5 }$ & 10.189 & SBIG (1) & & Gemini & \\\\\n456909420 & TYC 1208-01094-1 & $1779.4109 _{ - 0.0026 } ^ { + 0.0022 }$ & $5.78 _{ - 5.29 } ^ { + 5.95 }(s)$ & $0.078 _{ - 0.031 } ^ { + 0.045 }$ & $9.15 _{ - 3.61 } ^ { + 5.27 }$ & $0.973 _{ - 0.495 } ^ { + 0.063 }$ & $1.73 _{ - 0.27 } ^ { + 0.28 }$ & 10.941 & & & & \\\\\n458451774 & TWOMASS 12551793+4431260 & $1917.1875 _{ - 0.0019 } ^ { + 0.0019 }$ & $12.39 _{ - 6.34 } ^ { + 83.97 }(s)$ & $0.0752 _{ - 0.0054 } ^ { + 0.0211 }$ & $3.33 _{ - 0.26 } ^ { + 0.92 }$ & $0.61 _{ - 0.43 } ^ { + 0.32 }$ & $2.08 _{ - 0.19 } ^ { + 0.59 }$ & 13.713 & & & & \\\\\n48018596 & TYC 3548-00800-1 & $1713.4514 _{ - 0.0063 } ^ { + 0.0046 }$ & $100.1145 _{ - 0.0018 } ^ { + 0.0021 }$ & $0.049 _{ - 0.0081 } ^ { + 0.018 }$ & $7.88 _{ - 1.33 } ^ { + 2.9 }$ & $0.984 _{ - 0.028 } ^ { + 0.027 }$ & $2.83 _{ - 0.26 } ^ { + 0.29 }$ & 9.595 & & NRES (1) & Gemini & \\\\\n53309262 & TWOMASS 07475406+5741549 & $1863.1133 _{ - 0.0064 } ^ { + 0.0061 }$ & $294.8 _{ - 96.0 } ^ { + 327.0 }(s)$ & $0.1239 _{ - 0.0075 } ^ { + 0.0098 }$ & $5.38 _{ - 0.36 } ^ { + 0.46 }$ & $0.46 _{ - 0.31 } ^ { + 0.28 }$ & $6.74 _{ - 0.45 } ^ { + 0.62 }$ & 15.51 & & & & \\\\\n53843023 & TYC 6956-00758-1 & $1328.0335 _{ - 0.0054 } ^ { + 0.0057 }$ & $202.0 _{ - 189.0 } ^ { + 272.0 }(s)$ & $0.058 _{ - 0.02 } ^ { + 0.056 }$ & $5.14 _{ - 1.77 } ^ { + 4.99 }$ & $0.962 _{ - 0.597 } ^ { + 0.083 }$ & $4.25 _{ - 0.72 } ^ { + 0.66 }$ & 11.571 & & & & \\\\\n55525572* & TYC 8876-01059-1 & $1454.6713 _{ - 0.0066 } ^ { + 0.0065 }$ & $83.8951 _{ - 0.004 } ^ { + 0.004 }$ & $0.0343 _{ - 0.001 } ^ { + 0.0021 }$ & $7.31 _{ - 0.46 } ^ { + 0.56 }$ & $0.43 _{ - 0.29 } ^ { + 0.31 }$ & $13.54 _{ - 0.3 } ^ { + 0.51 }$ & 10.358 & & CHIRON (5) & Gemini & Confirmed planet \\citep{2020eisner} \\\\\n63698669* & TYC 6993-00729-1 & $1364.6226 _{ - 0.0074 } ^ { + 0.0067 }$ & $73.6 _{ - 26.8 } ^ { + 133.6 }(s)$ & $0.0248 _{ - 0.0019 } ^ { + 0.0023 }$ & $2.15 _{ - 0.2 } ^ { + 0.25 }$ & $0.42 _{ - 0.29 } ^ { + 0.35 }$ & $5.63 _{ - 0.32 } ^ { + 0.57 }$ & 10.701 & SBIG (1) & & & \\\\\n70887357* & TYC 5883-01412-1 & $1454.3341 _{ - 0.0016 } ^ { + 0.0015 }$ & $56.1 _{ - 15.3 } ^ { + 18.8 }(s)$ & $0.0605 _{ - 0.0027 } ^ { + 0.0027 }$ & $12.84 _{ - 0.86 } ^ { + 0.9 }$ & $0.917 _{ - 0.028 } ^ { + 0.016 }$ & $7.29 _{ - 0.18 } ^ { + 0.19 }$ & 9.293 & & & & \\\\\n7422496$\\dagger$ & HIP 25359 & $1470.3625 _{ - 0.0031 } ^ { + 0.0023 }$ & $61.4 _{ - 16.7 } ^ { + 49.0 }(s)$ & $0.0255 _{ - 0.001 } ^ { + 0.0011 }$ & $2.44 _{ - 0.15 } ^ { + 0.16 }$ & $0.37 _{ - 0.25 } ^ { + 0.29 }$ & $5.89 _{ - 0.15 } ^ { + 0.15 }$ & 9.36 & & MINERVA (4) & & SB 2 from MINERVA observations. \\\\\n82452140 & TYC 3076-00921-1 & $1964.292 _{ - 0.011 } ^ { + 0.011 }$ & $21.1338 _{ - 0.0052 } ^ { + 0.0066 }$ & $0.0266 _{ - 0.0019 } ^ { + 0.0027 }$ & $2.95 _{ - 0.25 } ^ { + 0.34 }$ & $0.42 _{ - 0.29 } ^ { + 0.36 }$ & $5.87 _{ - 0.62 } ^ { + 0.94 }$ & 10.616 & & & & \\\\\n88840705 & TYC 3091-00808-1 & $2026.6489 _{ - 0.001 } ^ { + 0.001 }$ & $260.6 _{ - 87.6 } ^ { + 142.2 }(s)$ & $0.109 _{ - 0.023 } ^ { + 0.027 }$ & $9.98 _{ - 2.28 } ^ { + 2.75 }$ & $1.001 _{ - 0.042 } ^ { + 0.037 }$ & $4.72 _{ - 0.13 } ^ { + 0.15 }$ & 9.443 & & & & \\\\\n91987762* & HIP 47288 & $1894.25381 _{ - 0.00051 } ^ { + 0.00047 }$ & $10.51 _{ - 3.48 } ^ { + 3.67 }(s)$ & $0.05459 _{ - 0.00106 } ^ { + 0.00097 }$ & $9.56 _{ - 0.56 } ^ { + 0.52 }$ & $0.771 _{ - 0.062 } ^ { + 0.033 }$ & $4.342 _{ - 0.073 } ^ { + 0.063 }$ & 7.87 & & NRES (4) & Gemini & \\\\\n95768667 & TYC 1434-00331-1 & $1918.3318 _{ - 0.0093 } ^ { + 0.0079 }$ & $26.9 _{ - 12.4 } ^ { + 72.3 }(s)$ & $0.0282 _{ - 0.0022 } ^ { + 0.0031 }$ & $3.54 _{ - 0.32 } ^ { + 0.43 }$ & $0.48 _{ - 0.33 } ^ { + 0.35 }$ & $5.4 _{ - 0.64 } ^ { + 0.76 }$ & 10.318 & & & & \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\textbf{Properties of PHT candidates (continued)}}\n\\label{tab:PHT-caniddates2}\n\\end{table}\n\\end{landscape}\n\n\n\\section{Conclusion}\n\\label{sec:condlusion}\n\nWe present the results from the analysis of the first 26 \\emph{TESS}\\ sectors. The outlined citizen science approach engages over 22 thousand registered citizen scientists who completed 12,617,038 classifications from December 2018 through August 2020 for the sectors observed during the first two years of the \\emph{TESS}\\ mission. We applied a systematic search for planetary candidates using visual vetting by multiple volunteers to identify \\emph{TESS}\\ targets that are most likely to host a planet. Between 8 and 15 volunteers have inspected each \\emph{TESS}\\ light curve and marked times of transit-like events using the PHT online interface. For each light curve, the markings from all the volunteers who saw that target were combined using an unsupervised machine learning method, known as DBSCAN, in order to identify likely transit-like events. Each of these identified events was given a transit score based on the number of volunteers who identified a given event and on the user weighting of each of those volunteers. Individual user weights were calculated based on the user's ability to identify simulated transit events, injected into real \\emph{TESS}\\ light curves, that are displayed on the PHT site alongside of the real data. The transit scores were then used to generate a ranked list of candidates that range from most likely to least likely to host a planet candidate. The top 500 highest ranked candidates were further vetted by the PHT science team. This stage of vetting primarily made use of the open source {\\sc latte} \\citep{LATTE2020} tool which generates a number of standard diagnostic plots that help identify promising candidates and weed out false positive signals. \n\nOn average we found around three high priority candidates per sector which were followed up using ground based telescopes, where possible. To date, PHT has statistically confirmed one planet, TOI-813 \\citep{2020eisner}: a Saturn-sized planet on an 84 day orbit around a subgiant host star. Other PHT identified planets listed in this paper are being followed up by other teams of astronomers, such as TOI-1899 (TIC 172370679) which was recently confirmed to be a warm Jupiter transiting an M-dwarf \\citep{canas2020}. The remaining candidates outlined in this paper require further follow-up observations to confirm their planetary nature.\n\nThe sensitivity of our transit search effort was assessed using synthetic data, as well as the known TOI and TCE candidates flagged by the SPOC pipeline. For simulated planets (where simulated signals are injected into real \\emph{TESS}\\ light curves) we have shown that the recovery efficiency of human vetting starts to decrease for transit-signals that have a SNR less than 7.5. The detection efficiency was further evaluated by the fractional recovery of the TOI and TCEs. We have shown that PHT is over 85 \\% complete in the recovery of planets that have a radius greater than 4 $R_{\\oplus}$, 51 \\% complete for radii between 3 and 4 $R_{\\oplus}$ and 49 \\% complete for radii between 2 and 3 $R_{\\oplus}$. Furthermore, we have shown that human vetting is not sensitive to the number of transits present in the light curve, meaning that they are equally likely to identify candidates on longer orbital periods as they are those with shorter orbital periods for periods greater than $\\sim$ 1 day. Planets with periods shorter than around 1 day exhibit over 20 transits within one \\emph{TESS}\\ sectors resulting in a decrease in identification by the volunteers. This is due to many volunteers only marking a random subset of these events, resulting in a lack of consensus on any given transit event and thus decreasing the overall transit score of these light curves. \n\nIn addition to searching for signals due to transiting exoplanets, PHT provides a platform that can be used to identify other stellar phenomena that may otherwise be difficult to identify with automated pipelines. Such phenomena, including eclipsing binaries, multiple stellar systems, dwarf novae, and stellar flares are often mentioned on the PHT discussion forums where volunteers can use searchable hashtags and comments to bring these systems to the attention of other citizen scientists as well as the PHT science team. All of the eclipsing binaries identified on the site, for example, are being used and vetted by the \\emph{TESS}\\ Eclipsing Binary Working Group (Prsa et al. in prep). Furthermore, we have investigated the nature of all of the targets that were identified as possible multiple stellar systems, as summarised in Table~\\ref{tab:PHT-multis}.\n\nOverall we have shown that large scale visual vetting can complement the findings \\textcolor{red}{from the major \\emph{TESS}\\ pipeline} by identifying longer period planets that may only exhibit a single transit event in their light curve, as well as in finding signals that are aperiodic or embedded in a strong varying stellar signal. The identification of planets around stars with variable signals allow us to potentially characterise the host-star (e.g., with asteroseismology or spot modulation). Additionally, the longer period planets are integral to our understanding of how planet systems form and evolve, as they allow us to investigate the evolution of planets that are farther away from their host star and therefore less dependent on stellar radiation. \\textcolor{red}{While automated pipelines specifically designed to identify single transit events in the \\emph{TESS}\\ data exist \\citep[e.g., ][]{Gill2020}, neither their methodology nor the full list of their findings are yet publicly available and thus we are unable to compare results.} \n\nThe planets that PHT finds have longer periods ($\\gtrsim$ 27 d) than those found in \\emph{TESS}\\ data using automated pipelines, and are more typical of the Kepler sample (25\\% of Kepler confirmed planets have periods greater than 27 days\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}). However, the Kepler planets are considerably fainter, and thus less amenable to ground-based follow-up or atmospheric characterisation from space (CHEOPS and JWST). Thus PHT helps to bridge the parameter spaces covered by these two missions, by identifying longer period planet candidates around bright, nearby stars, for which we can ultimately obtain precise planetary mass estimates. Although statistical characterisation of exo-planetary systems is no doubt important, precise mass measurements are key to developing our understanding of exoplanets and the physics which dictate their evolution. In particular, identification of this PHT sample provides follow-up targets to investigate the dependence of photo-evaporation on the mass of planets as well as on the planet radius, and will help our understanding of the photo-evaporation valley at longer orbital periods \\citep{Owen2013}. \n\nPHT will continue to operate throughout the \\emph{TESS}\\ extended mission, hopefully allowing us to identify even longer period planets as well as help verify some of the existing candidates with additional transits. \n\n\n\n\\begin{table*}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{cccccccccc}\n\\textbf{TIC} & \\textbf{Period (days)} & \\textbf{Epoch (\\textcolor{red}{BJD - 2457000})} & \\textbf{Depth (ppm)} & \\textbf{Comment} \\\\\n\\hline\n13968858 & $3.4850 \\pm 0.001$ & $ 1684.780 \\pm 0.005$ & 410000 & Candidate multiple system \\\\\n & $1.4380 \\pm 0.001$ & $ 1684.335 \\pm 0.005$ & 50000 & \\\\\n35655828 & $ 8.073 \\pm 0.01$ & $ 1550.94 \\pm 0.01 $ & 23000 & Confirmed blend \\\\\n & $ 1.220 \\pm 0.001 $ & $ 1545.540 \\pm 0.005 $ & 2800 & \\\\\n63291675 & $ 8.099 \\pm 0.003 $ & $ 1685.1 \\pm 0.01 $ & 60000 & Confirmed blend \\\\\n & $ 1.4635 \\pm 0.0005 $ & $ 1683.8 \\pm 0.1 $ & 7000 & \\\\\n63459761 & $4.3630 \\pm 0.003 $ & $ 1714.350 \\pm 0.005 $ & 160000 & Candidate multiple system \\\\\n & $4.235 \\pm $ 0.005 & $ 1715.130 \\pm 0.03$ & 35000 & \\\\\n104909909 & $1.3060 \\pm 0.0001$ & $ 1684.470 \\pm 0.005$ & 32000 & Candidate multiple system \\\\\n & $2.5750 \\pm 0.003$ & $ 1684.400 \\pm 0.005$ & 65000 & \\\\\n115980439 & $ 4.615 \\pm 0.002 $ & $ 1818.05 \\pm 0.01 $ & 95000 & Confirmed blend \\\\\n & $ 0.742 \\pm 0.005 $ & $ 1816.23 \\pm 0.02 $ & 2000 & \\\\\n120362128 & $ 3.286 \\pm 0.002 $ & $ 1684.425 \\pm 0.01 $ & 33000 & Candidate multiple system \\\\\n & $ - $ & $ 1701.275 \\pm 0.02 $ & 12000 & \\\\\n & $ - $ & $ 1702.09 \\pm 0.02 $ & 36000 & \\\\\n121945407 & $ 0.9056768 \\pm 0.00000002$ & $-1948.76377 \\pm 0.0000001$ & 2500 & Confirmed multiple system $^{(\\mathrm{a})}$ \\\\\n & $ 45.4711 \\pm 0.00002$ & $-1500.0038 \\pm 0.0004 $ & 7500 & \\\\\n122275115 & $ - $ & $ 1821.779 \\pm 0.01 $ & 155000 & Candidate multiple system \\\\\n & $ - $ & $ 1830.628 \\pm 0.01 $ & 63000 & \\\\\n & $ - $ & $ 1838.505 \\pm 0.01 $ & 123000 & \\\\\n229804573 & $1.4641 \\pm 0.0005$ & $ 1326.135 \\pm 0.005$ & 180000 & Candidate multiple system \\\\\n & $0.5283 \\pm 0.0001$ & $ 1378.114 \\pm 0.005$ & 9000 & \\\\\n252403752 & $ - $ & $ 1817.73 \\pm 0.01 $ & 2800 & Candidate multiple system \\\\\n & $ - $ & $ 1829.76 \\pm 0.01 $ & 23000 & \\\\\n & $ - $ & $ 1833.63 \\pm 0.01 $ & 5500 & \\\\\n258837989 & $0.8870 \\pm 0.001$ & $ 1599.350 \\pm 0.005$ & 64000 & Candidate multiple system \\\\\n & $3.0730 \\pm 0.001$ & $ 1598.430 \\pm 0.005$ & 25000 & \\\\\n266958963 & $1.5753 \\pm 0.0002$ & $ 1816.425 \\pm 0.001$ & 265000 & Candidate multiple system \\\\\n & $2.3685 \\pm 0.0001$ & $ 1817.790 \\pm 0.001$ & 75000 & \\\\\n278956474 & $5.488068 \\pm 0.000016 $ & $ 1355.400 \\pm 0.005$ & 93900 & Confirmed multiple system $^{(\\mathrm{b})}$ \\\\\n & $5.674256 \\pm \u22120.000030$ & $ 1330.690 \\pm 0.005$ & 30000 & \\\\\n284925600 & $ 1.24571 \\pm 0.00001 $ & $ 1765.248 \\pm 0.005 $ & 490000 & Confirmed blend \\\\\n & $ 0.31828 \\pm 0.00001 $ & $ 1764.75 \\pm 0.005 $ & 35000 & \\\\\n293954660 & $2.814 \\pm 0.001 $ & $ 1739.177 \\pm 0.03 $ & 272000 & Confirmed blend \\\\\n & $4.904 \\pm 0.03 $ & $ 1739.73 \\pm 0.01 $ & 9500 & \\\\\n312353805 & $4.951 \\pm 0.003 $ & $ 1817.73 \\pm 0.01 $ & 66000 & Confirmed blend \\\\\n & $12.89 \\pm 0.01 $ & $ 1822.28 \\pm 0.01$ & 19000 & \\\\\n318210930 & $ 1.3055432 \\pm 0.000000033$ & $ -653.21602 \\pm 0.0000013$ & 570000 & Confirmed multiple system $^{(\\mathrm{c})}$ \\\\\n & $ 0.22771622 \\pm 0.0000000035$& $ -732.071119 \\pm 0.00000026 $ & 220000 & \\\\\n336434532 & $ 3.888 \\pm 0.002 $ & $ 1713.66 \\pm 0.01 $ & 22900 & Confirmed blend \\\\\n & $ 0.949 \\pm 0.003 $ & $ 1712.81 \\pm 0.01 $ & 2900 & \\\\\n350622185 & $1.1686 \\pm 0.0001$ & $ 1326.140 \\pm 0.005$ & 200000 & Candidate multiple system \\\\\n & $5.2410 \\pm 0.0005$ & $ 1326.885 \\pm 0.05$ & 4000 & \\\\\n375422201 & $9.9649 \\pm 0.001$ & $ 1711.937 \\pm 0.005$ & 245000 & Candidate multiple system \\\\\n & $4.0750 \\pm 0.001$ & $ 1713.210 \\pm 0.01 $ & 39000 & \\\\\n376606423 & $ 0.8547 \\pm 0.0002 $ & $ 1900.766 \\pm 0.005 $ & 9700 & Candidate multiple system \\\\\n & $ - $ & $ 1908.085 \\pm 0.01 $ & 33000 & \\\\\n394177355 & $ 94.22454 \\pm 0.00040 $ & $ - $ & - & Confirmed multiple system $^{(\\mathrm{d})}$ \\\\\n & $ 8.6530941 \\pm 0.0000016$ & $-2038.99492 \\pm 0.00017 $ & 140000 & \\\\\n & $ 1.5222468 \\pm 0.0000025$ & $ -2039.1201 \\pm 0.0014 $ & - & \\\\\n & $ 1.43420486 \\pm 0.00000012 $ & $-2039.23941 \\pm 0.00007 $ & - & \\\\\n424508303 & $ 2.0832649 \\pm 0.0000029 $ & $-3144.8661 \\pm 0.0034 $ & 430000 & Confirmed multiple system $^{(\\mathrm{e})}$ \\\\\n & $ 1.4200401 \\pm 0.0000042 $ & $-3142.5639 \\pm 0.0054 $ & 250000 & \\\\\n441794509 & $ 4.6687 \\pm 0.0002 $ & $ 1958.895 \\pm 0.005 $ & 34000 & Candidate multiple system \\\\\n & $ 14.785 \\pm 0.002 $ & $ 1960.845 \\pm 0.005 $ & 17000 & \\\\\n470710327 & $ 9.9733 \\pm 0.0001 $ & $ 1766.27 \\pm 0.005 $ & 51000 & Confirmed multiple system $^{(\\mathrm{f})}$ \\\\\n & $ 1.104686 \\pm 0.00001 $ & $ 1785.53266 \\pm 0.000005$ & 42000 & \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\nNote -- $^{(\\mathrm{a})}$ KOI-6139, \\citet{Borkovits2013}; \n$^{(\\mathrm{b})}$ \\citet{2020Rowden}\n$^{(\\mathrm{c})}$ \\citet{Koo2014}; \n$^{(\\mathrm{d})}$ KOI-3156, \\citet{2017Helminiak};\n$^{(\\mathrm{e})}$ V994 Her; \\citet{Zasche2016}; \n$^{(\\mathrm{f})}$ Eisner et al. {\\it in prep.}\n}\n\n\\label{tab:PHT-multis}\n\n\\end{table*}\n\n\\section*{Data Availability}\n\nAll of the \\emph{TESS}\\ data used within this article are hosted and made publicly available by the Mikulski Archive for Space Telescopes (MAST, \\url{http:\/\/archive.stsci.edu\/tess\/}). Similarly, the Planet Hunters TESS classifications made by the citizen scientists can be found on the Planet Hunters Analysis Database (PHAD, \\url{https:\/\/mast.stsci.edu\/phad\/}), which is also hosted by MAST. All planet candidates and their properties presented in this article have been uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS, \\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}) website as community TOIs (cTOIs), under their corresponding TIC IDs. The ground-based follow-up observations of individual targets will be shared on reasonable request to the corresponding author.\n\nThe models of individual transit events and the data validation reports used for the vetting of the targets were both generated using publicly available open software codes, \\texttt{pyaneti}\\ and {\\sc latte}.\n\n\\section*{Acknowledgements} \n\nThis project works under the in \\textit{populum veritas est} philosophy, and for that reason we would like to thank all of the citizen scientists who have taken part in the Planet Hunters TESS project and enable us to find many interesting astrophysical systems. \n\nSome of the observations in the paper made use of the High-Resolution Imaging instruments `Alopeke and Zorro. `Alopeke and Zorro were funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. `Alopeke and Zorro were mounted on the Gemini North and South telescope of the international Gemini Observatory, a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci\\'{o}n y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog\\'{i}a e Innovaci\\'{o}n (Argentina), Minist\\'{e}rio da Ci\\^{e}ncia, Tecnologia, Inova\\c{c}\\~{o}es e Comunica\\c{c}\\~{o}es (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). The authors also acknowledge the very significant cultural role and sacred nature of Maunakea. We are most fortunate to have the opportunity to conduct observations from this mountain.\n\nThis project has also received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement N$^\\circ$730890. This material reflects only the authors views and the Commission is not liable for any use that may be made of the information contained therein. This work makes use of observations from the Las Cumbres Observatory global telescope network, including the NRES spectrograph and the SBIG and Sinistro photometric instruments. \n\nFurthermore, NLE thanks the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant N$^\\circ$1829740, the Brinson Foundation, and the Moore Foundation; her participation in the program has benefited this work. Finally, CJ acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement N$^\\circ$670519: MAMSIE), and from the Research Foundation Flanders (FWO) under grant agreement G0A2917N (BlackGEM). \n\nThis research made use of Astropy, a community-developed core Python package for Astronomy \\citep{astropy2013}, matplotlib \\citep{matplotlib}, pandas \\citep{pandas}, NumPy \\citep{numpy}, astroquery \\citep{ginsburg2019astroquery} and sklearn \\citep{pedregosa2011scikit}. \n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Planet candidate descriptions}\n\\label{appendixA}\n\nA short outline all of the planet candidates, and any conclusions drawn from follow-up observations (where available). A more in depth description of the ground-based data will be presented in a follow-up paper. Unless stated otherwise, these candidates are not TOIs at the time of writing. Candidates for which we have no additional information to complement the results presented in Table~\\ref{tab:PHT-caniddates} are not discussed further here.\n\n\\subsection{Single-transit planet candidates}\n\n\n\\textbf{TIC 103633672.} Single transit event identified in Sector 20. The single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary. We caution that there is a star on the same pixels, which is 0.1 mag brighter. We are unable to rule this star out as the cause for the transit-like signal.\n\n\\textbf{TIC 110996418.} Single transit event identified in Sector 10. We caution that there is a star on the same \\emph{TESS}\\ pixel, which is 2.4 magnitudes fainter than the target.\n\n\\textbf{TIC 128703021.} Single transit event identified in Sector 11. With a stellar radius of 1.6 $R_{\\odot}$ and a T$_{eff}$ of 6281 this host star is likely in the subgiant phase of its evolution. The 43 spectra obtained with MINERVA australis and the two obtained with LCO\/NRES are consistent with a planetary nature. Gemini speckle interferometry shows no nearby companion stars.\n\n\n\\textbf{TIC 142087638.} Single transit event identified in Sector 7. The best fit \\texttt{pyaneti}\\ model of the transit suggests an orbital period of only 3.14 d. As there are no additional transits seen in the light curve, this period is clearly not possible. We caution that the transit is most likely caused by a grazing object, and is therefore likely to be caused by a stellar companion. However, without further data we are unable to rule this candidate out as being planetary in nature.\n\n\\textbf{TIC 159159904.} Single transit event identified in Sector 22. The initial two observations obtained using LCO\/NRES show no sign of the candidate being a double lined spectroscopic binary.\n\n\n\\textbf{TIC 166184426.} Single transit event identified in Sector 11. Since the PHT discovery this cTOI has been become the priority 1 (1=highest, 5=lowest) target TOI 1955.01.\n\n\n\\textbf{TIC 172370679.} Single transit event identified in Sector 15. \\textcolor{black}{This candidate was independently discovered and verified using a BLS algorithm used to search for transiting planets around M-dwarfs. The candidate is now the confirmed planet TOI 1899 b \\citep{canas2020}.} \n\n\\textbf{TIC 174302697.} Single transit event identified in Sectors 16. With a stellar radius of 1.6 $R_{\\odot}$ and a T$_{eff}$ of 6750 this host star is likely in the subgiant phase of its evolution. \\textcolor{black}{This candidate was initially flagged as a TCE and but was erroneously discounted due to the pipeline mistaking the data glitch at the time of a momentum dump as a secondary eclipse.} Since the PHT discovery this cTOI has become the priority 3 target TOI 1896.01. \n\n\\textbf{TIC 192415680.} Single transit event identified in Sector 18. The two epochs of RV measurement obtained with OHP\/SOPHIE are consistent with a planetary scenario.\n\n\\textbf{TIC 192790476.} Single transit event identified in Sector 5. This target has been identified to be a wide binary with am angular separation of 72.40 arcseconds \\citep{andrews2017wideBinary} and a period of 162705 years \\citep{benavides2010new}. The star exhibits large scale variability on the order of around 10 d. The signal is consistent with that of spot modulations, which would suggest that this is a slowly rotating star.\n\n\n\n\\textbf{TIC 219466784.} Single transit event identified in Sector 22. We caution that there is a nearby companion located within the same \\emph{TESS}\\ pixel at an angular separation of 16.3 with a Vmag of 16.3\". Since the PHT discovery this cTOI has become the priority 2 target TOI 2007.01.\n\n\n\n\\textbf{TIC 229055790.} Single transit event identified in Sector 21. We note that the midpoint of the transit-like events coincides with a \\emph{TESS}\\ momentum dump, however, we believe the shape to be convincing enough to warrant further investigation. The two LCO\/NRES spectra show no sign of this being a spectroscopic binary.\n\n\\textbf{TIC 229608594.} Single transit event identified in Sector 24. Since the PHT discovery this cTOI has become the priority 3 target TOI 2298.01.\n\n\n\n\\textbf{TIC 233194447.} Single transit event identified in Sector 14. The transit-like event is shallow and asymmetric and we cannot definitively rule out systematics as the cause for the event without additional data. The initial two LCO\/NRES spectra show no sign of this target being a spectroscopic binary.\n\n\\textbf{TIC 237201858.} Single transit event identified in Sector 18. The single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary.\n\n\\textbf{TIC 243187830.} Single transit event identified in Sector 18. There are no nearby bright stars. \\textcolor{black}{This light curve was initially flagged as a TCE, however, the flagged events corresponded to stellar variability and not the same event identified by PHT.} The single LCO\/NRES spectrum shows no sign of this being a double lined spectroscopic binary. Since the PHT discovery this cTOI has become the priority 3 target TOI 2009.01.\n\n\\textbf{TIC 243417115.} Single transit event identified in Sector 11. We note that the best fit \\texttt{pyaneti}\\ model of the transit suggests an orbital period of only 1.81 d. As there are no additional transits seen in the light curve, this period is clearly not possible. We caution that the transit is most likely caused by a grazing object, and is therefore likely to be caused by a stellar companion. However, without further follow-up data we are unable to rule this candidate out as being planetary in nature.\n\n\n\\textbf{TIC 264544388.} Single transit event identified in Sector 19. The single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary. Apart from the single transit event, the light curve shows no obvious signals. A periodogram of the light curve, however, reveals a series of five significant peaks, nearly equidistantly spaced by $\\sim1.03$~d$^{-1}$. Additionally, a rotationally split quintuplet is visible at 7.34~d$^{-1}$, with a splitting of $\\sim0.12$~d$^{-1}$, suggesting an $\\ell=2$ p-mode pulsation. The Maelstrom code \\citep{hey2020maelstrom} revealed pulsation timing variations which are consistent with a long period planet. \\textcolor{black}{The short period signal, which was also identified by the periodogram, was flagged as a TCE, however, the single-transit event was not flagged as a TCE.} Since the PHT discovery this cTOI has become the priority 3 target TOI 1893.01.\n\n\\textbf{TIC 264766922.} Single transit event identified in Sector 8. With a stellar radius of 1.7 $R_{\\odot}$ and a T$_{eff}$ of 6913 K this host star is likely entering the subgiant phase of its evolution. The V-shape of this transit and the resultant high impact parameter suggests that the object is grazing. We can therefore not rule out that this candidate it a grazing eclipsing binary. There are clear p-mode pulsations at frequencies of 9.01 and 11.47 cycles per day, as well as possible g-mode pulsations. \\textcolor{black}{A very short period signal within this light curve was flagged as a TCE, however, the single transit event was ignored by the pipeline.}\n\n\\textbf{TIC 26547036.} Single transit event identified in Sector 14. The four LCO\/NRES observations are consistent with the target being a planetary body and show no sign of the signal being caused by a spectroscopic binary. We caution that there is a star on the same \\emph{TESS}\\ pixel, however, this star is 8.2 magnitudes fainter than the target, and therefore unable to be responsible for the transit event seen in the light curve. Gemini speckle interferometry reveal no additional nearby companion stars. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit event the pipeline identified further periodic signals that correspond to times of momentum dumps. Due to this, the candidate was never promoted to TOI status.}\n\n\n\\textbf{TIC 278990954.} Single transit event identified in Sector 12. With a stellar radius of 2.6 $R_{\\odot}$ and a T$_{eff}$ of 5761 K this host star is likely in the subgiant phase of its evolution. We note that there are two additional stars on the same pixel as TIC 278990954. These two stars are 2.7 and 3.7 magnitudes fainter in the v-band than the target and can't be ruled out as the cause for the transit-like event without additional follow-up data.\n\n\\textbf{TIC 280865159.} Single transit event identified in Sector 16. Gemini speckle interferometry revealed any nearby companion stars. Since the PHT discovery this cTOI has become the priority 3 target TOI 1894.01.\n\n\\textbf{TIC 284361752.} Single transit event identified in Sector 26. Since the PHT discovery this cTOI has become the priority 2 target TOI 2294.01.\n\n\n\\textbf{TIC 296737508.} Single transit event identified in Sector 8. The single LCO\/NRES and the single MINERVA australis spectra show no sign of this being a spectroscopic binary. The Sinistro snapshot image revealed no additional nearby companions.\n\n\\textbf{TIC 298663873.} Single transit event identified in Sector 19. The two LCO\/NRES spectra show no sign of this being a spectroscopic binary. With a stellar radius of 1.6 $R_{\\odot}$ and a T$_{eff}$ of 6750 this host star is likely in the subgiant phase of its evolution. Gemini speckle images obtained by other teams show no signs of there being nearby companion stars. Since the PHT discovery this cTOI has become the priority 3 target TOI 2180.01.\n\n\n\\textbf{TIC 303050301.} Single transit event identified in Sector 2. The variability of the light curve is consistent with spot modulation. A single LCO\/NRES spectrum shows no signs of this being a double lined spectroscopic binary.\n\n\\textbf{TIC 303317324.} Single transit event identified in Sector 2. We note that a second transit was later seen in Sector 29, however, as this work only covers sectors 1-26 of the primary \\emph{TESS}\\ mission, this candidates is considered a single-transit event in this work. \n\n\n\\textbf{TIC 304142124.} Single transit event identified in Sector 10.\\textcolor{black}{This target was independently identified as part of the Planet Finder Spectrograph, which uses precision RVs \\citep{diaz2020}. This candidate is know the confirmed planet HD 95338 b.}\n\n\n\n\n\n\n\n\\textbf{TIC 331644554.} Single transit event identified in Sector 16. There is a clear mono-periodic signal in the periodogram at around 11.2 cycles per day, which is consistent with p-mode pulsation.\n\n\\textbf{TIC 332657786.} Single transit event identified in Sector 8. We caution that there is a star on the adjacent \\emph{TESS}\\ pixel that is brighter in the V-band by 2.4 magnitudes. At this point we are unable to rule out this star as the cause of the transit-like signal. \n\n\n\\textbf{TIC 356700488.} Single transit event identified in Sector 16. There is a clear mono-periodic signal in the periodogram at around 1.2 cycles per day, which is consistent with either spot modulation or g-mode pulsation. However, there is no clear signal visible in the light curve that would allow us to differentiate between these two scenarios based on the morphology of the variation. Since the PHT discovery this cTOI has become the priority 3 target TOI 2098.01.\n\n\\textbf{TIC 356710041.} Single transit event identified in Sector 23. With a stellar radius of 2.8 $R_{\\odot}$ and a T$_{eff}$ of 5701 K this host star is likely in the subgiant phase of its evolution. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit event, the pipeline identified a further event that corresponds to the time of a momentum dump. Due to this the candidate failed the `odd-even test' and was initially discarded as a TOI.} Since the PHT discovery this cTOI has become the priority 3 target TOI 2065.01\n\n\\textbf{TIC 369532319.} Single transit event identified in Sector 16. Gemini speckle interferometry revealed no nearby companion stars.\n\n\n\\textbf{TIC 384159646.} Single transit event identified in Sector 12. The eight LCO\/NRES and six MINERVA australis spectra are consistent with this candidate being a planet. Both the SBIG snapshot and the Gemini speckle interferometry observations revealed no companion stars. Since the PHT discovery this cTOI has become the priority 3 target TOI 1895.01.\n\n\n\n\n\\textbf{TIC 418255064.} Single transit event identified in Sector 12. The Gemini speckle image shows no sign of nearby companions.\n\n\n\\textbf{TIC 422914082.} Single transit event identified in Sector 4. Single Sinistro snapshot image reveals no additional nearby stars.\n\n\\textbf{TIC 427344083.} Single transit event identified in Sector 24. We note that there is a star on the adjacent \\emph{TESS}\\ pixel to the target, which is 3.5 magnitude fainter in the V-band than the target star. We also caution that the V-shape of the transit and the high impact parameter suggest that this is a grazing transit. However, without additional follow-up observations we are unable to rule this candidate out as a planet.\n\n\n\\textbf{TIC 436873727.} Single transit event identified in Sector 18. The host star shows strong variability on the order of one day, which is consistent with spot modulations or g-mode pulsations. The periodogram reveals multi-periodic behaviour in the low frequency range consistent with g-mode pulsations. \n\n\\textbf{TIC 452920657.} Single transit event identified in Sector 17. The V-shape of this transit suggests that the object is grazing and future follow-up observations may reveal this to be an EB. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit event, the pipeline identified a further two event that corresponds to likely stellar variability. Due to this the candidate failed the `odd-even test' and was initially discarded as a TOI.}\n\n\\textbf{TIC 455737331.} Single transit event identified in Sector 17. We note that there is a star on the same \\emph{TESS}\\ pixel as the target, which is 4.5 magnitudes fainter in the V-band. Neither the SBIG snapshot nor the Gemini speckle interferometry revealed any further nearby companion stars.\n\n\\textbf{TIC 456909420.} Single transit event identified in Sector 17. We caution that the V-shape of the transit and the high impact parameter suggest that this is a grazing transit. However, without additional follow-up observations we are unable to rule this candidate out as a planet.\n\n\n\n\n\\textbf{TIC 53843023.} Single transit event identified in Sector 1. We caution that the high impact parameter returned by the best fit \\texttt{pyaneti}\\ model suggests that the transit event is caused by a grazing body. However, at this point we are unable to rule this candidate out as being planetary in nature.\n\n\\textbf{TIC 63698669.} Single transit event identified in Sector 2. The SBIG snapshot image revealed no nearby companions. \\textcolor{black}{This candidate was initially identified as a TCE, however, in addition to the single transit event, the pipeline identified a further 3 events the light curve. Due to these, additional events, which correspond to stellar variability, the candidate was not initially promoted to TOI status.} However, since the PHT discovery this cTOI has become TOI 1892.01.\n\n\\textbf{TIC 70887357.} Single transit event identified in Sector 5. With a stellar radius of 2.1 $R_{\\odot}$ and a T$_{eff}$ of 5463 K this host star is likely in the subgiant phase of its evolution. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit-event the pipeline identified a further signal, and thus failed the `odd-even' transit test.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 2008.01.\n\n\n\n\\textbf{TIC 91987762.} Single transit event identified in Sector 21. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit-event the pipeline identified a further signal, and thus failed the `odd-even' transit test.} Since the PHT discovery this cTOI has become the priority 3 target TOI 1898.01.\n\n\n\n\\subsection{Multi-transit and multi-planet candidates}\n\n\n\\textbf{TIC 160039081.} Multi-transit candidate with a period of 30.2 d. Single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary and a snapshot image using SBIG shows no nearby companions. The Gemini speckle images also show no additional nearby companions. Since the PHT discovery this cTOI has become the priority 1 target TOI 2082.01.\n\n\\textbf{TIC 167661160.} Multi-transit candidate with a period of 36.8 d. The nine LCO\/NRES and four MINERVA australis spectra have revealed this to be a long period eclipsing binary.\n\n\\textbf{TIC 179582003.} Multi-transit candidate with a period of 104.6 d. There is a clear mono-periodic signal in the periodogram at around 0.59 cycles per day, which is consistent with either spot modulation or g-mode pulsation. We caution that this candidate is located in a crowded field. With a stellar radius of 2.0 $R_{\\odot}$ and a T$_{eff}$ of 6115 K this host star is likely in the subgiant phase of its evolution.\n\n\\textbf{TIC 219501568.} Multi-transit candidate with a period of 16.6 d. With a stellar radius of 1.7 $R_{\\odot}$ and a T$_{eff}$ of 6690 K this host star is likely entering the subgiant phase of its evolution. \\textcolor{black}{This candidate was identified as a TCE, however, it was not initially promoted to TOI status as the signal was thought to be off-target by the automated pipeline.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 2259.01\n\n\\textbf{TIC 229742722.} Multi-transit candidate with a period of 63.48 d. Eight LCO\/NRES and four OHP\/SOPHIE observations are consistent with this candidate being a planet. Gemini speckle interferometry reveals no nearby companion stars. \\textcolor{black}{This candidate was flagged as a TCE in sector 20, where it only exhibits a single transit event. An additional event was identified at the time of a momentum dump, and as such it failed the `odd-even' test and was not initially promoted to TOI status.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 1895.01.\n\n\\textbf{TIC 235943205.} Multi-transit candidate with a period of 121.3 d. The LCO\/NRES and OHP\/SOPHIE observations remain consistent with a planetary nature of the signal. Since the PHT discovery this cTOI has become the priority 3 target TOI 2264.01.\n\n\n\\textbf{TIC 267542728.} Multi-transit event with period of 39.7 d. Observations obtained with Keck showed that the RV shifts are not consistent with a planetary body and are most likely due to an M-dwarf companion.\n\n\\textbf{TIC 274599700.} Multi-transit candidate with a period of 33.0 d. One of the two transit-like even is only half visible, with the other half of the event falling in a \\emph{TESS}\\ data gap.\n\n\n\n\\textbf{TIC 328933398.} Multi-planet candidates. The 2-minute cadence light curve shows two single transit events of different depths across two \\emph{TESS}\\ sectors, both of which are consistent with an independent planetary body. In addition to the short cadence data, this target was observed in an additional three sectors as part of the 30-minute cadence full frame images. These showed additional transit events for one of the planet candidates, with a period of 24.9 d. \\textcolor{black}{This light curve was initially flagged as containing a TCE event, however, the two 2-minute cadence single transit events were thought to belong to the same transiting planet. The TCE was initially discarded as the pipeline identified the events to be off-target.} However, since the PHT discovery these two cTOIs has become the priority 3 and 1 targets, TOI 1873.01 and TOI 1873.01, respectively.\n\n\\textbf{TIC 349488688.} Multi-planet candidate, with one single transit event and one multi-transit candidate with a period of 11 d. Two LCO\/NRES and two OHP\/SOPHIE spectra, along with ongoing HARPS North are consistent with both of these candidates being planetary in nature. \\textcolor{black}{The single transit event was initially identified as a TCE, however, in addition to the event it identified two other signals at the time of momentum dumps, and was therefore initially discarded by the pipeline as it failed the `odd-even' transit test.} However, since the PHT discovery the two-transit event has become the 1 targets, TOI 2319.01 (Eisner et al. in prep).\n\n\\textbf{TIC 385557214.} Multi-transit candidate with a period of 5.6 d. The prominent stellar variation seen in the light curve is likely due to spots or pulsation The high impact parameter returned by the best fit \\texttt{pyaneti}\\ modelling suggests that the transit is likely caused by a grazing object. Without further observations, however, we are unable to rule this candidate out as being planetary in nature. \\textcolor{black}{This candidate was flagged as a TCE but was not promoted to TOI status due to the other nearby stars.}\n\n\\textbf{TIC 408636441.} Multi-transit candidate with a period of 18.8 or 37.7 d. Due to \\emph{TESS}\\ data gaps, half of the period stated in Table~\\ref{tab:PHT-caniddates} is likely. The SBIG snapshot and Gemini speckle images show no signs of companion stars. \\textcolor{black}{This candidate was flagged as a TCE in sector 24, where it only exhibits a single transit event. An additional event was identified at the time of a momentum dump, and as such it failed the `odd-even' test and was not initially promoted to TOI status.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 1759.01.\n\n\\textbf{TIC 441642457.} Multi-transit candidate with a period of 79.8 d. \\textcolor{black}{This candidate was flagged as a TCE in sector 14, where it only exhibits a single transit event. An additional event was identified at the time of a momentum dump, and as such it failed the `odd-even' test and was not initially promoted to TOI status.} Since the PHT discovery this cTOI has become the priority 2 target TOI 2073.01.\n\n\\textbf{TIC 441765914.} Multi-transit candidate with a period of 161.6 d. Since the PHT discovery this cTOI has become the priority 1 target TOI 2088.01.\n\n\\textbf{TIC 48018596.} Multi-transit candidate with a period of 100.1 days (or a multiple thereof). The single LCO\/NRES spectrum shows no sign of this target being a double lined spectroscopic binary. Gemini speckle interferometry revealed no nearby companion stars. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the transit-events, the pipeline classified, what we consider stellar variability as an additional event. As such it failed the `odd-even' transit test and wasn't promoted to TOI status.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 2295.01.\n\n\\textbf{TIC 55525572.} Multi-transit candidate with a period of 83.9 d. Since the PHT discovery this cTOI has become the confirmed planet TOI 813 \\citep{2020eisner}.\n\n\\textbf{TIC 82452140.} Multi-transit candidate with a period of 21.1 d. Since the PHT discovery this cTOI has become the priority 2 target TOI 2289.01.\n\n\n\\section{Introduction}\n\nSince the first unambiguous discovery of an exoplanet in 1995 \\citep[][]{Mayor1995} over 4,000 more have been confirmed. Studies of their characteristics have unveiled an extremely wide range of planetary properties in terms of planetary mass, size, system architecture and orbital periods, greatly revolutionising our understanding of how these bodies form and evolve.\n\nThe transit method, whereby we observe a temporary decrease in the brightness of a star due to a planet passing in front of its host star, is to date the most successful method for planet detection, having discovered over 75\\% of the planets listed on the NASA Exoplanet Archive\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}. It yields a wealth of information including planet radius, orbital period, system orientation and potentially even atmospheric composition. Furthermore, when combined with Radial Velocity \\citep[RV; e.g.,][]{Mayor1995, Marcy1997} observations, which yield the planetary mass, we can infer planet densities, and thus their internal bulk compositions. Other indirect detection methods include radio pulsar timing \\citep[e.g.,][]{Wolszczan1992} and microlensing \\citep[e.g.,][]{Gaudi2012}.\n\n\nThe \\textit{Transiting Exoplanet Survey Satellite} mission \\citep[\\protect\\emph{TESS};][]{ricker15} is currently in its extended mission, searching for transiting planets orbiting bright ($V < 11$\\,mag) nearby stars. Over the course of the two year nominal mission, \\emph{TESS}\\ monitored around 85 per cent of the sky, split up into 26 rectangular sectors of 96 $\\times$ 24 deg each (13 per hemisphere). Each sector is monitored for $\\approx$ 27.4 continuous days, measuring the brightness of $\\approx$ 20,000 pre-selected stars every two minutes. In addition to these short cadence (SC) observations, the \\emph{TESS}\\ mission provides Full Frame Images (FFI) that span across all pixels of all CCDs and are taken at a cadence of 30 minutes. While most of the targets ($\\sim$ 63 per cent) will be observed for $\\approx$ 27.4 continuous days, around $\\sim$ 2 per cent of the targets at the ecliptic poles are located in the `continuous viewing zones' and will be continuously monitored for $\\sim$ 356 days.\n\nStars themselves are extremely complex, with phenomena ranging from outbursts to long and short term variability and oscillations, which manifest themselves in the light curves. These signals, as well as systematic effects and artifacts introduced by the telescope and instruments, mean that standard periodic search methods, such as the Box-Least-Squared method \\citep{bls2002} can struggle to identify certain transit events, especially if the observed signal is dominated by natural stellar variability. Standard detection pipelines also tend to bias the detection of short period planets, as they typically require a minimum of two transit events in order to gain the signal-to-noise ratio (SNR) required for detection.\n\nOne of the prime science goals of the \\emph{TESS}\\ mission is to further our understanding of the overall planet population, an active area of research that is strongly affected by observational and detection biases. In order for exoplanet population studies to be able to draw meaningful conclusions, they require a certain level of completeness in the sample of known exoplanets as well as a robust sample of validated planets spanning a wide range of parameter space. \\textcolor{red}{Due to this, we independently search the \\emph{TESS}\\ light curves for transiting planets via visual vetting in order to detect candidates that were either intentionally ignored by the main \\emph{TESS}\\ pipelines, which require at least two transits for a detection, missed because of stellar variability or instrumental artefacts, or were identified but subsequently erroneously discounted at the vetting stage, usually because the period found by the pipeline was incorrect. These candidates can help populate under-explored regions of parameter space and will, for example, benefit the study of planet occurrence rates around different stellar types as well as inform theories of physical processes involved with the formation and evolution of different types of exoplanets.}\n\nHuman brains excel in activities related to pattern recognition, making the task of identifying transiting events in light curves, even when the pattern is in the midst of a strong varying signal, ideally suited for visual vetting. Early citizen science projects, such as Planet Hunters \\citep[PH;][]{fischer12} and Exoplanet Explorers \\citep{Christiansen2018}, successfully harnessed the analytic power of a large number of volunteers and made substantial contributions to the field of exoplanet discoveries. The PH project, for example, showed that human vetting has a higher detection efficiency than automated detection algorithms for certain types of transits. In particular, they showed that citizen science can outperform on the detection of single (long-period) transits \\citep[e.g.,][]{wang13, schmitt14a}, aperiodic transits \\citep[e.g. circumbinary planets;][]{schwamb13} and planets around variable stars \\citep[e.g., young systems,][]{fischer12}. Both PH and Exoplanet Explorers, which are hosted by the world's largest citizen science platform Zooniverse \\citep{lintott08}, ensured easy access to \\textit{Kepler} and \\textit{K2} data by making them publicly available online in an immediately accessible graphical format that is easy to understand for non-specialists. The popularity of these projects is reflected in the number of participants, with PH attracting 144,466 volunteers from 137 different countries over 9 years of the project being active.\n\nFollowing the end of the \\textit{Kepler} mission and the launch of the \\emph{TESS}\\ satellite in 2018, PH was relaunched as the new citizen science project \\textit{Planet Hunters TESS} (PHT) \\footnote{\\url{www.planethunters.org}}, with the aim of identifying transit events in the \\emph{TESS}\\ data that were \\textcolor{red}{intentionally ignored or missed} by the main \\emph{TESS}\\ pipelines. \\textcolor{red}{Such a search complements other methods methods via its sensitivity to single-transit, and, therefore, longer period planets. Additionally, other dedicated non-citizen science based methods are also employed to look for single transit candidates \\citep[see e.g., the Bayesian transit fitting method by ][]{Gill2020, Osborn2016}}.\n\nCitizen science transit searches specialise in finding the rare events that the standard detection pipelines miss, however, these results are of limited use without an indication of the completeness of the search. Addressing the problem of completeness was therefore one of our highest priorities while designing PHT as discussed throughout this paper. \n\nThe layout of the remainder of the paper will be as follows. An overview of the Planet Hunters TESS project is found in Section~\\ref{sec:PHT}, followed by an in depth description of how the project identifies planet candidates in Section~\\ref{sec:method}. The recovery efficiency of the citizen science approach is assessed in Section~\\ref{sec:recovery_efficiency}, followed by a description of the in-depth vetting of candidates and ground based-follow up efforts in Section~\\ref{sec:vetting} and \\ref{sec:follow_up}, respectively. Planet Candidates and noteworthy systems identified by Planet Hunters TESS are outlined in Section~\\ref{sec:PHT_canidates}, followed by a discussion of the results in Section~\\ref{sec:condlusion}.\n\n\\section{Planet Hunters TESS}\n\\label{sec:PHT}\n\nThe PHT project works by displaying \\emph{TESS}\\ light curves (Figure~\\ref{fig:interface}), and asking volunteers to identify transit-like signals. Only the two-minute cadence targets, which are produced by the \\emph{TESS}\\ pipeline at the Science Processing Operations Center \\citep[SPOC,][]{Jenkins2018} and made publicly available by the Mikulski Archive for Space Telescopes (MAST)\\footnote{\\url{http:\/\/archive.stsci.edu\/tess\/}}, are searched by PHT. First-time visitors to the PHT site, or returning visitors who have not logged in are prompted to look through a short tutorial, which briefly explains the main aim of the project and shows examples of transit events and other stellar phenomena. Scientific explanation of the project can be found elsewhere on the site in the `field guide' and on the project's `About' page. \n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=\\textwidth]{Figures\/PHT_new_interface.png}\n \\caption{\n PHT user interface showing a simulated light curve. The transit events are highlighted with white partially-transparent columns that are drawn on using the mouse. Stellar information on the target star is available by clicking on `subject info' below the light curve.} \n \\label{fig:interface}\n\\end{figure*}\n\nAfter viewing the tutorial, volunteers are ready to participate in the project and are presented with \\emph{TESS}\\ light curves (known as `subjects') that need to be classified. The project was designed to be as simple as possible and therefore only asks one question: \\textit{`Do you see a transit?}'. Users identify transit-like events, and the time of their occurrence, by drawing a column over the event using the mouse button, as shown in Figure~\\ref{fig:interface}. There is no limit on the number of transit-like events that can be marked in a light curve. No markings indicate that there are no transit-like events present in the light curve. Once the subject has been analysed, users submit their classification and continue to view the next light curve by clicking `Done'. \n\nAlongside each light curve, users are offered information on the stellar properties of the target, such as the radius, effective temperature and magnitude (subject to availability, see \\cite{Stassun18}). However, in order to reduce biases in the classifications, the TESS Input Catalog (TIC) ID of the target star is not provided until after the subject classification has been submitted.\n\nIn addition to classifying the data, users are given the option to comment on light curves via the `Talk' discussion forum. Each light curve has its own discussion page to allow volunteers to discuss and comment, as well as to `tag' light curves using searchable hashtags, and to bring promising candidates to the attention of other users and the research team. The talk discussion forums complement the main PHT analysis and have been shown to yield interesting objects which may be challenging to detect using automated algorithms \\citep[e.g.,][]{eisner2019RN}. Unlike in the initial PH project, there are no questions in the main interface regarding stellar variability, however, volunteers are encouraged to mention astrophysical phenomenon or \\textit{unusual} features, such as eclipsing binaries or stellar flares, using the `Talk' discussion forum. \n\nThe subject TIC IDs are revealed on the subject discussion pages, allowing volunteers to carry out further analysis on specific targets of interest and to report and discuss their findings. This is extremely valuable for both other volunteers and the PHT science team, as it can speed up the process of identifying candidates as well as rule out false positives in a fast and effective manner. \n\nSince the launch of PHT on 6 December 2018, there has been one significant makeover to the user interface. The initial PHT user interface (UI1), which was used for sectors 1 through 9, split the \\emph{TESS}\\ light curves up into either three or four chunks (depending on the data gaps in each sector) which lasted around seven days each. This allowed for a more `zoomed' in view of the data, making it easier to identify transit-like events than when the full $\\sim$ 30 day light curves were shown. The results from a PHT beta project, which displayed only simulated data, showed that a more zoomed in view of the light curve was likely to yield a higher transit recovery rate.\n\nThe updated, and current, user interface (UI2) allows users to manually zoom in on the x-axis (time) of the data. Due to this additional feature, each target has been displayed as a single light curve as of Sector 10. In order to verify that the changes in interface did not affect our findings, all of the Sector 9 subjects were classified using both UI1 and UI2. We saw no significant change in the number of candidates recovered (see Section~\\ref{sec:recovery_efficiency} for a description of how we quantified detection efficiency).\n\n\n\\subsection{Simulated Data}\n\\label{subsec:sims} \n\nIn addition to the real data, volunteers are shown simulated light curves, which are generated by randomly injecting simulated transit signals, provided by the SPOC pipeline \\citep[][]{Jenkins2018}, into real \\emph{TESS}\\ light curves. The simulated data play an important role in assessing the sensitivity of the project, training the users and providing immediate feedback, and to gauge the relative abilities of individual users (see Sec~\\ref{subsec:weighting}). \n\nWe calculate a signal to noise ratio (SNR) of the injected signal by dividing the injected transit depth by the Root Mean Square Combined Differential Photometric Precision (RMS CDPP) of the light curve on 0.5-, 1- or 2-hr time scales (whichever is closest to the duration of the injected transit signal). Only simulations with a SNR greater than 7 in UI1 and greater than 4 for UI2 are shown to volunteers.\n\nSimulated light curves are randomly shown to the volunteers and classified in the exact same manner as the real data. The user is always notified after a simulated light curve has been classified and given feedback as to whether the injected signal was correctly identified or not. For each sector, we generate between one and two thousand simulated light curves, using the real data from that sector in order to ensure that the sector specific systematic effects and data gaps of the simulated data do not differ from the real data. The rate at which a volunteer is shown simulated light curves decreases from an initial rate of 30 per cent for the first 10 classifications, down to a rate of 1 per cent by the time that the user has classified 100 light curves. \n\n\n\\section{Identifying Candidates}\n\\label{sec:method}\n\nEach subject is seen by multiple volunteers, before it is `retired' from the site, and the classifications are combined (see Section~\\ref{subsec:DBscan}) in order to assess the likelihood of a transit event. For sectors 1 through 9, the subjects were retired after 8 classifications if the first 8 volunteers who saw the light curves did not mark any transit events, after 10 classifications if the first 10 volunteers all marked a transit event and after 15 classifications if there was not complete consensus amongst the users. As of Sector 9 with UI2, all subjects were classified by 15 volunteers, regardless of whether or not any transit-like events were marked. Sector 9, which was classified with both UI1 and UI2, was also classified with both retirement rules.\n\nThere were a total of 12,617,038 individual classifications completed across the project on the nominal mission data. 95.4 per cent of these classifications were made by 22,341 registered volunteers, with the rest made by unregistered volunteers. Around 25 per cent of the registered volunteers complete more than 100 classifications, 11.8 per cent more than 300, 8.4 per cent more than 500, 5.4 per cent more than 1000 and 1.1 per cent more than 10,000. The registered volunteers completed a mean and median of 647 and 33 classifications, respectively. Figure~\\ref{fig:user_count} shows the distribution in user effort for logged in users who made between 0 and 300 classifications. \n\nThe distribution in the number of classifications made by the registered volunteers is assessed using the Gini coefficient, which ranges from 0 (equal contributions from all users) to 1 (large disparity in the contributions). The Gini coefficients for individual sectors ranges from 0.84 to 0.91 with a mean of 0.87, while the Gini coefficient for the overall project (all of the sectors combined) is 0.94. The mean Gini coefficient among other astronomy Zooniverse projects lies at 0.82 \\citep{spiers2019}. We note that the only other Zooniverse project with an equally high Gini coefficient as PHT is \\textit{Supernova Hunters}, a project which, similarly to PHT and unlike most other Zooniverse projects, has periodic data releases that are accompanied by an e-newsletter sent to all project volunteers. Periodic e-newsletters have the effect of promoting the project to both regularly and irregularly participating volunteers, who may only complete a couple of classifications as they explore the task, as well as to returning users who complete a large number of classifications following every data release, increasing the disparity in user contributions (the Gini coefficient).\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.5\\textwidth]{Figures\/user_count.png}\n \\caption{\n The distribution of the number of classifications by the registered volunteers, using a bin size of 5 from 0 to 300 classifications. A total of 11.8 per cent of the registered volunteers completed more than 300 classifications.} \n \\label{fig:user_count}\n\\end{figure}\n\n\n\\subsection{User Weighting}\n\\label{subsec:weighting} \nUser weights are calculated for each individual volunteer in order to identify users who are more sensitive to detecting transit-like signals and those who are more likely to mark false positives. The weighting scheme is based on the weighting scheme described by \\cite{schwamb12}.\n\nUser weights are calculated independently for each observation sector, using the simulated light curves shown alongside the data from that sector. All users start off with a weighting of one, which is then increased or decreased when a simulated transit event is correctly or incorrectly identified, respectively. \n\nSimulated transits are deemed correctly identified, or `True', if the mid-point of a user's marking falls within the width of the simulated transit events. If none of the user's markings fall within this range, the simulated transit is deemed not identified, or `False'. If more than one of a user's markings coincide with the same simulated signal, it is only counted as being correct once, such that the total number of `True' markings cannot exceed the number of injected signals. For each classification, we record the number of `Extra' markings, which is the total number of markings made by the user minus the number of correctly identified simulated transits. \n\nEach simulated light curve, identified by superscript $i$ (where $i=1$, \\ldots, $N$) was seen by $K^{(i)}$ users (the mean value of $K^{(i)}$\nwas 10), and contained $T^{(i)}$ simulated transits (where $T^{(i)}$ depends on the period of the simulated transit signal and the duration of the light curve). For a specific light curve $i$, each user who saw the light curve is identified by a subscript $k$ (where $k=1$, \\ldots, $K^{(i)}$) and each injected transit by a subscript $t$ (where $t=1$, \\ldots, $T^{(i)}$). \n\nIn order to distinguish between users who are able to identify obvious transits and those who are also able to find those that are more difficult to see, we start by defining a `recoverability' $r^{(i)}_t$ for each injected transit $t$ in each light curve. This is defined empirically, as the number of users who identified the transit correctly divided by $K^{(i)}$ (the total number of users who saw the light curve in question).\n\nNext, we quantify the performance of each user on each light curve as follows (this performance is analogous to the `seed' defined in \\citealt{schwamb12}, but we define it slightly differently):\n\\begin{equation}\n p^{(i)}_{k} = C_{\\rm E} ~ \\frac{E^{(i)}_{k}}{\\langle E^{(i)} \\rangle} + \\sum_{t=1}^{T^{(i)}} \\begin{cases}\n C_{\\rm T} ~ \\left[ r^{(i)}_t \\right]^{-1}, & \\text{if $m^{(i)}_{t,k} = $`True'}\\\\\n C_{\\rm F} ~ r^{(i)}_t, & \\text{if $m^{(i)}_{t,k} = $`False'},\n \\end{cases}\n\\end{equation}\nwhere $m^{(i)}_{t,k}$ is the identification of transit $t$ by user $k$ in light curve $i$, which is either `True' or `False'; $E^{(i)}_{k}$ is the number of `Extra' markings made by user $k$ for light curve $i$, and $\\langle E^{(i)} \\rangle$ is the mean number of `Extra' markings made by all users who saw subject $i$. The parameters $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ control the impact of the `Extra', `True' and `False' markings on the overall user weightings, and are optimized empirically as discussed below in Section~\\ref{subsec:optimizesearch}. \n\nFollowing \\citealt{schwamb12}, we then assign a global `weight' $w_k$ to each user $k$, which is defined as:\n\\begin{equation}\n\\begin{split}\n\tw_k = I \\times (1 + \\log_{10} N_k)^{\\nicefrac{\\sum_i p^{(i)}_k}{N_k}}\n\\label{equ:weight}\n\\end{split}\n\\end{equation}\nwhere $I$ is an empirical normalization factor, such that the distribution of user weights remains centred on one, $N_k$ is the total number of simulated transit events that user $k$ assessed, and the sum over $i$ concerns only the light curves that user $k$ saw. \nWe limit the user weights to the range 0.05--3 \\emph{a posteriori}.\n\n\nWe experimented with a number of alternative ways to define the user weights, including the simpler $w_k=\\nicefrac{\\sum_i p^{(i)}_k}{N_k}$, but Eqn.~\\ref{equ:weight} was found to give the best results (see Section~\\ref{sec:recovery_efficiency} for how this was evaluated).\n\n\\subsection{Systematic Removal}\n\\label{subsec:sysrem} \nSystematic effects, for example caused by the spacecraft or background events, can result in spurious signals that affect a large subset of the data, resulting in an excess in markings of transit-like events at certain times within an observation sector. As the four \\emph{TESS}\\ cameras can yield unique systematic effects, the times of systematics were identified uniquely for each camera. The times were identified using a Kernel Density Estimation \\citep[KDE;][]{rosenblatt1956} with a cosine kernel and a bandwidth of 0.1 days, applied across all of the markings from that sector for each camera. Fig.~\\ref{fig:sys_rem} shows the KDE of all marked transit-events made during Sector 17 for TESS's cameras 1 (top panel) to 4 (bottom panel). The isolated spikes, or prominences, in the number of marked events, such as at T = 21-22 days in the bottom panel, are assumed to be caused by systematic effects that affect multiple light curves. Prominences are considered significant if they exceed a factor four times the standard deviation of the kernel output, which was empirically determined to be the highest cut-off to not miss clearly visible systematics. All user-markings within the full width at half maximum of these peaks are omitted from all further analysis. \\textcolor{red}{The KDE profiles for each Sector are provided as electronic supplementary material.}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.46\\textwidth]{Figures\/systematics_sec17.png}\n \\caption{\n Kernel density estimation of the user-markings made for Sector 17, for targets observed with TESS's observational Cameras 1 (top panel) to 4 (bottom panel). The orange vertical lines the indicate prominences that are at least four times greater than the standard deviation of the distribution. The black points underneath the figures show the mid-points of all of the volunteer-markings, where darker regions represent a higher density of markings.}\n \\label{fig:sys_rem}\n\\end{figure}\n\n\\subsection{Density Based Clustering}\n\\label{subsec:DBscan} \n\nThe times and likelihoods of transit-like events are determined by combining all of the classifications made for each subject and identifying times where multiple volunteers identified a signal. We do this using an unsupervised machine learning method, known as DBSCAN \\citep[][Density-Based Spatial Clustering of Applications with Noise]{ester1996DB}. DBSCAN is a non-parametric density based clustering algorithm that helps to distinguish between dense clusters of data and sparse noise. For a data point to belong to a cluster it must be closer than a given distance ($\\epsilon$) to at least a set minimum number of other points (minPoints). \n\nIn our case, the data points are one-dimensional arrays of times of transits events, as identified by the volunteers, and clusters are times where multiple volunteers identified the same event. For each cluster a `transit score' ($s_i$) is determined, which is the sum of the user weights of the volunteers who contribute to the given cluster divided by the sum of the user weights of volunteers who saw that light curve. These transit scores are used to rank subjects from most to least likely to contain a transit-like event. Subjects which contain multiple successful clusters with different scores are ranked by the highest transit score. \n\n\\subsection{Optimizing the search}\n\\label{subsec:optimizesearch}\n\nThe methodology described in Sections~\\ref{subsec:weighting} to \\ref{subsec:DBscan} has five free parameters: the number of markings required to constitute a cluster ($minPoints$), the maximum separation of markings required for members of a cluster ($\\epsilon$), and $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ used in the weighting scheme. The values of these parameters were optimized via a grid search, where $C_{\\rm E}$ and $C_{\\rm F}$ ranged from -5 to 0, $C_{\\rm T}$ ranged from 0 to 20, and $minPoints$ ranged from 1 to 8, all in steps of 1. ($\\epsilon$) ranged from 0.5 to 1.5 in steps of 0.5. This grid search was carried out on 4 sectors, two from UI1 and two from UI2, for various variations of Equation~\\ref{equ:weight}. \n\nThe success of each combination of parameters was assessed by the fractions of TOIs and TCEs that were recovered within the top highest ranked 500 candidates, as discussed in more detail Section~\\ref{sec:recovery_efficiency}. We found the most successful combination of parameters to be $minPoints$ = 4 markings, $\\epsilon$, = 1 day, $C_{\\rm T}$ = 3, $C_{\\rm F}$= -2 and $C_{\\rm E}$ = -2.\n\n\\subsection{MAST deliverables}\n\\label{subsec:deliverables}\n\nThe analysis described above is carried out both in real-time as classifications are made, as well as offline after all of the light curves of a given sector have been classified. When the real-time analysis identifies a successful DB cluster (i.e. when at least four citizen scientists identified a transit within a day of the \\emph{TESS}\\ data of one another), the potential candidate is automatically uploaded to the open access Planet Hunters Analysis Database (PHAD) \\footnote{\\url{https:\/\/mast.stsci.edu\/phad\/}} hosted by the Mikulski Archive for Space Telescopes (MAST) \\footnote{\\url{https:\/\/archive.stsci.edu\/}}. While PHAD does not list every single classification made on PHT, it does display all transit candidates which had significant consensus amongst the volunteers who saw that light curve, along with the user-weight-weighted transit scores. This analysis does not apply the systematics removal described in Section~\\ref{subsec:sysrem}. The aim of PHAD is to provide an open source database of potential planet candidates identified by PHT, and to credit the volunteers who identified said targets. \n\nThe offline analysis is carried out following the complete classifications of all of the data from a given \\emph{TESS}\\ sector. The combination of all of the classifications allows us to identify and remove times of systematics and calculate better calibrated and more representative user weights. The remainder of this paper will only discuss the results from the offline analysis.\n\n\\section{Recovery Efficiency}\n\\label{sec:recovery_efficiency}\n\\subsection{Recovery of simulated transits}\n\nThe recovery efficiency is, in part, assessed by analysing the recovery rate of the injected transit-like signals (see Section~\\ref{subsec:sims}). Figure~\\ref{fig:SIM_recovery} shows the median and mean transit scores (fraction of volunteers who correctly identified a given transit scaled by user weights) of the simulated transits within SNR bins ranging from 4 to 20 in steps of 0.5. Simulations with a SNR less than 4 were not shown on PHT. The figure highlights that transit signals with a SNR of 7.5 or greater are correctly identified by the vast majority of volunteers. \n\n\\textcolor{red}{As the simulated data solely consist of real light curves with synthetically injected transit signals, we do not have any light curves, simulated or otherwise, which we can guarantee do not contain any planetary transits (real or injected). As such, this prohibits us from using simulated data to infer an analogous false-positive rate.}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.47\\textwidth]{Figures\/SIMS_recovery.png}\n \\caption{The median (blue) and mean (orange) transit scores for injected transits with SNR ranges between 4 and 20. The mean and median are calculated in SNR bins with a width of 0.5, as indicated by the horizontal lines around each data point. \n }\n \\label{fig:SIM_recovery}\n\\end{figure}\n\n\\subsection{Recovery of TCEs and TOIs}\n\\label{subsec:TCE_TOI}\nThe recovery efficiency of PHT is assessed further using the planet candidates identified by the SPOC pipeline \\citep{Jenkins2018}. The SPOC pipeline extracts and processes all of the 2-minute cadence \\emph{TESS}\\ light curves prior to performing a large scale transit search. Data Validation (DV) reports, which include a range of transit diagnostic tests, are generated by the pipeline for around 1250 Threshold Crossing Events (TCEs), which were flagged as containing two or more transit-like features. Visual vetting is then performed by the \\emph{TESS}\\ science team on these targets, and promising candidates are added to the catalog of \\emph{TESS}\\ Objects of Interest (TOIs). Each sector yields around 80 TOIs \\textcolor{red}{and a mean of 1025 TCEs.}\n\nFig~\\ref{fig:TCE_TOI_recovery} shows the fraction of TOIs and TCEs (top and bottom panel respectively) that we recover with PHT as a function of the rank, where a higher rank corresponds to a lower transit score, for Sectors 1 through 26. TOIs and TCEs with R < 2 $R_{\\oplus}$ are not included in this analysis, as the initial PH showed that human vetting alone is unable to reliably recover planets smaller than 2 $R_{\\oplus}$ \\citep{schwamb12}. Planets smaller than 2 $R_{\\oplus}$ are, therefore, not the main focus of our search.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI-recovery_radlim2.png}\n \\caption{The fraction of recovered TOIs and TCEs (top and bottom panel respectively) with R > 2$R_{\\oplus}$ as a function of the rank, for sectors 1 to 26. The lines represent the results from different observation sectors.}\n \\label{fig:TCE_TOI_recovery}\n\\end{figure}\n\n\nFig~\\ref{fig:TCE_TOI_recovery} shows a steep increase in the fractional TOI recovery rate up to a rank of $\\sim$ 500. Within the 500 highest ranked PHT candidates for a given sector, we are able to recover between 46 and 62 \\% (mean of 53 \\%) of all of the TOIs (R > 2 $R_{\\oplus}$), a median 90 \\% of the TOIs where the SNR of the transit events are greater than 7.5 and median 88 \\% of TOIs where the SNR of the transit events are greater than 5.\n\nThe relation between planet recovery rate and the SNR of the transit events is further highlighted in Figure~\\ref{fig:TOI_properties}, which shows the SNR vs the orbital period of the recovered TOIs. The colour of the markers indicate the TOI's rank within a given sector, with the lighter colours representing a lower rank. The circles and crosses represent candidates at a rank lower and higher than 500, respectively. The figure shows that transit events with a SNR less than 3.5 are missed by the majority of volunteers, whereas events with a SNR greater than 5 are mostly recovered within the top 500 highest ranked candidates. \n\nThe steep increase in the fractional TOI recovery rate at lower ranks, as shown in figure~\\ref{fig:TCE_TOI_recovery}, is therefore due to the detection of the high SNR candidates that are identified by most, if not all, of the PHT volunteers who classified those targets. At a rank of around 500, the SNR of the TOIs tends towards the limit of what human vetting can detect and thus the identification of TOIs beyond a rank of 500 is more sporadic.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.49\\textwidth]{Figures\/TOI_recovery_properties.png}\n \\caption{The SNR vs orbital period of TOIs with R > 2$R_{\\oplus}$. The colour represents their rank within the sector, as determined by the weighted DB clustering algorithm. Circles indicate that they were identified at a rank < 500, while crosses indicate that they were not within the top 500 highest ranked candidates of a given sector.\n }\n \\label{fig:TOI_properties}\n\\end{figure}\n\nThe fractional TCE recovery rate (bottom panel of Figure~\\ref{fig:TCE_TOI_recovery}) is systematically lower than that of the TOIs. There are qualitative reasons as to why humans might not identify a TCE as opposed to a TOI, including that TCEs may be caused by artefacts or periodic stellar signals that the SPOC pipeline identified as a potential transit but that the human eye would either miss or be able to rule out as systematic effect. This leads to a lower recovery fraction of TCEs comparatively, an effect that is further amplified by the much larger number of TCEs.\n\nThe detection efficiency of PHT is estimated using the fractional recovery rate of TOIs for a range of radius and period bins, as shown in Figure~\\ref{fig:recovery_rank500_radius_period}. A TOI is considered to be recovered if its detection rank is less than 500 within the given sector. Out of the total 1913 TOIs, to date, \\textcolor{red}{PHT recovered 715 TOIs among the highest ranked candidates across the 26 sectors. This corresponds to a mean of 12.7~\\% of the top 500 ranked candidates per sector being TOIs. In comparison, the primary \\emph{TESS}\\ team on average visually vets 1025 TCEs per sector, out of which a mean of 17.3~\\% are promoted to TOI status.} We find that, independent of the orbital period, PHT is over 85~\\% complete in the recovery of TOIs with radii equal to or greater than 4 $R_{\\oplus}$. This agrees with the findings from the initial Planet Hunters project \\citep{schwamb12}. The detection efficiency decreases to 51~\\% for 3 - 4 $R_{\\oplus}$ TOIs, 49~\\% for 2 - 3 $R_{\\oplus}$ TOIs and to less than 40~\\% for TOIs with radii less than 2 $R_{\\oplus}$. Fig~\\ref{fig:recovery_rank500_radius_period} shows that the orbital period does not have a strong effect on the detection efficiency for periods greater than $\\sim$~1~day, which highlights that human vetting efficiency is independent of the number of transits present within a light curve. For periods shorter than around 1~day, the detection efficiency decreases even for larger planets, due to the high frequency of events seen in the light curve. For these light curves, many volunteers will only mark a subset of the transits, which may not overlap with the subset marked by other volunteers. Due to the methodology used to identify and rank the candidates, as described in Section~\\ref{sec:method}, this will actively disfavour the recovery of very short period planets. Although this obviously introduces biases in the detectability of very short period signals, the major detection pipelines are specifically designed to identify these types of planets and thus this does not present a serious detriment to our main science goal of finding planets that were \\textcolor{red}{intentionally ignored or missed} by the main automated pipelines.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{Figures\/TOI_recovery_grid.png}\n \\caption{TOI recovery rate as a function of planet radius and orbital period. A TOI is considered recovered if it is amongst the top 500 highest ranked candidates within a given sector. The logarithmically spaced grid ranges from 0.2 to 225 d and 0.6 to 55 $R{_\\oplus}$ for the orbital period and planet radius, respectively. The fraction of TOIs recovered using PHT is computed for each cell and represented by the colour the grid. Cells with less than 10 TOIs are considered incomplete for statistical analysis and are shown by the hatched lines. White cells contain no TOIs. The annotations for each cell indicate the number of recovered TOIs followed by the Poisson uncertainty in brackets. The filled in and empty grey circles indicated the recovered and not-recovered TOIs, respectively.}\n \\label{fig:recovery_rank500_radius_period}\n\\end{figure*}\n\n\nFinally, we assessed whether the detection efficiency varies across different sectors by assessing the fraction of recovered TOIs and TCEs within the highest ranked 500 candidates. We found the recovery of TOIs within the top 500 highest ranked candidates to remain relatively constant across all sectors, while the fraction of recovered TCEs in the top 500 highest ranked candidates increases in later sectors, as shown in Figure~\\ref{fig:recovery_rank500}). After applying a Spearman's rank test we find a positive correlation of 0.86 (pvalue = 5.9 $\\times$ $10^{-8}$) and 0.57 (pvalue = 0.003) between the observation sector and TCE and TOI recovery rates, respectively. These correlations suggest that the ability of users to detect transit-like events improves as they classify more subjects. The improvement of volunteers over time can also be seen in Fig~\\ref{fig:user_weights}, which shows the mean (unnormalized) user weight per sector for volunteers who completed one or more classifications in at least one sector (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors 26 sectors from the nominal \\emph{TESS}\\ mission (pink). The figure highlights an overall improvement in the mean user weight in later sectors, as well as a positive correlation between the overall increase in user weight and the number of sectors that volunteers have participated in.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI_rank500.png}\n \\caption{The fractional recovery rate of the TOIs (blue circles) and TCEs (teal squares) at a rank of 500 for each sector. Sector 1-9 (white background) represent southern hemisphere sectors classified with UI1, sectors 9-14 (light grey background) show the southern hemisphere sectors classified with UI2, and sectors 14-24 (dark grey background) show the northern hemisphere sectors classified with US2.}\n \\label{fig:recovery_rank500}\n\\end{figure}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.50\\textwidth]{Figures\/user_weights_sectors.png}\n \\caption{Mean user weights per sector. The solid lines show the user weights for the old user interface and the dashed line for the new interface, separated by the black line (Sector 9). The different coloured lines show the mean user weights calculated considering user who participated in any number of sectors (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors observed during the nominal \\emph{TESS}\\ mission (pink).}\n \\label{fig:user_weights}\n\\end{figure}\n\n\n\\section{Candidate vetting}\n\\label{sec:vetting}\n\nFor each observation sector the subjects are ranked according to their transit scores, and the 500 highest ranked targets (excluding TOIs) visually vetted by the PHT science team in order to identify potential candidates and rule out false positives. A vetting cut-off rank of 500 was chosen as we found this to maximise the number of found candidates while minimising the number of likely false positives. In the initial round of vetting, which is completed via a separate Zooniverse classification interface that is only accessible to the core science team, a minimum of three members of the team sort the highest ranked targets into either `keep for further analysis', `eclipsing binary' or `discard'. The sorting is based on the inspection of the full \\emph{TESS}\\ light curve of the target, with the times of the satellite momentum dumps indicated. Additionally, around the time of each likely transit event (i.e. time of successful DB clusters) we inspect the background flux and the x and y centroid positions. Stellar parameters are provided for each candidate, subject to availability, alongside links to the SPOC Data Validation (DV) reports for candidates that had been flagged as TCEs but were never promoted to TOIs status.\n\nCandidates where at least two of the reviewers indicated that the signal is consistent with a planetary transit are kept for further analysis. \\textcolor{red}{This constitute a $\\sim$~5~\\% retention rate of the 500 highest ranked candidates per sector between the initial citizen science classification stage and the PHT science team vetting stage. Considering that the known planets and TOIs are not included at this stage of vetting, it is not surprising that our retention rate is lower that the true-positive rates of TCEs (see Section~\\ref{subsec:TCE_TOI}). Furthermore, this false-positive rate is consistent with the the findings of the initial Planet Hunters project \\citep{schwamb12}.}\n\nThe rest of the 500 candidates were grouped into $\\sim$~37~\\% `eclipsing binary' and $\\sim$~58~\\% `discard'. The most common reasons for discarding light curves are due to events caused by momentum dumps and due to background events, such as background eclipsing binaries, that mimic transit-like signals in the light curve. The targets identified as eclipsing binaries are analysed further by the \\emph{TESS}\\ Eclipsing Binaries Working Group (Prsa et al, in prep).\n\n\n\n\nFor the second round of candidate vetting we generate our own data validation reports for all candidates classified as `keep for further analysis'. The reports are generated using the open source software {\\sc latte} \\citep[Lightcurve Analysis Tool for Transiting Exoplanets;][]{LATTE2020}, which includes a range of standard diagnostic plots that are specifically designed to help identify transit-like signals and weed out astrophysical false positives in \\emph{TESS}\\ data. In brief the diagnostics consist of:\n\n\\textbf{Momentum Dumps}. The times of the \\emph{TESS}\\ reaction wheel momentum dumps that can result in instrumental effects that mimic astrophysical signals.\n\n\\textbf{Background Flux}. The background flux to help identify trends caused by background events such as asteroids or fireflies \\citep{vanderspek2018tess} passing through the field of view.\n\n\\textbf{x and y centroid positions}. The CCD column and row local position of the target's flux-weighted centroid, and the CCD column and row motion which considers differential velocity aberration (DVA), pointing drift, and thermal effects. This can help identify signals caused by systematics due to the satellite. \n\\textbf{Aperture size test}. The target light curve around the time of the transit-like event extracted using two apertures of different sizes. This can help identify signals resulting from background eclipsing binaries.\n \n\\textbf{Pixel-level centroid analysis}. A comparison between the average in-transit and average out-of-transit flux, as well as the difference between them. This can help identify signals resulting from background eclipsing binaries.\n\n\\textbf{Nearby companion stars}. The location of nearby stars brighter than V-band magnitude 15 as queried from the Gaia Data Release 2 catalog \\citep{gaia2018gaia} and the DSS2 red field of view around the target star in order to identify nearby contaminating sources. \n\n\\textbf{Nearest neighbour light curves}. Normalized flux light curves of the five short-cadence \\emph{TESS}\\ stars with the smallest projected distances to the target star, used to identify alternative sources of the signal or systematic effects that affect multiple target stars. \n\n\\textbf{Pixel level light curves}. Individual light curves extracted for each pixel around the target. Used to identify signals resulting from background eclipsing binaries, background events and systematics.\n\n\\textbf{Box-Least-Squares fit}. Results from two consecutive BLS searches, where the identified signals from the initial search are removed prior to the second BLS search.\n\nThe {\\sc latte} validation reports are assessed by the PHT science team in order to identify planetary candidates that warrant further investigation. Around 10~\\% of the targets assessed at this stage of vetting are kept for further investigation, resulting in $\\sim$~3 promising planet candidates per observation sector. The discarded candidates can be loosely categorized into (background) eclipsing binaries ($\\sim$~40~\\%), systematic effects ($\\sim$~25~\\%), background events ($\\sim$~15~\\%) and other (stellar signals such as spots; $\\sim$~10~\\%).\n\n\nWe use \\texttt{pyaneti}\\ \\citep{pyaneti} to infer the planetary and orbital parameters of our most promising candidates. For multi-transit candidates we fit for seven parameters per planet, time of mid-transit $T_0$, orbital period $P$, impact parameter $b$, scaled semi-major axis $a\/R_\\star$, scaled planet radius $r_{\\rm p}\/R_\\star$, and two limb darkening coefficients following a \\citet{Mandel2002} quadratic limb darkening model, implemented with the $q_1$ and $q_2$ parametrization suggested by \\citet{Kipping2013}. Orbits were assumed to be circular.\nFor the mono-transit candidates, we fit the same parameters as for the multi-transit case, except for the orbital period and scaled semi-major axis which cannot be known for single transits. We follow \\citet{Osborn2016} to estimate the orbital period of the mono-transit candidates assuming circular orbits.\n\nWe note that some of our candidates are V-shaped, consistent with a grazing transit configuration. For these cases, we set uniform priors between 0 and 0.15 for $r_{\\rm p}\/R_\\star$ and between 0 and 1.15 for the impact parameter in order to avoid large radii caused by the $r_{\\rm p}\/R_\\star - b$ degeneracy. Thus, the $r_{\\rm p}\/R_\\star$ for these candidates should not be trusted. A full characterisation of these grazing transits is out of the scope of this manuscript.\n\nFigure~\\ref{fig:PHT_pyaneti} shows the \\emph{TESS}\\ transits together with the inferred model for each candidate. Table~\\ref{tab:PHT-caniddates} shows the inferred main parameters, the values and their uncertainties are given by the median and 68.3\\% credible interval of the posterior distributions.\n\n\n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_one.png}\n \\caption{All of the PHT candidates modelled using \\texttt{pyaneti}. The parameters of the best fits are summarised in Table~\\protect\\ref{tab:PHT-caniddates}. The blue and magenta fits show the multi and single transit event candidates, respectively.} \n \\label{fig:PHT_pyaneti}\n\\end{figure*}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_two.png}\n \\addtocounter{figure}{-1}\n \\caption{\\textbf{PHT candidates (continued)}} \n\\end{figure*}\n\n\nCandidates that pass all of our rounds of vetting are uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS) website\\footnote{\\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}} as community TOIs (cTOIs).\n\n\\section{Follow-up observations}\n\\label{sec:follow_up}\n\nMany astrophysical false positive scenarios can be ruled out from the detailed examination of the \\emph{TESS}\\ data, both from the light curves themselves and from the target pixel files. However, not all of the false positive scenarios can be ruled out from these data alone, due in part to the large \\emph{TESS}\\ pixels (20 arcsconds). Our third stage of vetting, therefore, consists of following up the candidates with ground based observations including photometry, reconnaissance spectroscopy and speckle imaging. The results from these observations will be discussed in detail in a dedicated follow-up paper. \n\n\\subsection{Photometry}\n\nWe make use of the LCO global network of fully robotic 0.4-m\/SBIG and 1.0-m\/Sinistro facilities \\citep{LCO2013} to observe additional transits, where the orbital period is known, in order to refine the ephemeris and confirm that the transit events are not due to a blended eclipsing binary in the vicinity of the main target. Snapshot images are taken of single transit event candidates in order to identify nearby contaminating sources. \n\n\n\\subsection{Spectroscopy}\n\nWe perform high-resolution optical spectroscopy using telescopes from across the globe in order to cover a wide range of RA and Dec:\n\\begin{itemize}\n\\item The Las Cumbres Observatory (LCO) telescopes with the Network of Robotic Echelle Spectrographs \\citep[NRES,][]{LCO2013}. These fibre-fed spectrographs, mounted on 1.0-m telescopes around the globe, have a resolution of R = 53,000 and a wavelength coverage of 380 to 860 nm. \n\n\\item The MINERVA Australis Telescope facility, located at Mount Kent Observatory in Queensland, Australia \\citep{addison2019}. This facility is made up of four 0.7m CDK700 telescopes, which individually feed light via optic fibre into a KiwiSpec high-resolution (R = 80,000) stabilised spectrograph \\citep{barnes2012} that covers wavelengths from 480 nm to 620 nm. \n\n\\item The CHIRON spectrograph mounted on the SMARTS 1.5-m telescope \\citep{Tokovinin2018}, located at the Cerro Tololo\nInter-American Observatory (CTIO) in Chile. The high resolution cross-dispersed echelle spectrometer is fiber-fed followed by an image slicer. It has a resolution of R = 80,000 and covers wavelengths ranging from 410 to 870 nm.\n\n\\item The SOPHIE echelle spectrograph mounted on the 1.93-m Haute-Provence Observatory (OHP), France\n\\citep{2008Perruchot,2009Bouchy}. The high resolution cross-dispersed stabilized echelle spectrometer is fed by two optical fibers. Observations were taken in high-resolution mode (R = 75,000) with a wavelength range of 387 to 694 nm.\n\n\\end{itemize}\n\nReconnaissance spectroscopy with these instruments allow us to extract stellar parameters, identify spectroscopic binaries, and place upper limits on the companion masses. Spectroscopic binaries and targets whose spectral type is incompatible with the initial planet hypothesis and\/or precludes precision RV observations (giant or early type stars) are not followed up further. Promising targets, however, are monitored in order to constrain their period and place limits on their mass. \n\n\\subsection{Speckle Imaging}\n\nFor our most promising candidates we perform high resolution speckle imaging using the `Alopeke instrument on the 8.1-m Frederick C. Gillett Gemini North telescope in Maunakea, Hawaii, USA, and its twin, Zorro, on the 8.1-m Gemini South telescope on Cerro Pach\\'{o}n, Chile \\citep{Matson2019, Howell2011}. Speckle interferometric observations provide extremely high resolution images reaching the diffraction limit of the telescope. We obtain simultaneous 562 nm and 832 nm rapid exposure (60 msec) images in succession that effectively `freeze out' atmospheric turbulence and through Fourier analysis are used to search for close companion stars at 5-8 magnitude contrast levels. This analysis, along with the reconstructed images, allow us to identify nearby companions and to quantify their light contribution to the TESS aperture and thus the transit signal.\n\n\n\\section{Planet candidates and Noteworthy Systems}\n\\label{sec:PHT_canidates}\n\\subsection{Planet candidate properties}\n\nIn this final part of the paper we discuss the 90 PHT candidates around 88 host stars that passed the initial two stages of vetting and that were uploaded to ExoFOP as cTOIs. At the time of discovery none of these candidates were TOIs. The properties of all of the PHT candidates are summarised in Table~\\ref{tab:PHT-caniddates}. Candidates that have been promoted to TOI status since their PHT discovery are highlighted with an asterisk following the TIC ID, and candidates that have been shown to be false positives, based on the ground-based follow-up observations, are marked with a dagger symbol ($\\dagger$). The majority (81\\%) of PHT candidates are single transit events, indicated by an `s' following the orbital period presented in the table. \\textcolor{red}{18 of the PHT candidates were flagged as TCEs by the \\emph{TESS}\\ pipeline, but not initially promoted to TOI status. The most common reasons for this was that the pipeline identified a single-transit event as well as times of systematics (often caused by momentum dumps), due to its two-transit minimum detection threshold. This resulted in the candidate being discarded on the basis of it not passing the `odd-even' transit depth test. Out of the 18 TCEs, 14 have become TOI's since the PHT discovery. More detail on the TCE candidates can be found in Appendix~\\ref{appendixA}.} \n\nAll planet parameters (columns 2 to 8) are derived from the \\texttt{pyaneti}\\ modelling as described in Section~\\ref{sec:vetting}. Finally, the table summarises the ground-based follow-up observations (see Sec~\\ref{sec:follow_up}) that have been obtained to date, where the bracketed numbers following the observing instruments indicate the number of epochs. Unless otherwise noted, the follow-up observations are consistent with a planetary scenario. More in depth descriptions of individual targets for which we have additional information to complement the results in Table~\\ref{tab:PHT-caniddates} can be found in Appendix~\\ref{appendixA}.\n\n\\subsection{Planet candidate analysis}\n\n\nThe majority of the TOIs (87.7\\%) have orbital periods shorter than 15 days due to the requirement of observing at least two transits included in all major pipelines \\textcolor{red}{combined with the observing strategy of \\emph{TESS}}. As visual vetting does not impose these limits, the candidates outlined in this paper are helping to populate the relatively under-explored long-period region of parameter space. This is highlighted in Figure~\\ref{fig:PHT_candidates}, which shows the transit depths vs the orbital periods of the PHT single transit candidates (orange circles) and the multi-transit candidates (magenta squares) compared to the TOIs (blue circles). Values of the orbital periods and transit depths were obtained via transit modelling using \\texttt{pyaneti} (see Section~\\ref{sec:vetting}). The orbital period of single transit events are poorly constrained, which is reflected by the large errorbars in Figure~\\ref{fig:PHT_candidates}. Figure~\\ref{fig:PHT_candidates} also highlights that with PHT we are able to recover a similar range of transit depths as the pipeline found TOIs, as was previously shown in Figure~\\ref{fig:recovery_rank500_radius_period}.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_candidate_period_depth_plot_errobrars.png}\n \\caption{The properties of the PHT single transit (orange circles) and multi transit (magenta squares) candidates compared to the properties TOIs (blue circles). All parameters (listed in Table~\\ref{fig:PHT_candidates}) were extracted using \\texttt{pyaneti}\\ modelling.}\n \\label{fig:PHT_candidates}\n\\end{figure}\n\nThe PHT candidates were further compared to the TOIs in terms of the properties of their host stars. Figure~\\ref{fig:eep} shows the effective temperature and stellar radii as taken from the TIC \\citep{Stassun18}, for TOIs (blue dots) and the PHT candidates (magenta circles). The solid and dashed lines indicate the main sequence and post-main sequence MIST stellar evolutionary tracks \\citep{choi2016}, respectively, for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. This shows that around 10\\% of the host stars are in the process of, or have recently evolved off the main sequence. The models assume solar metalicity, no stellar rotation and no additional internal mixing.\n\n\\textcolor{red}{Ground based follow-up spectroscopy has revealed that six of the PHT candidates listed in Table~\\ref{tab:PHT-caniddates} are astrophysical false positives. As the follow-up campaign of the targets is still underway, the true false-positive rate of the candidates to have made it through all stages of the vetting process, as outlined in the methodology, will be be assessed in future PHT papers once the true nature of more of the candidates has been independently verified.}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_eep.png}\n \\caption{Stellar evolution tracks showing main sequence (solid black lines) and post-main sequence (dashed grey lines) MIST stellar evolution for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. The blue dots show the TOIs and the magenta circles show the PHT candidates.} \n \\label{fig:eep}\n\\end{figure}\n\n\n\\subsection{Stellar systems}\n\\label{subsec:PHT_stars}\n\nIn addition to the planetary candidates, citizen science allows for the identification of interesting stellar systems and astrophysical phenomena, in particular where the signals are aperiodic or small compared to the dominant stellar signal. These include light curves that exhibit multiple transit-like signals, possibly as a result of a multiple stellar system or a blend of eclipsing binaries. We have investigated all light curves that were flagged as possible multi-stellar systems via the PHT discussion boards. Similar to the planet vetting, as described in Section~\\ref{sec:vetting}, we generated {\\sc latte} data validation reports in order to assess the nature of the signal. Additionally, we subjected these systems to an iterative signal removal process, whereby we phase-folded the light curve on the dominant orbital period, binned the light curve into between 200-500 phase bins, created an interpolation model, and then subtracted said signal in order to evaluate the individual transit signals. The period of each signal, as listed in Table~\\ref{tab:PHT-multis}, was determined by phase folding the light curve at a number of trial periods and assessing by eye the best fit period and corresponding uncertainty.\n\nDue to the large \\emph{TESS}\\ pixels, blends are expected to be common. We searched for blends by generating phase folded light curves for each pixel around the source of the target in order to better locate the source of each signal. Shifts in the \\emph{TESS}\\ x and y centroid positions were also found to be good indicators of visually separated sources. Nearby sources with a magnitude difference greater than 5 mags were ruled out as possible contaminators. We consider a candidate to be a confirmed blend when the centroids are separated by more than 1 \\emph{TESS}\\ pixel, as this corresponds to an angular separation > 21 arcseconds meaning that the systems are highly unlikely to be gravitationally bound. Systems where the signal appears to be coming from the same \\emph{TESS}\\ pixel and that show no clear centroid shifts are considered to be candidate multiple systems. We note that blends are still possible, however, without further investigation we cannot conclusively rule these out as possible multi stellar systems. \n\nAll of the systems are summarised in Table~\\ref{tab:PHT-multis}. Out of the 26 systems, 6 are confirmed multiple systems which have either been published or are being prepared for publication; 7 are visually separated eclipsing binaries (confirmed blends); and 13 are candidate multiple system. Additional observations will be required to determine whether or not these candidate multiple systems are in fact gravitationally bound or photometric blends as a results of the large \\emph{TESS}\\ pixels or due to a line of sight happenstance. \n\n\\begin{landscape}\n\\begin{table}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period} & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n & \\textbf{Name} & \\textbf{(\\textcolor{red}{BJD - 2457000})} & \\textbf{(days)} & & $(R_{\\oplus})$ & \\textbf{Parameter} & \\textbf{(hours)} & & & & & \\\\\n\\hline\n101641905 & TWOMASS 11412617+3441004 & $1917.26335 _{ - 0.00072 } ^ { + 0.00071 }$ & $14.52 _{ - 5.25 } ^ { + 6.21 }(s)$ & $0.1135 _{ - 0.0064 } ^ { + 0.0032 }$ & $9.76 _{ - 0.69 } ^ { + 0.65 }$ & $0.691 _{ - 0.183 } ^ { + 0.077 }$ & $3.163 _{ - 0.088 } ^ { + 0.093 }$ & 12.196 & & & & \\\\\n103633672* & TYC 4387-00923-1 & $1850.3211 _{ - 0.00077 } ^ { + 0.00135 }$ & $90.9 _{ - 23.7 } ^ { + 46.4 }(s)$ & $0.0395 _{ - 0.0013 } ^ { + 0.0013 }$ & $3.45 _{ - 0.24 } ^ { + 0.26 }$ & $0.3 _{ - 0.21 } ^ { + 0.26 }$ & $6.7 _{ - 0.11 } ^ { + 0.12 }$ & 10.586 & & NRES (1) & & \\\\\n110996418 & TWOMASS 12344723-1019107 & $1580.6406 _{ - 0.0038 } ^ { + 0.0037 }$ & $5.18 _{ - 2.93 } ^ { + 6.86 }(s)$ & $0.1044 _{ - 0.0067 } ^ { + 0.008 }$ & $12.7 _{ - 0.99 } ^ { + 1.15 }$ & $0.44 _{ - 0.3 } ^ { + 0.3 }$ & $3.53 _{ - 0.27 } ^ { + 0.36 }$ & 13.945 & & & & \\\\\n128703021 & HIP 71639 & $1601.8442 _{ - 0.00108 } ^ { + 0.00093 }$ & $26.0 _{ - 8.22 } ^ { + 22.35 }(s)$ & $0.0254 _{ - 0.00049 } ^ { + 0.00072 }$ & $4.44 _{ - 0.2 } ^ { + 0.23 }$ & $0.47 _{ - 0.3 } ^ { + 0.22 }$ & $7.283 _{ - 0.091 } ^ { + 0.141 }$ & 6.06 & & NRES (2);MINERVA (34) & Gemini & \\\\\n138126035 & TYC 1450-00833-1 & $1954.3229 _{ - 0.0041 } ^ { + 0.0067 }$ & $28.8 _{ - 14.0 } ^ { + 203.2 }(s)$ & $0.0375 _{ - 0.0026 } ^ { + 0.0069 }$ & $4.01 _{ - 0.35 } ^ { + 0.74 }$ & $0.58 _{ - 0.38 } ^ { + 0.35 }$ & $4.65 _{ - 0.32 } ^ { + 0.85 }$ & 10.349 & & & & \\\\\n142087638 & TYC 9189-00274-1 & $1512.1673 _{ - 0.0043 } ^ { + 0.0034 }$ & $3.14 _{ - 1.41 } ^ { + 12.04 }(s)$ & $0.0469 _{ - 0.0035 } ^ { + 0.0063 }$ & $6.05 _{ - 0.54 } ^ { + 0.89 }$ & $0.5 _{ - 0.35 } ^ { + 0.36 }$ & $2.72 _{ - 0.23 } ^ { + 0.5 }$ & 11.526 & & & & \\\\\n159159904 & HIP 64812 & $1918.6109 _{ - 0.0067 } ^ { + 0.0091 }$ & $584.0 _{ - 215.0 } ^ { + 1724.0 }(s)$ & $0.0237 _{ - 0.0011 } ^ { + 0.0026 }$ & $3.12 _{ - 0.22 } ^ { + 0.36 }$ & $0.49 _{ - 0.34 } ^ { + 0.35 }$ & $15.11 _{ - 0.54 } ^ { + 0.7 }$ & 9.2 & & NRES (2) & & \\\\\n160039081* & HIP 78892 & $1752.9261 _{ - 0.0045 } ^ { + 0.005 }$ & $30.19918 _{ - 0.00099 } ^ { + 0.00094 }$ & $0.0211 _{ - 0.0013 } ^ { + 0.0035 }$ & $2.67 _{ - 0.21 } ^ { + 0.43 }$ & $0.52 _{ - 0.34 } ^ { + 0.36 }$ & $4.93 _{ - 0.27 } ^ { + 0.37 }$ & 8.35 & SBIG (1) & NRES (1);SOPHIE (4) & Gemini & \\\\\n162631539 & HIP 80264 & $1978.2794 _{ - 0.0044 } ^ { + 0.0051 }$ & $17.32 _{ - 6.66 } ^ { + 52.35 }(s)$ & $0.0195 _{ - 0.0011 } ^ { + 0.0024 }$ & $2.94 _{ - 0.24 } ^ { + 0.38 }$ & $0.48 _{ - 0.33 } ^ { + 0.36 }$ & $5.54 _{ - 0.33 } ^ { + 0.41 }$ & 7.42 & & & & \\\\\n166184426* & TWOMASS 13442500-4020122 & $1600.4409 _{ - 0.003 } ^ { + 0.0036 }$ & $16.3325 _{ - 0.0066 } ^ { + 0.0052 }$ & $0.0545 _{ - 0.0031 } ^ { + 0.0039 }$ & $1.85 _{ - 0.12 } ^ { + 0.15 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.98 _{ - 0.22 } ^ { + 0.17 }$ & 12.911 & & & & \\\\\n167661160 & TYC 7054-01577-1 & $1442.0703 _{ - 0.0028 } ^ { + 0.004 }$ & $36.802 _{ - 0.07 } ^ { + 0.069 }$ & $0.0307 _{ - 0.0014 } ^ { + 0.0024 }$ & $4.07 _{ - 0.32 } ^ { + 0.43 }$ & $0.37 _{ - 0.26 } ^ { + 0.33 }$ & $5.09 _{ - 0.23 } ^ { + 0.21 }$ & 9.927 & & NRES (9);MINERVA (4) & & EB from MINERVA observations \\\\\n172370679* & TWOMASS 19574239+4008357 & $1711.95923 _{ - 0.00099 } ^ { + 0.001 }$ & $32.84 _{ - 4.17 } ^ { + 5.59 }(s)$ & $0.1968 _{ - 0.0032 } ^ { + 0.0022 }$ & $13.24 _{ - 0.43 } ^ { + 0.43 }$ & $0.22 _{ - 0.15 } ^ { + 0.14 }$ & $4.999 _{ - 0.097 } ^ { + 0.111 }$ & 14.88 & & & & Confirmed planet \\citep{canas2020}. \\\\\n174302697* & TYC 3641-01789-1 & $1743.7267 _{ - 0.00092 } ^ { + 0.00093 }$ & $498.2 _{ - 80.0 } ^ { + 95.3 }(s)$ & $0.07622 _{ - 0.00068 } ^ { + 0.00063 }$ & $13.34 _{ - 0.57 } ^ { + 0.58 }$ & $0.642 _{ - 0.029 } ^ { + 0.024 }$ & $17.71 _{ - 0.12 } ^ { + 0.13 }$ & 9.309 & SBIG (1) & & & \\\\\n179582003 & TYC 9166-00745-1 & $1518.4688 _{ - 0.0016 } ^ { + 0.0016 }$ & $104.6137 _{ - 0.0022 } ^ { + 0.0022 }$ & $0.06324 _{ - 0.0008 } ^ { + 0.0008 }$ & $7.51 _{ - 0.35 } ^ { + 0.35 }$ & $0.21 _{ - 0.15 } ^ { + 0.19 }$ & $9.073 _{ - 0.084 } ^ { + 0.097 }$ & 10.806 & & & & \\\\\n192415680 & TYC 2859-00682-1 & $1796.0265 _{ - 0.0012 } ^ { + 0.0013 }$ & $18.47 _{ - 6.34 } ^ { + 21.73 }(s)$ & $0.0478 _{ - 0.0017 } ^ { + 0.0027 }$ & $4.43 _{ - 0.33 } ^ { + 0.38 }$ & $0.45 _{ - 0.31 } ^ { + 0.31 }$ & $3.94 _{ - 0.1 } ^ { + 0.12 }$ & 9.838 & SBIG (1) & SOPHIE (2) & & \\\\\n192790476 & TYC 7595-00649-1 & $1452.3341 _{ - 0.0014 } ^ { + 0.002 }$ & $16.09 _{ - 5.73 } ^ { + 15.49 }(s)$ & $0.0438 _{ - 0.0018 } ^ { + 0.0026 }$ & $3.24 _{ - 0.34 } ^ { + 0.37 }$ & $0.37 _{ - 0.25 } ^ { + 0.3 }$ & $3.395 _{ - 0.099 } ^ { + 0.192 }$ & 10.772 & & & & \\\\\n206361691$\\dagger$ & HIP 117250 & $1363.2224 _{ - 0.0082 } ^ { + 0.009 }$ & $237.7 _{ - 81.0 } ^ { + 314.4 }(s)$ & $0.01762 _{ - 0.00088 } ^ { + 0.00125 }$ & $2.69 _{ - 0.19 } ^ { + 0.25 }$ & $0.43 _{ - 0.28 } ^ { + 0.32 }$ & $13.91 _{ - 0.53 } ^ { + 0.52 }$ & 8.88 & & CHIRON (2) & & SB2 from CHIRON \\\\\n207501148 & TYC 3881-00527-1 & $2007.7273 _{ - 0.0011 } ^ { + 0.0011 }$ & $39.9 _{ - 10.3 } ^ { + 14.3 }(s)$ & $0.0981 _{ - 0.0047 } ^ { + 0.011 }$ & $13.31 _{ - 0.95 } ^ { + 1.56 }$ & $0.9 _{ - 0.03 } ^ { + 0.039 }$ & $4.73 _{ - 0.14 } ^ { + 0.14 }$ & 10.385 & & & & \\\\\n219466784* & TYC 4409-00437-1 & $1872.6879 _{ - 0.0097 } ^ { + 0.0108 }$ & $318.0 _{ - 147.0 } ^ { + 1448.0 }(s)$ & $0.0332 _{ - 0.0024 } ^ { + 0.0048 }$ & $3.26 _{ - 0.31 } ^ { + 0.49 }$ & $0.55 _{ - 0.39 } ^ { + 0.34 }$ & $10.06 _{ - 0.81 } ^ { + 1.12 }$ & 11.099 & & & & \\\\\n219501568 & HIP 79876 & $1961.7879 _{ - 0.0018 } ^ { + 0.002 }$ & $16.5931 _{ - 0.0017 } ^ { + 0.0015 }$ & $0.0221 _{ - 0.0012 } ^ { + 0.0015 }$ & $4.22 _{ - 0.3 } ^ { + 0.35 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.615 _{ - 0.077 } ^ { + 0.093 }$ & 8.38 & & & & \\\\\n229055790 & TYC 7492-01197-1 & $1337.866 _{ - 0.0022 } ^ { + 0.0019 }$ & $48.0 _{ - 12.8 } ^ { + 48.4 }(s)$ & $0.0304 _{ - 0.00097 } ^ { + 0.00115 }$ & $3.52 _{ - 0.2 } ^ { + 0.24 }$ & $0.37 _{ - 0.26 } ^ { + 0.32 }$ & $6.53 _{ - 0.11 } ^ { + 0.14 }$ & 9.642 & & NRES (2) & & \\\\\n229608594 & TWOMASS 18180283+7428005 & $1960.0319 _{ - 0.0037 } ^ { + 0.0045 }$ & $152.4 _{ - 54.1 } ^ { + 152.6 }(s)$ & $0.0474 _{ - 0.0023 } ^ { + 0.0024 }$ & $3.42 _{ - 0.34 } ^ { + 0.36 }$ & $0.38 _{ - 0.26 } ^ { + 0.3 }$ & $6.98 _{ - 0.23 } ^ { + 0.37 }$ & 12.302 & & & & \\\\\n229742722* & TYC 4434-00596-1 & $1689.688 _{ - 0.025 } ^ { + 0.02 }$ & $29.0 _{ - 16.4 } ^ { + 66.3 }(s)$ & $0.019 _{ - 0.0028 } ^ { + 0.0029 }$ & $2.9 _{ - 0.44 } ^ { + 0.48 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $4.27 _{ - 0.09 } ^ { + 0.11 }$ & 10.33 & & NRES (8);SOPHIE (4) & Gemini & \\\\\n233194447 & TYC 4211-00650-1 & $1770.4924 _{ - 0.0065 } ^ { + 0.0107 }$ & $373.0 _{ - 101.0 } ^ { + 284.0 }(s)$ & $0.02121 _{ - 0.00073 } ^ { + 0.001 }$ & $5.08 _{ - 0.28 } ^ { + 0.33 }$ & $0.34 _{ - 0.24 } ^ { + 0.29 }$ & $24.45 _{ - 0.47 } ^ { + 0.5 }$ & 9.178 & & NRES (2) & Gemini & \\\\\n235943205 & TYC 4588-00127-1 & $1827.0267 _{ - 0.004 } ^ { + 0.0034 }$ & $121.3394 _{ - 0.0063 } ^ { + 0.0065 }$ & $0.0402 _{ - 0.0016 } ^ { + 0.0019 }$ & $4.2 _{ - 0.25 } ^ { + 0.29 }$ & $0.4 _{ - 0.27 } ^ { + 0.28 }$ & $6.37 _{ - 0.2 } ^ { + 0.3 }$ & 11.076 & & NRES (1);SOPHIE (2) & & \\\\\n237201858 & TYC 4452-00759-1 & $1811.5032 _{ - 0.0069 } ^ { + 0.0067 }$ & $129.7 _{ - 41.5 } ^ { + 146.8 }(s)$ & $0.0258 _{ - 0.0013 } ^ { + 0.0015 }$ & $4.12 _{ - 0.27 } ^ { + 0.3 }$ & $0.4 _{ - 0.28 } ^ { + 0.31 }$ & $10.94 _{ - 0.37 } ^ { + 0.53 }$ & 10.344 & & NRES (1) & & \\\\\n243187830* & HIP 5286 & $1783.7671 _{ - 0.0017 } ^ { + 0.0019 }$ & $4.05 _{ - 1.53 } ^ { + 9.21 }(s)$ & $0.0268 _{ - 0.0015 } ^ { + 0.0027 }$ & $2.06 _{ - 0.17 } ^ { + 0.23 }$ & $0.47 _{ - 0.32 } ^ { + 0.34 }$ & $2.02 _{ - 0.12 } ^ { + 0.15 }$ & 8.407 & SBIG (1) & & & \\\\\n243417115 & TYC 8262-02120-1 & $1614.4796 _{ - 0.0028 } ^ { + 0.0022 }$ & $1.81 _{ - 0.73 } ^ { + 3.45 }(s)$ & $0.0523 _{ - 0.0035 } ^ { + 0.005 }$ & $5.39 _{ - 0.47 } ^ { + 0.64 }$ & $0.47 _{ - 0.33 } ^ { + 0.34 }$ & $2.03 _{ - 0.16 } ^ { + 0.23 }$ & 11.553 & & & & \\\\\n256429408 & TYC 4462-01942-1 & $1962.16 _{ - 0.0022 } ^ { + 0.0023 }$ & $382.0 _{ - 132.0 } ^ { + 265.0 }(s)$ & $0.03582 _{ - 0.00086 } ^ { + 0.00094 }$ & $6.12 _{ - 0.29 } ^ { + 0.3 }$ & $0.51 _{ - 0.36 } ^ { + 0.18 }$ & $16.96 _{ - 0.2 } ^ { + 0.24 }$ & 8.898 & & & & \\\\\n264544388* & TYC 4607-01275-1 & $1824.8438 _{ - 0.0076 } ^ { + 0.0078 }$ & $7030.0 _{ - 6260.0 } ^ { + 3330.0 }(s)$ & $0.0288 _{ - 0.0029 } ^ { + 0.0018 }$ & $4.58 _{ - 0.43 } ^ { + 0.35 }$ & $0.936 _{ - 0.363 } ^ { + 0.011 }$ & $19.13 _{ - 1.35 } ^ { + 0.84 }$ & 8.758 & & NRES (1) & & \\\\\n264766922 & TYC 8565-01780-1 & $1538.69518 _{ - 0.00091 } ^ { + 0.00091 }$ & $3.28 _{ - 0.94 } ^ { + 1.25 }(s)$ & $0.0933 _{ - 0.0063 } ^ { + 0.0176 }$ & $16.95 _{ - 1.33 } ^ { + 3.19 }$ & $0.908 _{ - 0.039 } ^ { + 0.048 }$ & $2.73 _{ - 0.11 } ^ { + 0.11 }$ & 10.747 & & & & \\\\\n26547036* & TYC 3921-01563-1 & $1712.30464 _{ - 0.00041 } ^ { + 0.0004 }$ & $73.0 _{ - 13.6 } ^ { + 16.5 }(s)$ & $0.10034 _{ - 0.0007 } ^ { + 0.00078 }$ & $11.75 _{ - 0.59 } ^ { + 0.58 }$ & $0.17 _{ - 0.12 } ^ { + 0.11 }$ & $8.681 _{ - 0.049 } ^ { + 0.052 }$ & 9.849 & & NRES (4) & Gemini & \\\\\n267542728$\\dagger$ & TYC 4583-01499-1 & $1708.4956 _{ - 0.0073 } ^ { + 0.0085 }$ & $39.7382 _{ - 0.0023 } ^ { + 0.0023 }$ & $0.03267 _{ - 0.00089 } ^ { + 0.00175 }$ & $18.46 _{ - 0.94 } ^ { + 1.14 }$ & $0.38 _{ - 0.26 } ^ { + 0.27 }$ & $24.16 _{ - 0.39 } ^ { + 0.45 }$ & 11.474 & & & & EB from HIRES RVs. \\\\\n270371513$\\dagger$ & HIP 10047 & $1426.2967 _{ - 0.0023 } ^ { + 0.002 }$ & $0.39 _{ - 0.17 } ^ { + 1.79 }(s)$ & $0.024 _{ - 0.0015 } ^ { + 0.0032 }$ & $4.8 _{ - 0.38 } ^ { + 0.64 }$ & $0.5 _{ - 0.34 } ^ { + 0.39 }$ & $1.93 _{ - 0.16 } ^ { + 0.19 }$ & 6.98515 & & MINERVA (20) & & SB 2 from MINERVA observations. \\\\\n274599700 & TWOMASS 17011885+5131455 & $2002.1202 _{ - 0.0024 } ^ { + 0.0024 }$ & $32.9754 _{ - 0.005 } ^ { + 0.005 }$ & $0.0847 _{ - 0.0021 } ^ { + 0.0018 }$ & $13.25 _{ - 0.83 } ^ { + 0.83 }$ & $0.37 _{ - 0.24 } ^ { + 0.19 }$ & $8.2 _{ - 0.18 } ^ { + 0.21 }$ & 12.411 & & & & \\\\\n278990954 & TYC 8548-00717-1 & $1650.0191 _{ - 0.0086 } ^ { + 0.0105 }$ & $18.45 _{ - 8.66 } ^ { + 230.7 }(s)$ & $0.034 _{ - 0.0024 } ^ { + 0.0115 }$ & $9.65 _{ - 0.92 } ^ { + 3.13 }$ & $0.58 _{ - 0.4 } ^ { + 0.36 }$ & $10.62 _{ - 0.66 } ^ { + 2.46 }$ & 10.749 & & & & \\\\\n280865159* & TYC 9384-01533-1 & $1387.0749 _{ - 0.0045 } ^ { + 0.0044 }$ & $1045.0 _{ - 249.0 } ^ { + 536.0 }(s)$ & $0.0406 _{ - 0.0011 } ^ { + 0.0014 }$ & $4.75 _{ - 0.26 } ^ { + 0.28 }$ & $0.35 _{ - 0.24 } ^ { + 0.23 }$ & $19.08 _{ - 0.32 } ^ { + 0.36 }$ & 11.517 & & & Gemini & \\\\\n284361752 & TYC 3924-01678-1 & $2032.093 _{ - 0.0078 } ^ { + 0.008 }$ & $140.6 _{ - 46.6 } ^ { + 159.1 }(s)$ & $0.0259 _{ - 0.0014 } ^ { + 0.0017 }$ & $3.62 _{ - 0.26 } ^ { + 0.31 }$ & $0.4 _{ - 0.27 } ^ { + 0.34 }$ & $8.98 _{ - 0.66 } ^ { + 0.86 }$ & 10.221 & & & & \\\\\n288240183 & TYC 4634-01225-1 & $1896.941 _{ - 0.0051 } ^ { + 0.0047 }$ & $119.0502 _{ - 0.0091 } ^ { + 0.0089 }$ & $0.02826 _{ - 0.00089 } ^ { + 0.00119 }$ & $4.28 _{ - 0.35 } ^ { + 0.36 }$ & $0.55 _{ - 0.37 } ^ { + 0.25 }$ & $17.49 _{ - 0.36 } ^ { + 0.6 }$ & 9.546 & & & & \\\\\n29169215 & TWOMASS 09011787+4727085 & $1872.5047 _{ - 0.0032 } ^ { + 0.0036 }$ & $14.89 _{ - 6.12 } ^ { + 24.84 }(s)$ & $0.0403 _{ - 0.0025 } ^ { + 0.0033 }$ & $3.28 _{ - 0.37 } ^ { + 0.45 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $3.56 _{ - 0.21 } ^ { + 0.32 }$ & 11.828 & & & & \\\\\n293649602 & TYC 8103-00266-1 & $1511.2109 _{ - 0.004 } ^ { + 0.0037 }$ & $12.85 _{ - 5.34 } ^ { + 42.21 }(s)$ & $0.04 _{ - 0.0024 } ^ { + 0.0039 }$ & $4.66 _{ - 0.36 } ^ { + 0.5 }$ & $0.5 _{ - 0.35 } ^ { + 0.34 }$ & $4.1 _{ - 0.31 } ^ { + 0.56 }$ & 10.925 & & & & \\\\\n296737508 & TYC 5472-01060-1 & $1538.0036 _{ - 0.0015 } ^ { + 0.0016 }$ & $18.27 _{ - 5.06 } ^ { + 17.45 }(s)$ & $0.0425 _{ - 0.0014 } ^ { + 0.0019 }$ & $5.33 _{ - 0.22 } ^ { + 0.27 }$ & $0.44 _{ - 0.3 } ^ { + 0.26 }$ & $5.13 _{ - 0.13 } ^ { + 0.15 }$ & 9.772 & Sinistro (1) & NRES (1);MINERVA (1) & Gemini & \\\\\n298663873 & TYC 3913-01781-1 & $1830.76819 _{ - 0.00099 } ^ { + 0.00099 }$ & $479.9 _{ - 89.4 } ^ { + 109.4 }(s)$ & $0.06231 _{ - 0.00034 } ^ { + 0.00045 }$ & $11.07 _{ - 0.57 } ^ { + 0.57 }$ & $0.16 _{ - 0.11 } ^ { + 0.13 }$ & $23.99 _{ - 0.093 } ^ { + 0.1 }$ & 9.162 & & NRES (2) & Gemini & Dalba et al. (in prep) \\\\\n303050301 & TYC 6979-01108-1 & $1366.1301 _{ - 0.0022 } ^ { + 0.0023 }$ & $281.0 _{ - 170.0 } ^ { + 264.0 }(s)$ & $0.0514 _{ - 0.0027 } ^ { + 0.0018 }$ & $4.85 _{ - 0.32 } ^ { + 0.32 }$ & $0.73 _{ - 0.48 } ^ { + 0.1 }$ & $7.91 _{ - 0.31 } ^ { + 0.36 }$ & 10.048 & & NRES (1) & Gemini & \\\\\n303317324 & TYC 6983-00438-1 & $1365.1845 _{ - 0.0023 } ^ { + 0.0028 }$ & $69.0 _{ - 25.5 } ^ { + 78.1 }(s)$ & $0.0365 _{ - 0.0013 } ^ { + 0.0016 }$ & $2.88 _{ - 0.3 } ^ { + 0.31 }$ & $0.39 _{ - 0.26 } ^ { + 0.32 }$ & $5.78 _{ - 0.18 } ^ { + 0.24 }$ & 10.799 & & & & \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\emph{Note} -- Candidates that have become TOIs following the PHT discovery are marked with an asterisk (*). The `s' following the orbital period indicates that the candidates is a single transit event. The ground-based follow-up observations are summarized in columns 10-12, where the bracketed numbers correspond the number of epochs obtained with each instrument. See Section~\\ref{sec:follow_up} for description of each instrument. The $\\dagger$ symbol indicates candidates that have been shown to be astrophysical false positives based on the ground based follow-up observations.}\n\\label{tab:PHT-caniddates}\n\\end{table}\n\\end{landscape}\n\n\\begin{landscape}\n\\begin{table}\n\\addtocounter{table}{-1}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period} & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n & \\textbf{Name} & \\textbf{(\\textcolor{red}{BJD - 2457000})} & \\textbf{(days)} & & $(R_{\\oplus})$ & \\textbf{Parameter} & \\textbf{(hours)} & & & & & \\\\\n\\hline\n303586471$\\dagger$ & HIP 115828 & $1363.7692 _{ - 0.0033 } ^ { + 0.0027 }$ & $13.85 _{ - 4.19 } ^ { + 18.2 }(s)$ & $0.0214 _{ - 0.001 } ^ { + 0.0014 }$ & $2.52 _{ - 0.16 } ^ { + 0.2 }$ & $0.4 _{ - 0.27 } ^ { + 0.33 }$ & $4.23 _{ - 0.19 } ^ { + 0.16 }$ & 8.27 & & MINERVA (11) & & SB 2 from MINERVA observations. \\\\\n304142124* & HIP 53719 & $1585.28023 _{ - 0.0008 } ^ { + 0.0008 }$ & $42.8 _{ - 10.0 } ^ { + 18.2 }(s)$ & $0.04311 _{ - 0.00093 } ^ { + 0.00153 }$ & $4.1 _{ - 0.23 } ^ { + 0.24 }$ & $0.33 _{ - 0.21 } ^ { + 0.21 }$ & $5.66 _{ - 0.067 } ^ { + 0.09 }$ & 8.62 & & NRES (1);MINERVA (4) & & Confirmed planet \\citep{diaz2020} \\\\\n304339227 & TYC 9290-01087-1 & $1673.3242 _{ - 0.009 } ^ { + 0.0128 }$ & $111.9 _{ - 72.2 } ^ { + 4844.1 }(s)$ & $0.0253 _{ - 0.0024 } ^ { + 0.0481 }$ & $3.27 _{ - 0.61 } ^ { + 5.72 }$ & $0.67 _{ - 0.47 } ^ { + 0.36 }$ & $7.44 _{ - 0.86 } ^ { + 2.84 }$ & 9.169 & & & & \\\\\n307958020 & TYC 4191-00309-1 & $1864.82 _{ - 0.014 } ^ { + 0.013 }$ & $169.0 _{ - 107.0 } ^ { + 10194.0 }(s)$ & $0.0223 _{ - 0.0022 } ^ { + 0.0543 }$ & $3.92 _{ - 0.52 } ^ { + 9.27 }$ & $0.71 _{ - 0.53 } ^ { + 0.33 }$ & $12.48 _{ - 1.1 } ^ { + 5.41 }$ & 9.017 & & & & \\\\\n308301091 & TYC 2081-01273-1 & $2030.3691 _{ - 0.0024 } ^ { + 0.0026 }$ & $29.24 _{ - 8.49 } ^ { + 22.46 }(s)$ & $0.0362 _{ - 0.0013 } ^ { + 0.0014 }$ & $5.41 _{ - 0.34 } ^ { + 0.35 }$ & $0.35 _{ - 0.25 } ^ { + 0.29 }$ & $6.57 _{ - 0.14 } ^ { + 0.19 }$ & 10.273 & & & & \\\\\n313006381 & HIP 45012 & $1705.687 _{ - 0.0081 } ^ { + 0.0045 }$ & $21.56 _{ - 8.9 } ^ { + 54.15 }(s)$ & $0.0261 _{ - 0.0017 } ^ { + 0.0027 }$ & $2.34 _{ - 0.2 } ^ { + 0.27 }$ & $0.45 _{ - 0.3 } ^ { + 0.38 }$ & $3.85 _{ - 0.51 } ^ { + 0.31 }$ & 9.39 & & & & \\\\\n323295479* & TYC 9506-01881-1 & $1622.9258 _{ - 0.00083 } ^ { + 0.00087 }$ & $117.8 _{ - 25.8 } ^ { + 30.9 }(s)$ & $0.0981 _{ - 0.0021 } ^ { + 0.0023 }$ & $11.35 _{ - 0.67 } ^ { + 0.66 }$ & $0.839 _{ - 0.024 } ^ { + 0.019 }$ & $6.7 _{ - 0.14 } ^ { + 0.15 }$ & 10.595 & & & & \\\\\n328933398.01* & TYC 4634-01435-1 & $1880.9878 _{ - 0.0039 } ^ { + 0.0042 }$ & $24.9335 _{ - 0.0046 } ^ { + 0.005 }$ & $0.0437 _{ - 0.0022 } ^ { + 0.0023 }$ & $4.62 _{ - 0.32 } ^ { + 0.33 }$ & $0.38 _{ - 0.25 } ^ { + 0.27 }$ & $5.02 _{ - 0.22 } ^ { + 0.27 }$ & 11.215 & & & & Potential multi-planet system. \\\\\n328933398.02* & TYC 4634-01435-1 & $1848.6557 _{ - 0.0053 } ^ { + 0.0072 }$ & $50.5 _{ - 22.4 } ^ { + 77.1 }(s)$ & $0.0296 _{ - 0.0028 } ^ { + 0.0033 }$ & $3.14 _{ - 0.33 } ^ { + 0.39 }$ & $0.41 _{ - 0.28 } ^ { + 0.35 }$ & $5.99 _{ - 0.8 } ^ { + 0.77 }$ & 11.215 & & & & \\\\\n331644554 & TYC 3609-00469-1 & $1757.0354 _{ - 0.0031 } ^ { + 0.0033 }$ & $947.0 _{ - 215.0 } ^ { + 274.0 }(s)$ & $0.12 _{ - 0.025 } ^ { + 0.021 }$ & $21.84 _{ - 4.57 } ^ { + 3.86 }$ & $1.018 _{ - 0.036 } ^ { + 0.028 }$ & $10.93 _{ - 0.34 } ^ { + 0.35 }$ & 9.752 & & & & \\\\\n332657786 & TWOMASS 09595797-1609323 & $1536.7659 _{ - 0.0015 } ^ { + 0.0015 }$ & $63.76 _{ - 9.52 } ^ { + 11.13 }(s)$ & $0.14961 _{ - 0.00064 } ^ { + 0.00029 }$ & $3.83 _{ - 0.12 } ^ { + 0.12 }$ & $0.059 _{ - 0.041 } ^ { + 0.064 }$ & $3.333 _{ - 0.095 } ^ { + 0.096 }$ & 15.99 & & & & \\\\\n336075472 & TYC 3526-00332-1 & $2028.1762 _{ - 0.0043 } ^ { + 0.0037 }$ & $61.9 _{ - 24.0 } ^ { + 95.6 }(s)$ & $0.0402 _{ - 0.0022 } ^ { + 0.0033 }$ & $3.09 _{ - 0.34 } ^ { + 0.4 }$ & $0.43 _{ - 0.29 } ^ { + 0.32 }$ & $5.39 _{ - 0.23 } ^ { + 0.37 }$ & 11.842 & & & & \\\\\n349488688.01 & TYC 1529-00224-1 & $1994.283 _{ - 0.0038 } ^ { + 0.0033 }$ & $11.6254 _{ - 0.005 } ^ { + 0.0052 }$ & $0.02195 _{ - 0.00096 } ^ { + 0.00122 }$ & $3.44 _{ - 0.18 } ^ { + 0.21 }$ & $0.39 _{ - 0.27 } ^ { + 0.3 }$ & $5.58 _{ - 0.15 } ^ { + 0.18 }$ & 8.855 & & NRES (2);SOPHIE (2) & & Potential multi-planet system. \\\\\n349488688.02 & TYC 1529-00224-1 & $2002.77063 _{ - 0.00097 } ^ { + 0.00103 }$ & $15.35 _{ - 1.94 } ^ { + 4.15 }(s)$ & $0.03688 _{ - 0.00067 } ^ { + 0.00069 }$ & $5.78 _{ - 0.18 } ^ { + 0.18 }$ & $0.24 _{ - 0.16 } ^ { + 0.21 }$ & $6.291 _{ - 0.058 } ^ { + 0.074 }$ & 8.855 & & NRES (2);SOPHIE (2) & & \\\\\n356700488* & TYC 4420-01295-1 & $1756.638 _{ - 0.013 } ^ { + 0.011 }$ & $184.5 _{ - 64.7 } ^ { + 333.1 }(s)$ & $0.0173 _{ - 0.0011 } ^ { + 0.0015 }$ & $2.92 _{ - 0.2 } ^ { + 0.28 }$ & $0.44 _{ - 0.3 } ^ { + 0.34 }$ & $11.76 _{ - 0.65 } ^ { + 1.03 }$ & 8.413 & & & & \\\\\n356710041* & TYC 1993-00419-1 & $1932.2939 _{ - 0.0019 } ^ { + 0.0019 }$ & $29.6 _{ - 14.0 } ^ { + 19.0 }(s)$ & $0.0496 _{ - 0.0021 } ^ { + 0.0011 }$ & $14.82 _{ - 0.85 } ^ { + 0.84 }$ & $0.66 _{ - 0.42 } ^ { + 0.11 }$ & $12.76 _{ - 0.24 } ^ { + 0.24 }$ & 9.646 & & & & \\\\\n369532319 & TYC 2743-01716-1 & $1755.8158 _{ - 0.006 } ^ { + 0.0051 }$ & $35.4 _{ - 12.0 } ^ { + 51.6 }(s)$ & $0.0316 _{ - 0.0023 } ^ { + 0.0028 }$ & $3.43 _{ - 0.3 } ^ { + 0.37 }$ & $0.41 _{ - 0.29 } ^ { + 0.34 }$ & $5.5 _{ - 0.32 } ^ { + 0.32 }$ & 10.594 & & & Gemini & \\\\\n369779127 & TYC 9510-00090-1 & $1643.9403 _{ - 0.0046 } ^ { + 0.0058 }$ & $9.93 _{ - 3.38 } ^ { + 19.74 }(s)$ & $0.0288 _{ - 0.0015 } ^ { + 0.0033 }$ & $4.89 _{ - 0.31 } ^ { + 0.56 }$ & $0.46 _{ - 0.31 } ^ { + 0.33 }$ & $5.64 _{ - 0.38 } ^ { + 0.33 }$ & 9.279 & & & & \\\\\n384159646* & TYC 9454-00957-1 & $1630.39405 _{ - 0.00079 } ^ { + 0.00079 }$ & $11.68 _{ - 2.75 } ^ { + 4.21 }(s)$ & $0.0658 _{ - 0.0012 } ^ { + 0.0011 }$ & $9.87 _{ - 0.45 } ^ { + 0.44 }$ & $0.27 _{ - 0.18 } ^ { + 0.21 }$ & $5.152 _{ - 0.069 } ^ { + 0.087 }$ & 10.158 & SBIG (1) & NRES (8);MINERVA (6) & Gemini & \\\\\n385557214 & TYC 1807-00046-1 & $1791.58399 _{ - 0.00068 } ^ { + 0.0007 }$ & $5.62451 _{ - 0.0004 } ^ { + 0.00043 }$ & $0.096 _{ - 0.019 } ^ { + 0.032 }$ & $8.32 _{ - 2.06 } ^ { + 2.77 }$ & $0.95 _{ - 0.075 } ^ { + 0.053 }$ & $1.221 _{ - 0.094 } ^ { + 0.058 }$ & 10.856 & & & & \\\\\n388134787 & TYC 4260-00427-1 & $1811.034 _{ - 0.015 } ^ { + 0.017 }$ & $246.0 _{ - 127.0 } ^ { + 6209.0 }(s)$ & $0.0265 _{ - 0.0024 } ^ { + 0.023 }$ & $2.57 _{ - 0.28 } ^ { + 2.19 }$ & $0.55 _{ - 0.39 } ^ { + 0.44 }$ & $8.85 _{ - 1.13 } ^ { + 1.84 }$ & 10.95 & & NRES (1) & Gemini & \\\\\n404518509 & HIP 16038 & $1431.2696 _{ - 0.0037 } ^ { + 0.0035 }$ & $26.83 _{ - 9.46 } ^ { + 56.14 }(s)$ & $0.0259 _{ - 0.0013 } ^ { + 0.0022 }$ & $2.94 _{ - 0.21 } ^ { + 0.29 }$ & $0.47 _{ - 0.31 } ^ { + 0.34 }$ & $5.02 _{ - 0.23 } ^ { + 0.28 }$ & 9.17 & & & & \\\\\n408636441* & TYC 4266-00736-1 & $1745.4668 _{ - 0.0016 } ^ { + 0.0015 }$ & $37.695 _{ - 0.0034 } ^ { + 0.0033 }$ & $0.0485 _{ - 0.0019 } ^ { + 0.0023 }$ & $3.32 _{ - 0.16 } ^ { + 0.19 }$ & $0.39 _{ - 0.27 } ^ { + 0.29 }$ & $3.63 _{ - 0.1 } ^ { + 0.14 }$ & 11.93 & SBIG (1) & & Gemini & Half of the period likely. \\\\\n418255064 & TWOMASS 13063680-8037015 & $1629.3304 _{ - 0.0018 } ^ { + 0.0018 }$ & $25.37 _{ - 7.06 } ^ { + 15.41 }(s)$ & $0.0732 _{ - 0.0029 } ^ { + 0.0031 }$ & $5.57 _{ - 0.36 } ^ { + 0.38 }$ & $0.37 _{ - 0.25 } ^ { + 0.25 }$ & $3.83 _{ - 0.13 } ^ { + 0.14 }$ & 12.478 & SBIG (1) & & Gemini & \\\\\n420645189$\\dagger$ & TYC 4508-00478-1 & $1837.4767 _{ - 0.0018 } ^ { + 0.0017 }$ & $250.2 _{ - 66.6 } ^ { + 99.4 }(s)$ & $0.0784 _{ - 0.0033 } ^ { + 0.0046 }$ & $8.82 _{ - 0.55 } ^ { + 0.7 }$ & $0.892 _{ - 0.026 } ^ { + 0.028 }$ & $6.95 _{ - 0.27 } ^ { + 0.3 }$ & 10.595 & & MINERVA (1) & & SB 2 from MINERVA observations. \\\\\n422914082 & TYC 0046-00133-1 & $1431.5538 _{ - 0.0014 } ^ { + 0.0017 }$ & $12.91 _{ - 3.91 } ^ { + 8.97 }(s)$ & $0.0418 _{ - 0.0015 } ^ { + 0.0016 }$ & $3.96 _{ - 0.32 } ^ { + 0.35 }$ & $0.36 _{ - 0.25 } ^ { + 0.28 }$ & $4.07 _{ - 0.09 } ^ { + 0.126 }$ & 11.026 & Sinistro (1) & NRES (1) & & \\\\\n427344083 & TWOMASS 22563609+7040518 & $1961.8967 _{ - 0.0031 } ^ { + 0.0036 }$ & $7.77 _{ - 5.6 } ^ { + 9.65 }(s)$ & $0.107 _{ - 0.016 } ^ { + 0.025 }$ & $12.27 _{ - 1.87 } ^ { + 2.9 }$ & $0.834 _{ - 0.484 } ^ { + 0.094 }$ & $2.88 _{ - 0.3 } ^ { + 0.42 }$ & 13.404 & & & & \\\\\n436873727 & HIP 13224 & $1803.83679 _{ - 0.00058 } ^ { + 0.00056 }$ & $19.26 _{ - 5.95 } ^ { + 6.73 }(s)$ & $0.05246 _{ - 0.00061 } ^ { + 0.00059 }$ & $10.02 _{ - 0.43 } ^ { + 0.41 }$ & $0.767 _{ - 0.057 } ^ { + 0.038 }$ & $5.462 _{ - 0.081 } ^ { + 0.074 }$ & 7.51 & & & & \\\\ \n441642457* & TYC 3858-00452-1 & $1745.5102 _{ - 0.0108 } ^ { + 0.0097 }$ & $79.8072 _{ - 0.0071 } ^ { + 0.0076 }$ & $0.0281 _{ - 0.0024 } ^ { + 0.0033 }$ & $3.55 _{ - 0.34 } ^ { + 0.46 }$ & $0.934 _{ - 0.023 } ^ { + 0.026 }$ & $6.9 _{ - 0.39 } ^ { + 0.6 }$ & 9.996 & & & & \\\\\n441765914* & TWOMASS 17253007+7552562 & $1769.6154 _{ - 0.0058 } ^ { + 0.0093 }$ & $161.6 _{ - 58.2 } ^ { + 1460.1 }(s)$ & $0.0411 _{ - 0.0024 } ^ { + 0.0119 }$ & $3.6 _{ - 0.3 } ^ { + 1.01 }$ & $0.45 _{ - 0.32 } ^ { + 0.48 }$ & $7.44 _{ - 0.36 } ^ { + 1.08 }$ & 11.638 & & & & \\\\\n452920657 & TWOMASS 00332018+5906355 & $1810.5765 _{ - 0.0031 } ^ { + 0.003 }$ & $53.2 _{ - 29.0 } ^ { + 34.3 }(s)$ & $0.135 _{ - 0.016 } ^ { + 0.012 }$ & $9.71 _{ - 1.16 } ^ { + 0.9 }$ & $0.73 _{ - 0.48 } ^ { + 0.11 }$ & $4.6 _{ - 0.26 } ^ { + 0.29 }$ & 14.167 & SBIG (1) & & & \\\\\n455737331 & TYC 2779-00785-1 & $1780.7084 _{ - 0.008 } ^ { + 0.0073 }$ & $50.4 _{ - 17.6 } ^ { + 75.0 }(s)$ & $0.0257 _{ - 0.0016 } ^ { + 0.002 }$ & $3.05 _{ - 0.24 } ^ { + 0.29 }$ & $0.43 _{ - 0.29 } ^ { + 0.33 }$ & $6.6 _{ - 0.43 } ^ { + 0.5 }$ & 10.189 & SBIG (1) & & Gemini & \\\\\n456909420 & TYC 1208-01094-1 & $1779.4109 _{ - 0.0026 } ^ { + 0.0022 }$ & $5.78 _{ - 5.29 } ^ { + 5.95 }(s)$ & $0.078 _{ - 0.031 } ^ { + 0.045 }$ & $9.15 _{ - 3.61 } ^ { + 5.27 }$ & $0.973 _{ - 0.495 } ^ { + 0.063 }$ & $1.73 _{ - 0.27 } ^ { + 0.28 }$ & 10.941 & & & & \\\\\n458451774 & TWOMASS 12551793+4431260 & $1917.1875 _{ - 0.0019 } ^ { + 0.0019 }$ & $12.39 _{ - 6.34 } ^ { + 83.97 }(s)$ & $0.0752 _{ - 0.0054 } ^ { + 0.0211 }$ & $3.33 _{ - 0.26 } ^ { + 0.92 }$ & $0.61 _{ - 0.43 } ^ { + 0.32 }$ & $2.08 _{ - 0.19 } ^ { + 0.59 }$ & 13.713 & & & & \\\\\n48018596 & TYC 3548-00800-1 & $1713.4514 _{ - 0.0063 } ^ { + 0.0046 }$ & $100.1145 _{ - 0.0018 } ^ { + 0.0021 }$ & $0.049 _{ - 0.0081 } ^ { + 0.018 }$ & $7.88 _{ - 1.33 } ^ { + 2.9 }$ & $0.984 _{ - 0.028 } ^ { + 0.027 }$ & $2.83 _{ - 0.26 } ^ { + 0.29 }$ & 9.595 & & NRES (1) & Gemini & \\\\\n53309262 & TWOMASS 07475406+5741549 & $1863.1133 _{ - 0.0064 } ^ { + 0.0061 }$ & $294.8 _{ - 96.0 } ^ { + 327.0 }(s)$ & $0.1239 _{ - 0.0075 } ^ { + 0.0098 }$ & $5.38 _{ - 0.36 } ^ { + 0.46 }$ & $0.46 _{ - 0.31 } ^ { + 0.28 }$ & $6.74 _{ - 0.45 } ^ { + 0.62 }$ & 15.51 & & & & \\\\\n53843023 & TYC 6956-00758-1 & $1328.0335 _{ - 0.0054 } ^ { + 0.0057 }$ & $202.0 _{ - 189.0 } ^ { + 272.0 }(s)$ & $0.058 _{ - 0.02 } ^ { + 0.056 }$ & $5.14 _{ - 1.77 } ^ { + 4.99 }$ & $0.962 _{ - 0.597 } ^ { + 0.083 }$ & $4.25 _{ - 0.72 } ^ { + 0.66 }$ & 11.571 & & & & \\\\\n55525572* & TYC 8876-01059-1 & $1454.6713 _{ - 0.0066 } ^ { + 0.0065 }$ & $83.8951 _{ - 0.004 } ^ { + 0.004 }$ & $0.0343 _{ - 0.001 } ^ { + 0.0021 }$ & $7.31 _{ - 0.46 } ^ { + 0.56 }$ & $0.43 _{ - 0.29 } ^ { + 0.31 }$ & $13.54 _{ - 0.3 } ^ { + 0.51 }$ & 10.358 & & CHIRON (5) & Gemini & Confirmed planet \\citep{2020eisner} \\\\\n63698669* & TYC 6993-00729-1 & $1364.6226 _{ - 0.0074 } ^ { + 0.0067 }$ & $73.6 _{ - 26.8 } ^ { + 133.6 }(s)$ & $0.0248 _{ - 0.0019 } ^ { + 0.0023 }$ & $2.15 _{ - 0.2 } ^ { + 0.25 }$ & $0.42 _{ - 0.29 } ^ { + 0.35 }$ & $5.63 _{ - 0.32 } ^ { + 0.57 }$ & 10.701 & SBIG (1) & & & \\\\\n70887357* & TYC 5883-01412-1 & $1454.3341 _{ - 0.0016 } ^ { + 0.0015 }$ & $56.1 _{ - 15.3 } ^ { + 18.8 }(s)$ & $0.0605 _{ - 0.0027 } ^ { + 0.0027 }$ & $12.84 _{ - 0.86 } ^ { + 0.9 }$ & $0.917 _{ - 0.028 } ^ { + 0.016 }$ & $7.29 _{ - 0.18 } ^ { + 0.19 }$ & 9.293 & & & & \\\\\n7422496$\\dagger$ & HIP 25359 & $1470.3625 _{ - 0.0031 } ^ { + 0.0023 }$ & $61.4 _{ - 16.7 } ^ { + 49.0 }(s)$ & $0.0255 _{ - 0.001 } ^ { + 0.0011 }$ & $2.44 _{ - 0.15 } ^ { + 0.16 }$ & $0.37 _{ - 0.25 } ^ { + 0.29 }$ & $5.89 _{ - 0.15 } ^ { + 0.15 }$ & 9.36 & & MINERVA (4) & & SB 2 from MINERVA observations. \\\\\n82452140 & TYC 3076-00921-1 & $1964.292 _{ - 0.011 } ^ { + 0.011 }$ & $21.1338 _{ - 0.0052 } ^ { + 0.0066 }$ & $0.0266 _{ - 0.0019 } ^ { + 0.0027 }$ & $2.95 _{ - 0.25 } ^ { + 0.34 }$ & $0.42 _{ - 0.29 } ^ { + 0.36 }$ & $5.87 _{ - 0.62 } ^ { + 0.94 }$ & 10.616 & & & & \\\\\n88840705 & TYC 3091-00808-1 & $2026.6489 _{ - 0.001 } ^ { + 0.001 }$ & $260.6 _{ - 87.6 } ^ { + 142.2 }(s)$ & $0.109 _{ - 0.023 } ^ { + 0.027 }$ & $9.98 _{ - 2.28 } ^ { + 2.75 }$ & $1.001 _{ - 0.042 } ^ { + 0.037 }$ & $4.72 _{ - 0.13 } ^ { + 0.15 }$ & 9.443 & & & & \\\\\n91987762* & HIP 47288 & $1894.25381 _{ - 0.00051 } ^ { + 0.00047 }$ & $10.51 _{ - 3.48 } ^ { + 3.67 }(s)$ & $0.05459 _{ - 0.00106 } ^ { + 0.00097 }$ & $9.56 _{ - 0.56 } ^ { + 0.52 }$ & $0.771 _{ - 0.062 } ^ { + 0.033 }$ & $4.342 _{ - 0.073 } ^ { + 0.063 }$ & 7.87 & & NRES (4) & Gemini & \\\\\n95768667 & TYC 1434-00331-1 & $1918.3318 _{ - 0.0093 } ^ { + 0.0079 }$ & $26.9 _{ - 12.4 } ^ { + 72.3 }(s)$ & $0.0282 _{ - 0.0022 } ^ { + 0.0031 }$ & $3.54 _{ - 0.32 } ^ { + 0.43 }$ & $0.48 _{ - 0.33 } ^ { + 0.35 }$ & $5.4 _{ - 0.64 } ^ { + 0.76 }$ & 10.318 & & & & \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\textbf{Properties of PHT candidates (continued)}}\n\\label{tab:PHT-caniddates2}\n\\end{table}\n\\end{landscape}\n\n\n\\section{Conclusion}\n\\label{sec:condlusion}\n\nWe present the results from the analysis of the first 26 \\emph{TESS}\\ sectors. The outlined citizen science approach engages over 22 thousand registered citizen scientists who completed 12,617,038 classifications from December 2018 through August 2020 for the sectors observed during the first two years of the \\emph{TESS}\\ mission. We applied a systematic search for planetary candidates using visual vetting by multiple volunteers to identify \\emph{TESS}\\ targets that are most likely to host a planet. Between 8 and 15 volunteers have inspected each \\emph{TESS}\\ light curve and marked times of transit-like events using the PHT online interface. For each light curve, the markings from all the volunteers who saw that target were combined using an unsupervised machine learning method, known as DBSCAN, in order to identify likely transit-like events. Each of these identified events was given a transit score based on the number of volunteers who identified a given event and on the user weighting of each of those volunteers. Individual user weights were calculated based on the user's ability to identify simulated transit events, injected into real \\emph{TESS}\\ light curves, that are displayed on the PHT site alongside of the real data. The transit scores were then used to generate a ranked list of candidates that range from most likely to least likely to host a planet candidate. The top 500 highest ranked candidates were further vetted by the PHT science team. This stage of vetting primarily made use of the open source {\\sc latte} \\citep{LATTE2020} tool which generates a number of standard diagnostic plots that help identify promising candidates and weed out false positive signals. \n\nOn average we found around three high priority candidates per sector which were followed up using ground based telescopes, where possible. To date, PHT has statistically confirmed one planet, TOI-813 \\citep{2020eisner}: a Saturn-sized planet on an 84 day orbit around a subgiant host star. Other PHT identified planets listed in this paper are being followed up by other teams of astronomers, such as TOI-1899 (TIC 172370679) which was recently confirmed to be a warm Jupiter transiting an M-dwarf \\citep{canas2020}. The remaining candidates outlined in this paper require further follow-up observations to confirm their planetary nature.\n\nThe sensitivity of our transit search effort was assessed using synthetic data, as well as the known TOI and TCE candidates flagged by the SPOC pipeline. For simulated planets (where simulated signals are injected into real \\emph{TESS}\\ light curves) we have shown that the recovery efficiency of human vetting starts to decrease for transit-signals that have a SNR less than 7.5. The detection efficiency was further evaluated by the fractional recovery of the TOI and TCEs. We have shown that PHT is over 85 \\% complete in the recovery of planets that have a radius greater than 4 $R_{\\oplus}$, 51 \\% complete for radii between 3 and 4 $R_{\\oplus}$ and 49 \\% complete for radii between 2 and 3 $R_{\\oplus}$. Furthermore, we have shown that human vetting is not sensitive to the number of transits present in the light curve, meaning that they are equally likely to identify candidates on longer orbital periods as they are those with shorter orbital periods for periods greater than $\\sim$ 1 day. Planets with periods shorter than around 1 day exhibit over 20 transits within one \\emph{TESS}\\ sectors resulting in a decrease in identification by the volunteers. This is due to many volunteers only marking a random subset of these events, resulting in a lack of consensus on any given transit event and thus decreasing the overall transit score of these light curves. \n\nIn addition to searching for signals due to transiting exoplanets, PHT provides a platform that can be used to identify other stellar phenomena that may otherwise be difficult to identify with automated pipelines. Such phenomena, including eclipsing binaries, multiple stellar systems, dwarf novae, and stellar flares are often mentioned on the PHT discussion forums where volunteers can use searchable hashtags and comments to bring these systems to the attention of other citizen scientists as well as the PHT science team. All of the eclipsing binaries identified on the site, for example, are being used and vetted by the \\emph{TESS}\\ Eclipsing Binary Working Group (Prsa et al. in prep). Furthermore, we have investigated the nature of all of the targets that were identified as possible multiple stellar systems, as summarised in Table~\\ref{tab:PHT-multis}.\n\nOverall we have shown that large scale visual vetting can complement the findings \\textcolor{red}{from the major \\emph{TESS}\\ pipeline} by identifying longer period planets that may only exhibit a single transit event in their light curve, as well as in finding signals that are aperiodic or embedded in a strong varying stellar signal. The identification of planets around stars with variable signals allow us to potentially characterise the host-star (e.g., with asteroseismology or spot modulation). Additionally, the longer period planets are integral to our understanding of how planet systems form and evolve, as they allow us to investigate the evolution of planets that are farther away from their host star and therefore less dependent on stellar radiation. \\textcolor{red}{While automated pipelines specifically designed to identify single transit events in the \\emph{TESS}\\ data exist \\citep[e.g., ][]{Gill2020}, neither their methodology nor the full list of their findings are yet publicly available and thus we are unable to compare results.} \n\nThe planets that PHT finds have longer periods ($\\gtrsim$ 27 d) than those found in \\emph{TESS}\\ data using automated pipelines, and are more typical of the Kepler sample (25\\% of Kepler confirmed planets have periods greater than 27 days\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}). However, the Kepler planets are considerably fainter, and thus less amenable to ground-based follow-up or atmospheric characterisation from space (CHEOPS and JWST). Thus PHT helps to bridge the parameter spaces covered by these two missions, by identifying longer period planet candidates around bright, nearby stars, for which we can ultimately obtain precise planetary mass estimates. Although statistical characterisation of exo-planetary systems is no doubt important, precise mass measurements are key to developing our understanding of exoplanets and the physics which dictate their evolution. In particular, identification of this PHT sample provides follow-up targets to investigate the dependence of photo-evaporation on the mass of planets as well as on the planet radius, and will help our understanding of the photo-evaporation valley at longer orbital periods \\citep{Owen2013}. \n\nPHT will continue to operate throughout the \\emph{TESS}\\ extended mission, hopefully allowing us to identify even longer period planets as well as help verify some of the existing candidates with additional transits. \n\n\n\n\\begin{table*}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{cccccccccc}\n\\textbf{TIC} & \\textbf{Period (days)} & \\textbf{Epoch (\\textcolor{red}{BJD - 2457000})} & \\textbf{Depth (ppm)} & \\textbf{Comment} \\\\\n\\hline\n13968858 & $3.4850 \\pm 0.001$ & $ 1684.780 \\pm 0.005$ & 410000 & Candidate multiple system \\\\\n & $1.4380 \\pm 0.001$ & $ 1684.335 \\pm 0.005$ & 50000 & \\\\\n35655828 & $ 8.073 \\pm 0.01$ & $ 1550.94 \\pm 0.01 $ & 23000 & Confirmed blend \\\\\n & $ 1.220 \\pm 0.001 $ & $ 1545.540 \\pm 0.005 $ & 2800 & \\\\\n63291675 & $ 8.099 \\pm 0.003 $ & $ 1685.1 \\pm 0.01 $ & 60000 & Confirmed blend \\\\\n & $ 1.4635 \\pm 0.0005 $ & $ 1683.8 \\pm 0.1 $ & 7000 & \\\\\n63459761 & $4.3630 \\pm 0.003 $ & $ 1714.350 \\pm 0.005 $ & 160000 & Candidate multiple system \\\\\n & $4.235 \\pm $ 0.005 & $ 1715.130 \\pm 0.03$ & 35000 & \\\\\n104909909 & $1.3060 \\pm 0.0001$ & $ 1684.470 \\pm 0.005$ & 32000 & Candidate multiple system \\\\\n & $2.5750 \\pm 0.003$ & $ 1684.400 \\pm 0.005$ & 65000 & \\\\\n115980439 & $ 4.615 \\pm 0.002 $ & $ 1818.05 \\pm 0.01 $ & 95000 & Confirmed blend \\\\\n & $ 0.742 \\pm 0.005 $ & $ 1816.23 \\pm 0.02 $ & 2000 & \\\\\n120362128 & $ 3.286 \\pm 0.002 $ & $ 1684.425 \\pm 0.01 $ & 33000 & Candidate multiple system \\\\\n & $ - $ & $ 1701.275 \\pm 0.02 $ & 12000 & \\\\\n & $ - $ & $ 1702.09 \\pm 0.02 $ & 36000 & \\\\\n121945407 & $ 0.9056768 \\pm 0.00000002$ & $-1948.76377 \\pm 0.0000001$ & 2500 & Confirmed multiple system $^{(\\mathrm{a})}$ \\\\\n & $ 45.4711 \\pm 0.00002$ & $-1500.0038 \\pm 0.0004 $ & 7500 & \\\\\n122275115 & $ - $ & $ 1821.779 \\pm 0.01 $ & 155000 & Candidate multiple system \\\\\n & $ - $ & $ 1830.628 \\pm 0.01 $ & 63000 & \\\\\n & $ - $ & $ 1838.505 \\pm 0.01 $ & 123000 & \\\\\n229804573 & $1.4641 \\pm 0.0005$ & $ 1326.135 \\pm 0.005$ & 180000 & Candidate multiple system \\\\\n & $0.5283 \\pm 0.0001$ & $ 1378.114 \\pm 0.005$ & 9000 & \\\\\n252403752 & $ - $ & $ 1817.73 \\pm 0.01 $ & 2800 & Candidate multiple system \\\\\n & $ - $ & $ 1829.76 \\pm 0.01 $ & 23000 & \\\\\n & $ - $ & $ 1833.63 \\pm 0.01 $ & 5500 & \\\\\n258837989 & $0.8870 \\pm 0.001$ & $ 1599.350 \\pm 0.005$ & 64000 & Candidate multiple system \\\\\n & $3.0730 \\pm 0.001$ & $ 1598.430 \\pm 0.005$ & 25000 & \\\\\n266958963 & $1.5753 \\pm 0.0002$ & $ 1816.425 \\pm 0.001$ & 265000 & Candidate multiple system \\\\\n & $2.3685 \\pm 0.0001$ & $ 1817.790 \\pm 0.001$ & 75000 & \\\\\n278956474 & $5.488068 \\pm 0.000016 $ & $ 1355.400 \\pm 0.005$ & 93900 & Confirmed multiple system $^{(\\mathrm{b})}$ \\\\\n & $5.674256 \\pm \u22120.000030$ & $ 1330.690 \\pm 0.005$ & 30000 & \\\\\n284925600 & $ 1.24571 \\pm 0.00001 $ & $ 1765.248 \\pm 0.005 $ & 490000 & Confirmed blend \\\\\n & $ 0.31828 \\pm 0.00001 $ & $ 1764.75 \\pm 0.005 $ & 35000 & \\\\\n293954660 & $2.814 \\pm 0.001 $ & $ 1739.177 \\pm 0.03 $ & 272000 & Confirmed blend \\\\\n & $4.904 \\pm 0.03 $ & $ 1739.73 \\pm 0.01 $ & 9500 & \\\\\n312353805 & $4.951 \\pm 0.003 $ & $ 1817.73 \\pm 0.01 $ & 66000 & Confirmed blend \\\\\n & $12.89 \\pm 0.01 $ & $ 1822.28 \\pm 0.01$ & 19000 & \\\\\n318210930 & $ 1.3055432 \\pm 0.000000033$ & $ -653.21602 \\pm 0.0000013$ & 570000 & Confirmed multiple system $^{(\\mathrm{c})}$ \\\\\n & $ 0.22771622 \\pm 0.0000000035$& $ -732.071119 \\pm 0.00000026 $ & 220000 & \\\\\n336434532 & $ 3.888 \\pm 0.002 $ & $ 1713.66 \\pm 0.01 $ & 22900 & Confirmed blend \\\\\n & $ 0.949 \\pm 0.003 $ & $ 1712.81 \\pm 0.01 $ & 2900 & \\\\\n350622185 & $1.1686 \\pm 0.0001$ & $ 1326.140 \\pm 0.005$ & 200000 & Candidate multiple system \\\\\n & $5.2410 \\pm 0.0005$ & $ 1326.885 \\pm 0.05$ & 4000 & \\\\\n375422201 & $9.9649 \\pm 0.001$ & $ 1711.937 \\pm 0.005$ & 245000 & Candidate multiple system \\\\\n & $4.0750 \\pm 0.001$ & $ 1713.210 \\pm 0.01 $ & 39000 & \\\\\n376606423 & $ 0.8547 \\pm 0.0002 $ & $ 1900.766 \\pm 0.005 $ & 9700 & Candidate multiple system \\\\\n & $ - $ & $ 1908.085 \\pm 0.01 $ & 33000 & \\\\\n394177355 & $ 94.22454 \\pm 0.00040 $ & $ - $ & - & Confirmed multiple system $^{(\\mathrm{d})}$ \\\\\n & $ 8.6530941 \\pm 0.0000016$ & $-2038.99492 \\pm 0.00017 $ & 140000 & \\\\\n & $ 1.5222468 \\pm 0.0000025$ & $ -2039.1201 \\pm 0.0014 $ & - & \\\\\n & $ 1.43420486 \\pm 0.00000012 $ & $-2039.23941 \\pm 0.00007 $ & - & \\\\\n424508303 & $ 2.0832649 \\pm 0.0000029 $ & $-3144.8661 \\pm 0.0034 $ & 430000 & Confirmed multiple system $^{(\\mathrm{e})}$ \\\\\n & $ 1.4200401 \\pm 0.0000042 $ & $-3142.5639 \\pm 0.0054 $ & 250000 & \\\\\n441794509 & $ 4.6687 \\pm 0.0002 $ & $ 1958.895 \\pm 0.005 $ & 34000 & Candidate multiple system \\\\\n & $ 14.785 \\pm 0.002 $ & $ 1960.845 \\pm 0.005 $ & 17000 & \\\\\n470710327 & $ 9.9733 \\pm 0.0001 $ & $ 1766.27 \\pm 0.005 $ & 51000 & Confirmed multiple system $^{(\\mathrm{f})}$ \\\\\n & $ 1.104686 \\pm 0.00001 $ & $ 1785.53266 \\pm 0.000005$ & 42000 & \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\nNote -- $^{(\\mathrm{a})}$ KOI-6139, \\citet{Borkovits2013}; \n$^{(\\mathrm{b})}$ \\citet{2020Rowden}\n$^{(\\mathrm{c})}$ \\citet{Koo2014}; \n$^{(\\mathrm{d})}$ KOI-3156, \\citet{2017Helminiak};\n$^{(\\mathrm{e})}$ V994 Her; \\citet{Zasche2016}; \n$^{(\\mathrm{f})}$ Eisner et al. {\\it in prep.}\n}\n\n\\label{tab:PHT-multis}\n\n\\end{table*}\n\n\\section*{Data Availability}\n\nAll of the \\emph{TESS}\\ data used within this article are hosted and made publicly available by the Mikulski Archive for Space Telescopes (MAST, \\url{http:\/\/archive.stsci.edu\/tess\/}). Similarly, the Planet Hunters TESS classifications made by the citizen scientists can be found on the Planet Hunters Analysis Database (PHAD, \\url{https:\/\/mast.stsci.edu\/phad\/}), which is also hosted by MAST. All planet candidates and their properties presented in this article have been uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS, \\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}) website as community TOIs (cTOIs), under their corresponding TIC IDs. The ground-based follow-up observations of individual targets will be shared on reasonable request to the corresponding author.\n\nThe models of individual transit events and the data validation reports used for the vetting of the targets were both generated using publicly available open software codes, \\texttt{pyaneti}\\ and {\\sc latte}.\n\n\\section*{Acknowledgements} \n\nThis project works under the in \\textit{populum veritas est} philosophy, and for that reason we would like to thank all of the citizen scientists who have taken part in the Planet Hunters TESS project and enable us to find many interesting astrophysical systems. \n\nSome of the observations in the paper made use of the High-Resolution Imaging instruments `Alopeke and Zorro. `Alopeke and Zorro were funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. `Alopeke and Zorro were mounted on the Gemini North and South telescope of the international Gemini Observatory, a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci\\'{o}n y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog\\'{i}a e Innovaci\\'{o}n (Argentina), Minist\\'{e}rio da Ci\\^{e}ncia, Tecnologia, Inova\\c{c}\\~{o}es e Comunica\\c{c}\\~{o}es (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). The authors also acknowledge the very significant cultural role and sacred nature of Maunakea. We are most fortunate to have the opportunity to conduct observations from this mountain.\n\nThis project has also received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement N$^\\circ$730890. This material reflects only the authors views and the Commission is not liable for any use that may be made of the information contained therein. This work makes use of observations from the Las Cumbres Observatory global telescope network, including the NRES spectrograph and the SBIG and Sinistro photometric instruments. \n\nFurthermore, NLE thanks the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant N$^\\circ$1829740, the Brinson Foundation, and the Moore Foundation; her participation in the program has benefited this work. Finally, CJ acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement N$^\\circ$670519: MAMSIE), and from the Research Foundation Flanders (FWO) under grant agreement G0A2917N (BlackGEM). \n\nThis research made use of Astropy, a community-developed core Python package for Astronomy \\citep{astropy2013}, matplotlib \\citep{matplotlib}, pandas \\citep{pandas}, NumPy \\citep{numpy}, astroquery \\citep{ginsburg2019astroquery} and sklearn \\citep{pedregosa2011scikit}. \n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Introduction}\n\nSince the first unambiguous discovery of an exoplanet in 1995 \\citep[][]{Mayor1995} over 4,000 more have been confirmed. Studies of their characteristics have unveiled an extremely wide range of planetary properties in terms of planetary mass, size, system architecture and orbital periods, greatly revolutionising our understanding of how these bodies form and evolve.\n\nThe transit method, whereby we observe a temporary decrease in the brightness of a star due to a planet passing in front of its host star, is to date the most successful method for planet detection, having discovered over 75\\% of the planets listed on the NASA Exoplanet Archive\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}. It yields a wealth of information including planet radius, orbital period, system orientation and potentially even atmospheric composition. Furthermore, when combined with Radial Velocity \\citep[RV; e.g.,][]{Mayor1995, Marcy1997} observations, which yield the planetary mass, we can infer planet densities, and thus their internal bulk compositions. Other indirect detection methods include radio pulsar timing \\citep[e.g.,][]{Wolszczan1992} and microlensing \\citep[e.g.,][]{Gaudi2012}.\n\n\nThe \\textit{Transiting Exoplanet Survey Satellite} mission \\citep[\\protect\\emph{TESS};][]{ricker15} is currently in its extended mission, searching for transiting planets orbiting bright ($V < 11$\\,mag) nearby stars. Over the course of the two year nominal mission, \\emph{TESS}\\ monitored around 85 per cent of the sky, split up into 26 rectangular sectors of 96 $\\times$ 24 deg each (13 per hemisphere). Each sector is monitored for $\\approx$ 27.4 continuous days, measuring the brightness of $\\approx$ 20,000 pre-selected stars every two minutes. In addition to these short cadence (SC) observations, the \\emph{TESS}\\ mission provides Full Frame Images (FFI) that span across all pixels of all CCDs and are taken at a cadence of 30 minutes. While most of the targets ($\\sim$ 63 per cent) will be observed for $\\approx$ 27.4 continuous days, around $\\sim$ 2 per cent of the targets at the ecliptic poles are located in the `continuous viewing zones' and will be continuously monitored for $\\sim$ 356 days.\n\nStars themselves are extremely complex, with phenomena ranging from outbursts to long and short term variability and oscillations, which manifest themselves in the light curves. These signals, as well as systematic effects and artifacts introduced by the telescope and instruments, mean that standard periodic search methods, such as the Box-Least-Squared method \\citep{bls2002} can struggle to identify certain transit events, especially if the observed signal is dominated by natural stellar variability. Standard detection pipelines also tend to bias the detection of short period planets, as they typically require a minimum of two transit events in order to gain the signal-to-noise ratio (SNR) required for detection.\n\nOne of the prime science goals of the \\emph{TESS}\\ mission is to further our understanding of the overall planet population, an active area of research that is strongly affected by observational and detection biases. In order for exoplanet population studies to be able to draw meaningful conclusions, they require a certain level of completeness in the sample of known exoplanets as well as a robust sample of validated planets spanning a wide range of parameter space. \\textcolor{black}{Due to this, we independently search the \\emph{TESS}\\ light curves for transiting planets via visual vetting in order to detect candidates that were either intentionally ignored by the main \\emph{TESS}\\ pipelines, which require at least two transits for a detection, missed because of stellar variability or instrumental artefacts, or were identified but subsequently erroneously discounted at the vetting stage, usually because the period found by the pipeline was incorrect. These candidates can help populate under-explored regions of parameter space and will, for example, benefit the study of planet occurrence rates around different stellar types as well as inform theories of physical processes involved with the formation and evolution of different types of exoplanets.}\n\nHuman brains excel in activities related to pattern recognition, making the task of identifying transiting events in light curves, even when the pattern is in the midst of a strong varying signal, ideally suited for visual vetting. Early citizen science projects, such as Planet Hunters \\citep[PH;][]{fischer12} and Exoplanet Explorers \\citep{Christiansen2018}, successfully harnessed the analytic power of a large number of volunteers and made substantial contributions to the field of exoplanet discoveries. The PH project, for example, showed that human vetting has a higher detection efficiency than automated detection algorithms for certain types of transits. In particular, they showed that citizen science can outperform on the detection of single (long-period) transits \\citep[e.g.,][]{wang13, schmitt14a}, aperiodic transits \\citep[e.g. circumbinary planets;][]{schwamb13} and planets around variable stars \\citep[e.g., young systems,][]{fischer12}. Both PH and Exoplanet Explorers, which are hosted by the world's largest citizen science platform Zooniverse \\citep{lintott08}, ensured easy access to \\textit{Kepler} and \\textit{K2} data by making them publicly available online in an immediately accessible graphical format that is easy to understand for non-specialists. The popularity of these projects is reflected in the number of participants, with PH attracting 144,466 volunteers from 137 different countries over 9 years of the project being active.\n\nFollowing the end of the \\textit{Kepler} mission and the launch of the \\emph{TESS}\\ satellite in 2018, PH was relaunched as the new citizen science project \\textit{Planet Hunters TESS} (PHT) \\footnote{\\url{www.planethunters.org}}, with the aim of identifying transit events in the \\emph{TESS}\\ data that were \\textcolor{black}{intentionally ignored or missed} by the main \\emph{TESS}\\ pipelines. \\textcolor{black}{Such a search complements other methods methods via its sensitivity to single-transit, and, therefore, longer period planets. Additionally, other dedicated non-citizen science based methods are also employed to look for single transit candidates \\citep[see e.g., the Bayesian transit fitting method by ][]{Gill2020, Osborn2016}}.\n\nCitizen science transit searches specialise in finding the rare events that the standard detection pipelines miss, however, these results are of limited use without an indication of the completeness of the search. Addressing the problem of completeness was therefore one of our highest priorities while designing PHT as discussed throughout this paper. \n\nThe layout of the remainder of the paper will be as follows. An overview of the Planet Hunters TESS project is found in Section~\\ref{sec:PHT}, followed by an in depth description of how the project identifies planet candidates in Section~\\ref{sec:method}. The recovery efficiency of the citizen science approach is assessed in Section~\\ref{sec:recovery_efficiency}, followed by a description of the in-depth vetting of candidates and ground based-follow up efforts in Section~\\ref{sec:vetting} and \\ref{sec:follow_up}, respectively. Planet Candidates and noteworthy systems identified by Planet Hunters TESS are outlined in Section~\\ref{sec:PHT_canidates}, followed by a discussion of the results in Section~\\ref{sec:condlusion}.\n\n\\section{Planet Hunters TESS}\n\\label{sec:PHT}\n\nThe PHT project works by displaying \\emph{TESS}\\ light curves (Figure~\\ref{fig:interface}), and asking volunteers to identify transit-like signals. Only the two-minute cadence targets, which are produced by the \\emph{TESS}\\ pipeline at the Science Processing Operations Center \\citep[SPOC,][]{Jenkins2018} and made publicly available by the Mikulski Archive for Space Telescopes (MAST)\\footnote{\\url{http:\/\/archive.stsci.edu\/tess\/}}, are searched by PHT. First-time visitors to the PHT site, or returning visitors who have not logged in are prompted to look through a short tutorial, which briefly explains the main aim of the project and shows examples of transit events and other stellar phenomena. Scientific explanation of the project can be found elsewhere on the site in the `field guide' and on the project's `About' page. \n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=\\textwidth]{Figures\/PHT_new_interface.png}\n \\caption{\n PHT user interface showing a simulated light curve. The transit events are highlighted with white partially-transparent columns that are drawn on using the mouse. Stellar information on the target star is available by clicking on `subject info' below the light curve.} \n \\label{fig:interface}\n\\end{figure*}\n\nAfter viewing the tutorial, volunteers are ready to participate in the project and are presented with \\emph{TESS}\\ light curves (known as `subjects') that need to be classified. The project was designed to be as simple as possible and therefore only asks one question: \\textit{`Do you see a transit?}'. Users identify transit-like events, and the time of their occurrence, by drawing a column over the event using the mouse button, as shown in Figure~\\ref{fig:interface}. There is no limit on the number of transit-like events that can be marked in a light curve. No markings indicate that there are no transit-like events present in the light curve. Once the subject has been analysed, users submit their classification and continue to view the next light curve by clicking `Done'. \n\nAlongside each light curve, users are offered information on the stellar properties of the target, such as the radius, effective temperature and magnitude (subject to availability, see \\cite{Stassun18}). However, in order to reduce biases in the classifications, the TESS Input Catalog (TIC) ID of the target star is not provided until after the subject classification has been submitted.\n\nIn addition to classifying the data, users are given the option to comment on light curves via the `Talk' discussion forum. Each light curve has its own discussion page to allow volunteers to discuss and comment, as well as to `tag' light curves using searchable hashtags, and to bring promising candidates to the attention of other users and the research team. The talk discussion forums complement the main PHT analysis and have been shown to yield interesting objects which may be challenging to detect using automated algorithms \\citep[e.g.,][]{eisner2019RN}. Unlike in the initial PH project, there are no questions in the main interface regarding stellar variability, however, volunteers are encouraged to mention astrophysical phenomenon or \\textit{unusual} features, such as eclipsing binaries or stellar flares, using the `Talk' discussion forum. \n\nThe subject TIC IDs are revealed on the subject discussion pages, allowing volunteers to carry out further analysis on specific targets of interest and to report and discuss their findings. This is extremely valuable for both other volunteers and the PHT science team, as it can speed up the process of identifying candidates as well as rule out false positives in a fast and effective manner. \n\nSince the launch of PHT on 6 December 2018, there has been one significant makeover to the user interface. The initial PHT user interface (UI1), which was used for sectors 1 through 9, split the \\emph{TESS}\\ light curves up into either three or four chunks (depending on the data gaps in each sector) which lasted around seven days each. This allowed for a more `zoomed' in view of the data, making it easier to identify transit-like events than when the full $\\sim$ 30 day light curves were shown. The results from a PHT beta project, which displayed only simulated data, showed that a more zoomed in view of the light curve was likely to yield a higher transit recovery rate.\n\nThe updated, and current, user interface (UI2) allows users to manually zoom in on the x-axis (time) of the data. Due to this additional feature, each target has been displayed as a single light curve as of Sector 10. In order to verify that the changes in interface did not affect our findings, all of the Sector 9 subjects were classified using both UI1 and UI2. We saw no significant change in the number of candidates recovered (see Section~\\ref{sec:recovery_efficiency} for a description of how we quantified detection efficiency).\n\n\n\\subsection{Simulated Data}\n\\label{subsec:sims} \n\nIn addition to the real data, volunteers are shown simulated light curves, which are generated by randomly injecting simulated transit signals, provided by the SPOC pipeline \\citep[][]{Jenkins2018}, into real \\emph{TESS}\\ light curves. The simulated data play an important role in assessing the sensitivity of the project, training the users and providing immediate feedback, and to gauge the relative abilities of individual users (see Sec~\\ref{subsec:weighting}). \n\nWe calculate a signal to noise ratio (SNR) of the injected signal by dividing the injected transit depth by the Root Mean Square Combined Differential Photometric Precision (RMS CDPP) of the light curve on 0.5-, 1- or 2-hr time scales (whichever is closest to the duration of the injected transit signal). Only simulations with a SNR greater than 7 in UI1 and greater than 4 for UI2 are shown to volunteers.\n\nSimulated light curves are randomly shown to the volunteers and classified in the exact same manner as the real data. The user is always notified after a simulated light curve has been classified and given feedback as to whether the injected signal was correctly identified or not. For each sector, we generate between one and two thousand simulated light curves, using the real data from that sector in order to ensure that the sector specific systematic effects and data gaps of the simulated data do not differ from the real data. The rate at which a volunteer is shown simulated light curves decreases from an initial rate of 30 per cent for the first 10 classifications, down to a rate of 1 per cent by the time that the user has classified 100 light curves. \n\n\n\\section{Identifying Candidates}\n\\label{sec:method}\n\nEach subject is seen by multiple volunteers, before it is `retired' from the site, and the classifications are combined (see Section~\\ref{subsec:DBscan}) in order to assess the likelihood of a transit event. For sectors 1 through 9, the subjects were retired after 8 classifications if the first 8 volunteers who saw the light curves did not mark any transit events, after 10 classifications if the first 10 volunteers all marked a transit event and after 15 classifications if there was not complete consensus amongst the users. As of Sector 9 with UI2, all subjects were classified by 15 volunteers, regardless of whether or not any transit-like events were marked. Sector 9, which was classified with both UI1 and UI2, was also classified with both retirement rules.\n\nThere were a total of 12,617,038 individual classifications completed across the project on the nominal mission data. 95.4 per cent of these classifications were made by 22,341 registered volunteers, with the rest made by unregistered volunteers. Around 25 per cent of the registered volunteers complete more than 100 classifications, 11.8 per cent more than 300, 8.4 per cent more than 500, 5.4 per cent more than 1000 and 1.1 per cent more than 10,000. The registered volunteers completed a mean and median of 647 and 33 classifications, respectively. Figure~\\ref{fig:user_count} shows the distribution in user effort for logged in users who made between 0 and 300 classifications. \n\nThe distribution in the number of classifications made by the registered volunteers is assessed using the Gini coefficient, which ranges from 0 (equal contributions from all users) to 1 (large disparity in the contributions). The Gini coefficients for individual sectors ranges from 0.84 to 0.91 with a mean of 0.87, while the Gini coefficient for the overall project (all of the sectors combined) is 0.94. The mean Gini coefficient among other astronomy Zooniverse projects lies at 0.82 \\citep{spiers2019}. We note that the only other Zooniverse project with an equally high Gini coefficient as PHT is \\textit{Supernova Hunters}, a project which, similarly to PHT and unlike most other Zooniverse projects, has periodic data releases that are accompanied by an e-newsletter sent to all project volunteers. Periodic e-newsletters have the effect of promoting the project to both regularly and irregularly participating volunteers, who may only complete a couple of classifications as they explore the task, as well as to returning users who complete a large number of classifications following every data release, increasing the disparity in user contributions (the Gini coefficient).\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.5\\textwidth]{Figures\/user_count.png}\n \\caption{\n The distribution of the number of classifications by the registered volunteers, using a bin size of 5 from 0 to 300 classifications. A total of 11.8 per cent of the registered volunteers completed more than 300 classifications.} \n \\label{fig:user_count}\n\\end{figure}\n\n\n\\subsection{User Weighting}\n\\label{subsec:weighting} \nUser weights are calculated for each individual volunteer in order to identify users who are more sensitive to detecting transit-like signals and those who are more likely to mark false positives. The weighting scheme is based on the weighting scheme described by \\cite{schwamb12}.\n\nUser weights are calculated independently for each observation sector, using the simulated light curves shown alongside the data from that sector. All users start off with a weighting of one, which is then increased or decreased when a simulated transit event is correctly or incorrectly identified, respectively. \n\nSimulated transits are deemed correctly identified, or `True', if the mid-point of a user's marking falls within the width of the simulated transit events. If none of the user's markings fall within this range, the simulated transit is deemed not identified, or `False'. If more than one of a user's markings coincide with the same simulated signal, it is only counted as being correct once, such that the total number of `True' markings cannot exceed the number of injected signals. For each classification, we record the number of `Extra' markings, which is the total number of markings made by the user minus the number of correctly identified simulated transits. \n\nEach simulated light curve, identified by superscript $i$ (where $i=1$, \\ldots, $N$) was seen by $K^{(i)}$ users (the mean value of $K^{(i)}$\nwas 10), and contained $T^{(i)}$ simulated transits (where $T^{(i)}$ depends on the period of the simulated transit signal and the duration of the light curve). For a specific light curve $i$, each user who saw the light curve is identified by a subscript $k$ (where $k=1$, \\ldots, $K^{(i)}$) and each injected transit by a subscript $t$ (where $t=1$, \\ldots, $T^{(i)}$). \n\nIn order to distinguish between users who are able to identify obvious transits and those who are also able to find those that are more difficult to see, we start by defining a `recoverability' $r^{(i)}_t$ for each injected transit $t$ in each light curve. This is defined empirically, as the number of users who identified the transit correctly divided by $K^{(i)}$ (the total number of users who saw the light curve in question).\n\nNext, we quantify the performance of each user on each light curve as follows (this performance is analogous to the `seed' defined in \\citealt{schwamb12}, but we define it slightly differently):\n\\begin{equation}\n p^{(i)}_{k} = C_{\\rm E} ~ \\frac{E^{(i)}_{k}}{\\langle E^{(i)} \\rangle} + \\sum_{t=1}^{T^{(i)}} \\begin{cases}\n C_{\\rm T} ~ \\left[ r^{(i)}_t \\right]^{-1}, & \\text{if $m^{(i)}_{t,k} = $`True'}\\\\\n C_{\\rm F} ~ r^{(i)}_t, & \\text{if $m^{(i)}_{t,k} = $`False'},\n \\end{cases}\n\\end{equation}\nwhere $m^{(i)}_{t,k}$ is the identification of transit $t$ by user $k$ in light curve $i$, which is either `True' or `False'; $E^{(i)}_{k}$ is the number of `Extra' markings made by user $k$ for light curve $i$, and $\\langle E^{(i)} \\rangle$ is the mean number of `Extra' markings made by all users who saw subject $i$. The parameters $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ control the impact of the `Extra', `True' and `False' markings on the overall user weightings, and are optimized empirically as discussed below in Section~\\ref{subsec:optimizesearch}. \n\nFollowing \\citealt{schwamb12}, we then assign a global `weight' $w_k$ to each user $k$, which is defined as:\n\\begin{equation}\n\\begin{split}\n\tw_k = I \\times (1 + \\log_{10} N_k)^{\\nicefrac{\\sum_i p^{(i)}_k}{N_k}}\n\\label{equ:weight}\n\\end{split}\n\\end{equation}\nwhere $I$ is an empirical normalization factor, such that the distribution of user weights remains centred on one, $N_k$ is the total number of simulated transit events that user $k$ assessed, and the sum over $i$ concerns only the light curves that user $k$ saw. \nWe limit the user weights to the range 0.05--3 \\emph{a posteriori}.\n\n\nWe experimented with a number of alternative ways to define the user weights, including the simpler $w_k=\\nicefrac{\\sum_i p^{(i)}_k}{N_k}$, but Eqn.~\\ref{equ:weight} was found to give the best results (see Section~\\ref{sec:recovery_efficiency} for how this was evaluated).\n\n\\subsection{Systematic Removal}\n\\label{subsec:sysrem} \nSystematic effects, for example caused by the spacecraft or background events, can result in spurious signals that affect a large subset of the data, resulting in an excess in markings of transit-like events at certain times within an observation sector. As the four \\emph{TESS}\\ cameras can yield unique systematic effects, the times of systematics were identified uniquely for each camera. The times were identified using a Kernel Density Estimation \\citep[KDE;][]{rosenblatt1956} with a cosine kernel and a bandwidth of 0.1 days, applied across all of the markings from that sector for each camera. Fig.~\\ref{fig:sys_rem} shows the KDE of all marked transit-events made during Sector 17 for TESS's cameras 1 (top panel) to 4 (bottom panel). The isolated spikes, or prominences, in the number of marked events, such as at T = 21-22 days in the bottom panel, are assumed to be caused by systematic effects that affect multiple light curves. Prominences are considered significant if they exceed a factor four times the standard deviation of the kernel output, which was empirically determined to be the highest cut-off to not miss clearly visible systematics. All user-markings within the full width at half maximum of these peaks are omitted from all further analysis. \\textcolor{black}{The KDE profiles for each Sector are provided as electronic supplementary material.}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.46\\textwidth]{Figures\/systematics_sec17.png}\n \\caption{\n Kernel density estimation of the user-markings made for Sector 17, for targets observed with TESS's observational Cameras 1 (top panel) to 4 (bottom panel). The orange vertical lines the indicate prominences that are at least four times greater than the standard deviation of the distribution. The black points underneath the figures show the mid-points of all of the volunteer-markings, where darker regions represent a higher density of markings.}\n \\label{fig:sys_rem}\n\\end{figure}\n\n\\subsection{Density Based Clustering}\n\\label{subsec:DBscan} \n\nThe times and likelihoods of transit-like events are determined by combining all of the classifications made for each subject and identifying times where multiple volunteers identified a signal. We do this using an unsupervised machine learning method, known as DBSCAN \\citep[][Density-Based Spatial Clustering of Applications with Noise]{ester1996DB}. DBSCAN is a non-parametric density based clustering algorithm that helps to distinguish between dense clusters of data and sparse noise. For a data point to belong to a cluster it must be closer than a given distance ($\\epsilon$) to at least a set minimum number of other points (minPoints). \n\nIn our case, the data points are one-dimensional arrays of times of transits events, as identified by the volunteers, and clusters are times where multiple volunteers identified the same event. For each cluster a `transit score' ($s_i$) is determined, which is the sum of the user weights of the volunteers who contribute to the given cluster divided by the sum of the user weights of volunteers who saw that light curve. These transit scores are used to rank subjects from most to least likely to contain a transit-like event. Subjects which contain multiple successful clusters with different scores are ranked by the highest transit score. \n\n\\subsection{Optimizing the search}\n\\label{subsec:optimizesearch}\n\nThe methodology described in Sections~\\ref{subsec:weighting} to \\ref{subsec:DBscan} has five free parameters: the number of markings required to constitute a cluster ($minPoints$), the maximum separation of markings required for members of a cluster ($\\epsilon$), and $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ used in the weighting scheme. The values of these parameters were optimized via a grid search, where $C_{\\rm E}$ and $C_{\\rm F}$ ranged from -5 to 0, $C_{\\rm T}$ ranged from 0 to 20, and $minPoints$ ranged from 1 to 8, all in steps of 1. ($\\epsilon$) ranged from 0.5 to 1.5 in steps of 0.5. This grid search was carried out on 4 sectors, two from UI1 and two from UI2, for various variations of Equation~\\ref{equ:weight}. \n\nThe success of each combination of parameters was assessed by the fractions of TOIs and TCEs that were recovered within the top highest ranked 500 candidates, as discussed in more detail Section~\\ref{sec:recovery_efficiency}. We found the most successful combination of parameters to be $minPoints$ = 4 markings, $\\epsilon$, = 1 day, $C_{\\rm T}$ = 3, $C_{\\rm F}$= -2 and $C_{\\rm E}$ = -2.\n\n\\subsection{MAST deliverables}\n\\label{subsec:deliverables}\n\nThe analysis described above is carried out both in real-time as classifications are made, as well as offline after all of the light curves of a given sector have been classified. When the real-time analysis identifies a successful DB cluster (i.e. when at least four citizen scientists identified a transit within a day of the \\emph{TESS}\\ data of one another), the potential candidate is automatically uploaded to the open access Planet Hunters Analysis Database (PHAD) \\footnote{\\url{https:\/\/mast.stsci.edu\/phad\/}} hosted by the Mikulski Archive for Space Telescopes (MAST) \\footnote{\\url{https:\/\/archive.stsci.edu\/}}. While PHAD does not list every single classification made on PHT, it does display all transit candidates which had significant consensus amongst the volunteers who saw that light curve, along with the user-weight-weighted transit scores. This analysis does not apply the systematics removal described in Section~\\ref{subsec:sysrem}. The aim of PHAD is to provide an open source database of potential planet candidates identified by PHT, and to credit the volunteers who identified said targets. \n\nThe offline analysis is carried out following the complete classifications of all of the data from a given \\emph{TESS}\\ sector. The combination of all of the classifications allows us to identify and remove times of systematics and calculate better calibrated and more representative user weights. The remainder of this paper will only discuss the results from the offline analysis.\n\n\\section{Recovery Efficiency}\n\\label{sec:recovery_efficiency}\n\\subsection{Recovery of simulated transits}\n\nThe recovery efficiency is, in part, assessed by analysing the recovery rate of the injected transit-like signals (see Section~\\ref{subsec:sims}). Figure~\\ref{fig:SIM_recovery} shows the median and mean transit scores (fraction of volunteers who correctly identified a given transit scaled by user weights) of the simulated transits within SNR bins ranging from 4 to 20 in steps of 0.5. Simulations with a SNR less than 4 were not shown on PHT. The figure highlights that transit signals with a SNR of 7.5 or greater are correctly identified by the vast majority of volunteers. \n\n\\textcolor{black}{As the simulated data solely consist of real light curves with synthetically injected transit signals, we do not have any light curves, simulated or otherwise, which we can guarantee do not contain any planetary transits (real or injected). As such, this prohibits us from using simulated data to infer an analogous false-positive rate.}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.47\\textwidth]{Figures\/SIMS_recovery.png}\n \\caption{The median (blue) and mean (orange) transit scores for injected transits with SNR ranges between 4 and 20. The mean and median are calculated in SNR bins with a width of 0.5, as indicated by the horizontal lines around each data point. \n }\n \\label{fig:SIM_recovery}\n\\end{figure}\n\n\\subsection{Recovery of TCEs and TOIs}\n\\label{subsec:TCE_TOI}\nThe recovery efficiency of PHT is assessed further using the planet candidates identified by the SPOC pipeline \\citep{Jenkins2018}. The SPOC pipeline extracts and processes all of the 2-minute cadence \\emph{TESS}\\ light curves prior to performing a large scale transit search. Data Validation (DV) reports, which include a range of transit diagnostic tests, are generated by the pipeline for around 1250 Threshold Crossing Events (TCEs), which were flagged as containing two or more transit-like features. Visual vetting is then performed by the \\emph{TESS}\\ science team on these targets, and promising candidates are added to the catalog of \\emph{TESS}\\ Objects of Interest (TOIs). Each sector yields around 80 TOIs \\textcolor{black}{and a mean of 1025 TCEs.}\n\nFig~\\ref{fig:TCE_TOI_recovery} shows the fraction of TOIs and TCEs (top and bottom panel respectively) that we recover with PHT as a function of the rank, where a higher rank corresponds to a lower transit score, for Sectors 1 through 26. TOIs and TCEs with R < 2 $R_{\\oplus}$ are not included in this analysis, as the initial PH showed that human vetting alone is unable to reliably recover planets smaller than 2 $R_{\\oplus}$ \\citep{schwamb12}. Planets smaller than 2 $R_{\\oplus}$ are, therefore, not the main focus of our search.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI-recovery_radlim2.png}\n \\caption{The fraction of recovered TOIs and TCEs (top and bottom panel respectively) with R > 2$R_{\\oplus}$ as a function of the rank, for sectors 1 to 26. The lines represent the results from different observation sectors.}\n \\label{fig:TCE_TOI_recovery}\n\\end{figure}\n\n\nFig~\\ref{fig:TCE_TOI_recovery} shows a steep increase in the fractional TOI recovery rate up to a rank of $\\sim$ 500. Within the 500 highest ranked PHT candidates for a given sector, we are able to recover between 46 and 62 \\% (mean of 53 \\%) of all of the TOIs (R > 2 $R_{\\oplus}$), a median 90 \\% of the TOIs where the SNR of the transit events are greater than 7.5 and median 88 \\% of TOIs where the SNR of the transit events are greater than 5.\n\nThe relation between planet recovery rate and the SNR of the transit events is further highlighted in Figure~\\ref{fig:TOI_properties}, which shows the SNR vs the orbital period of the recovered TOIs. The colour of the markers indicate the TOI's rank within a given sector, with the lighter colours representing a lower rank. The circles and crosses represent candidates at a rank lower and higher than 500, respectively. The figure shows that transit events with a SNR less than 3.5 are missed by the majority of volunteers, whereas events with a SNR greater than 5 are mostly recovered within the top 500 highest ranked candidates. \n\nThe steep increase in the fractional TOI recovery rate at lower ranks, as shown in figure~\\ref{fig:TCE_TOI_recovery}, is therefore due to the detection of the high SNR candidates that are identified by most, if not all, of the PHT volunteers who classified those targets. At a rank of around 500, the SNR of the TOIs tends towards the limit of what human vetting can detect and thus the identification of TOIs beyond a rank of 500 is more sporadic.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.49\\textwidth]{Figures\/TOI_recovery_properties.png}\n \\caption{The SNR vs orbital period of TOIs with R > 2$R_{\\oplus}$. The colour represents their rank within the sector, as determined by the weighted DB clustering algorithm. Circles indicate that they were identified at a rank < 500, while crosses indicate that they were not within the top 500 highest ranked candidates of a given sector.\n }\n \\label{fig:TOI_properties}\n\\end{figure}\n\nThe fractional TCE recovery rate (bottom panel of Figure~\\ref{fig:TCE_TOI_recovery}) is systematically lower than that of the TOIs. There are qualitative reasons as to why humans might not identify a TCE as opposed to a TOI, including that TCEs may be caused by artefacts or periodic stellar signals that the SPOC pipeline identified as a potential transit but that the human eye would either miss or be able to rule out as systematic effect. This leads to a lower recovery fraction of TCEs comparatively, an effect that is further amplified by the much larger number of TCEs.\n\nThe detection efficiency of PHT is estimated using the fractional recovery rate of TOIs for a range of radius and period bins, as shown in Figure~\\ref{fig:recovery_rank500_radius_period}. A TOI is considered to be recovered if its detection rank is less than 500 within the given sector. Out of the total 1913 TOIs, to date, \\textcolor{black}{PHT recovered 715 TOIs among the highest ranked candidates across the 26 sectors. This corresponds to a mean of 12.7~\\% of the top 500 ranked candidates per sector being TOIs. In comparison, the primary \\emph{TESS}\\ team on average visually vets 1025 TCEs per sector, out of which a mean of 17.3~\\% are promoted to TOI status.} We find that, independent of the orbital period, PHT is over 85~\\% complete in the recovery of TOIs with radii equal to or greater than 4 $R_{\\oplus}$. This agrees with the findings from the initial Planet Hunters project \\citep{schwamb12}. The detection efficiency decreases to 51~\\% for 3 - 4 $R_{\\oplus}$ TOIs, 49~\\% for 2 - 3 $R_{\\oplus}$ TOIs and to less than 40~\\% for TOIs with radii less than 2 $R_{\\oplus}$. Fig~\\ref{fig:recovery_rank500_radius_period} shows that the orbital period does not have a strong effect on the detection efficiency for periods greater than $\\sim$~1~day, which highlights that human vetting efficiency is independent of the number of transits present within a light curve. For periods shorter than around 1~day, the detection efficiency decreases even for larger planets, due to the high frequency of events seen in the light curve. For these light curves, many volunteers will only mark a subset of the transits, which may not overlap with the subset marked by other volunteers. Due to the methodology used to identify and rank the candidates, as described in Section~\\ref{sec:method}, this will actively disfavour the recovery of very short period planets. Although this obviously introduces biases in the detectability of very short period signals, the major detection pipelines are specifically designed to identify these types of planets and thus this does not present a serious detriment to our main science goal of finding planets that were \\textcolor{black}{intentionally ignored or missed} by the main automated pipelines.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{Figures\/TOI_recovery_grid.png}\n \\caption{TOI recovery rate as a function of planet radius and orbital period. A TOI is considered recovered if it is amongst the top 500 highest ranked candidates within a given sector. The logarithmically spaced grid ranges from 0.2 to 225 d and 0.6 to 55 $R{_\\oplus}$ for the orbital period and planet radius, respectively. The fraction of TOIs recovered using PHT is computed for each cell and represented by the colour the grid. Cells with less than 10 TOIs are considered incomplete for statistical analysis and are shown by the hatched lines. White cells contain no TOIs. The annotations for each cell indicate the number of recovered TOIs followed by the Poisson uncertainty in brackets. The filled in and empty grey circles indicated the recovered and not-recovered TOIs, respectively.}\n \\label{fig:recovery_rank500_radius_period}\n\\end{figure*}\n\n\nFinally, we assessed whether the detection efficiency varies across different sectors by assessing the fraction of recovered TOIs and TCEs within the highest ranked 500 candidates. We found the recovery of TOIs within the top 500 highest ranked candidates to remain relatively constant across all sectors, while the fraction of recovered TCEs in the top 500 highest ranked candidates increases in later sectors, as shown in Figure~\\ref{fig:recovery_rank500}). After applying a Spearman's rank test we find a positive correlation of 0.86 (pvalue = 5.9 $\\times$ $10^{-8}$) and 0.57 (pvalue = 0.003) between the observation sector and TCE and TOI recovery rates, respectively. These correlations suggest that the ability of users to detect transit-like events improves as they classify more subjects. The improvement of volunteers over time can also be seen in Fig~\\ref{fig:user_weights}, which shows the mean (unnormalized) user weight per sector for volunteers who completed one or more classifications in at least one sector (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors 26 sectors from the nominal \\emph{TESS}\\ mission (pink). The figure highlights an overall improvement in the mean user weight in later sectors, as well as a positive correlation between the overall increase in user weight and the number of sectors that volunteers have participated in.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI_rank500.png}\n \\caption{The fractional recovery rate of the TOIs (blue circles) and TCEs (teal squares) at a rank of 500 for each sector. Sector 1-9 (white background) represent southern hemisphere sectors classified with UI1, sectors 9-14 (light grey background) show the southern hemisphere sectors classified with UI2, and sectors 14-24 (dark grey background) show the northern hemisphere sectors classified with US2.}\n \\label{fig:recovery_rank500}\n\\end{figure}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.50\\textwidth]{Figures\/user_weights_sectors.png}\n \\caption{Mean user weights per sector. The solid lines show the user weights for the old user interface and the dashed line for the new interface, separated by the black line (Sector 9). The different coloured lines show the mean user weights calculated considering user who participated in any number of sectors (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors observed during the nominal \\emph{TESS}\\ mission (pink).}\n \\label{fig:user_weights}\n\\end{figure}\n\n\n\\section{Candidate vetting}\n\\label{sec:vetting}\n\nFor each observation sector the subjects are ranked according to their transit scores, and the 500 highest ranked targets (excluding TOIs) visually vetted by the PHT science team in order to identify potential candidates and rule out false positives. A vetting cut-off rank of 500 was chosen as we found this to maximise the number of found candidates while minimising the number of likely false positives. In the initial round of vetting, which is completed via a separate Zooniverse classification interface that is only accessible to the core science team, a minimum of three members of the team sort the highest ranked targets into either `keep for further analysis', `eclipsing binary' or `discard'. The sorting is based on the inspection of the full \\emph{TESS}\\ light curve of the target, with the times of the satellite momentum dumps indicated. Additionally, around the time of each likely transit event (i.e. time of successful DB clusters) we inspect the background flux and the x and y centroid positions. Stellar parameters are provided for each candidate, subject to availability, alongside links to the SPOC Data Validation (DV) reports for candidates that had been flagged as TCEs but were never promoted to TOIs status.\n\nCandidates where at least two of the reviewers indicated that the signal is consistent with a planetary transit are kept for further analysis. \\textcolor{black}{This constitute a $\\sim$~5~\\% retention rate of the 500 highest ranked candidates per sector between the initial citizen science classification stage and the PHT science team vetting stage. Considering that the known planets and TOIs are not included at this stage of vetting, it is not surprising that our retention rate is lower that the true-positive rates of TCEs (see Section~\\ref{subsec:TCE_TOI}). Furthermore, this false-positive rate is consistent with the the findings of the initial Planet Hunters project \\citep{schwamb12}.}\n\nThe rest of the 500 candidates were grouped into $\\sim$~37~\\% `eclipsing binary' and $\\sim$~58~\\% `discard'. The most common reasons for discarding light curves are due to events caused by momentum dumps and due to background events, such as background eclipsing binaries, that mimic transit-like signals in the light curve. The targets identified as eclipsing binaries are analysed further by the \\emph{TESS}\\ Eclipsing Binaries Working Group (Prsa et al, in prep).\n\n\n\n\nFor the second round of candidate vetting we generate our own data validation reports for all candidates classified as `keep for further analysis'. The reports are generated using the open source software {\\sc latte} \\citep[Lightcurve Analysis Tool for Transiting Exoplanets;][]{LATTE2020}, which includes a range of standard diagnostic plots that are specifically designed to help identify transit-like signals and weed out astrophysical false positives in \\emph{TESS}\\ data. In brief the diagnostics consist of:\n\n\\textbf{Momentum Dumps}. The times of the \\emph{TESS}\\ reaction wheel momentum dumps that can result in instrumental effects that mimic astrophysical signals.\n\n\\textbf{Background Flux}. The background flux to help identify trends caused by background events such as asteroids or fireflies \\citep{vanderspek2018tess} passing through the field of view.\n\n\\textbf{x and y centroid positions}. The CCD column and row local position of the target's flux-weighted centroid, and the CCD column and row motion which considers differential velocity aberration (DVA), pointing drift, and thermal effects. This can help identify signals caused by systematics due to the satellite. \n\\textbf{Aperture size test}. The target light curve around the time of the transit-like event extracted using two apertures of different sizes. This can help identify signals resulting from background eclipsing binaries.\n \n\\textbf{Pixel-level centroid analysis}. A comparison between the average in-transit and average out-of-transit flux, as well as the difference between them. This can help identify signals resulting from background eclipsing binaries.\n\n\\textbf{Nearby companion stars}. The location of nearby stars brighter than V-band magnitude 15 as queried from the Gaia Data Release 2 catalog \\citep{gaia2018gaia} and the DSS2 red field of view around the target star in order to identify nearby contaminating sources. \n\n\\textbf{Nearest neighbour light curves}. Normalized flux light curves of the five short-cadence \\emph{TESS}\\ stars with the smallest projected distances to the target star, used to identify alternative sources of the signal or systematic effects that affect multiple target stars. \n\n\\textbf{Pixel level light curves}. Individual light curves extracted for each pixel around the target. Used to identify signals resulting from background eclipsing binaries, background events and systematics.\n\n\\textbf{Box-Least-Squares fit}. Results from two consecutive BLS searches, where the identified signals from the initial search are removed prior to the second BLS search.\n\nThe {\\sc latte} validation reports are assessed by the PHT science team in order to identify planetary candidates that warrant further investigation. Around 10~\\% of the targets assessed at this stage of vetting are kept for further investigation, resulting in $\\sim$~3 promising planet candidates per observation sector. The discarded candidates can be loosely categorized into (background) eclipsing binaries ($\\sim$~40~\\%), systematic effects ($\\sim$~25~\\%), background events ($\\sim$~15~\\%) and other (stellar signals such as spots; $\\sim$~10~\\%).\n\n\nWe use \\texttt{pyaneti}\\ \\citep{pyaneti} to infer the planetary and orbital parameters of our most promising candidates. For multi-transit candidates we fit for seven parameters per planet, time of mid-transit $T_0$, orbital period $P$, impact parameter $b$, scaled semi-major axis $a\/R_\\star$, scaled planet radius $r_{\\rm p}\/R_\\star$, and two limb darkening coefficients following a \\citet{Mandel2002} quadratic limb darkening model, implemented with the $q_1$ and $q_2$ parametrization suggested by \\citet{Kipping2013}. Orbits were assumed to be circular.\nFor the mono-transit candidates, we fit the same parameters as for the multi-transit case, except for the orbital period and scaled semi-major axis which cannot be known for single transits. We follow \\citet{Osborn2016} to estimate the orbital period of the mono-transit candidates assuming circular orbits.\n\nWe note that some of our candidates are V-shaped, consistent with a grazing transit configuration. For these cases, we set uniform priors between 0 and 0.15 for $r_{\\rm p}\/R_\\star$ and between 0 and 1.15 for the impact parameter in order to avoid large radii caused by the $r_{\\rm p}\/R_\\star - b$ degeneracy. Thus, the $r_{\\rm p}\/R_\\star$ for these candidates should not be trusted. A full characterisation of these grazing transits is out of the scope of this manuscript.\n\nFigure~\\ref{fig:PHT_pyaneti} shows the \\emph{TESS}\\ transits together with the inferred model for each candidate. Table~\\ref{tab:PHT-caniddates} shows the inferred main parameters, the values and their uncertainties are given by the median and 68.3\\% credible interval of the posterior distributions.\n\n\n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_one.png}\n \\caption{All of the PHT candidates modelled using \\texttt{pyaneti}. The parameters of the best fits are summarised in Table~\\protect\\ref{tab:PHT-caniddates}. The blue and magenta fits show the multi and single transit event candidates, respectively.} \n \\label{fig:PHT_pyaneti}\n\\end{figure*}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_two.png}\n \\addtocounter{figure}{-1}\n \\caption{\\textbf{PHT candidates (continued)}} \n\\end{figure*}\n\n\nCandidates that pass all of our rounds of vetting are uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS) website\\footnote{\\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}} as community TOIs (cTOIs).\n\n\\section{Follow-up observations}\n\\label{sec:follow_up}\n\nMany astrophysical false positive scenarios can be ruled out from the detailed examination of the \\emph{TESS}\\ data, both from the light curves themselves and from the target pixel files. However, not all of the false positive scenarios can be ruled out from these data alone, due in part to the large \\emph{TESS}\\ pixels (20 arcsconds). Our third stage of vetting, therefore, consists of following up the candidates with ground based observations including photometry, reconnaissance spectroscopy and speckle imaging. The results from these observations will be discussed in detail in a dedicated follow-up paper. \n\n\\subsection{Photometry}\n\nWe make use of the LCO global network of fully robotic 0.4-m\/SBIG and 1.0-m\/Sinistro facilities \\citep{LCO2013} to observe additional transits, where the orbital period is known, in order to refine the ephemeris and confirm that the transit events are not due to a blended eclipsing binary in the vicinity of the main target. Snapshot images are taken of single transit event candidates in order to identify nearby contaminating sources. \n\n\n\\subsection{Spectroscopy}\n\nWe perform high-resolution optical spectroscopy using telescopes from across the globe in order to cover a wide range of RA and Dec:\n\\begin{itemize}\n\\item The Las Cumbres Observatory (LCO) telescopes with the Network of Robotic Echelle Spectrographs \\citep[NRES,][]{LCO2013}. These fibre-fed spectrographs, mounted on 1.0-m telescopes around the globe, have a resolution of R = 53,000 and a wavelength coverage of 380 to 860 nm. \n\n\\item The MINERVA Australis Telescope facility, located at Mount Kent Observatory in Queensland, Australia \\citep{addison2019}. This facility is made up of four 0.7m CDK700 telescopes, which individually feed light via optic fibre into a KiwiSpec high-resolution (R = 80,000) stabilised spectrograph \\citep{barnes2012} that covers wavelengths from 480 nm to 620 nm. \n\n\\item The CHIRON spectrograph mounted on the SMARTS 1.5-m telescope \\citep{Tokovinin2018}, located at the Cerro Tololo\nInter-American Observatory (CTIO) in Chile. The high resolution cross-dispersed echelle spectrometer is fiber-fed followed by an image slicer. It has a resolution of R = 80,000 and covers wavelengths ranging from 410 to 870 nm.\n\n\\item The SOPHIE echelle spectrograph mounted on the 1.93-m Haute-Provence Observatory (OHP), France\n\\citep{2008Perruchot,2009Bouchy}. The high resolution cross-dispersed stabilized echelle spectrometer is fed by two optical fibers. Observations were taken in high-resolution mode (R = 75,000) with a wavelength range of 387 to 694 nm.\n\n\\end{itemize}\n\nReconnaissance spectroscopy with these instruments allow us to extract stellar parameters, identify spectroscopic binaries, and place upper limits on the companion masses. Spectroscopic binaries and targets whose spectral type is incompatible with the initial planet hypothesis and\/or precludes precision RV observations (giant or early type stars) are not followed up further. Promising targets, however, are monitored in order to constrain their period and place limits on their mass. \n\n\\subsection{Speckle Imaging}\n\nFor our most promising candidates we perform high resolution speckle imaging using the `Alopeke instrument on the 8.1-m Frederick C. Gillett Gemini North telescope in Maunakea, Hawaii, USA, and its twin, Zorro, on the 8.1-m Gemini South telescope on Cerro Pach\\'{o}n, Chile \\citep{Matson2019, Howell2011}. Speckle interferometric observations provide extremely high resolution images reaching the diffraction limit of the telescope. We obtain simultaneous 562 nm and 832 nm rapid exposure (60 msec) images in succession that effectively `freeze out' atmospheric turbulence and through Fourier analysis are used to search for close companion stars at 5-8 magnitude contrast levels. This analysis, along with the reconstructed images, allow us to identify nearby companions and to quantify their light contribution to the TESS aperture and thus the transit signal.\n\n\n\\section{Planet candidates and Noteworthy Systems}\n\\label{sec:PHT_canidates}\n\\subsection{Planet candidate properties}\n\nIn this final part of the paper we discuss the 90 PHT candidates around 88 host stars that passed the initial two stages of vetting and that were uploaded to ExoFOP as cTOIs. At the time of discovery none of these candidates were TOIs. The properties of all of the PHT candidates are summarised in Table~\\ref{tab:PHT-caniddates}. Candidates that have been promoted to TOI status since their PHT discovery are highlighted with an asterisk following the TIC ID, and candidates that have been shown to be false positives, based on the ground-based follow-up observations, are marked with a dagger symbol ($\\dagger$). The majority (81\\%) of PHT candidates are single transit events, indicated by an `s' following the orbital period presented in the table. \\textcolor{black}{18 of the PHT candidates were flagged as TCEs by the \\emph{TESS}\\ pipeline, but not initially promoted to TOI status. The most common reasons for this was that the pipeline identified a single-transit event as well as times of systematics (often caused by momentum dumps), due to its two-transit minimum detection threshold. This resulted in the candidate being discarded on the basis of it not passing the `odd-even' transit depth test. Out of the 18 TCEs, 14 have become TOI's since the PHT discovery. More detail on the TCE candidates can be found in Appendix~\\ref{appendixA}.} \n\nAll planet parameters (columns 2 to 8) are derived from the \\texttt{pyaneti}\\ modelling as described in Section~\\ref{sec:vetting}. Finally, the table summarises the ground-based follow-up observations (see Sec~\\ref{sec:follow_up}) that have been obtained to date, where the bracketed numbers following the observing instruments indicate the number of epochs. Unless otherwise noted, the follow-up observations are consistent with a planetary scenario. More in depth descriptions of individual targets for which we have additional information to complement the results in Table~\\ref{tab:PHT-caniddates} can be found in Appendix~\\ref{appendixA}.\n\n\\subsection{Planet candidate analysis}\n\n\nThe majority of the TOIs (87.7\\%) have orbital periods shorter than 15 days due to the requirement of observing at least two transits included in all major pipelines \\textcolor{black}{combined with the observing strategy of \\emph{TESS}}. As visual vetting does not impose these limits, the candidates outlined in this paper are helping to populate the relatively under-explored long-period region of parameter space. This is highlighted in Figure~\\ref{fig:PHT_candidates}, which shows the transit depths vs the orbital periods of the PHT single transit candidates (orange circles) and the multi-transit candidates (magenta squares) compared to the TOIs (blue circles). Values of the orbital periods and transit depths were obtained via transit modelling using \\texttt{pyaneti} (see Section~\\ref{sec:vetting}). The orbital period of single transit events are poorly constrained, which is reflected by the large errorbars in Figure~\\ref{fig:PHT_candidates}. Figure~\\ref{fig:PHT_candidates} also highlights that with PHT we are able to recover a similar range of transit depths as the pipeline found TOIs, as was previously shown in Figure~\\ref{fig:recovery_rank500_radius_period}.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_candidate_period_depth_plot_errobrars.png}\n \\caption{The properties of the PHT single transit (orange circles) and multi transit (magenta squares) candidates compared to the properties of the TOIs (blue circles). All parameters (listed in Table~\\ref{fig:PHT_candidates}) were extracted using \\texttt{pyaneti}\\ modelling.}\n \\label{fig:PHT_candidates}\n\\end{figure}\n\nThe PHT candidates were further compared to the TOIs in terms of the properties of their host stars. Figure~\\ref{fig:eep} shows the effective temperature and stellar radii as taken from the TIC \\citep{Stassun18}, for TOIs (blue dots) and the PHT candidates (magenta circles). The solid and dashed lines indicate the main sequence and post-main sequence MIST stellar evolutionary tracks \\citep{choi2016}, respectively, for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. This shows that around 10\\% of the host stars are in the process of, or have recently evolved off the main sequence. The models assume solar metalicity, no stellar rotation and no additional internal mixing.\n\n\\textcolor{black}{Ground based follow-up spectroscopy has revealed that six of the PHT candidates listed in Table~\\ref{tab:PHT-caniddates} are astrophysical false positives. As the follow-up campaign of the targets is still underway, the true false-positive rate of the candidates to have made it through all stages of the vetting process, as outlined in the methodology, will be be assessed in future PHT papers once the true nature of more of the candidates has been independently verified.}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_eep.png}\n \\caption{Stellar evolution tracks showing main sequence (solid black lines) and post-main sequence (dashed grey lines) MIST stellar evolution for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. The blue dots show the TOIs and the magenta circles show the PHT candidates.} \n \\label{fig:eep}\n\\end{figure}\n\n\n\\subsection{Stellar systems}\n\\label{subsec:PHT_stars}\n\nIn addition to the planetary candidates, citizen science allows for the identification of interesting stellar systems and astrophysical phenomena, in particular where the signals are aperiodic or small compared to the dominant stellar signal. These include light curves that exhibit multiple transit-like signals, possibly as a result of a multiple stellar system or a blend of eclipsing binaries. We have investigated all light curves that were flagged as possible multi-stellar systems via the PHT discussion boards. Similar to the planet vetting, as described in Section~\\ref{sec:vetting}, we generated {\\sc latte} data validation reports in order to assess the nature of the signal. Additionally, we subjected these systems to an iterative signal removal process, whereby we phase-folded the light curve on the dominant orbital period, binned the light curve into between 200-500 phase bins, created an interpolation model, and then subtracted said signal in order to evaluate the individual transit signals. The period of each signal, as listed in Table~\\ref{tab:PHT-multis}, was determined by phase folding the light curve at a number of trial periods and assessing by eye the best fit period and corresponding uncertainty.\n\nDue to the large \\emph{TESS}\\ pixels, blends are expected to be common. We searched for blends by generating phase folded light curves for each pixel around the source of the target in order to better locate the source of each signal. Shifts in the \\emph{TESS}\\ x and y centroid positions were also found to be good indicators of visually separated sources. Nearby sources with a magnitude difference greater than 5 mags were ruled out as possible contaminators. We consider a candidate to be a confirmed blend when the centroids are separated by more than 1 \\emph{TESS}\\ pixel, as this corresponds to an angular separation > 21 arcseconds meaning that the systems are highly unlikely to be gravitationally bound. Systems where the signal appears to be coming from the same \\emph{TESS}\\ pixel and that show no clear centroid shifts are considered to be candidate multiple systems. We note that blends are still possible, however, without further investigation we cannot conclusively rule these out as possible multi stellar systems. \n\nAll of the systems are summarised in Table~\\ref{tab:PHT-multis}. Out of the 26 systems, 6 are confirmed multiple systems which have either been published or are being prepared for publication; 7 are visually separated eclipsing binaries (confirmed blends); and 13 are candidate multiple system. Additional observations will be required to determine whether or not these candidate multiple systems are in fact gravitationally bound or photometric blends as a results of the large \\emph{TESS}\\ pixels or due to a line of sight happenstance. \n\n\\begin{landscape}\n\\begin{table}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period} & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n & \\textbf{Name} & \\textbf{(\\textcolor{black}{BJD - 2457000})} & \\textbf{(days)} & & $(R_{\\oplus})$ & \\textbf{Parameter} & \\textbf{(hours)} & & & & & \\\\\n\\hline\n101641905 & TWOMASS 11412617+3441004 & $1917.26335 _{ - 0.00072 } ^ { + 0.00071 }$ & $14.52 _{ - 5.25 } ^ { + 6.21 }(s)$ & $0.1135 _{ - 0.0064 } ^ { + 0.0032 }$ & $9.76 _{ - 0.69 } ^ { + 0.65 }$ & $0.691 _{ - 0.183 } ^ { + 0.077 }$ & $3.163 _{ - 0.088 } ^ { + 0.093 }$ & 12.196 & & & & \\\\\n103633672* & TYC 4387-00923-1 & $1850.3211 _{ - 0.00077 } ^ { + 0.00135 }$ & $90.9 _{ - 23.7 } ^ { + 46.4 }(s)$ & $0.0395 _{ - 0.0013 } ^ { + 0.0013 }$ & $3.45 _{ - 0.24 } ^ { + 0.26 }$ & $0.3 _{ - 0.21 } ^ { + 0.26 }$ & $6.7 _{ - 0.11 } ^ { + 0.12 }$ & 10.586 & & NRES (1) & & \\\\\n110996418 & TWOMASS 12344723-1019107 & $1580.6406 _{ - 0.0038 } ^ { + 0.0037 }$ & $5.18 _{ - 2.93 } ^ { + 6.86 }(s)$ & $0.1044 _{ - 0.0067 } ^ { + 0.008 }$ & $12.7 _{ - 0.99 } ^ { + 1.15 }$ & $0.44 _{ - 0.3 } ^ { + 0.3 }$ & $3.53 _{ - 0.27 } ^ { + 0.36 }$ & 13.945 & & & & \\\\\n128703021 & HIP 71639 & $1601.8442 _{ - 0.00108 } ^ { + 0.00093 }$ & $26.0 _{ - 8.22 } ^ { + 22.35 }(s)$ & $0.0254 _{ - 0.00049 } ^ { + 0.00072 }$ & $4.44 _{ - 0.2 } ^ { + 0.23 }$ & $0.47 _{ - 0.3 } ^ { + 0.22 }$ & $7.283 _{ - 0.091 } ^ { + 0.141 }$ & 6.06 & & NRES (2);MINERVA (34) & Gemini & \\\\\n138126035 & TYC 1450-00833-1 & $1954.3229 _{ - 0.0041 } ^ { + 0.0067 }$ & $28.8 _{ - 14.0 } ^ { + 203.2 }(s)$ & $0.0375 _{ - 0.0026 } ^ { + 0.0069 }$ & $4.01 _{ - 0.35 } ^ { + 0.74 }$ & $0.58 _{ - 0.38 } ^ { + 0.35 }$ & $4.65 _{ - 0.32 } ^ { + 0.85 }$ & 10.349 & & & & \\\\\n142087638 & TYC 9189-00274-1 & $1512.1673 _{ - 0.0043 } ^ { + 0.0034 }$ & $3.14 _{ - 1.41 } ^ { + 12.04 }(s)$ & $0.0469 _{ - 0.0035 } ^ { + 0.0063 }$ & $6.05 _{ - 0.54 } ^ { + 0.89 }$ & $0.5 _{ - 0.35 } ^ { + 0.36 }$ & $2.72 _{ - 0.23 } ^ { + 0.5 }$ & 11.526 & & & & \\\\\n159159904 & HIP 64812 & $1918.6109 _{ - 0.0067 } ^ { + 0.0091 }$ & $584.0 _{ - 215.0 } ^ { + 1724.0 }(s)$ & $0.0237 _{ - 0.0011 } ^ { + 0.0026 }$ & $3.12 _{ - 0.22 } ^ { + 0.36 }$ & $0.49 _{ - 0.34 } ^ { + 0.35 }$ & $15.11 _{ - 0.54 } ^ { + 0.7 }$ & 9.2 & & NRES (2) & & \\\\\n160039081* & HIP 78892 & $1752.9261 _{ - 0.0045 } ^ { + 0.005 }$ & $30.19918 _{ - 0.00099 } ^ { + 0.00094 }$ & $0.0211 _{ - 0.0013 } ^ { + 0.0035 }$ & $2.67 _{ - 0.21 } ^ { + 0.43 }$ & $0.52 _{ - 0.34 } ^ { + 0.36 }$ & $4.93 _{ - 0.27 } ^ { + 0.37 }$ & 8.35 & SBIG (1) & NRES (1);SOPHIE (4) & Gemini & \\\\\n162631539 & HIP 80264 & $1978.2794 _{ - 0.0044 } ^ { + 0.0051 }$ & $17.32 _{ - 6.66 } ^ { + 52.35 }(s)$ & $0.0195 _{ - 0.0011 } ^ { + 0.0024 }$ & $2.94 _{ - 0.24 } ^ { + 0.38 }$ & $0.48 _{ - 0.33 } ^ { + 0.36 }$ & $5.54 _{ - 0.33 } ^ { + 0.41 }$ & 7.42 & & & & \\\\\n166184426* & TWOMASS 13442500-4020122 & $1600.4409 _{ - 0.003 } ^ { + 0.0036 }$ & $16.3325 _{ - 0.0066 } ^ { + 0.0052 }$ & $0.0545 _{ - 0.0031 } ^ { + 0.0039 }$ & $1.85 _{ - 0.12 } ^ { + 0.15 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.98 _{ - 0.22 } ^ { + 0.17 }$ & 12.911 & & & & \\\\\n167661160$\\dagger$ & TYC 7054-01577-1 & $1442.0703 _{ - 0.0028 } ^ { + 0.004 }$ & $36.802 _{ - 0.07 } ^ { + 0.069 }$ & $0.0307 _{ - 0.0014 } ^ { + 0.0024 }$ & $4.07 _{ - 0.32 } ^ { + 0.43 }$ & $0.37 _{ - 0.26 } ^ { + 0.33 }$ & $5.09 _{ - 0.23 } ^ { + 0.21 }$ & 9.927 & & NRES (9);MINERVA (4) & & EB from MINERVA observations \\\\\n172370679* & TWOMASS 19574239+4008357 & $1711.95923 _{ - 0.00099 } ^ { + 0.001 }$ & $32.84 _{ - 4.17 } ^ { + 5.59 }(s)$ & $0.1968 _{ - 0.0032 } ^ { + 0.0022 }$ & $13.24 _{ - 0.43 } ^ { + 0.43 }$ & $0.22 _{ - 0.15 } ^ { + 0.14 }$ & $4.999 _{ - 0.097 } ^ { + 0.111 }$ & 14.88 & & & & Confirmed planet \\citep{canas2020}. \\\\\n174302697* & TYC 3641-01789-1 & $1743.7267 _{ - 0.00092 } ^ { + 0.00093 }$ & $498.2 _{ - 80.0 } ^ { + 95.3 }(s)$ & $0.07622 _{ - 0.00068 } ^ { + 0.00063 }$ & $13.34 _{ - 0.57 } ^ { + 0.58 }$ & $0.642 _{ - 0.029 } ^ { + 0.024 }$ & $17.71 _{ - 0.12 } ^ { + 0.13 }$ & 9.309 & SBIG (1) & & & \\\\\n179582003 & TYC 9166-00745-1 & $1518.4688 _{ - 0.0016 } ^ { + 0.0016 }$ & $104.6137 _{ - 0.0022 } ^ { + 0.0022 }$ & $0.06324 _{ - 0.0008 } ^ { + 0.0008 }$ & $7.51 _{ - 0.35 } ^ { + 0.35 }$ & $0.21 _{ - 0.15 } ^ { + 0.19 }$ & $9.073 _{ - 0.084 } ^ { + 0.097 }$ & 10.806 & & & & \\\\\n192415680 & TYC 2859-00682-1 & $1796.0265 _{ - 0.0012 } ^ { + 0.0013 }$ & $18.47 _{ - 6.34 } ^ { + 21.73 }(s)$ & $0.0478 _{ - 0.0017 } ^ { + 0.0027 }$ & $4.43 _{ - 0.33 } ^ { + 0.38 }$ & $0.45 _{ - 0.31 } ^ { + 0.31 }$ & $3.94 _{ - 0.1 } ^ { + 0.12 }$ & 9.838 & SBIG (1) & SOPHIE (2) & & \\\\\n192790476 & TYC 7595-00649-1 & $1452.3341 _{ - 0.0014 } ^ { + 0.002 }$ & $16.09 _{ - 5.73 } ^ { + 15.49 }(s)$ & $0.0438 _{ - 0.0018 } ^ { + 0.0026 }$ & $3.24 _{ - 0.34 } ^ { + 0.37 }$ & $0.37 _{ - 0.25 } ^ { + 0.3 }$ & $3.395 _{ - 0.099 } ^ { + 0.192 }$ & 10.772 & & & & \\\\\n206361691$\\dagger$ & HIP 117250 & $1363.2224 _{ - 0.0082 } ^ { + 0.009 }$ & $237.7 _{ - 81.0 } ^ { + 314.4 }(s)$ & $0.01762 _{ - 0.00088 } ^ { + 0.00125 }$ & $2.69 _{ - 0.19 } ^ { + 0.25 }$ & $0.43 _{ - 0.28 } ^ { + 0.32 }$ & $13.91 _{ - 0.53 } ^ { + 0.52 }$ & 8.88 & & CHIRON (2) & & SB2 from CHIRON \\\\\n207501148 & TYC 3881-00527-1 & $2007.7273 _{ - 0.0011 } ^ { + 0.0011 }$ & $39.9 _{ - 10.3 } ^ { + 14.3 }(s)$ & $0.0981 _{ - 0.0047 } ^ { + 0.011 }$ & $13.31 _{ - 0.95 } ^ { + 1.56 }$ & $0.9 _{ - 0.03 } ^ { + 0.039 }$ & $4.73 _{ - 0.14 } ^ { + 0.14 }$ & 10.385 & & & & \\\\\n219466784* & TYC 4409-00437-1 & $1872.6879 _{ - 0.0097 } ^ { + 0.0108 }$ & $318.0 _{ - 147.0 } ^ { + 1448.0 }(s)$ & $0.0332 _{ - 0.0024 } ^ { + 0.0048 }$ & $3.26 _{ - 0.31 } ^ { + 0.49 }$ & $0.55 _{ - 0.39 } ^ { + 0.34 }$ & $10.06 _{ - 0.81 } ^ { + 1.12 }$ & 11.099 & & & & \\\\\n219501568 & HIP 79876 & $1961.7879 _{ - 0.0018 } ^ { + 0.002 }$ & $16.5931 _{ - 0.0017 } ^ { + 0.0015 }$ & $0.0221 _{ - 0.0012 } ^ { + 0.0015 }$ & $4.22 _{ - 0.3 } ^ { + 0.35 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.615 _{ - 0.077 } ^ { + 0.093 }$ & 8.38 & & & & \\\\\n229055790 & TYC 7492-01197-1 & $1337.866 _{ - 0.0022 } ^ { + 0.0019 }$ & $48.0 _{ - 12.8 } ^ { + 48.4 }(s)$ & $0.0304 _{ - 0.00097 } ^ { + 0.00115 }$ & $3.52 _{ - 0.2 } ^ { + 0.24 }$ & $0.37 _{ - 0.26 } ^ { + 0.32 }$ & $6.53 _{ - 0.11 } ^ { + 0.14 }$ & 9.642 & & NRES (2) & & \\\\\n229608594 & TWOMASS 18180283+7428005 & $1960.0319 _{ - 0.0037 } ^ { + 0.0045 }$ & $152.4 _{ - 54.1 } ^ { + 152.6 }(s)$ & $0.0474 _{ - 0.0023 } ^ { + 0.0024 }$ & $3.42 _{ - 0.34 } ^ { + 0.36 }$ & $0.38 _{ - 0.26 } ^ { + 0.3 }$ & $6.98 _{ - 0.23 } ^ { + 0.37 }$ & 12.302 & & & & \\\\\n229742722* & TYC 4434-00596-1 & $1689.688 _{ - 0.025 } ^ { + 0.02 }$ & $29.0 _{ - 16.4 } ^ { + 66.3 }(s)$ & $0.019 _{ - 0.0028 } ^ { + 0.0029 }$ & $2.9 _{ - 0.44 } ^ { + 0.48 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $4.27 _{ - 0.09 } ^ { + 0.11 }$ & 10.33 & & NRES (8);SOPHIE (4) & Gemini & \\\\\n233194447 & TYC 4211-00650-1 & $1770.4924 _{ - 0.0065 } ^ { + 0.0107 }$ & $373.0 _{ - 101.0 } ^ { + 284.0 }(s)$ & $0.02121 _{ - 0.00073 } ^ { + 0.001 }$ & $5.08 _{ - 0.28 } ^ { + 0.33 }$ & $0.34 _{ - 0.24 } ^ { + 0.29 }$ & $24.45 _{ - 0.47 } ^ { + 0.5 }$ & 9.178 & & NRES (2) & Gemini & \\\\\n235943205 & TYC 4588-00127-1 & $1827.0267 _{ - 0.004 } ^ { + 0.0034 }$ & $121.3394 _{ - 0.0063 } ^ { + 0.0065 }$ & $0.0402 _{ - 0.0016 } ^ { + 0.0019 }$ & $4.2 _{ - 0.25 } ^ { + 0.29 }$ & $0.4 _{ - 0.27 } ^ { + 0.28 }$ & $6.37 _{ - 0.2 } ^ { + 0.3 }$ & 11.076 & & NRES (1);SOPHIE (2) & & \\\\\n237201858 & TYC 4452-00759-1 & $1811.5032 _{ - 0.0069 } ^ { + 0.0067 }$ & $129.7 _{ - 41.5 } ^ { + 146.8 }(s)$ & $0.0258 _{ - 0.0013 } ^ { + 0.0015 }$ & $4.12 _{ - 0.27 } ^ { + 0.3 }$ & $0.4 _{ - 0.28 } ^ { + 0.31 }$ & $10.94 _{ - 0.37 } ^ { + 0.53 }$ & 10.344 & & NRES (1) & & \\\\\n243187830* & HIP 5286 & $1783.7671 _{ - 0.0017 } ^ { + 0.0019 }$ & $4.05 _{ - 1.53 } ^ { + 9.21 }(s)$ & $0.0268 _{ - 0.0015 } ^ { + 0.0027 }$ & $2.06 _{ - 0.17 } ^ { + 0.23 }$ & $0.47 _{ - 0.32 } ^ { + 0.34 }$ & $2.02 _{ - 0.12 } ^ { + 0.15 }$ & 8.407 & SBIG (1) & & & \\\\\n243417115 & TYC 8262-02120-1 & $1614.4796 _{ - 0.0028 } ^ { + 0.0022 }$ & $1.81 _{ - 0.73 } ^ { + 3.45 }(s)$ & $0.0523 _{ - 0.0035 } ^ { + 0.005 }$ & $5.39 _{ - 0.47 } ^ { + 0.64 }$ & $0.47 _{ - 0.33 } ^ { + 0.34 }$ & $2.03 _{ - 0.16 } ^ { + 0.23 }$ & 11.553 & & & & \\\\\n256429408 & TYC 4462-01942-1 & $1962.16 _{ - 0.0022 } ^ { + 0.0023 }$ & $382.0 _{ - 132.0 } ^ { + 265.0 }(s)$ & $0.03582 _{ - 0.00086 } ^ { + 0.00094 }$ & $6.12 _{ - 0.29 } ^ { + 0.3 }$ & $0.51 _{ - 0.36 } ^ { + 0.18 }$ & $16.96 _{ - 0.2 } ^ { + 0.24 }$ & 8.898 & & & & \\\\\n264544388* & TYC 4607-01275-1 & $1824.8438 _{ - 0.0076 } ^ { + 0.0078 }$ & $7030.0 _{ - 6260.0 } ^ { + 3330.0 }(s)$ & $0.0288 _{ - 0.0029 } ^ { + 0.0018 }$ & $4.58 _{ - 0.43 } ^ { + 0.35 }$ & $0.936 _{ - 0.363 } ^ { + 0.011 }$ & $19.13 _{ - 1.35 } ^ { + 0.84 }$ & 8.758 & & NRES (1) & & \\\\\n264766922 & TYC 8565-01780-1 & $1538.69518 _{ - 0.00091 } ^ { + 0.00091 }$ & $3.28 _{ - 0.94 } ^ { + 1.25 }(s)$ & $0.0933 _{ - 0.0063 } ^ { + 0.0176 }$ & $16.95 _{ - 1.33 } ^ { + 3.19 }$ & $0.908 _{ - 0.039 } ^ { + 0.048 }$ & $2.73 _{ - 0.11 } ^ { + 0.11 }$ & 10.747 & & & & \\\\\n26547036* & TYC 3921-01563-1 & $1712.30464 _{ - 0.00041 } ^ { + 0.0004 }$ & $73.0 _{ - 13.6 } ^ { + 16.5 }(s)$ & $0.10034 _{ - 0.0007 } ^ { + 0.00078 }$ & $11.75 _{ - 0.59 } ^ { + 0.58 }$ & $0.17 _{ - 0.12 } ^ { + 0.11 }$ & $8.681 _{ - 0.049 } ^ { + 0.052 }$ & 9.849 & & NRES (4) & Gemini & \\\\\n267542728$\\dagger$ & TYC 4583-01499-1 & $1708.4956 _{ - 0.0073 } ^ { + 0.0085 }$ & $39.7382 _{ - 0.0023 } ^ { + 0.0023 }$ & $0.03267 _{ - 0.00089 } ^ { + 0.00175 }$ & $18.46 _{ - 0.94 } ^ { + 1.14 }$ & $0.38 _{ - 0.26 } ^ { + 0.27 }$ & $24.16 _{ - 0.39 } ^ { + 0.45 }$ & 11.474 & & & & EB from HIRES RVs. \\\\\n270371513$\\dagger$ & HIP 10047 & $1426.2967 _{ - 0.0023 } ^ { + 0.002 }$ & $0.39 _{ - 0.17 } ^ { + 1.79 }(s)$ & $0.024 _{ - 0.0015 } ^ { + 0.0032 }$ & $4.8 _{ - 0.38 } ^ { + 0.64 }$ & $0.5 _{ - 0.34 } ^ { + 0.39 }$ & $1.93 _{ - 0.16 } ^ { + 0.19 }$ & 6.98515 & & MINERVA (20) & & SB 2 from MINERVA observations. \\\\\n274599700 & TWOMASS 17011885+5131455 & $2002.1202 _{ - 0.0024 } ^ { + 0.0024 }$ & $32.9754 _{ - 0.005 } ^ { + 0.005 }$ & $0.0847 _{ - 0.0021 } ^ { + 0.0018 }$ & $13.25 _{ - 0.83 } ^ { + 0.83 }$ & $0.37 _{ - 0.24 } ^ { + 0.19 }$ & $8.2 _{ - 0.18 } ^ { + 0.21 }$ & 12.411 & & & & \\\\\n278990954 & TYC 8548-00717-1 & $1650.0191 _{ - 0.0086 } ^ { + 0.0105 }$ & $18.45 _{ - 8.66 } ^ { + 230.7 }(s)$ & $0.034 _{ - 0.0024 } ^ { + 0.0115 }$ & $9.65 _{ - 0.92 } ^ { + 3.13 }$ & $0.58 _{ - 0.4 } ^ { + 0.36 }$ & $10.62 _{ - 0.66 } ^ { + 2.46 }$ & 10.749 & & & & \\\\\n280865159* & TYC 9384-01533-1 & $1387.0749 _{ - 0.0045 } ^ { + 0.0044 }$ & $1045.0 _{ - 249.0 } ^ { + 536.0 }(s)$ & $0.0406 _{ - 0.0011 } ^ { + 0.0014 }$ & $4.75 _{ - 0.26 } ^ { + 0.28 }$ & $0.35 _{ - 0.24 } ^ { + 0.23 }$ & $19.08 _{ - 0.32 } ^ { + 0.36 }$ & 11.517 & & & Gemini & \\\\\n284361752 & TYC 3924-01678-1 & $2032.093 _{ - 0.0078 } ^ { + 0.008 }$ & $140.6 _{ - 46.6 } ^ { + 159.1 }(s)$ & $0.0259 _{ - 0.0014 } ^ { + 0.0017 }$ & $3.62 _{ - 0.26 } ^ { + 0.31 }$ & $0.4 _{ - 0.27 } ^ { + 0.34 }$ & $8.98 _{ - 0.66 } ^ { + 0.86 }$ & 10.221 & & & & \\\\\n288240183 & TYC 4634-01225-1 & $1896.941 _{ - 0.0051 } ^ { + 0.0047 }$ & $119.0502 _{ - 0.0091 } ^ { + 0.0089 }$ & $0.02826 _{ - 0.00089 } ^ { + 0.00119 }$ & $4.28 _{ - 0.35 } ^ { + 0.36 }$ & $0.55 _{ - 0.37 } ^ { + 0.25 }$ & $17.49 _{ - 0.36 } ^ { + 0.6 }$ & 9.546 & & & & \\\\\n29169215 & TWOMASS 09011787+4727085 & $1872.5047 _{ - 0.0032 } ^ { + 0.0036 }$ & $14.89 _{ - 6.12 } ^ { + 24.84 }(s)$ & $0.0403 _{ - 0.0025 } ^ { + 0.0033 }$ & $3.28 _{ - 0.37 } ^ { + 0.45 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $3.56 _{ - 0.21 } ^ { + 0.32 }$ & 11.828 & & & & \\\\\n293649602 & TYC 8103-00266-1 & $1511.2109 _{ - 0.004 } ^ { + 0.0037 }$ & $12.85 _{ - 5.34 } ^ { + 42.21 }(s)$ & $0.04 _{ - 0.0024 } ^ { + 0.0039 }$ & $4.66 _{ - 0.36 } ^ { + 0.5 }$ & $0.5 _{ - 0.35 } ^ { + 0.34 }$ & $4.1 _{ - 0.31 } ^ { + 0.56 }$ & 10.925 & & & & \\\\\n296737508 & TYC 5472-01060-1 & $1538.0036 _{ - 0.0015 } ^ { + 0.0016 }$ & $18.27 _{ - 5.06 } ^ { + 17.45 }(s)$ & $0.0425 _{ - 0.0014 } ^ { + 0.0019 }$ & $5.33 _{ - 0.22 } ^ { + 0.27 }$ & $0.44 _{ - 0.3 } ^ { + 0.26 }$ & $5.13 _{ - 0.13 } ^ { + 0.15 }$ & 9.772 & Sinistro (1) & NRES (1);MINERVA (1) & Gemini & \\\\\n298663873 & TYC 3913-01781-1 & $1830.76819 _{ - 0.00099 } ^ { + 0.00099 }$ & $479.9 _{ - 89.4 } ^ { + 109.4 }(s)$ & $0.06231 _{ - 0.00034 } ^ { + 0.00045 }$ & $11.07 _{ - 0.57 } ^ { + 0.57 }$ & $0.16 _{ - 0.11 } ^ { + 0.13 }$ & $23.99 _{ - 0.093 } ^ { + 0.1 }$ & 9.162 & & NRES (2) & Gemini & Dalba et al. (in prep) \\\\\n303050301 & TYC 6979-01108-1 & $1366.1301 _{ - 0.0022 } ^ { + 0.0023 }$ & $281.0 _{ - 170.0 } ^ { + 264.0 }(s)$ & $0.0514 _{ - 0.0027 } ^ { + 0.0018 }$ & $4.85 _{ - 0.32 } ^ { + 0.32 }$ & $0.73 _{ - 0.48 } ^ { + 0.1 }$ & $7.91 _{ - 0.31 } ^ { + 0.36 }$ & 10.048 & & NRES (1) & Gemini & \\\\\n303317324 & TYC 6983-00438-1 & $1365.1845 _{ - 0.0023 } ^ { + 0.0028 }$ & $69.0 _{ - 25.5 } ^ { + 78.1 }(s)$ & $0.0365 _{ - 0.0013 } ^ { + 0.0016 }$ & $2.88 _{ - 0.3 } ^ { + 0.31 }$ & $0.39 _{ - 0.26 } ^ { + 0.32 }$ & $5.78 _{ - 0.18 } ^ { + 0.24 }$ & 10.799 & & & & \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\emph{Note} -- Candidates that have become TOIs following the PHT discovery are marked with an asterisk (*). The `s' following the orbital period indicates that the candidates is a single transit event. The ground-based follow-up observations are summarized in columns 10-12, where the bracketed numbers correspond the number of epochs obtained with each instrument. See Section~\\ref{sec:follow_up} for description of each instrument. The $\\dagger$ symbol indicates candidates that have been shown to be astrophysical false positives based on the ground based follow-up observations.}\n\\label{tab:PHT-caniddates}\n\\end{table}\n\\end{landscape}\n\n\\begin{landscape}\n\\begin{table}\n\\addtocounter{table}{-1}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period} & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n & \\textbf{Name} & \\textbf{(\\textcolor{black}{BJD - 2457000})} & \\textbf{(days)} & & $(R_{\\oplus})$ & \\textbf{Parameter} & \\textbf{(hours)} & & & & & \\\\\n\\hline\n303586471$\\dagger$ & HIP 115828 & $1363.7692 _{ - 0.0033 } ^ { + 0.0027 }$ & $13.85 _{ - 4.19 } ^ { + 18.2 }(s)$ & $0.0214 _{ - 0.001 } ^ { + 0.0014 }$ & $2.52 _{ - 0.16 } ^ { + 0.2 }$ & $0.4 _{ - 0.27 } ^ { + 0.33 }$ & $4.23 _{ - 0.19 } ^ { + 0.16 }$ & 8.27 & & MINERVA (11) & & SB 2 from MINERVA observations. \\\\\n304142124* & HIP 53719 & $1585.28023 _{ - 0.0008 } ^ { + 0.0008 }$ & $42.8 _{ - 10.0 } ^ { + 18.2 }(s)$ & $0.04311 _{ - 0.00093 } ^ { + 0.00153 }$ & $4.1 _{ - 0.23 } ^ { + 0.24 }$ & $0.33 _{ - 0.21 } ^ { + 0.21 }$ & $5.66 _{ - 0.067 } ^ { + 0.09 }$ & 8.62 & & NRES (1);MINERVA (4) & & Confirmed planet \\citep{diaz2020} \\\\\n304339227 & TYC 9290-01087-1 & $1673.3242 _{ - 0.009 } ^ { + 0.0128 }$ & $111.9 _{ - 72.2 } ^ { + 4844.1 }(s)$ & $0.0253 _{ - 0.0024 } ^ { + 0.0481 }$ & $3.27 _{ - 0.61 } ^ { + 5.72 }$ & $0.67 _{ - 0.47 } ^ { + 0.36 }$ & $7.44 _{ - 0.86 } ^ { + 2.84 }$ & 9.169 & & & & \\\\\n307958020 & TYC 4191-00309-1 & $1864.82 _{ - 0.014 } ^ { + 0.013 }$ & $169.0 _{ - 107.0 } ^ { + 10194.0 }(s)$ & $0.0223 _{ - 0.0022 } ^ { + 0.0543 }$ & $3.92 _{ - 0.52 } ^ { + 9.27 }$ & $0.71 _{ - 0.53 } ^ { + 0.33 }$ & $12.48 _{ - 1.1 } ^ { + 5.41 }$ & 9.017 & & & & \\\\\n308301091 & TYC 2081-01273-1 & $2030.3691 _{ - 0.0024 } ^ { + 0.0026 }$ & $29.24 _{ - 8.49 } ^ { + 22.46 }(s)$ & $0.0362 _{ - 0.0013 } ^ { + 0.0014 }$ & $5.41 _{ - 0.34 } ^ { + 0.35 }$ & $0.35 _{ - 0.25 } ^ { + 0.29 }$ & $6.57 _{ - 0.14 } ^ { + 0.19 }$ & 10.273 & & & & \\\\\n313006381 & HIP 45012 & $1705.687 _{ - 0.0081 } ^ { + 0.0045 }$ & $21.56 _{ - 8.9 } ^ { + 54.15 }(s)$ & $0.0261 _{ - 0.0017 } ^ { + 0.0027 }$ & $2.34 _{ - 0.2 } ^ { + 0.27 }$ & $0.45 _{ - 0.3 } ^ { + 0.38 }$ & $3.85 _{ - 0.51 } ^ { + 0.31 }$ & 9.39 & & & & \\\\\n323295479* & TYC 9506-01881-1 & $1622.9258 _{ - 0.00083 } ^ { + 0.00087 }$ & $117.8 _{ - 25.8 } ^ { + 30.9 }(s)$ & $0.0981 _{ - 0.0021 } ^ { + 0.0023 }$ & $11.35 _{ - 0.67 } ^ { + 0.66 }$ & $0.839 _{ - 0.024 } ^ { + 0.019 }$ & $6.7 _{ - 0.14 } ^ { + 0.15 }$ & 10.595 & & & & \\\\\n328933398.01* & TYC 4634-01435-1 & $1880.9878 _{ - 0.0039 } ^ { + 0.0042 }$ & $24.9335 _{ - 0.0046 } ^ { + 0.005 }$ & $0.0437 _{ - 0.0022 } ^ { + 0.0023 }$ & $4.62 _{ - 0.32 } ^ { + 0.33 }$ & $0.38 _{ - 0.25 } ^ { + 0.27 }$ & $5.02 _{ - 0.22 } ^ { + 0.27 }$ & 11.215 & & & & Potential multi-planet system. \\\\\n328933398.02* & TYC 4634-01435-1 & $1848.6557 _{ - 0.0053 } ^ { + 0.0072 }$ & $50.5 _{ - 22.4 } ^ { + 77.1 }(s)$ & $0.0296 _{ - 0.0028 } ^ { + 0.0033 }$ & $3.14 _{ - 0.33 } ^ { + 0.39 }$ & $0.41 _{ - 0.28 } ^ { + 0.35 }$ & $5.99 _{ - 0.8 } ^ { + 0.77 }$ & 11.215 & & & & \\\\\n331644554 & TYC 3609-00469-1 & $1757.0354 _{ - 0.0031 } ^ { + 0.0033 }$ & $947.0 _{ - 215.0 } ^ { + 274.0 }(s)$ & $0.12 _{ - 0.025 } ^ { + 0.021 }$ & $21.84 _{ - 4.57 } ^ { + 3.86 }$ & $1.018 _{ - 0.036 } ^ { + 0.028 }$ & $10.93 _{ - 0.34 } ^ { + 0.35 }$ & 9.752 & & & & \\\\\n332657786 & TWOMASS 09595797-1609323 & $1536.7659 _{ - 0.0015 } ^ { + 0.0015 }$ & $63.76 _{ - 9.52 } ^ { + 11.13 }(s)$ & $0.14961 _{ - 0.00064 } ^ { + 0.00029 }$ & $3.83 _{ - 0.12 } ^ { + 0.12 }$ & $0.059 _{ - 0.041 } ^ { + 0.064 }$ & $3.333 _{ - 0.095 } ^ { + 0.096 }$ & 15.99 & & & & \\\\\n336075472 & TYC 3526-00332-1 & $2028.1762 _{ - 0.0043 } ^ { + 0.0037 }$ & $61.9 _{ - 24.0 } ^ { + 95.6 }(s)$ & $0.0402 _{ - 0.0022 } ^ { + 0.0033 }$ & $3.09 _{ - 0.34 } ^ { + 0.4 }$ & $0.43 _{ - 0.29 } ^ { + 0.32 }$ & $5.39 _{ - 0.23 } ^ { + 0.37 }$ & 11.842 & & & & \\\\\n349488688.01 & TYC 1529-00224-1 & $1994.283 _{ - 0.0038 } ^ { + 0.0033 }$ & $11.6254 _{ - 0.005 } ^ { + 0.0052 }$ & $0.02195 _{ - 0.00096 } ^ { + 0.00122 }$ & $3.44 _{ - 0.18 } ^ { + 0.21 }$ & $0.39 _{ - 0.27 } ^ { + 0.3 }$ & $5.58 _{ - 0.15 } ^ { + 0.18 }$ & 8.855 & & NRES (2);SOPHIE (2) & & Potential multi-planet system. \\\\\n349488688.02 & TYC 1529-00224-1 & $2002.77063 _{ - 0.00097 } ^ { + 0.00103 }$ & $15.35 _{ - 1.94 } ^ { + 4.15 }(s)$ & $0.03688 _{ - 0.00067 } ^ { + 0.00069 }$ & $5.78 _{ - 0.18 } ^ { + 0.18 }$ & $0.24 _{ - 0.16 } ^ { + 0.21 }$ & $6.291 _{ - 0.058 } ^ { + 0.074 }$ & 8.855 & & NRES (2);SOPHIE (2) & & \\\\\n356700488* & TYC 4420-01295-1 & $1756.638 _{ - 0.013 } ^ { + 0.011 }$ & $184.5 _{ - 64.7 } ^ { + 333.1 }(s)$ & $0.0173 _{ - 0.0011 } ^ { + 0.0015 }$ & $2.92 _{ - 0.2 } ^ { + 0.28 }$ & $0.44 _{ - 0.3 } ^ { + 0.34 }$ & $11.76 _{ - 0.65 } ^ { + 1.03 }$ & 8.413 & & & & \\\\\n356710041* & TYC 1993-00419-1 & $1932.2939 _{ - 0.0019 } ^ { + 0.0019 }$ & $29.6 _{ - 14.0 } ^ { + 19.0 }(s)$ & $0.0496 _{ - 0.0021 } ^ { + 0.0011 }$ & $14.82 _{ - 0.85 } ^ { + 0.84 }$ & $0.66 _{ - 0.42 } ^ { + 0.11 }$ & $12.76 _{ - 0.24 } ^ { + 0.24 }$ & 9.646 & & & & \\\\\n369532319 & TYC 2743-01716-1 & $1755.8158 _{ - 0.006 } ^ { + 0.0051 }$ & $35.4 _{ - 12.0 } ^ { + 51.6 }(s)$ & $0.0316 _{ - 0.0023 } ^ { + 0.0028 }$ & $3.43 _{ - 0.3 } ^ { + 0.37 }$ & $0.41 _{ - 0.29 } ^ { + 0.34 }$ & $5.5 _{ - 0.32 } ^ { + 0.32 }$ & 10.594 & & & Gemini & \\\\\n369779127 & TYC 9510-00090-1 & $1643.9403 _{ - 0.0046 } ^ { + 0.0058 }$ & $9.93 _{ - 3.38 } ^ { + 19.74 }(s)$ & $0.0288 _{ - 0.0015 } ^ { + 0.0033 }$ & $4.89 _{ - 0.31 } ^ { + 0.56 }$ & $0.46 _{ - 0.31 } ^ { + 0.33 }$ & $5.64 _{ - 0.38 } ^ { + 0.33 }$ & 9.279 & & & & \\\\\n384159646* & TYC 9454-00957-1 & $1630.39405 _{ - 0.00079 } ^ { + 0.00079 }$ & $11.68 _{ - 2.75 } ^ { + 4.21 }(s)$ & $0.0658 _{ - 0.0012 } ^ { + 0.0011 }$ & $9.87 _{ - 0.45 } ^ { + 0.44 }$ & $0.27 _{ - 0.18 } ^ { + 0.21 }$ & $5.152 _{ - 0.069 } ^ { + 0.087 }$ & 10.158 & SBIG (1) & NRES (8);MINERVA (6) & Gemini & \\\\\n385557214 & TYC 1807-00046-1 & $1791.58399 _{ - 0.00068 } ^ { + 0.0007 }$ & $5.62451 _{ - 0.0004 } ^ { + 0.00043 }$ & $0.096 _{ - 0.019 } ^ { + 0.032 }$ & $8.32 _{ - 2.06 } ^ { + 2.77 }$ & $0.95 _{ - 0.075 } ^ { + 0.053 }$ & $1.221 _{ - 0.094 } ^ { + 0.058 }$ & 10.856 & & & & \\\\\n388134787 & TYC 4260-00427-1 & $1811.034 _{ - 0.015 } ^ { + 0.017 }$ & $246.0 _{ - 127.0 } ^ { + 6209.0 }(s)$ & $0.0265 _{ - 0.0024 } ^ { + 0.023 }$ & $2.57 _{ - 0.28 } ^ { + 2.19 }$ & $0.55 _{ - 0.39 } ^ { + 0.44 }$ & $8.85 _{ - 1.13 } ^ { + 1.84 }$ & 10.95 & & NRES (1) & Gemini & \\\\\n404518509 & HIP 16038 & $1431.2696 _{ - 0.0037 } ^ { + 0.0035 }$ & $26.83 _{ - 9.46 } ^ { + 56.14 }(s)$ & $0.0259 _{ - 0.0013 } ^ { + 0.0022 }$ & $2.94 _{ - 0.21 } ^ { + 0.29 }$ & $0.47 _{ - 0.31 } ^ { + 0.34 }$ & $5.02 _{ - 0.23 } ^ { + 0.28 }$ & 9.17 & & & & \\\\\n408636441* & TYC 4266-00736-1 & $1745.4668 _{ - 0.0016 } ^ { + 0.0015 }$ & $37.695 _{ - 0.0034 } ^ { + 0.0033 }$ & $0.0485 _{ - 0.0019 } ^ { + 0.0023 }$ & $3.32 _{ - 0.16 } ^ { + 0.19 }$ & $0.39 _{ - 0.27 } ^ { + 0.29 }$ & $3.63 _{ - 0.1 } ^ { + 0.14 }$ & 11.93 & SBIG (1) & & Gemini & Half of the period likely. \\\\\n418255064 & TWOMASS 13063680-8037015 & $1629.3304 _{ - 0.0018 } ^ { + 0.0018 }$ & $25.37 _{ - 7.06 } ^ { + 15.41 }(s)$ & $0.0732 _{ - 0.0029 } ^ { + 0.0031 }$ & $5.57 _{ - 0.36 } ^ { + 0.38 }$ & $0.37 _{ - 0.25 } ^ { + 0.25 }$ & $3.83 _{ - 0.13 } ^ { + 0.14 }$ & 12.478 & SBIG (1) & & Gemini & \\\\\n420645189$\\dagger$ & TYC 4508-00478-1 & $1837.4767 _{ - 0.0018 } ^ { + 0.0017 }$ & $250.2 _{ - 66.6 } ^ { + 99.4 }(s)$ & $0.0784 _{ - 0.0033 } ^ { + 0.0046 }$ & $8.82 _{ - 0.55 } ^ { + 0.7 }$ & $0.892 _{ - 0.026 } ^ { + 0.028 }$ & $6.95 _{ - 0.27 } ^ { + 0.3 }$ & 10.595 & & MINERVA (1) & & SB 2 from MINERVA observations. \\\\\n422914082 & TYC 0046-00133-1 & $1431.5538 _{ - 0.0014 } ^ { + 0.0017 }$ & $12.91 _{ - 3.91 } ^ { + 8.97 }(s)$ & $0.0418 _{ - 0.0015 } ^ { + 0.0016 }$ & $3.96 _{ - 0.32 } ^ { + 0.35 }$ & $0.36 _{ - 0.25 } ^ { + 0.28 }$ & $4.07 _{ - 0.09 } ^ { + 0.126 }$ & 11.026 & Sinistro (1) & NRES (1) & & \\\\\n427344083 & TWOMASS 22563609+7040518 & $1961.8967 _{ - 0.0031 } ^ { + 0.0036 }$ & $7.77 _{ - 5.6 } ^ { + 9.65 }(s)$ & $0.107 _{ - 0.016 } ^ { + 0.025 }$ & $12.27 _{ - 1.87 } ^ { + 2.9 }$ & $0.834 _{ - 0.484 } ^ { + 0.094 }$ & $2.88 _{ - 0.3 } ^ { + 0.42 }$ & 13.404 & & & & \\\\\n436873727 & HIP 13224 & $1803.83679 _{ - 0.00058 } ^ { + 0.00056 }$ & $19.26 _{ - 5.95 } ^ { + 6.73 }(s)$ & $0.05246 _{ - 0.00061 } ^ { + 0.00059 }$ & $10.02 _{ - 0.43 } ^ { + 0.41 }$ & $0.767 _{ - 0.057 } ^ { + 0.038 }$ & $5.462 _{ - 0.081 } ^ { + 0.074 }$ & 7.51 & & & & \\\\ \n441642457* & TYC 3858-00452-1 & $1745.5102 _{ - 0.0108 } ^ { + 0.0097 }$ & $79.8072 _{ - 0.0071 } ^ { + 0.0076 }$ & $0.0281 _{ - 0.0024 } ^ { + 0.0033 }$ & $3.55 _{ - 0.34 } ^ { + 0.46 }$ & $0.934 _{ - 0.023 } ^ { + 0.026 }$ & $6.9 _{ - 0.39 } ^ { + 0.6 }$ & 9.996 & & & & \\\\\n441765914* & TWOMASS 17253007+7552562 & $1769.6154 _{ - 0.0058 } ^ { + 0.0093 }$ & $161.6 _{ - 58.2 } ^ { + 1460.1 }(s)$ & $0.0411 _{ - 0.0024 } ^ { + 0.0119 }$ & $3.6 _{ - 0.3 } ^ { + 1.01 }$ & $0.45 _{ - 0.32 } ^ { + 0.48 }$ & $7.44 _{ - 0.36 } ^ { + 1.08 }$ & 11.638 & & & & \\\\\n452920657 & TWOMASS 00332018+5906355 & $1810.5765 _{ - 0.0031 } ^ { + 0.003 }$ & $53.2 _{ - 29.0 } ^ { + 34.3 }(s)$ & $0.135 _{ - 0.016 } ^ { + 0.012 }$ & $9.71 _{ - 1.16 } ^ { + 0.9 }$ & $0.73 _{ - 0.48 } ^ { + 0.11 }$ & $4.6 _{ - 0.26 } ^ { + 0.29 }$ & 14.167 & SBIG (1) & & & \\\\\n455737331 & TYC 2779-00785-1 & $1780.7084 _{ - 0.008 } ^ { + 0.0073 }$ & $50.4 _{ - 17.6 } ^ { + 75.0 }(s)$ & $0.0257 _{ - 0.0016 } ^ { + 0.002 }$ & $3.05 _{ - 0.24 } ^ { + 0.29 }$ & $0.43 _{ - 0.29 } ^ { + 0.33 }$ & $6.6 _{ - 0.43 } ^ { + 0.5 }$ & 10.189 & SBIG (1) & & Gemini & \\\\\n456909420 & TYC 1208-01094-1 & $1779.4109 _{ - 0.0026 } ^ { + 0.0022 }$ & $5.78 _{ - 5.29 } ^ { + 5.95 }(s)$ & $0.078 _{ - 0.031 } ^ { + 0.045 }$ & $9.15 _{ - 3.61 } ^ { + 5.27 }$ & $0.973 _{ - 0.495 } ^ { + 0.063 }$ & $1.73 _{ - 0.27 } ^ { + 0.28 }$ & 10.941 & & & & \\\\\n458451774 & TWOMASS 12551793+4431260 & $1917.1875 _{ - 0.0019 } ^ { + 0.0019 }$ & $12.39 _{ - 6.34 } ^ { + 83.97 }(s)$ & $0.0752 _{ - 0.0054 } ^ { + 0.0211 }$ & $3.33 _{ - 0.26 } ^ { + 0.92 }$ & $0.61 _{ - 0.43 } ^ { + 0.32 }$ & $2.08 _{ - 0.19 } ^ { + 0.59 }$ & 13.713 & & & & \\\\\n48018596 & TYC 3548-00800-1 & $1713.4514 _{ - 0.0063 } ^ { + 0.0046 }$ & $100.1145 _{ - 0.0018 } ^ { + 0.0021 }$ & $0.049 _{ - 0.0081 } ^ { + 0.018 }$ & $7.88 _{ - 1.33 } ^ { + 2.9 }$ & $0.984 _{ - 0.028 } ^ { + 0.027 }$ & $2.83 _{ - 0.26 } ^ { + 0.29 }$ & 9.595 & & NRES (1) & Gemini & \\\\\n53309262 & TWOMASS 07475406+5741549 & $1863.1133 _{ - 0.0064 } ^ { + 0.0061 }$ & $294.8 _{ - 96.0 } ^ { + 327.0 }(s)$ & $0.1239 _{ - 0.0075 } ^ { + 0.0098 }$ & $5.38 _{ - 0.36 } ^ { + 0.46 }$ & $0.46 _{ - 0.31 } ^ { + 0.28 }$ & $6.74 _{ - 0.45 } ^ { + 0.62 }$ & 15.51 & & & & \\\\\n53843023 & TYC 6956-00758-1 & $1328.0335 _{ - 0.0054 } ^ { + 0.0057 }$ & $202.0 _{ - 189.0 } ^ { + 272.0 }(s)$ & $0.058 _{ - 0.02 } ^ { + 0.056 }$ & $5.14 _{ - 1.77 } ^ { + 4.99 }$ & $0.962 _{ - 0.597 } ^ { + 0.083 }$ & $4.25 _{ - 0.72 } ^ { + 0.66 }$ & 11.571 & & & & \\\\\n55525572* & TYC 8876-01059-1 & $1454.6713 _{ - 0.0066 } ^ { + 0.0065 }$ & $83.8951 _{ - 0.004 } ^ { + 0.004 }$ & $0.0343 _{ - 0.001 } ^ { + 0.0021 }$ & $7.31 _{ - 0.46 } ^ { + 0.56 }$ & $0.43 _{ - 0.29 } ^ { + 0.31 }$ & $13.54 _{ - 0.3 } ^ { + 0.51 }$ & 10.358 & & CHIRON (5) & Gemini & Confirmed planet \\citep{2020eisner} \\\\\n63698669* & TYC 6993-00729-1 & $1364.6226 _{ - 0.0074 } ^ { + 0.0067 }$ & $73.6 _{ - 26.8 } ^ { + 133.6 }(s)$ & $0.0248 _{ - 0.0019 } ^ { + 0.0023 }$ & $2.15 _{ - 0.2 } ^ { + 0.25 }$ & $0.42 _{ - 0.29 } ^ { + 0.35 }$ & $5.63 _{ - 0.32 } ^ { + 0.57 }$ & 10.701 & SBIG (1) & & & \\\\\n70887357* & TYC 5883-01412-1 & $1454.3341 _{ - 0.0016 } ^ { + 0.0015 }$ & $56.1 _{ - 15.3 } ^ { + 18.8 }(s)$ & $0.0605 _{ - 0.0027 } ^ { + 0.0027 }$ & $12.84 _{ - 0.86 } ^ { + 0.9 }$ & $0.917 _{ - 0.028 } ^ { + 0.016 }$ & $7.29 _{ - 0.18 } ^ { + 0.19 }$ & 9.293 & & & & \\\\\n7422496$\\dagger$ & HIP 25359 & $1470.3625 _{ - 0.0031 } ^ { + 0.0023 }$ & $61.4 _{ - 16.7 } ^ { + 49.0 }(s)$ & $0.0255 _{ - 0.001 } ^ { + 0.0011 }$ & $2.44 _{ - 0.15 } ^ { + 0.16 }$ & $0.37 _{ - 0.25 } ^ { + 0.29 }$ & $5.89 _{ - 0.15 } ^ { + 0.15 }$ & 9.36 & & MINERVA (4) & & SB 2 from MINERVA observations. \\\\\n82452140 & TYC 3076-00921-1 & $1964.292 _{ - 0.011 } ^ { + 0.011 }$ & $21.1338 _{ - 0.0052 } ^ { + 0.0066 }$ & $0.0266 _{ - 0.0019 } ^ { + 0.0027 }$ & $2.95 _{ - 0.25 } ^ { + 0.34 }$ & $0.42 _{ - 0.29 } ^ { + 0.36 }$ & $5.87 _{ - 0.62 } ^ { + 0.94 }$ & 10.616 & & & & \\\\\n88840705 & TYC 3091-00808-1 & $2026.6489 _{ - 0.001 } ^ { + 0.001 }$ & $260.6 _{ - 87.6 } ^ { + 142.2 }(s)$ & $0.109 _{ - 0.023 } ^ { + 0.027 }$ & $9.98 _{ - 2.28 } ^ { + 2.75 }$ & $1.001 _{ - 0.042 } ^ { + 0.037 }$ & $4.72 _{ - 0.13 } ^ { + 0.15 }$ & 9.443 & & & & \\\\\n91987762* & HIP 47288 & $1894.25381 _{ - 0.00051 } ^ { + 0.00047 }$ & $10.51 _{ - 3.48 } ^ { + 3.67 }(s)$ & $0.05459 _{ - 0.00106 } ^ { + 0.00097 }$ & $9.56 _{ - 0.56 } ^ { + 0.52 }$ & $0.771 _{ - 0.062 } ^ { + 0.033 }$ & $4.342 _{ - 0.073 } ^ { + 0.063 }$ & 7.87 & & NRES (4) & Gemini & \\\\\n95768667 & TYC 1434-00331-1 & $1918.3318 _{ - 0.0093 } ^ { + 0.0079 }$ & $26.9 _{ - 12.4 } ^ { + 72.3 }(s)$ & $0.0282 _{ - 0.0022 } ^ { + 0.0031 }$ & $3.54 _{ - 0.32 } ^ { + 0.43 }$ & $0.48 _{ - 0.33 } ^ { + 0.35 }$ & $5.4 _{ - 0.64 } ^ { + 0.76 }$ & 10.318 & & & & \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\textbf{Properties of PHT candidates (continued)}}\n\\label{tab:PHT-caniddates2}\n\\end{table}\n\\end{landscape}\n\n\n\\section{Conclusion}\n\\label{sec:condlusion}\n\nWe present the results from the analysis of the first 26 \\emph{TESS}\\ sectors. The outlined citizen science approach engages over 22 thousand registered citizen scientists who completed 12,617,038 classifications from December 2018 through August 2020 for the sectors observed during the first two years of the \\emph{TESS}\\ mission. We applied a systematic search for planetary candidates using visual vetting by multiple volunteers to identify \\emph{TESS}\\ targets that are most likely to host a planet. Between 8 and 15 volunteers have inspected each \\emph{TESS}\\ light curve and marked times of transit-like events using the PHT online interface. For each light curve, the markings from all the volunteers who saw that target were combined using an unsupervised machine learning method, known as DBSCAN, in order to identify likely transit-like events. Each of these identified events was given a transit score based on the number of volunteers who identified a given event and on the user weighting of each of those volunteers. Individual user weights were calculated based on the user's ability to identify simulated transit events, injected into real \\emph{TESS}\\ light curves, that are displayed on the PHT site alongside of the real data. The transit scores were then used to generate a ranked list of candidates that range from most likely to least likely to host a planet candidate. The top 500 highest ranked candidates were further vetted by the PHT science team. This stage of vetting primarily made use of the open source {\\sc latte} \\citep{LATTE2020} tool which generates a number of standard diagnostic plots that help identify promising candidates and weed out false positive signals. \n\nOn average we found around three high priority candidates per sector which were followed up using ground based telescopes, where possible. To date, PHT has statistically confirmed one planet, TOI-813 \\citep{2020eisner}: a Saturn-sized planet on an 84 day orbit around a subgiant host star. Other PHT identified planets listed in this paper are being followed up by other teams of astronomers, such as TOI-1899 (TIC 172370679) which was recently confirmed to be a warm Jupiter transiting an M-dwarf \\citep{canas2020}. The remaining candidates outlined in this paper require further follow-up observations to confirm their planetary nature.\n\nThe sensitivity of our transit search effort was assessed using synthetic data, as well as the known TOI and TCE candidates flagged by the SPOC pipeline. For simulated planets (where simulated signals are injected into real \\emph{TESS}\\ light curves) we have shown that the recovery efficiency of human vetting starts to decrease for transit-signals that have a SNR less than 7.5. The detection efficiency was further evaluated by the fractional recovery of the TOI and TCEs. We have shown that PHT is over 85 \\% complete in the recovery of planets that have a radius greater than 4 $R_{\\oplus}$, 51 \\% complete for radii between 3 and 4 $R_{\\oplus}$ and 49 \\% complete for radii between 2 and 3 $R_{\\oplus}$. Furthermore, we have shown that human vetting is not sensitive to the number of transits present in the light curve, meaning that they are equally likely to identify candidates on longer orbital periods as they are those with shorter orbital periods for periods greater than $\\sim$ 1 day. Planets with periods shorter than around 1 day exhibit over 20 transits within one \\emph{TESS}\\ sectors resulting in a decrease in identification by the volunteers. This is due to many volunteers only marking a random subset of these events, resulting in a lack of consensus on any given transit event and thus decreasing the overall transit score of these light curves. \n\nIn addition to searching for signals due to transiting exoplanets, PHT provides a platform that can be used to identify other stellar phenomena that may otherwise be difficult to identify with automated pipelines. Such phenomena, including eclipsing binaries, multiple stellar systems, dwarf novae, and stellar flares are often mentioned on the PHT discussion forums where volunteers can use searchable hashtags and comments to bring these systems to the attention of other citizen scientists as well as the PHT science team. All of the eclipsing binaries identified on the site, for example, are being used and vetted by the \\emph{TESS}\\ Eclipsing Binary Working Group (Prsa et al. in prep). Furthermore, we have investigated the nature of all of the targets that were identified as possible multiple stellar systems, as summarised in Table~\\ref{tab:PHT-multis}.\n\nOverall we have shown that large scale visual vetting can complement the findings \\textcolor{black}{from the major \\emph{TESS}\\ pipeline} by identifying longer period planets that may only exhibit a single transit event in their light curve, as well as in finding signals that are aperiodic or embedded in a strong varying stellar signal. The identification of planets around stars with variable signals allow us to potentially characterise the host-star (e.g., with asteroseismology or spot modulation). Additionally, the longer period planets are integral to our understanding of how planet systems form and evolve, as they allow us to investigate the evolution of planets that are farther away from their host star and therefore less dependent on stellar radiation. \\textcolor{black}{While automated pipelines specifically designed to identify single transit events in the \\emph{TESS}\\ data exist \\citep[e.g., ][]{Gill2020}, neither their methodology nor the full list of their findings are yet publicly available and thus we are unable to compare results.} \n\nThe planets that PHT finds have longer periods ($\\gtrsim$ 27 d) than those found in \\emph{TESS}\\ data using automated pipelines, and are more typical of the Kepler sample (25\\% of Kepler confirmed planets have periods greater than 27 days\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}). However, the Kepler planets are considerably fainter, and thus less amenable to ground-based follow-up or atmospheric characterisation from space (CHEOPS and JWST). Thus PHT helps to bridge the parameter spaces covered by these two missions, by identifying longer period planet candidates around bright, nearby stars, for which we can ultimately obtain precise planetary mass estimates. Although statistical characterisation of exo-planetary systems is no doubt important, precise mass measurements are key to developing our understanding of exoplanets and the physics which dictate their evolution. In particular, identification of this PHT sample provides follow-up targets to investigate the dependence of photo-evaporation on the mass of planets as well as on the planet radius, and will help our understanding of the photo-evaporation valley at longer orbital periods \\citep{Owen2013}. \n\nPHT will continue to operate throughout the \\emph{TESS}\\ extended mission, hopefully allowing us to identify even longer period planets as well as help verify some of the existing candidates with additional transits. \n\n\n\n\\begin{table*}\n\\resizebox{0.95\\textwidth}{!}{\n\\begin{tabular}{cccccccccc}\n\\textbf{TIC} & \\textbf{Period (days)} & \\textbf{Epoch (\\textcolor{black}{BJD - 2457000})} & \\textbf{Depth (ppm)} & \\textbf{Comment} \\\\\n\\hline\n13968858 & $3.4850 \\pm 0.001$ & $ 1684.780 \\pm 0.005$ & 410000 & Candidate multiple system \\\\\n & $1.4380 \\pm 0.001$ & $ 1684.335 \\pm 0.005$ & 50000 & \\\\\n35655828 & $ 8.073 \\pm 0.01$ & $ 1550.94 \\pm 0.01 $ & 23000 & Confirmed blend \\\\\n & $ 1.220 \\pm 0.001 $ & $ 1545.540 \\pm 0.005 $ & 2800 & \\\\\n63291675 & $ 8.099 \\pm 0.003 $ & $ 1685.1 \\pm 0.01 $ & 60000 & Confirmed blend \\\\\n & $ 1.4635 \\pm 0.0005 $ & $ 1683.8 \\pm 0.1 $ & 7000 & \\\\\n63459761 & $4.3630 \\pm 0.003 $ & $ 1714.350 \\pm 0.005 $ & 160000 & Candidate multiple system \\\\\n & $4.235 \\pm $ 0.005 & $ 1715.130 \\pm 0.03$ & 35000 & \\\\\n104909909 & $1.3060 \\pm 0.0001$ & $ 1684.470 \\pm 0.005$ & 32000 & Candidate multiple system \\\\\n & $2.5750 \\pm 0.003$ & $ 1684.400 \\pm 0.005$ & 65000 & \\\\\n115980439 & $ 4.615 \\pm 0.002 $ & $ 1818.05 \\pm 0.01 $ & 95000 & Confirmed blend \\\\\n & $ 0.742 \\pm 0.005 $ & $ 1816.23 \\pm 0.02 $ & 2000 & \\\\\n120362128 & $ 3.286 \\pm 0.002 $ & $ 1684.425 \\pm 0.01 $ & 33000 & Candidate multiple system \\\\\n & $ - $ & $ 1701.275 \\pm 0.02 $ & 12000 & \\\\\n & $ - $ & $ 1702.09 \\pm 0.02 $ & 36000 & \\\\\n121945407 & $ 0.9056768 \\pm 0.00000002$ & $-1948.76377 \\pm 0.0000001$ & 2500 & Confirmed multiple system $^{(\\mathrm{a})}$ \\\\\n & $ 45.4711 \\pm 0.00002$ & $-1500.0038 \\pm 0.0004 $ & 7500 & \\\\\n122275115 & $ - $ & $ 1821.779 \\pm 0.01 $ & 155000 & Candidate multiple system \\\\\n & $ - $ & $ 1830.628 \\pm 0.01 $ & 63000 & \\\\\n & $ - $ & $ 1838.505 \\pm 0.01 $ & 123000 & \\\\\n229804573 & $1.4641 \\pm 0.0005$ & $ 1326.135 \\pm 0.005$ & 180000 & Candidate multiple system \\\\\n & $0.5283 \\pm 0.0001$ & $ 1378.114 \\pm 0.005$ & 9000 & \\\\\n252403752 & $ - $ & $ 1817.73 \\pm 0.01 $ & 2800 & Candidate multiple system \\\\\n & $ - $ & $ 1829.76 \\pm 0.01 $ & 23000 & \\\\\n & $ - $ & $ 1833.63 \\pm 0.01 $ & 5500 & \\\\\n258837989 & $0.8870 \\pm 0.001$ & $ 1599.350 \\pm 0.005$ & 64000 & Candidate multiple system \\\\\n & $3.0730 \\pm 0.001$ & $ 1598.430 \\pm 0.005$ & 25000 & \\\\\n266958963 & $1.5753 \\pm 0.0002$ & $ 1816.425 \\pm 0.001$ & 265000 & Candidate multiple system \\\\\n & $2.3685 \\pm 0.0001$ & $ 1817.790 \\pm 0.001$ & 75000 & \\\\\n278956474 & $5.488068 \\pm 0.000016 $ & $ 1355.400 \\pm 0.005$ & 93900 & Confirmed multiple system $^{(\\mathrm{b})}$ \\\\\n & $5.674256 \\pm -0.000030$ & $ 1330.690 \\pm 0.005$ & 30000 & \\\\\n284925600 & $ 1.24571 \\pm 0.00001 $ & $ 1765.248 \\pm 0.005 $ & 490000 & Confirmed blend \\\\\n & $ 0.31828 \\pm 0.00001 $ & $ 1764.75 \\pm 0.005 $ & 35000 & \\\\\n293954660 & $2.814 \\pm 0.001 $ & $ 1739.177 \\pm 0.03 $ & 272000 & Confirmed blend \\\\\n & $4.904 \\pm 0.03 $ & $ 1739.73 \\pm 0.01 $ & 9500 & \\\\\n312353805 & $4.951 \\pm 0.003 $ & $ 1817.73 \\pm 0.01 $ & 66000 & Confirmed blend \\\\\n & $12.89 \\pm 0.01 $ & $ 1822.28 \\pm 0.01$ & 19000 & \\\\\n318210930 & $ 1.3055432 \\pm 0.000000033$ & $ -653.21602 \\pm 0.0000013$ & 570000 & Confirmed multiple system $^{(\\mathrm{c})}$ \\\\\n & $ 0.22771622 \\pm 0.0000000035$& $ -732.071119 \\pm 0.00000026 $ & 220000 & \\\\\n336434532 & $ 3.888 \\pm 0.002 $ & $ 1713.66 \\pm 0.01 $ & 22900 & Confirmed blend \\\\\n & $ 0.949 \\pm 0.003 $ & $ 1712.81 \\pm 0.01 $ & 2900 & \\\\\n350622185 & $1.1686 \\pm 0.0001$ & $ 1326.140 \\pm 0.005$ & 200000 & Candidate multiple system \\\\\n & $5.2410 \\pm 0.0005$ & $ 1326.885 \\pm 0.05$ & 4000 & \\\\\n375422201 & $9.9649 \\pm 0.001$ & $ 1711.937 \\pm 0.005$ & 245000 & Candidate multiple system \\\\\n & $4.0750 \\pm 0.001$ & $ 1713.210 \\pm 0.01 $ & 39000 & \\\\\n376606423 & $ 0.8547 \\pm 0.0002 $ & $ 1900.766 \\pm 0.005 $ & 9700 & Candidate multiple system \\\\\n & $ - $ & $ 1908.085 \\pm 0.01 $ & 33000 & \\\\\n394177355 & $ 94.22454 \\pm 0.00040 $ & $ - $ & - & Confirmed multiple system $^{(\\mathrm{d})}$ \\\\\n & $ 8.6530941 \\pm 0.0000016$ & $-2038.99492 \\pm 0.00017 $ & 140000 & \\\\\n & $ 1.5222468 \\pm 0.0000025$ & $ -2039.1201 \\pm 0.0014 $ & - & \\\\\n & $ 1.43420486 \\pm 0.00000012 $ & $-2039.23941 \\pm 0.00007 $ & - & \\\\\n424508303 & $ 2.0832649 \\pm 0.0000029 $ & $-3144.8661 \\pm 0.0034 $ & 430000 & Confirmed multiple system $^{(\\mathrm{e})}$ \\\\\n & $ 1.4200401 \\pm 0.0000042 $ & $-3142.5639 \\pm 0.0054 $ & 250000 & \\\\\n441794509 & $ 4.6687 \\pm 0.0002 $ & $ 1958.895 \\pm 0.005 $ & 34000 & Candidate multiple system \\\\\n & $ 14.785 \\pm 0.002 $ & $ 1960.845 \\pm 0.005 $ & 17000 & \\\\\n470710327 & $ 9.9733 \\pm 0.0001 $ & $ 1766.27 \\pm 0.005 $ & 51000 & Confirmed multiple system $^{(\\mathrm{f})}$ \\\\\n & $ 1.104686 \\pm 0.00001 $ & $ 1785.53266 \\pm 0.000005$ & 42000 & \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\nNote -- $^{(\\mathrm{a})}$ KOI-6139, \\citet{Borkovits2013}; \n$^{(\\mathrm{b})}$ \\citet{2020Rowden}\n$^{(\\mathrm{c})}$ \\citet{Koo2014}; \n$^{(\\mathrm{d})}$ KOI-3156, \\citet{2017Helminiak};\n$^{(\\mathrm{e})}$ V994 Her; \\citet{Zasche2016}; \n$^{(\\mathrm{f})}$ Eisner et al. {\\it in prep.}\n}\n\n\\label{tab:PHT-multis}\n\n\\end{table*}\n\n\\section*{Data Availability}\n\nAll of the \\emph{TESS}\\ data used within this article are hosted and made publicly available by the Mikulski Archive for Space Telescopes (MAST, \\url{http:\/\/archive.stsci.edu\/tess\/}). Similarly, the Planet Hunters TESS classifications made by the citizen scientists can be found on the Planet Hunters Analysis Database (PHAD, \\url{https:\/\/mast.stsci.edu\/phad\/}), which is also hosted by MAST. All planet candidates and their properties presented in this article have been uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS, \\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}) website as community TOIs (cTOIs), under their corresponding TIC IDs. The ground-based follow-up observations of individual targets will be shared on reasonable request to the corresponding author.\n\nThe models of individual transit events and the data validation reports used for the vetting of the targets were both generated using publicly available open software codes, \\texttt{pyaneti}\\ and {\\sc latte}.\n\n\\section*{Acknowledgements} \n\nThis project works under the in \\textit{populum veritas est} philosophy, and for that reason we would like to thank all of the citizen scientists who have taken part in the Planet Hunters TESS project and enable us to find many interesting astrophysical systems. \n\nSome of the observations in the paper made use of the High-Resolution Imaging instruments `Alopeke and Zorro. `Alopeke and Zorro were funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. `Alopeke and Zorro were mounted on the Gemini North and South telescope of the international Gemini Observatory, a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci\\'{o}n y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog\\'{i}a e Innovaci\\'{o}n (Argentina), Minist\\'{e}rio da Ci\\^{e}ncia, Tecnologia, Inova\\c{c}\\~{o}es e Comunica\\c{c}\\~{o}es (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). The authors also acknowledge the very significant cultural role and sacred nature of Maunakea. We are most fortunate to have the opportunity to conduct observations from this mountain.\n\nThis project has also received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement N$^\\circ$730890. This material reflects only the authors views and the Commission is not liable for any use that may be made of the information contained therein. This work makes use of observations from the Las Cumbres Observatory global telescope network, including the NRES spectrograph and the SBIG and Sinistro photometric instruments. \n\nFurthermore, NLE thanks the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant N$^\\circ$1829740, the Brinson Foundation, and the Moore Foundation; her participation in the program has benefited this work. Finally, CJ acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement N$^\\circ$670519: MAMSIE), and from the Research Foundation Flanders (FWO) under grant agreement G0A2917N (BlackGEM). \n\nThis research made use of Astropy, a community-developed core Python package for Astronomy \\citep{astropy2013}, matplotlib \\citep{matplotlib}, pandas \\citep{pandas}, NumPy \\citep{numpy}, astroquery \\citep{ginsburg2019astroquery} and sklearn \\citep{pedregosa2011scikit}. \n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Planet candidate descriptions}\n\\label{appendixA}\n\nA short outline all of the planet candidates, and any conclusions drawn from follow-up observations (where available). A more in depth description of the ground-based data will be presented in a follow-up paper. Unless stated otherwise, these candidates are not TOIs at the time of writing. Candidates for which we have no additional information to complement the results presented in Table~\\ref{tab:PHT-caniddates} are not discussed further here.\n\n\\subsection{Single-transit planet candidates}\n\n\n\\textbf{TIC 103633672.} Single transit event identified in Sector 20. The single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary. We caution that there is a star on the same pixels, which is 0.1 mag brighter. We are unable to rule this star out as the cause for the transit-like signal.\n\n\\textbf{TIC 110996418.} Single transit event identified in Sector 10. We caution that there is a star on the same \\emph{TESS}\\ pixel, which is 2.4 magnitudes fainter than the target.\n\n\\textbf{TIC 128703021.} Single transit event identified in Sector 11. With a stellar radius of 1.6 $R_{\\odot}$ and a T$_{eff}$ of 6281 this host star is likely in the subgiant phase of its evolution. The 43 spectra obtained with MINERVA australis and the two obtained with LCO\/NRES are consistent with a planetary nature. Gemini speckle interferometry shows no nearby companion stars.\n\n\n\\textbf{TIC 142087638.} Single transit event identified in Sector 7. The best fit \\texttt{pyaneti}\\ model of the transit suggests an orbital period of only 3.14 d. As there are no additional transits seen in the light curve, this period is clearly not possible. We caution that the transit is most likely caused by a grazing object, and is therefore likely to be caused by a stellar companion. However, without further data we are unable to rule this candidate out as being planetary in nature.\n\n\\textbf{TIC 159159904.} Single transit event identified in Sector 22. The initial two observations obtained using LCO\/NRES show no sign of the candidate being a double lined spectroscopic binary.\n\n\n\\textbf{TIC 166184426.} Single transit event identified in Sector 11. Since the PHT discovery this cTOI has been become the priority 1 (1=highest, 5=lowest) target TOI 1955.01.\n\n\n\\textbf{TIC 172370679.} Single transit event identified in Sector 15. \\textcolor{black}{This candidate was independently discovered and verified using a BLS algorithm used to search for transiting planets around M-dwarfs. The candidate is now the confirmed planet TOI 1899 b \\citep{canas2020}.} \n\n\\textbf{TIC 174302697.} Single transit event identified in Sectors 16. With a stellar radius of 1.6 $R_{\\odot}$ and a T$_{eff}$ of 6750 this host star is likely in the subgiant phase of its evolution. \\textcolor{black}{This candidate was initially flagged as a TCE and but was erroneously discounted due to the pipeline mistaking the data glitch at the time of a momentum dump as a secondary eclipse.} Since the PHT discovery this cTOI has become the priority 3 target TOI 1896.01. \n\n\\textbf{TIC 192415680.} Single transit event identified in Sector 18. The two epochs of RV measurement obtained with OHP\/SOPHIE are consistent with a planetary scenario.\n\n\\textbf{TIC 192790476.} Single transit event identified in Sector 5. This target has been identified to be a wide binary with am angular separation of 72.40 arcseconds \\citep{andrews2017wideBinary} and a period of 162705 years \\citep{benavides2010new}. The star exhibits large scale variability on the order of around 10 d. The signal is consistent with that of spot modulations, which would suggest that this is a slowly rotating star.\n\n\n\n\\textbf{TIC 219466784.} Single transit event identified in Sector 22. We caution that there is a nearby companion located within the same \\emph{TESS}\\ pixel at an angular separation of 16.3 with a Vmag of 16.3\". Since the PHT discovery this cTOI has become the priority 2 target TOI 2007.01.\n\n\n\n\\textbf{TIC 229055790.} Single transit event identified in Sector 21. We note that the midpoint of the transit-like events coincides with a \\emph{TESS}\\ momentum dump, however, we believe the shape to be convincing enough to warrant further investigation. The two LCO\/NRES spectra show no sign of this being a spectroscopic binary.\n\n\\textbf{TIC 229608594.} Single transit event identified in Sector 24. Since the PHT discovery this cTOI has become the priority 3 target TOI 2298.01.\n\n\n\n\\textbf{TIC 233194447.} Single transit event identified in Sector 14. The transit-like event is shallow and asymmetric and we cannot definitively rule out systematics as the cause for the event without additional data. The initial two LCO\/NRES spectra show no sign of this target being a spectroscopic binary.\n\n\\textbf{TIC 237201858.} Single transit event identified in Sector 18. The single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary.\n\n\\textbf{TIC 243187830.} Single transit event identified in Sector 18. There are no nearby bright stars. \\textcolor{black}{This light curve was initially flagged as a TCE, however, the flagged events corresponded to stellar variability and not the same event identified by PHT.} The single LCO\/NRES spectrum shows no sign of this being a double lined spectroscopic binary. Since the PHT discovery this cTOI has become the priority 3 target TOI 2009.01.\n\n\\textbf{TIC 243417115.} Single transit event identified in Sector 11. We note that the best fit \\texttt{pyaneti}\\ model of the transit suggests an orbital period of only 1.81 d. As there are no additional transits seen in the light curve, this period is clearly not possible. We caution that the transit is most likely caused by a grazing object, and is therefore likely to be caused by a stellar companion. However, without further follow-up data we are unable to rule this candidate out as being planetary in nature.\n\n\n\\textbf{TIC 264544388.} Single transit event identified in Sector 19. The single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary. Apart from the single transit event, the light curve shows no obvious signals. A periodogram of the light curve, however, reveals a series of five significant peaks, nearly equidistantly spaced by $\\sim1.03$~d$^{-1}$. Additionally, a rotationally split quintuplet is visible at 7.34~d$^{-1}$, with a splitting of $\\sim0.12$~d$^{-1}$, suggesting an $\\ell=2$ p-mode pulsation. The Maelstrom code \\citep{hey2020maelstrom} revealed pulsation timing variations which are consistent with a long period planet. \\textcolor{black}{The short period signal, which was also identified by the periodogram, was flagged as a TCE, however, the single-transit event was not flagged as a TCE.} Since the PHT discovery this cTOI has become the priority 3 target TOI 1893.01.\n\n\\textbf{TIC 264766922.} Single transit event identified in Sector 8. With a stellar radius of 1.7 $R_{\\odot}$ and a T$_{eff}$ of 6913 K this host star is likely entering the subgiant phase of its evolution. The V-shape of this transit and the resultant high impact parameter suggests that the object is grazing. We can therefore not rule out that this candidate it a grazing eclipsing binary. There are clear p-mode pulsations at frequencies of 9.01 and 11.47 cycles per day, as well as possible g-mode pulsations. \\textcolor{black}{A very short period signal within this light curve was flagged as a TCE, however, the single transit event was ignored by the pipeline.}\n\n\\textbf{TIC 26547036.} Single transit event identified in Sector 14. The four LCO\/NRES observations are consistent with the target being a planetary body and show no sign of the signal being caused by a spectroscopic binary. We caution that there is a star on the same \\emph{TESS}\\ pixel, however, this star is 8.2 magnitudes fainter than the target, and therefore unable to be responsible for the transit event seen in the light curve. Gemini speckle interferometry reveal no additional nearby companion stars. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit event the pipeline identified further periodic signals that correspond to times of momentum dumps. Due to this, the candidate was never promoted to TOI status.}\n\n\n\\textbf{TIC 278990954.} Single transit event identified in Sector 12. With a stellar radius of 2.6 $R_{\\odot}$ and a T$_{eff}$ of 5761 K this host star is likely in the subgiant phase of its evolution. We note that there are two additional stars on the same pixel as TIC 278990954. These two stars are 2.7 and 3.7 magnitudes fainter in the v-band than the target and can't be ruled out as the cause for the transit-like event without additional follow-up data.\n\n\\textbf{TIC 280865159.} Single transit event identified in Sector 16. Gemini speckle interferometry revealed any nearby companion stars. Since the PHT discovery this cTOI has become the priority 3 target TOI 1894.01.\n\n\\textbf{TIC 284361752.} Single transit event identified in Sector 26. Since the PHT discovery this cTOI has become the priority 2 target TOI 2294.01.\n\n\n\\textbf{TIC 296737508.} Single transit event identified in Sector 8. The single LCO\/NRES and the single MINERVA australis spectra show no sign of this being a spectroscopic binary. The Sinistro snapshot image revealed no additional nearby companions.\n\n\\textbf{TIC 298663873.} Single transit event identified in Sector 19. The two LCO\/NRES spectra show no sign of this being a spectroscopic binary. With a stellar radius of 1.6 $R_{\\odot}$ and a T$_{eff}$ of 6750 this host star is likely in the subgiant phase of its evolution. Gemini speckle images obtained by other teams show no signs of there being nearby companion stars. Since the PHT discovery this cTOI has become the priority 3 target TOI 2180.01.\n\n\n\\textbf{TIC 303050301.} Single transit event identified in Sector 2. The variability of the light curve is consistent with spot modulation. A single LCO\/NRES spectrum shows no signs of this being a double lined spectroscopic binary.\n\n\\textbf{TIC 303317324.} Single transit event identified in Sector 2. We note that a second transit was later seen in Sector 29, however, as this work only covers sectors 1-26 of the primary \\emph{TESS}\\ mission, this candidates is considered a single-transit event in this work. \n\n\n\\textbf{TIC 304142124.} Single transit event identified in Sector 10.\\textcolor{black}{This target was independently identified as part of the Planet Finder Spectrograph, which uses precision RVs \\citep{diaz2020}. This candidate is know the confirmed planet HD 95338 b.}\n\n\n\n\n\n\n\n\\textbf{TIC 331644554.} Single transit event identified in Sector 16. There is a clear mono-periodic signal in the periodogram at around 11.2 cycles per day, which is consistent with p-mode pulsation.\n\n\\textbf{TIC 332657786.} Single transit event identified in Sector 8. We caution that there is a star on the adjacent \\emph{TESS}\\ pixel that is brighter in the V-band by 2.4 magnitudes. At this point we are unable to rule out this star as the cause of the transit-like signal. \n\n\n\\textbf{TIC 356700488.} Single transit event identified in Sector 16. There is a clear mono-periodic signal in the periodogram at around 1.2 cycles per day, which is consistent with either spot modulation or g-mode pulsation. However, there is no clear signal visible in the light curve that would allow us to differentiate between these two scenarios based on the morphology of the variation. Since the PHT discovery this cTOI has become the priority 3 target TOI 2098.01.\n\n\\textbf{TIC 356710041.} Single transit event identified in Sector 23. With a stellar radius of 2.8 $R_{\\odot}$ and a T$_{eff}$ of 5701 K this host star is likely in the subgiant phase of its evolution. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit event, the pipeline identified a further event that corresponds to the time of a momentum dump. Due to this the candidate failed the `odd-even test' and was initially discarded as a TOI.} Since the PHT discovery this cTOI has become the priority 3 target TOI 2065.01\n\n\\textbf{TIC 369532319.} Single transit event identified in Sector 16. Gemini speckle interferometry revealed no nearby companion stars.\n\n\n\\textbf{TIC 384159646.} Single transit event identified in Sector 12. The eight LCO\/NRES and six MINERVA australis spectra are consistent with this candidate being a planet. Both the SBIG snapshot and the Gemini speckle interferometry observations revealed no companion stars. Since the PHT discovery this cTOI has become the priority 3 target TOI 1895.01.\n\n\n\n\n\\textbf{TIC 418255064.} Single transit event identified in Sector 12. The Gemini speckle image shows no sign of nearby companions.\n\n\n\\textbf{TIC 422914082.} Single transit event identified in Sector 4. Single Sinistro snapshot image reveals no additional nearby stars.\n\n\\textbf{TIC 427344083.} Single transit event identified in Sector 24. We note that there is a star on the adjacent \\emph{TESS}\\ pixel to the target, which is 3.5 magnitude fainter in the V-band than the target star. We also caution that the V-shape of the transit and the high impact parameter suggest that this is a grazing transit. However, without additional follow-up observations we are unable to rule this candidate out as a planet.\n\n\n\\textbf{TIC 436873727.} Single transit event identified in Sector 18. The host star shows strong variability on the order of one day, which is consistent with spot modulations or g-mode pulsations. The periodogram reveals multi-periodic behaviour in the low frequency range consistent with g-mode pulsations. \n\n\\textbf{TIC 452920657.} Single transit event identified in Sector 17. The V-shape of this transit suggests that the object is grazing and future follow-up observations may reveal this to be an EB. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit event, the pipeline identified a further two event that corresponds to likely stellar variability. Due to this the candidate failed the `odd-even test' and was initially discarded as a TOI.}\n\n\\textbf{TIC 455737331.} Single transit event identified in Sector 17. We note that there is a star on the same \\emph{TESS}\\ pixel as the target, which is 4.5 magnitudes fainter in the V-band. Neither the SBIG snapshot nor the Gemini speckle interferometry revealed any further nearby companion stars.\n\n\\textbf{TIC 456909420.} Single transit event identified in Sector 17. We caution that the V-shape of the transit and the high impact parameter suggest that this is a grazing transit. However, without additional follow-up observations we are unable to rule this candidate out as a planet.\n\n\n\n\n\\textbf{TIC 53843023.} Single transit event identified in Sector 1. We caution that the high impact parameter returned by the best fit \\texttt{pyaneti}\\ model suggests that the transit event is caused by a grazing body. However, at this point we are unable to rule this candidate out as being planetary in nature.\n\n\\textbf{TIC 63698669.} Single transit event identified in Sector 2. The SBIG snapshot image revealed no nearby companions. \\textcolor{black}{This candidate was initially identified as a TCE, however, in addition to the single transit event, the pipeline identified a further 3 events the light curve. Due to these, additional events, which correspond to stellar variability, the candidate was not initially promoted to TOI status.} However, since the PHT discovery this cTOI has become TOI 1892.01.\n\n\\textbf{TIC 70887357.} Single transit event identified in Sector 5. With a stellar radius of 2.1 $R_{\\odot}$ and a T$_{eff}$ of 5463 K this host star is likely in the subgiant phase of its evolution. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit-event the pipeline identified a further signal, and thus failed the `odd-even' transit test.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 2008.01.\n\n\n\n\\textbf{TIC 91987762.} Single transit event identified in Sector 21. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit-event the pipeline identified a further signal, and thus failed the `odd-even' transit test.} Since the PHT discovery this cTOI has become the priority 3 target TOI 1898.01.\n\n\n\n\\subsection{Multi-transit and multi-planet candidates}\n\n\n\\textbf{TIC 160039081.} Multi-transit candidate with a period of 30.2 d. Single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary and a snapshot image using SBIG shows no nearby companions. The Gemini speckle images also show no additional nearby companions. Since the PHT discovery this cTOI has become the priority 1 target TOI 2082.01.\n\n\\textbf{TIC 167661160.} Multi-transit candidate with a period of 36.8 d. The nine LCO\/NRES and four MINERVA australis spectra have revealed this to be a long period eclipsing binary.\n\n\\textbf{TIC 179582003.} Multi-transit candidate with a period of 104.6 d. There is a clear mono-periodic signal in the periodogram at around 0.59 cycles per day, which is consistent with either spot modulation or g-mode pulsation. We caution that this candidate is located in a crowded field. With a stellar radius of 2.0 $R_{\\odot}$ and a T$_{eff}$ of 6115 K this host star is likely in the subgiant phase of its evolution.\n\n\\textbf{TIC 219501568.} Multi-transit candidate with a period of 16.6 d. With a stellar radius of 1.7 $R_{\\odot}$ and a T$_{eff}$ of 6690 K this host star is likely entering the subgiant phase of its evolution. \\textcolor{black}{This candidate was identified as a TCE, however, it was not initially promoted to TOI status as the signal was thought to be off-target by the automated pipeline.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 2259.01\n\n\\textbf{TIC 229742722.} Multi-transit candidate with a period of 63.48 d. Eight LCO\/NRES and four OHP\/SOPHIE observations are consistent with this candidate being a planet. Gemini speckle interferometry reveals no nearby companion stars. \\textcolor{black}{This candidate was flagged as a TCE in sector 20, where it only exhibits a single transit event. An additional event was identified at the time of a momentum dump, and as such it failed the `odd-even' test and was not initially promoted to TOI status.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 1895.01.\n\n\\textbf{TIC 235943205.} Multi-transit candidate with a period of 121.3 d. The LCO\/NRES and OHP\/SOPHIE observations remain consistent with a planetary nature of the signal. Since the PHT discovery this cTOI has become the priority 3 target TOI 2264.01.\n\n\n\\textbf{TIC 267542728.} Multi-transit event with period of 39.7 d. Observations obtained with Keck showed that the RV shifts are not consistent with a planetary body and are most likely due to an M-dwarf companion.\n\n\\textbf{TIC 274599700.} Multi-transit candidate with a period of 33.0 d. One of the two transit-like even is only half visible, with the other half of the event falling in a \\emph{TESS}\\ data gap.\n\n\n\n\\textbf{TIC 328933398.} Multi-planet candidates. The 2-minute cadence light curve shows two single transit events of different depths across two \\emph{TESS}\\ sectors, both of which are consistent with an independent planetary body. In addition to the short cadence data, this target was observed in an additional three sectors as part of the 30-minute cadence full frame images. These showed additional transit events for one of the planet candidates, with a period of 24.9 d. \\textcolor{black}{This light curve was initially flagged as containing a TCE event, however, the two 2-minute cadence single transit events were thought to belong to the same transiting planet. The TCE was initially discarded as the pipeline identified the events to be off-target.} However, since the PHT discovery these two cTOIs has become the priority 3 and 1 targets, TOI 1873.01 and TOI 1873.01, respectively.\n\n\\textbf{TIC 349488688.} Multi-planet candidate, with one single transit event and one multi-transit candidate with a period of 11 d. Two LCO\/NRES and two OHP\/SOPHIE spectra, along with ongoing HARPS North are consistent with both of these candidates being planetary in nature. \\textcolor{black}{The single transit event was initially identified as a TCE, however, in addition to the event it identified two other signals at the time of momentum dumps, and was therefore initially discarded by the pipeline as it failed the `odd-even' transit test.} However, since the PHT discovery the two-transit event has become the 1 targets, TOI 2319.01 (Eisner et al. in prep).\n\n\\textbf{TIC 385557214.} Multi-transit candidate with a period of 5.6 d. The prominent stellar variation seen in the light curve is likely due to spots or pulsation The high impact parameter returned by the best fit \\texttt{pyaneti}\\ modelling suggests that the transit is likely caused by a grazing object. Without further observations, however, we are unable to rule this candidate out as being planetary in nature. \\textcolor{black}{This candidate was flagged as a TCE but was not promoted to TOI status due to the other nearby stars.}\n\n\\textbf{TIC 408636441.} Multi-transit candidate with a period of 18.8 or 37.7 d. Due to \\emph{TESS}\\ data gaps, half of the period stated in Table~\\ref{tab:PHT-caniddates} is likely. The SBIG snapshot and Gemini speckle images show no signs of companion stars. \\textcolor{black}{This candidate was flagged as a TCE in sector 24, where it only exhibits a single transit event. An additional event was identified at the time of a momentum dump, and as such it failed the `odd-even' test and was not initially promoted to TOI status.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 1759.01.\n\n\\textbf{TIC 441642457.} Multi-transit candidate with a period of 79.8 d. \\textcolor{black}{This candidate was flagged as a TCE in sector 14, where it only exhibits a single transit event. An additional event was identified at the time of a momentum dump, and as such it failed the `odd-even' test and was not initially promoted to TOI status.} Since the PHT discovery this cTOI has become the priority 2 target TOI 2073.01.\n\n\\textbf{TIC 441765914.} Multi-transit candidate with a period of 161.6 d. Since the PHT discovery this cTOI has become the priority 1 target TOI 2088.01.\n\n\\textbf{TIC 48018596.} Multi-transit candidate with a period of 100.1 days (or a multiple thereof). The single LCO\/NRES spectrum shows no sign of this target being a double lined spectroscopic binary. Gemini speckle interferometry revealed no nearby companion stars. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the transit-events, the pipeline classified, what we consider stellar variability as an additional event. As such it failed the `odd-even' transit test and wasn't promoted to TOI status.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 2295.01.\n\n\\textbf{TIC 55525572.} Multi-transit candidate with a period of 83.9 d. Since the PHT discovery this cTOI has become the confirmed planet TOI 813 \\citep{2020eisner}.\n\n\\textbf{TIC 82452140.} Multi-transit candidate with a period of 21.1 d. Since the PHT discovery this cTOI has become the priority 2 target TOI 2289.01.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\nIn machine learning, we often want to learn a model to not only to perform a specified task well, but also to have some sort of meaningful structure to, for example, make the model easier to understand and implement. This desire for both accuracy and meaningful structure is usually expressed in a regularized optimization problem, such as one of the form:\n\\begin{align}\n\\minimize{\\mathbf{x}\\in \\mathcal{X}}f(\\mathbf{x}) + \\lambda g(\\mathbf{x}). \\label{eq:regularized_problem}\n\\end{align}\nIn this problem, $\\mathbf{x}$ is a choice of model parameters from a parameter space $\\mathcal{X}$, $f:\\mathcal{X}\\rightarrow\\mathbb{R}$ is a function that describes the misfit of the model with the selected parameters to the given task (such as empirical risk), $g:\\mathcal{X}\\rightarrow\\mathbb{R}$ is a function that expresses the deviation of our selected model parameters from their desired structure, and $\\lambda\\in\\mathbb{R}_{\\geq 0}$ is a tradeoff parameter.\n\nProblem \\eqref{eq:regularized_problem} becomes difficult when the desired model structure is an inherently binary or discrete property, while the model parameters are continuous values $\\mathbf{x}$ chosen from a continuum $\\mathcal{X}$. A prime example of this issue arises in feature selection for sparse regression, where we seek a linear predictor $\\mathbf{x}^*\\in \\mathcal{X}\\subseteq\\mathbb{R}^n$ such that:\n\\begin{align}\n\\mathbf{x}^*\\in\\textrm{arg}\\min{\\mathbf{x}\\in \\mathcal{X}} \\Vert \\mathbf{Ax}-\\mathbf{b}\\Vert_2^2 + \\lambda \\Vert \\mathbf{x}\\Vert_0,\\label{eq:sparse_regression}\n\\end{align}\nfor some $\\mathbf{A}\\in\\mathbb{R}^{m\\times n}$ and $\\mathbf{b}\\in\\mathbb{R}^m$, with $\\Vert \\mathbf{x} \\Vert_2$ the standard Euclidean norm on $\\mathbb{R}^m$, and $\\Vert \\mathbf{x}\\Vert_0$ the $\\ell_0$ pseudo-norm that counts the number of nonzero entries in the predictor $\\mathbf{x}$. The desired structure, in this case, is sparsity of the predictor $\\mathbf{x}\\in \\mathcal{X}$. Sparsity, however, only depends on the choice of a finite set of zero entries in the model parameters $\\mathbf{x}$, whereas the model is defined by a choice of continuous values for $\\mathbf{x}\\in\\mathcal{X}$.\n\nProblems containing this mixed dependence on both continuous and discrete properties of the model parameters, such as \\eqref{eq:sparse_regression}, are notoriously difficult, and even NP-Hard in general \\cite{rauhut2010compressive}. A typical workaround is to replace the function describing model structure, $g$ in problem \\eqref{eq:regularized_problem}, with a continuous relaxation that is more amenable to optimization. One of the more celebrated instances of this approach is the relaxation of the $\\ell_0$ pseudo-norm to the convex $\\ell_1$ norm $\\Vert x\\Vert_1$, which instead sums the absolute values of the vector $x$. While this relaxation may still encourage some of the intended structure, the minimizer for the relaxed problem often does not correspond to the minimizer for the initially specified problem \\cite{bach2012structured}. Moreover, the more well-known conditions for sparse recovery in regression problems, such as Restricted Isometry Properties \\cite{candes2005decoding}, Null Space Properties \\cite{rauhut2010compressive}, and Irrepresentability Conditions \\cite{zhao2006model}, do not give this form of guarantee for more general discrete functions $g$.\n\nIn contrast, the goal of this work is to identify conditions that allow us to directly solve the originally posed regularized model-fitting problem \\eqref{eq:regularized_problem} exactly and efficiently. To this end, we leverage submodularity, a property of functions that guarantees that they can be minimized exactly and efficiently.\n\nTraditionally, submodularity is defined for functions on binary or discrete spaces. For arbitrary functions over a discrete space, computing the minimizer is typically NP-Hard. When the function is submodular, however, exact minimization is a polynomial time operation \\cite{schrijver2003combinatorial}. Recently, the definition of submodularity and the associated optimization guarantees have been extended to continuous functions \\cite{bach2019submodular}. In particular, if a continuous function is submodular, it can also be minimized exactly in polynomial time.\n\nA natural next question to ask is if submodularity still defines the boundary between easy and hard minimization problems when the function $f$ in \\eqref{eq:regularized_problem} is continuous, but the function $g$ has finite co-domain. Our work explores precisely this boundary, and identifies sufficient conditions, based on the submodularity of both functions, under which the exact solution of problem \\eqref{eq:regularized_problem} can be efficiently computed.\n\nExploiting submodularity in these mixed scenarios is not a new idea, given its utility in combinatorial optimization problems. Notable examples include establishing approximation guarantees for greedy algorithms to perform well on sparsity-constrained optimization problems \\cite{elenberg2018restricted}, or in producing tight convex relaxations for set-function descriptions of desired sparsity patterns \\cite{bach2012structured}. In more recent work, \\cite{bach2019submodular} shows that if a continuous function is submodular, it can be discretized into a discrete submodular function, which can then be minimized exactly and in polynomial time. However, this discretization is only valid for compact subsets of continuous spaces and necessarily introduces discretization error into the produced solution. \n\nIn a line of work similar to this one, authors in \\cite{eloptimal} propose converting the mixed problem to a purely discrete one without discretizing. They then advocate using a specific submodular set function minimization algorithm for solving the discrete problem, and give approximation guarantees under the assumption that the functions are not exactly submodular. This approach mirrors the one proposed herein, but we instead propose conditions under which their approach is exact. Moreover, our guarantees are independent of which submodular function minimization algorithm is employed, allowing the use of more specialized algorithms than the specific one suggested by \\cite{eloptimal}.\n\nOur work makes several technical contributions, summarized as:\n\\begin{enumerate}[label=(\\roman*)]\n\\item We identify sufficient conditions, based on submodularity, under which the regularized model fitting problem \\eqref{eq:regularized_problem} can be solved efficiently and exactly;\n\\item We extend this theory to accommodate simple continuous and discrete constraints on the model parameter for some problem classes;\n\\item We highlight the utility of exact solutions to these problems for adversarial and robust optimization scenarios;\n\\item We numerically validate the correctness of our theory with proof-of-concept examples from sparse regression.\n\\end{enumerate}\n\n\\section{Submodular Functions on Lattices}\nIn this work, we consider optimization problems defined on two sets--an uncountably infinite set, typically $\\mathbb{R}^n$ or a subset thereof referred to as a \\emph{continuous set}, and a countable set, typically finite and referred to as a \\emph{discrete set}. Because we would like to tractably solve optimization problems defined on both continuous and discrete sets, we study a structure that enables efficient optimization in both cases: submodularity.\n\nSubmodularity is typically defined for set functions, which are functions that map any subset of a finite set $V$ to a real number, i.e., $f:2^V\\rightarrow\\mathbb{R}$. More generally, however, submodularity is a property of functions on \\emph{lattices} defined over continuous or discrete sets.\n\nWe let $\\latone$ be a set equipped with a partial order of its elements, denoted by $\\leqone$. For any two elements $\\mathbf{x},\\mathbf{x'}\\in\\latone$ we define their least upper bound, or \\emph{join} as:\n\\begin{align}\n\\mathbf{x}\\joinone\\mathbf{x'} &= \\inf\\{\\mathbf{y}\\in\\latone~:~\\mathbf{x}\\leq \\mathbf{y},~\\mathbf{x'}\\leq \\mathbf{y}\\}.\\label{eq:join_def}\n\\end{align}\nDually, we define their greatest lower bound, or \\emph{meet} as:\n\\begin{align}\n\\mathbf{x}\\meetone\\mathbf{x'}&=\\sup\\left\\{\\mathbf{y}\\in\\latone~:~\\mathbf{y}\\leq\\mathbf{x},~\\mathbf{y}\\leq\\mathbf{x'}\\right\\}.\\label{eq:meet_def}\n\\end{align}\nIf for every $\\mathbf{x},\\mathbf{x'}\\in\\latone$, their join, $\\mathbf{x}\\joinone\\mathbf{x'}$, and their meet, $\\mathbf{x}\\meetone\\mathbf{x'}$, exist and are in $\\latone$, then the set $\\latone$ and its order define a \\emph{lattice}. We write the lattice and its partial order together as $(\\latone,\\leqone)$, but will often write just $\\latone$ when the order is clear from context. If a subset $\\mathcal{S}\\subseteq\\latone$ is such that for every $\\mathbf{x},\\mathbf{x'}\\in \\mathcal{S}$, both $\\mathbf{x}\\joinone\\mathbf{x'}$ and $\\mathbf{x}\\meetone\\mathbf{x'}$ are in $\\mathcal{S}$, the subset $\\mathcal{S}$ is called a \\emph{sublattice} of $\\latone$ \\cite{davey2002introduction}.\n\nAs an example, consider a finite set of elements $V$. Then its power set, $2^V$ (the set of all its possible subsets), forms a lattice when ordered by set inclusion, $(2^V,\\subseteq)$. Under this order, the join of any two elements $X,X'\\subseteq V$ is $X\\cup X'\\subseteq V$, and dually, their meet is $X\\cap X'\\subseteq V$.\n\nWe can also endow continuous sets with partial orders, thereby defining lattices. Recent work has brought attention to $\\mathbb{R}^n$, where we define a partial order, denoted by $\\leqone$, as follows:\n\\begin{align}\n\\mathbf{x}\\leqone\\mathbf{x'}\\quad\\Leftrightarrow\\quad\\mathbf{x}_i \\leq \\mathbf{x}_i'\\quad\\text{for all }i=1,2,...,n,\\label{eq:rn_order}\n\\end{align}\nwhere $\\leq$ denotes the usual order on $\\mathbb{R}$.\n\n\nUnder this order, the join and meet operation for any $\\mathbf{x},\\mathbf{x'}\\in\\mathbb{R}^n$ are element-wise maximum and minimum, meaning:\n\\begin{align}\n(\\mathbf{x}\\joinone\\mathbf{x'})_i &= \\max\\{\\mathbf{x}_i,\\mathbf{x}_i'\\},\\4all i=1,2,...,n,\\label{eq:rn_join} \\\\\n(\\mathbf{x}\\meetone\\mathbf{x'})_i &= \\min\\{\\mathbf{x}_i,\\mathbf{x}_i'\\},\\4all i=1,2,...,n.\\label{eq:rn_meet}\n\\end{align}\n\nGiven a lattice $\\latone$, consider a function $f:\\latone\\rightarrow\\mathbb{R}$. The function $f$ is \\emph{submodular} on the lattice $\\latone$ when the following inequality holds for all $\\mathbf{x},\\mathbf{x}'\\in\\latone$:\n\\begin{align}\nf(\\mathbf{x})+f(\\mathbf{x'}) \\geq f(\\mathbf{x}\\joinone\\mathbf{x'}) + f(\\mathbf{x}\\meetone\\mathbf{x'}).\\label{eq:lat_fn_submodular}\n\\end{align}\nThe function $f$ is \\emph{monotone} when the implication:\n\\begin{align}\n\\mathbf{x}\\leqone\\mathbf{x'}\\quad \\implies \\quad f(\\mathbf{x}) \\leq f(\\mathbf{x'}),\\label{eq:lat_fn_monotone}\n\\end{align}\nis satisfied.\n\nWhen working with the lattice $(2^V,\\subseteq)$, the submodular inequality \\eqref{eq:lat_fn_submodular} becomes:\n\\begin{align}\nf(A) + f(B) \\geq f(A\\cup B) + f(A\\cap B) \\quad \\4all A,B\\subseteq V.\\label{eq:set_fn_submodular}\n\\end{align}\nSimilarly, the monotonicity implication \\eqref{eq:lat_fn_monotone} becomes:\n\\begin{align}\nA\\subseteq B \\quad\\implies\\quad f(A) \\leq f(B).\\label{eq:set_fn_monotone}\n\\end{align}\nMinimizing or maximizing an arbitrary set function is NP-Hard in general. However, if the set function is submodular it can be exactly minimized and approximately maximized, up to a constant-factor approximation ratio, in polynomial time \\cite{schrijver2003combinatorial,nemhauser1978analysis}. The computational tractability of submodular optimization for set functions has a variety of applications in fields such as sparse regression, summarization, and sensor placement \\cite{elenberg2018restricted,hui2011class,krause2006near}.\n\nWhen working with the lattice $(\\mathbb{R}^n,\\leqone)$, a function $f:\\mathbb{R}^n\\rightarrow\\mathbb{R}$ is submodular when:\n\\begin{align}\nf(\\mathbf{x}) + f(\\mathbf{x'}) \\geq f(\\max\\{\\mathbf{x},\\mathbf{x'}\\}) + f(\\min\\{\\mathbf{x},\\mathbf{x'}\\})\\quad\\4all \\mathbf{x},\\mathbf{x'}\\in\\mathbb{R}^n,\n\\end{align}\nwhere the maximum and minimum operations are performed element-wise, as expressed in \\eqref{eq:rn_join} and \\eqref{eq:rn_meet}. When $f$ is twice differentiable, submodularity on $\\mathbb{R}^n$ is equivalent (see, e.g. \\cite{topkis1998supermodularity,bach2019submodular}) to the condition:\n\\begin{align}\n\\frac{\\partial^2f}{\\partial \\mathbf{x}_i\\partial\\mathbf{x}_j} &\\leq 0 \\quad \\4all i\\neq j.\\label{eq:submodular_hessian}\n\\end{align}\nPerhaps surprisingly, the guarantees associated with submodular set function optimization extend to functions that are submodular on $\\mathbb{R}^n$. In particular, submodular functions on $\\mathbb{R}^n$ can be minimized up to arbitrary precision in polynomial time (see \\cite{bach2019submodular}), and can be approximately maximized with constant-factor approximation ratios, as shown in \\cite{bian2016guaranteed,bian2017non}.\n\n\\section{Problem Formulation}\nIn this section, we bridge continuous and discrete submodular function minimization with one unified problem statement. We do this by drawing inspiration from the field of structured sparsity, where the choice of zero entries in real-valued decision variables is viewed as a coupled discrete and continuous problem \\cite{bach2013learning,bach2011shaping}.\n\nTo highlight the connection with structured sparsity problems, for $n\\in\\mathbb{Z}_{>0}$, we denote by $[n]$ the set $\\{1,2,...,n\\}$, and by $2^{[n]}$ the set of all possible subsets of $[n]$. Define the map $\\mathrm{supp}:\\mathbb{R}^n\\rightarrow 2^{[n]}$ as:\n\\begin{align}\n\\supp{\\mathbf{x}} &= \\{i\\in [n]\\mid \\mathbf{x}_i\\neq 0\\}.\\label{eq:supp}\n\\end{align}\nIn other words, $\\mathrm{supp}$ returns the set of indices corresponding to the nonzero entries of the vector $\\mathbf{x}$. Consider the functions $f:\\mathbb{R}^n\\rightarrow\\mathbb{R}$ and $g:2^{[n]}\\rightarrow\\mathbb{R}$. In structured sparsity, problems of the form:\n\\begin{align}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n}~f(\\mathbf{x}) + g(\\supp{\\mathbf{x}}), \\label{eq:cont_discrete_opt_problem}\n\\end{align}\nare of particular interest, where the preferences in discrete selections (the zero entries of $\\mathbf{x}$) are expressed through the function $g$. As a particularly special case, if we let $f(\\mathbf{x}) = \\Vert \\mathbf{D}\\mathbf{x}-\\mathbf{b}\\Vert_2^2$ with $\\mathbf{D}\\in\\mathbb{R}^{m\\times n}$ and $\\mathbf{b}\\in\\mathbb{R}^m$ and define $g(A) = \\vert A\\vert$ as the cardinality of the set $A$, \\eqref{eq:cont_discrete_opt_problem} becomes:\n\\begin{align}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n}~\\Vert\\mathbf{D}\\mathbf{x}-\\mathbf{b}\\Vert_2^2 + \\Vert\\mathbf{x}\\Vert_0, \\tag{CS}\\label{eq:compressed_sensing}\n\\end{align}\nwhere $\\Vert\\cdot\\Vert_0$ denotes the $\\ell_0$ pseudo-norm. The problem \\eqref{eq:compressed_sensing} is a form of the well-studied compressed sensing problem, which is NP-Hard in general \\cite{rauhut2010compressive}.\n\nBuilding on the idea of making continuous decisions through the choice of $\\mathbf{x}$ in \\eqref{eq:cont_discrete_opt_problem}, and discrete decisions through the choice of the zero entries of $\\mathbf{x}$, we consider two lattices, $(\\latone,\\leqone)$ and $(\\lattwo,\\leqtwo)$, related by a map $\\eta:\\latone\\rightarrow\\lattwo$. We let $f:\\latone\\rightarrow\\mathbb{R}$ be a function describing the cost of assignments of variables in $\\latone$, and similarly let $g:\\lattwo\\rightarrow\\mathbb{R}$ describe the associated cost of choices in $\\lattwo$. Then, we seek the optimal point $\\mathbf{x}^*\\in\\latone$ in the problem:\n\\begin{align}\n\\minimize{\\mathbf{x}\\in\\latone}~f(\\mathbf{x})+g(\\eta(\\mathbf{x})).\\tag{P}\\label{eq:lattice_opt_problem}\n\\end{align}\nAlthough we will eventually let $\\latone$ describe continuous choices and $\\lattwo$ describe associated discrete ones, none of our results rely on the cardinality of the lattices $\\latone$ and $\\lattwo$.\n\nIntuitively, this problem asks for the $\\mathbf{x}\\in\\latone$ which incurs minimum cost, as measured by $f(\\mathbf{x})$, and in $\\lattwo$, as measured by $g(\\eta(\\mathbf{x}))$. Given that the special case of \\eqref{eq:compressed_sensing} is already hard in general, with no additional structure on $f$, $g$ and $\\eta$, our more general problem is hopelessly difficult. To provide this structure, we make some assumptions on these functions.\n\\begin{assumptions}\nConsider the lattices $(\\latone,\\leqone)$ and $(\\lattwo,\\leqtwo)$ and the maps $\\eta:\\latone\\rightarrow\\lattwo$, $f:\\latone\\rightarrow\\mathbb{R}$ and $g:\\lattwo\\rightarrow\\mathbb{R}$. We make the following assumptions:\n\\begin{enumerate}\n\\item The functions $f$ and $g$ are submodular on the lattices $\\latone$ and $\\lattwo$, respectively,\n\\item The function $g$ is monotone on $\\lattwo$,\n\\item For all $\\mathbf{x},\\mathbf{x}'\\in\\latone$:\n\\begin{align*}\n\\eta(\\mathbf{x}\\joinone\\mathbf{x}') \\leqtwo \\eta(\\mathbf{x})\\jointwo\\eta(\\mathbf{x}'), \\quad \\eta(\\mathbf{x}\\meetone\\mathbf{x}')\\leqtwo \\eta(\\mathbf{x})\\meettwo\\eta(\\mathbf{x}').\n\\end{align*}\n\\end{enumerate}\n\\end{assumptions}\n\n\\begin{remark}\nIf the map $\\eta:\\latone\\rightarrow\\lattwo$ satisfies Assumption 3, it is necessarily an order-preserving join-homomorphism, meaning it maintains the order and joins of elements in $\\latone$. (Prop. 2.19 in \\cite{davey2002introduction}) Explicitly, Assumption 3 is equivalent to the condition that for any $\\mathbf{x}, \\mathbf{x}'\\in\\latone$:\n\\begin{gather*}\n\\mathbf{x}\\leqone \\mathbf{x}' \\Rightarrow \\eta(\\mathbf{x})\\leqtwo\\eta(\\mathbf{x}'),\\\\\n\\eta(\\mathbf{x}\\joinone\\mathbf{x}') = \\eta(\\mathbf{x})\\jointwo\\eta(\\mathbf{x}').\n\\end{gather*} Despite this equivalence, we leave Assumption 3 as written above for clarity in future proofs.\n\\end{remark}\n\\begin{comment}\nNote that the function $\\supp:\\mathbb{R}^n_{\\geq 0}\\rightarrow 2^{[n]}$ satisfies Assumption 3 with equality, and that with this particular choice of lattices, we recovered a form of the compressed sensing problem \\eqref{eq:compressed_sensing}. This problem and its variants are well-studied and difficult to solve exactly in general (see \\cite{rauhut2010compressive}), but ties to submodularity on $\\mathbb{R}^n_{\\geq 0}$ established in \\cite{elenberg2018restricted} provide approximation guarantees for solving \\eqref{eq:compressed_sensing} with greedy algorithms. In this work, however, we focus on conditions under which we can solve the problem exactly.\n\\end{comment}\nWe highlighted the lattices $(\\mathbb{R}^n,\\leqone)$ and $(2^{[n]},\\subseteq)$ in the context of compressed sensing problems such as \\eqref{eq:compressed_sensing}, but for the map $\\mathrm{supp}:\\mathbb{R}^n\\rightarrow 2^{[n]}$ to satisfy Assumption 3, we must restrict the domain of $f$ to only only the first orthant, $(\\mathbb{R}^n_{\\geq 0},\\leqone)$. As mentioned in \\cite{bian2017non}, this issue can often be resolved by considering an appropriate \\emph{orthant conic lattice}, which views $\\mathbb{R}^n$ as a product of $n$ copies of $\\mathbb{R}$ and selects a different order for each copy. Alternatively, any least-squares problem such as \\eqref{eq:compressed_sensing} can be lifted to a non-negative least-squares problem, allowing us to satisfy Assumption 3 with the map $\\mathrm{supp}$, but potentially no longer satisfying Assumption 1 (see Appendix \\ref{apdx:NNLS}).\n\n\\begin{comment}\nOne common method for solving problems such as \\eqref{eq:compressed_sensing} is to replace the function $g:2^{[n]}\\rightarrow\\mathbb{R}$ with a tight convex surrogate function \\cite{bach2013learning}. Conveniently, when $g$ is submodular, this convex surrogate function is easily computed through the Lov\\`asz extension, and is amenable to first-order methods for convex optimization \\cite{lovasz1983submodular}. Despite this convenience, the resulting minimizer for the new relaxed problem may not correspond to the minimizer of the original problem \\cite{bach2011shaping}.\n\nUnder Assumptions 1-3, it can be shown that the combined function $f + g \\circ \\mathrm{supp}~:\\mathbb{R}^n_{\\geq 0} \\rightarrow\\mathbb{R}$ is submodular on $\\mathbb{R}^n_{\\geq 0}$. We can then directly apply the algorithms developed in \\cite{bach2019submodular} for submodular function minimization on $\\mathbb{R}^n$. However, these algorithms rely on discretizing the space $\\mathbb{R}^n_{\\geq 0}$ and minimizing the resulting discrete function, necessarily introducing some error into the final result. Moreover, the discretization process only produces finite lattices when the set to be discretized is compact. Since we may often have to work on $\\mathbb{R}^n_{\\geq 0}$, it may be nontrivial or expensive to find a compact subset of $\\mathbb{R}^n_{\\geq 0}$ that is guaranteed to contain the minimizer of $f + g\\circ\\mathrm{supp}$.\n\nAnother recent approach to this problem performs a discrete parameterization of the function $f$, then uses the Lov\\`asz extension to compute approximate subgradients that are used for projected subgradient descent \\cite{eloptimal}. While this algorithm provides approximation guarantees, subgradient descent can be slow to converge in practice. Moreover, as we will show, this method amounts to a special instantiation of our framework, where rather than giving approximation guarantees, we know the algorithm produces the globally optimal solution.\n\\end{comment}\n\\section{Solving an Equivalent Problem}\nIn this section, we outline our approach for solving the problem \\eqref{eq:lattice_opt_problem} which uses a related optimization problem defined on a single lattice. We then prove that this related problem is also a submodular function minimization problem, and that by solving it we recover a solution to \\eqref{eq:lattice_opt_problem}. Finally, we highlight some conditions under which solving this related problem is a polynomial time operation.\n\n\n\\subsection{The Equivalent Submodular Minimization Problem}\nAs expressed above, the problem \\eqref{eq:lattice_opt_problem} asks for the best selection of $\\mathbf{x}\\in\\latone$ and associated $\\eta(\\mathbf{x})\\in\\lattwo$. Our key observation is that we could instead look for the choice of $\\mathbf{y}\\in\\lattwo$ and associated $\\mathbf{x}\\in\\latone$, leading to the problem:\n\\begin{align*}\n\\minimize{\\mathbf{y}\\in\\lattwo}~g(\\mathbf{y}) + \\underset{\\substack{\\mathbf{x}\\in\\latone\\\\ \\eta(\\mathbf{x}) = \\mathbf{y}}}{\\min}~f(\\mathbf{x}).\n\\end{align*}\nIn the special case of \\eqref{eq:compressed_sensing} explored earlier, this equivalent problem becomes:\n\\begin{align*}\n\\minimize{S\\in 2^{[n]}}~\\vert S\\vert + \\underset{\\substack{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}\\\\ \\supp{\\mathbf{x}} = S}}{\\min}~\\Vert\\mathbf{A}\\mathbf{x}-\\mathbf{b}\\Vert_2^2.\n\\end{align*}\nWhile this new problem is clearly the same as \\eqref{eq:compressed_sensing}, the innermost minimization is over the set of $\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}$ such that $\\supp{\\mathbf{x}} = S$, or equivalently, $\\mathbf{x}_i \\neq 0$ for all $i\\in S$, and $\\mathbf{x}_i = 0$ for all $i\\notin S$. This feasible set is not a closed subset of $\\mathbb{R}^n_{\\geq 0}$, and thus the corresponding minimizer of this innermost problem may not exist \\cite{borwein2000convex}.\n\nWith this issue in mind, we instead consider a slight relaxation of the above problem:\n\\begin{align*}\n\\minimize{\\mathbf{y}\\in\\lattwo}~g(\\mathbf{y}) + H(\\mathbf{y}), \\tag{P-R}\\label{eq:lattice_opt_prob_relax}\n\\end{align*}\nwhere we have defined the function $H:\\lattwo\\rightarrow\\mathbb{R}$ as:\n\\begin{align}\nH(\\mathbf{y}) &= \\underset{\\substack{\\mathbf{x}\\in\\latone\\\\ \\eta(\\mathbf{x}) \\leqtwo \\mathbf{y}}}{\\min}~f(\\mathbf{x}).\\label{eq:H_definition}\n\\end{align}\nIn the special case of \\eqref{eq:compressed_sensing}, this relaxation produces the problem:\n\\begin{align}\n\\minimize{S\\in 2^{[n]}}~\\vert S\\vert + \\underset{\\substack{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}\\\\ \\supp{\\mathbf{x}} \\subseteq S}}{\\min}~\\Vert\\mathbf{A}\\mathbf{x}-\\mathbf{b}\\Vert_2^2,\\tag{CS-R}\n\\end{align}\nwhere the innermost minimization is instead over the $\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}$ such that $\\mathbf{x}_i = 0$ for all $i\\notin S$, which is now a closed subset of $\\mathbb{R}^n_{\\geq 0}$.\n\nOur goal is to prove that under Assumptions 1-3, the new relaxed problem \\eqref{eq:lattice_opt_prob_relax} can be efficiently solved, and that by solving it we can recover the corresponding minimizer for \\eqref{eq:lattice_opt_problem}. As established above, minimizing functions on lattices is efficient when the functions are submodular, so we show that our new relaxed problem \\eqref{eq:lattice_opt_prob_relax} is a submodular function minimization problem on $\\lattwo$.\n\n\\begin{theorem}\\label{thm:main_result}\nUnder Assumptions 1-3, the function $g + H:\\lattwo\\rightarrow\\mathbb{R}$ is submodular on $\\lattwo$, and therefore the relaxed problem \\eqref{eq:lattice_opt_prob_relax} is a submodular function minimization problem on the lattice $\\lattwo$. Moreover, let $\\mathbf{y}^*\\in\\lattwo$ be the minimizer for the problem \\eqref{eq:lattice_opt_prob_relax}, and let $\\mathbf{x}^*\\in\\latone$ be such that:\n\\begin{align*}\n\\mathbf{x}^*\\in\\underset{\\substack{\\mathbf{x}\\in\\latone \\\\ \\eta(\\mathbf{x})\\leqtwo\\mathbf{y}^*}}{\\mathrm{argmin}}~f(\\mathbf{x}).\n\\end{align*}\nThen $\\mathbf{x}^*$ is a minimizer for the problem \\eqref{eq:lattice_opt_problem}.\n\\end{theorem}\n\nTo prove this result, we require a few technical lemmas.\n\n\\begin{lemma}\\label{lem:sublattice}\nLet $(\\latone,\\leqone)$ and $(\\lattwo,\\leqtwo)$ be lattices with the map $\\eta:\\latone\\rightarrow\\lattwo$ satisfying Assumption 3. Then the set:\n\\begin{align}\n\\mathcal{D} &= \\left\\lbrace(\\mathbf{x},\\mathbf{y})\\in \\latone\\times\\lattwo \\mid\\eta(\\mathbf{x})\\leqtwo \\mathbf{y}\\right\\rbrace, \\label{eq:sublattice_D}\n\\end{align}\nis a sublattice of the product lattice, $\\latone\\times \\lattwo$.\n\\end{lemma}\n\\begin{proof}\nTo prove that $\\mathcal{D}\\subseteq\\latone\\times\\lattwo$ is a sublattice, we have to show that the join and meet (as defined on the product lattice) of any two elements in $\\mathcal{D}$ is also in $\\mathcal{D}$. The join on the product lattice of any two $(\\mathbf{x},\\mathbf{y}),(\\mathbf{x}',\\mathbf{y}')\\in \\mathcal{D}$ is denoted by $\\vee_{\\mathcal{D}}$, and defined as:\n\\begin{align*}\n(\\mathbf{x},\\mathbf{y})\\vee_{\\mathcal{D}}(\\mathbf{x}',\\mathbf{y}') &= (\\mathbf{x}\\joinone\\mathbf{x}',\\mathbf{y}\\jointwo\\mathbf{y}').\n\\end{align*}\nThen, we note:\n\\begin{align*}\n\\eta(\\mathbf{x}\\joinone\\mathbf{x}')&\\leqtwo\\eta(\\mathbf{x})\\jointwo\\eta(\\mathbf{x}') \\leqtwo \\mathbf{y}\\jointwo\\mathbf{y}',\n\\end{align*}\nwhere we first used Assumption 3, then the fact that $(\\mathbf{x},\\mathbf{y}),(\\mathbf{x}',\\mathbf{y}')\\in \\mathcal{D}$. Therefore, the pair $(\\mathbf{x}\\joinone\\mathbf{x}',\\mathbf{y}\\jointwo\\mathbf{y}')$ is also in $D$.\n\nBecause $(\\mathbf{x},\\mathbf{y})$ and $(\\mathbf{x}',\\mathbf{y}')$ were arbitrary, this holds for all of $\\mathcal{D}$. A dual analysis follows for the meet operation, proving $\\mathcal{D}$ is a sublattice of the product lattice $\\latone\\times\\lattwo$.\n\\end{proof}\n\nThe sublattice $\\mathcal{D}$ is useful as the only pairs of $(\\mathbf{x},\\mathbf{y})\\in\\latone\\times\\lattwo$ of interest in our optimization problem \\eqref{eq:lattice_opt_prob_relax} are those that are in $\\mathcal{D}$. The following theorem then uses this sublattice to prove that $H$ is submodular. The result is adapted from an established theorem in literature, but we include its proof here for completeness.\\\\\n\\begin{theorem}\\label{thm:topkis}\n(Adapted from \\textit{Theorem 2.7.6 in \\cite{topkis1998supermodularity}}) Let $f:\\latone\\rightarrow\\mathbb{R}$, $g:\\lattwo\\rightarrow\\mathbb{R}$, and $\\eta:\\latone\\rightarrow\\lattwo$ be maps satisfying Assumptions 1 and 3. Then the function $g + H:\\lattwo\\rightarrow\\mathbb{R}$, with $H$ defined as in \\eqref{eq:H_definition}, is submodular on $\\lattwo$.\n\\end{theorem}\n\\begin{proof}\nTo prove this statement, we take two points $\\mathbf{y},\\mathbf{y}'\\in\\lattwo$ and compare the values of the function $g + H$, verifying the submodular inequality \\eqref{eq:lat_fn_submodular}. We note that for any $\\mathbf{y},\\mathbf{y'}\\in\\lattwo$, there are corresponding $\\mathbf{z},\\mathbf{z}'\\in\\latone$ such that:\n\\begin{align}\n\\begin{aligned}\n\\mathbf{z}&\\in\\textrm{arg}\\min{\\substack{\\mathbf{x}\\in\\latone \\\\ \\eta(\\mathbf{x})\\leqtwo \\mathbf{y}}}~f(\\mathbf{x}) \\quad \\Rightarrow\\quad H(\\mathbf{y}) = f(\\mathbf{z}),\\\\\n\\mathbf{z}'&\\in\\textrm{arg}\\min{\\substack{\\mathbf{x}\\in\\latone \\\\ \\eta(\\mathbf{x})\\leqtwo \\mathbf{y}'}}~f(\\mathbf{x})\\quad \\Rightarrow \\quad H(\\mathbf{y}') = f(\\mathbf{z}').\n\\end{aligned}\\label{eq:z_optimality}\n\\end{align}\nBy definition, $(\\mathbf{z},\\mathbf{y})$ and $(\\mathbf{z}',\\mathbf{y}')$ are both in $\\mathcal{D}$. Then, it follows:\n\\begin{align*}\ng(\\mathbf{y}) + H(\\mathbf{y}) + g(\\mathbf{y}') + H(\\mathbf{y}') &= g(\\mathbf{y}) + f(\\mathbf{z}) + g(\\mathbf{y}') + f(\\mathbf{z}') \\\\\n&\\geq g(\\mathbf{y}\\jointwo\\mathbf{y'}) + g(\\mathbf{y}\\meettwo\\mathbf{y}') + f(\\mathbf{z}\\joinone\\mathbf{z}') + f(\\mathbf{z}\\meetone\\mathbf{z}'),\n\\end{align*}\nwhere we first used \\eqref{eq:z_optimality} and then the submodularity of $f$ and $g$ on their respective sublattices.\n\nBy Lemma \\ref{lem:sublattice}, $\\mathcal{D}$ is a sublattice of $\\latone\\times\\lattwo$, and so the pairs $(\\mathbf{z}\\joinone\\mathbf{z}',\\mathbf{y}\\jointwo\\mathbf{y}')$ and $(\\mathbf{z}\\meetone\\mathbf{z}',\\mathbf{y}\\meettwo\\mathbf{y}')$ are also in $\\mathcal{D}$, meaning:\n\\begin{align*}\n\\eta(\\mathbf{x}\\joinone\\mathbf{x}') &\\leqtwo \\mathbf{y}\\jointwo\\mathbf{y}', \\\\ \\eta(\\mathbf{x}\\meetone\\mathbf{x}')&\\leqtwo\\mathbf{y}\\meettwo\\mathbf{y}'.\n\\end{align*}\nTherefore $\\mathbf{x}\\joinone\\mathbf{x}'$ and $\\mathbf{x}\\meetone\\mathbf{x}'$ are feasible points in the minimization defining $H(\\mathbf{y}\\jointwo\\mathbf{y}')$ and $H(\\mathbf{y}\\meettwo\\mathbf{y}')$ in \\eqref{eq:H_definition}. We then have:\n\\begin{align*}\ng(\\mathbf{y}) + H(\\mathbf{y}) + g(\\mathbf{y}') + H(\\mathbf{y}') &\\geq g(\\mathbf{y}\\jointwo\\mathbf{y'}) + g(\\mathbf{y}\\meettwo\\mathbf{y}') + f(\\mathbf{z}\\joinone\\mathbf{z}') + f(\\mathbf{z}\\meetone\\mathbf{z}') \\\\\n&\\geq g(\\mathbf{y}\\jointwo\\mathbf{y'}) + g(\\mathbf{y}\\meettwo\\mathbf{y}') + \\underset{\\substack{\\mathbf{x}\\in\\latone\\\\ \\eta(\\mathbf{x}) \\leqtwo \\mathbf{y}\\jointwo\\mathbf{y}'}}{\\min}\\hspace{-2.5mm}f(\\mathbf{x}) + \\underset{\\substack{\\mathbf{x}\\in\\latone\\\\ \\eta(\\mathbf{x}) \\leqtwo \\mathbf{y}\\meettwo\\mathbf{y}'}}{\\min}\\hspace{-2.5mm}f(\\mathbf{x})\\\\\n&= g(\\mathbf{y}\\jointwo\\mathbf{y'}) + H(\\mathbf{y}\\jointwo\\mathbf{y}') + g(\\mathbf{y}\\meettwo\\mathbf{y}') + H(\\mathbf{y}\\meettwo\\mathbf{y}').\n\\end{align*}\nThe final equality in this sequence provides the right-hand side of the submodular inequality \\eqref{eq:lat_fn_submodular} for $g + H$, as desired.\n\\end{proof}\n\nBecause $g+H$ is submodular on $\\lattwo$, we know that minimizing it over all $\\mathbf{y}\\in\\lattwo$, i.e., solving \\eqref{eq:lattice_opt_prob_relax}, is an instance of submodular function minimization. What remains is to show that solving this relaxed problem allows us to also solve to the original problem, \\eqref{eq:lattice_opt_problem}. We verify this in the following lemma.\n\n\\begin{comment}\nThe key insight in this lemma is noting that the only time the minimizer of the relaxed problem may differ from the original is when the minimizer, $\\mathbf{y}^*\\in\\lattwo$ corresponds to some $\\mathbf{z}^*\\in\\latone$ such that:\n\\begin{align*}\n\\mathbf{z}^*\\in \\textrm{arg}\\min{\\substack{\\mathbf{x}\\in\\latone \\\\ \\eta(\\mathbf{x})\\leqtwo \\mathbf{y}^*}}~f(\\mathbf{x})\n\\end{align*}\nbut $\\eta(\\mathbf{z}^*) = \\tilde{\\mathbf{y}} \\sleqtwo \\mathbf{y}^*$, with strict inequality. However, because $g$ is monotone we can simply select $\\tilde{\\mathbf{y}}\\in\\lattwo$ as a minimizer as it must have equal cost to $\\mathbf{y}^*$.\n\\end{comment}\n\n\\begin{lemma}\\label{lem:minimizers}\nLet $\\mathbf{y}^*\\in\\lattwo$ be the minimizer for the relaxed problem \\eqref{eq:lattice_opt_prob_relax}, and let $\\mathbf{x}^*\\in\\latone$ be such that:\n\\begin{align*}\n\\mathbf{x}^*\\in\\underset{\\substack{\\mathbf{x}\\in\\latone\\\\\\eta(\\mathbf{x})\\leqtwo\\mathbf{y}^*}}{\\mathrm{argmin}}~f(\\mathbf{x}).\n\\end{align*}\nIf $g$ satisfies Assumption 2, then $\\mathbf{x}^*$ is a minimizer for the problem \\eqref{eq:lattice_opt_problem}.\n\\end{lemma}\n\\begin{proof}\nTo prove this lemma, we consider an optimal $\\mathbf{z}^*\\in\\latone$ for problem \\eqref{eq:lattice_opt_problem} and verify that the proposed minimizer, $\\mathbf{x}^*\\in\\latone$, has the same cost.\n\nWe first note that by the optimality of $\\mathbf{z}^*$ in problem \\eqref{eq:lattice_opt_problem}:\n\\begin{align}\nf(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) \\leq f(\\mathbf{x}^*) + g(\\eta(\\mathbf{x}^*)). \\label{eq:optimality}\n\\end{align}\n\nAdditionally, we have:\n\\begin{align*}\n\\begin{array}{clc}\nf(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) &\\geq \\underset{\\substack{\\mathbf{x}\\in\\latone\\\\\\eta(\\mathbf{x})\\leqtwo\\eta(\\mathbf{z}^*)}}{\\min}~f(\\mathbf{x}) + g(\\eta(\\mathbf{z}^*))&\\quad\\text{(by definition of minimum, as $\\mathbf{z}^*$ is feasible)} \\\\\n&= H(\\eta(\\mathbf{z}^*)) + g(\\eta(\\mathbf{z}^*))&\\quad\\text{(by definition of $H$)} \\\\\n&\\geq H(\\mathbf{y}^*) + g(\\mathbf{y}^*)&\\quad\\text{(by optimality of $\\mathbf{y}^*$ in \\eqref{eq:lattice_opt_prob_relax})}\\\\\n&= f(\\mathbf{x}^*) + g(\\mathbf{y}^*)&\\quad\\text{(by definition of $\\mathbf{x}^*$).}\n\\end{array}\n\\end{align*}\nThis sequence of inequalities implies:\n\\begin{align}\nf(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) & \\geq f(\\mathbf{x}^*) + g(\\mathbf{y}^*). \\label{eq:lemma_two_ineq}\n\\end{align}\nBy construction, there are exactly two possible relationships between $\\mathbf{x}^*$ and $\\mathbf{y}^*$.\n\t\n\\textit{Case 1:} $\\eta(\\mathbf{x}^*) = \\mathbf{y}^*$. In this case, \\eqref{eq:lemma_two_ineq} becomes:\n\\begin{align}\nf(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) & \\geq f(\\mathbf{x}^*) + g(\\mathbf{y}^*)\\nonumber \\\\\n&= f(\\mathbf{x}^*) + g(\\eta(\\mathbf{x}^*)).\\label{eq:lemma_two_ineq_2}\n\\end{align}\nThen, combining inequality \\eqref{eq:lemma_two_ineq_2} with \\eqref{eq:optimality}, we have that:\n\\begin{gather*}\nf(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) \\geq f(\\mathbf{x}^*) + g(\\eta(\\mathbf{x}^*)) \\geq f(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) \\\\\n\\Rightarrow f(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) = f(\\mathbf{x}^*) + g(\\eta(\\mathbf{x}^*)),\n\\end{gather*}\nand therefore $\\mathbf{x}^*$ is also a minimizer for problem \\eqref{eq:lattice_opt_problem}.\n\n\\textit{Case 2:} $\\eta(\\mathbf{x}^*)\\sleqtwo\\mathbf{y}^*$. In this case, because $g$ is monotone, $g(\\mathbf{y}^*) \\geq g(\\eta(\\mathbf{x}^*))$. Using this fact, we can lower bound the right-hand side of \\eqref{eq:lemma_two_ineq}:\n\\begin{align*}\nf(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) &\\geq f(\\mathbf{x}^*) + g(\\mathbf{y}^*) \\\\\n&\\geq f(\\mathbf{x}^*) + g(\\eta(\\mathbf{x}^*)).\n\\end{align*}\nAt this point, we have obtained \\eqref{eq:lemma_two_ineq_2}, and we can follow the argument used in Case 1. In both cases then, $\\mathbf{x}^*$ is a minimizer in problem \\eqref{eq:lattice_opt_problem}.\n\\end{proof}\n\nThis series of results gives rise to Theorem \\ref{thm:main_result}, which provides sufficient conditions under which we can transform problem \\eqref{eq:lattice_opt_problem}, an optimization problem on two lattices, into problem \\eqref{eq:lattice_opt_prob_relax}, a submodular function minimization problem on a single lattice.\n\n\\begin{proof} \\textit{(Theorem \\ref{thm:main_result})}\\\\ Under Assumptions 1 and 3, Theorem \\ref{thm:topkis} states that the function $g + H:\\lattwo\\rightarrow\\mathbb{R}$ is submodular on the lattice $\\lattwo$. Therefore, solving \\eqref{eq:lattice_opt_prob_relax} is a submodular function minimization problem over $\\lattwo$, and the first part of the theorem is proved.\n\nUnder Assumption 2, by Lemma \\ref{lem:minimizers}, given the minimizer $\\mathbf{y}^*$ of \\eqref{eq:lattice_opt_prob_relax}, the point $\\mathbf{x}^*\\in\\latone$ defined by:\n\\begin{align*}\n\\mathbf{x}^*\\in\\underset{\\substack{\\mathbf{x}\\in\\latone\\\\\\eta(\\mathbf{x})\\leqtwo\\mathbf{y}^*}}{\\mathrm{argmin}}~f(\\mathbf{x}),\n\\end{align*} is a minimizer in the original problem \\eqref{eq:lattice_opt_problem}.\n\\end{proof}\n\n\n\\subsection{Solving \\eqref{eq:lattice_opt_prob_relax} in Polynomial Time\\label{sec:polytime}}\nDespite the efficiency of submodular function minimization, we can only truly solve \\eqref{eq:lattice_opt_prob_relax} in polynomial time if evaluating $g$ and $H$ is also a polynomial time operation. The function $H$, however, is implicitly defined through an optimization problem on $\\latone$ \\eqref{eq:H_definition}. Solving the problem \\eqref{eq:lattice_opt_prob_relax} in polynomial time then requires solving these smaller optimization problems defining $H$ efficiently.\n\nWe are particularly interested in joint continuous and discrete optimization, as illustrated by the example of $(\\latone,\\leqone) = (\\mathbb{R}^n_{\\geq 0},\\leqtwo)$ and $(\\lattwo,\\leqtwo) = (2^{[n]},\\subseteq)$ connected by the map \\mbox{$\\mathrm{supp}:\\mathbb{R}^n_{\\geq 0}\\rightarrow 2^{[n]}$} as expressed in \\eqref{eq:supp}. In this case, evaluating $H$ requires solving the optimization problem:\n\\begin{align}\n\\minimize{\\substack{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}\\\\\\supp{\\mathbf{x}}\\subseteq A}} f(\\mathbf{x}),\\label{eq:h_cont_discrete}\n\\end{align}\nfor any $A\\in 2^{[n]}$.\n\n\\begin{comment}\nTo solve problem \\eqref{eq:lattice_opt_problem}, a submodular function minimization on $\\lattwo$, algorithms heavily rely on computing the Lov\\`asz extension of $g + H$ \\cite{fujishige2011submodular,schrijver2003combinatorial}. Because $g + H$ is submodular, this computation has complexity $O(n\\log n + nEO)$, where $EO$ is the complexity of evaluating $g + H$. However, $EO$ is the complexity of solving the sub-problem in \\eqref{eq:h_cont_discrete}, thus our choice of algorithm for this sub-problem directly impacts efficiency of computing the Lov\\`asz extension, and thereby the submodular set function minimization.\n\\end{comment}\n\nOne interesting note in the sub-problem \\eqref{eq:h_cont_discrete} is that for any $A\\in 2^{[n]}$, the feasible set is a convex subset of $\\mathbb{R}^n_{\\geq 0}$. If the function $f:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ was convex, then we could use any generic convex optimization routine to solve \\eqref{eq:h_cont_discrete}. We already assumed that $f$ is submodular on $\\mathbb{R}^n_{\\geq 0}$, but submodular functions are neither a subset nor a superset of convex functions, so we could also require that $f$ is convex, making evaluating $H$ and solving \\eqref{eq:lattice_opt_problem} efficient.\n\n\\begin{corollary}\\label{cor:polytime}\nLet $f:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ be a submodular and convex function on $(\\mathbb{R}^n_{\\geq 0},\\leq)$, $\\lattwo$ be a finite lattice, $g:\\lattwo\\rightarrow\\mathbb{R}$ be a monotone submodular set function, and let $\\eta:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\lattwo$ satisfy Assumption 3. Further assume that for every $\\mathbf{y}\\in\\lattwo$, the set of $\\mathbf{x}\\in\\latone$ such that $\\eta(\\mathbf{x})\\leqtwo\\mathbf{y}$ is a convex subset of $\\mathbb{R}^n_{\\geq 0}$. Then the problem:\n\\begin{align*}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}}~f(\\mathbf{x}) + g(\\eta(\\mathbf{x})),\n\\end{align*}\ncan be solved in polynomial time.\n\\end{corollary}\n\\begin{proof}\nIt follows from the corollary's assumptions that Assumptions 1, 2, and 3 are satisfied by the lattices $(\\mathbb{R}^n_{\\geq 0},\\leq)$, $(\\lattwo,\\leqtwo)$, and the functions $\\eta:\\mathbb{R}^n_{\\geq 0}\\rightarrow \\lattwo$, $f:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ and $g:\\lattwo\\rightarrow\\mathbb{R}$. By Theorem \\ref{thm:main_result}, we can solve the problem \\eqref{eq:lattice_opt_problem} by instead minimizing the submodular function $H$ over $\\lattwo$, i.e., problem \\eqref{eq:lattice_opt_prob_relax}. Submodular function minimization has polynomial complexity in both $\\vert\\lattwo\\vert$, which is finite by assumption, and the number of function evaluations of $H$. Because $f$ is convex, evaluating $H$ is also a polynomial time operation, and the complexity of solving \\eqref{eq:lattice_opt_prob_relax} is polynomial.\n\\end{proof}\nOur theory is agnostic to the choice of subroutines both for evaluating $H$ and solving the set function minimization problem. If we assume $f$ is convex, evaluate it through convex optimization, and use projected subgradient descent on the Lov\\`asz extension of $H$ as the algorithm for solving the set function minimization, we recover exactly the approach proposed by \\cite{eloptimal}.\n\n\\begin{comment}\n\n\\subsubsection{Continuous Submodularity Alone}\nBy Assumption 1, $f$ is submodular on the lattice $\\mathbb{R}^n_{\\geq 0}$. Moreover, given any $A\\in 2^{[n]}$, the set:\n\\begin{align*}\nS = \\{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}\\mid \\supp{\\mathbf{x}}\\subseteq A\\},\n\\end{align*}\nis a lattice ideal (and also a sublattice) of $\\mathbb{R}^n_{\\geq 0}$ \\cite{davey2002introduction}. Given this structure, we can directly apply algorithms for continuous submodular function minimization provided in \\cite{bach2019submodular} to solve the sub-problem in \\eqref{eq:h_cont_discrete}.\n\nNo additional structure in $f$ is required in this case, but the algorithms for continuous submodular function minimization require discretizing the domain of the function, $S\\subseteq\\mathbb{R}^n_{\\geq 0}$. However, for any $A\\in 2^n$ with $A\\neq \\emptyset$, the ideal $S$ is unbounded and cannot be perfectly discretized. Moreover, discretizing $S$ enough to produce high accuracy evaluations of $H$ can drastically increase the running times.\n\nIf the set $S$ is replaced with a bounded sublattice $S_\\alpha = \\{\\mathbf{x}\\in[0,\\alpha]^n\\mid \\supp{\\mathbf{x}}\\subseteq A\\}$ and is then discretized into $k$ values at each index, and the function $f$ is $L$-Lipschitz continuous function, the complexity of reaching $\\epsilon$ suboptimality using the algorithms in \\cite{bach2019submodular} is $O\\left(\\left(\\frac{2L\\alpha n}{\\epsilon}\\right)^3\\log\\left(\\frac{2L\\alpha n}{\\epsilon}\\right)\\right)$. Plugging this complexity into the expression for the Lov\\`asz extension gives an overall complexity of order $O((n + n^4)\\log n)$.\n\\end{comment}\n\nConvexity of $f$ is not the only assumption that leads to tractable evaluations of $H$. As an alternative, we could consider a nonconvex quadratic form for $f:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$:\n\\begin{align}\nf(\\mathbf{x}) &= \\mathbf{x}^T\\mathbf{Q}\\mathbf{x} + \\mathbf{p}^T\\mathbf{x},\\label{eq:quadratic_f}\n\\end{align}\nwith $\\mathbf{Q}\\in\\mathbb{R}^{n\\times n}$ and $\\mathbf{p}\\in\\mathbb{R}^n$. The assumption that this quadratic function is submodular on $\\mathbb{R}^n_{\\geq 0}$ is equivalent to the condition:\n\\begin{align*}\n\\frac{\\partial^2 f}{\\partial \\mathbf{x}_i\\partial\\mathbf{x}_j} &= \\mathbf{Q}_{ij} \\leq 0, \\quad \\text{for all }i\\neq j.\n\\end{align*}\nMoreover, our sub-problem instance \\eqref{eq:h_cont_discrete} is a constrained, nonconvex quadratic program:\n\\begin{align}\n\\begin{array}{cc}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n} & \\mathbf{x}^T\\mathbf{Q}\\mathbf{x} + 2\\mathbf{p}^T\\mathbf{x} \\\\\n\\text{subject to}& \\mathbf{x} \\geq 0 \\\\\n& \\mathbf{x}_i = 0, ~i\\notin A.\n\\end{array}\\label{eq:quadratic_h}\n\\end{align}\nResearchers \\cite{kim2003exact} have established that nonconvex quadratic programs satisfying submodularity admit tight semidefinite program relaxations. In particular, we have the following theorem:\n\\begin{theorem}\n\\textit{Theorem 3.1 in \\cite{kim2003exact}}) Let $\\mathbf{Q}\\in\\mathbb{R}^{n\\times n}$ have nonpositive off-diagonal entries. Let $\\mathrm{tr}:\\mathbb{R}^{n\\times n}\\rightarrow\\mathbb{R}$ denote the trace of a matrix, $\\mathrm{diag}:\\mathbb{R}^{n\\times n}\\rightarrow\\mathbb{R}^n$ denote the diagonal entries of the matrix, and let $\\succeq$ indicate the positive semidefiniteness of a symmetric matrix. Further, for any $A\\in 2^{[n]}$, let $\\mathbf{Z}_{A^c}$ denote the rows and columns of $\\mathbf{Z}$ with indices not in the set $A$. Consider the semi-definite program:\n\\begin{align*}\n\\begin{array}{cc}\n\\minimize{\\substack{\\mathbf{z}\\in\\mathbb{R}^n \\\\ \\mathbf{Z}\\in \\mathbb{S}^n}} & \\tr{\\mathbf{QZ}} + 2\\mathbf{p}^T\\mathbf{z} \\\\\n\\text{subject to} & \\tr{\\mathbf{Z}_{A^c}} \\leq 0 \\\\\n& \\diag{\\mathbf{Z}} \\geq 0 \\\\\n& \\begin{bmatrix}\n1 & \\mathbf{z}^T \\\\\n\\mathbf{z} & \\mathbf{Z}\n\\end{bmatrix} \\succeq 0,\n\\end{array}\n\\end{align*}\nGiven the solution $(\\mathbf{Z}^*,\\mathbf{z}^*)$ to this SDP, the vector $\\mathbf{x}^*_i = \\sqrt{\\mathbf{Z}^*_{ii}}$, $i=1,...,n$ is a minimizer for the non-convex quadratic program \\eqref{eq:quadratic_h}.\n\\end{theorem}\n\nBecause semi-definite programs can be solved in polynomial time, we could use this relaxation to evaluate $H$ for any subset $A\\in 2^{[n]}$ in polynomial time. As before, this ability would produce an identical statement to Corollary \\ref{cor:polytime}, but for functions $f$ of the form \\eqref{eq:quadratic_f} that satisfy submodularity.\n\n\n\\begin{comment}\n\\begin{corollary}\nLet $f:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ be a submodular function on $(\\mathbb{R}^n_{\\geq 0},\\leq)$ and $g:2^{[n]}\\rightarrow\\mathbb{R}$ be a monotone submodular set function. Let $\\mathrm{supp}:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ be the support map as expressed in \\eqref{eq:supp}. If $f$ is also convex, or a nonconvex quadratic form, the problem:\n\\begin{align*}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}}~f(\\mathbf{x}) + g(\\supp{\\mathbf{x}})\n\\end{align*}\ncan be solved in polynomial time.\n\\end{corollary}\n\\begin{proof}\nFor the lattices $(\\mathbb{R}^n_{\\geq 0},\\leq)$ and $(2^{[n]},\\subseteq)$, the function $\\mathrm{supp}:\\mathbb{R}^n_{\\geq 0}\\rightarrow 2^{[n]}$ satisfies Assumption 3. As the functions $f$ and $g$ satisfy Assumptions 1 and 2, by Theorem \\ref{thm:main_result}, solving the problem:\n\\begin{align}\n\\minimize{A\\in 2^{[n]}} g(A) + H(A)\\label{eq:set_fn_min}\n\\end{align}\nwith $H$ as defined in \\eqref{eq:H_definition} is a submodular set function minimization that gives the solution to the original discrete-continuous problem.\n\nThe complexity of submodular set function minimization is polynomial in $n$ and the number of function evaluations. By the assumptions on $f$, evaluating $g + H$ is also a polynomial time operation, meaning the overall complexity of solving \\eqref{eq:set_fn_min} is polynomial.\n\\end{proof}\n\\end{comment}\n\n\\section{Constrained Optimization}\nIn this section, we extend our framework both theoretically and algorithmically to accommodate constraints for the specific case of the lattices $(\\mathbb{R}^n_{\\geq 0}, \\leqone)$ and $(2^{[n]},\\subseteq)$, connected by the support map $\\mathrm{supp}:\\mathbb{R}^n_{\\geq 0}\\rightarrow 2^{[n]}$.\n\nIn many problems, we may be interested in optimization over a feasible subset $C\\subset\\mathbb{R}^n_{\\geq 0}$ that is strictly contained in $\\mathbb{R}^n_{\\geq 0}$. Unfortunately, submodular function minimization and maximization subject to constraints is NP-Hard in general \\cite{fujishige2011submodular}. This difficulty arises because arbitrary subsets of a lattice rarely define sublattices.\n\nOne simple class of problems whose feasible sets do not define sublattices are problems with \\emph{budget constraints}:\n\\begin{align}\n\\begin{array}{cc}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}} & f(\\mathbf{x}) + g(\\supp{\\mathbf{x}}) \\\\\n\\text{subject to} & \\sum_{i=1}^nW_i(\\mathbf{x}_i) \\leq B,\n\\end{array}\\label{eq:constrained_case}\n\\end{align}\nwith $W_i:\\mathbb{R}{\\geq 0}\\rightarrow\\mathbb{R}$ strictly increasing functions for $i = 1,2,...,n$ and $B \\in \\mathbb{R}_{>0}$ a ``budget''.\n\n\\begin{comment}\n\\blue{JB: this small ``example'' section could be removed if it's not helpful.}\n\nFor instance, consider the feasible set $C = \\{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}\\mid \\sum_{i=1}^n \\mathbf{x}_i \\leq B\\}$ with $n \\geq 2$. Consider the two points $\\mathbf{x},\\mathbf{x}'\\in C$ where $\\mathbf{x}_i = B$ and $\\mathbf{x}'_j = B$, $i\\neq j$ and note:\n\\begin{align*}\n\\sum_{i=1}^n(\\mathbf{x}\\joinone\\mathbf{x}')_i &= \\sum_{i=1}^n\\max\\{\\mathbf{x}_i,\\mathbf{x}_i'\\} = 2B \\nleq B.\n\\end{align*}\nTherefore $\\mathbf{x}\\joinone\\mathbf{x}' \\notin C$, thus $C$ is not a sublattice.\n\nWe may also consider discrete budget constraints, where $W(\\mathbf{x}) = \\sum_{j\\in\\supp{\\mathbf{x}}}\\mathbf{w}_j$ for some $\\mathbf{w}\\in\\mathbb{R}^n_{\\geq 0}$. We refer to this as a support knapsack, as we ``pay'' some cost $\\mathbf{w}_j$ to set $\\mathbf{x}_j \\neq 0$ for each $j= 1,2,...,n$.\n\nIn this case, the feasible set is then $C =\\{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}\\mid \\sum_{j\\in\\supp{\\mathbf{x}}}\\mathbf{w}_j\\leq B\\}$, which is again not necessarily a sublattice of $\\mathbb{R}^n$.\n\n\\blue{(End small example section.)}\n\\end{comment}\n\nWhen confronted with constrained optimization problems such as \\eqref{eq:constrained_case}, one common approach is to add a Lagrange multiplier $\\mu\\in\\mathbb{R}_{\\geq 0}$ and instead solve the unconstrained problem:\n\\begin{align}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}}&~f(\\mathbf{x}) + g(\\supp{\\mathbf{x}}) + \\mu \\sum_{i=1}^nW_i(\\mathbf{x}_i).\\label{eq:regularized_case}\n\\end{align}\nFor the correct choice of $\\mu\\in\\mathbb{R}_{\\geq 0}$, solving the regularized problem \\eqref{eq:regularized_case} is equivalent to solving the constrained problem \\eqref{eq:constrained_case}. Determining the $\\mu$ that renders the two problems equivalent, however, is typically a difficult task.\n\n\n\nOur work in this section relies on the following result that relates parameterized families of submodular set function minimization problems to a single convex optimization problem.\n\n\\begin{theorem}(Proposition 8.4 in \\cite{bach2013learning})\\label{thm:bach_convex_submod} Let $g:2^{[n]}\\rightarrow\\mathbb{R}$ be a submodular set function, and $g_L:\\mathbb{R}^n\\rightarrow\\mathbb{R}$ its Lov\\`asz extension (which is therefore convex). If, for some $\\epsilon > 0$, $\\psi_i:\\mathbb{R}_{\\geq \\epsilon}\\rightarrow\\mathbb{R}$ is a strictly increasing function on its domain for all $i=1,2,...,n$, then the minimizer $\\mathbf{u}^*\\in\\mathbb{R}^n$ of the convex optimization problem:\n\\begin{align}\\label{eq:single_convex}\n\\minimize{\\mathbf{u}\\in\\mathbb{R}^n_{\\geq 0}}~h_L(\\mathbf{u})+\\sum_{i=1}^n\\int_{\\epsilon}^{\\epsilon+\\mathbf{u}_i}\\psi_i(\\mu)d\\mu,\n\\end{align}\nis such that the set $A^\\mu = \\{i\\in [n]~:~\\mathbf{u}_i^* > \\mu\\}$ is the minimizer with smallest cardinality for the submodular set function minimization problem:\n\\begin{align}\\label{eq:family_submodular}\n\\minimize{A\\in 2^{[n]}}~h(A) + \\sum_{i\\in A}\\psi_i(\\mu),\n\\end{align}\nfor any $\\mu\\in\\mathbb{R}_{\\geq \\epsilon}$.\n\\end{theorem}\n\nIn the following subsections we identify classes of problems that allow the regularized problem \\eqref{eq:regularized_case} to be expressed in the form given by \\eqref{eq:family_submodular}. Theorem \\ref{thm:bach_convex_submod} then provides a single convex optimization problem we can solve to recover the solution to \\eqref{eq:regularized_case} for all possible values of the regularization strength $\\mu$. In prior work, this same theory was applied to purely discrete submodular minimization problems \\cite{fujishige2011submodular}, and purely continuous submodular minimization problems \\cite{staib2019robust}, but our work lies between these two extremes.\n\n\\subsection{Support Knapsack Constraints}\nWe first consider a knapsack constraint, meaning the function $W$ has the form:\n\\begin{align*}\nW(\\mathbf{x}) &= \\sum_{j\\in\\supp{\\mathbf{x}}}\\mathbf{w}_j,\n\\end{align*}\nfor some $\\mathbf{w}\\in\\mathbb{R}^n_{>0}$. The regularized problem \\eqref{eq:regularized_case} in this case is:\n\\begin{align*}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}}&~f(\\mathbf{x}) + g(\\supp{\\mathbf{x}}) + \\mu \\sum_{j\\in\\supp{\\mathbf{x}}}\\mathbf{w}_j.\n\\end{align*}\nBecause $W$ can be viewed as a set function in this case, the corresponding relaxed problem \\eqref{eq:lattice_opt_prob_relax} becomes:\n\\begin{align}\n\\minimize{A\\in 2^{[n]}}~g(A) + H(A) + \\sum_{j\\in A}\\psi_j(\\mu),\\label{eq:regularized_support_knapsack}\n\\end{align}\nwhere we have defined $\\psi_j(\\mu) = \\mu\\mathbf{w}_j$ for each $j=1,2,...,n$. Because $\\mathbf{w}_j>0$ for all $j$, these functions are strictly increasing, and we have a problem precisely in the form \\eqref{eq:family_submodular}. By Theorem \\ref{thm:bach_convex_submod}, we can solve the convex optimization problem:\n\\begin{align*}\n\\minimize{\\mathbf{u}\\in\\mathbb{R}_{\\geq \\epsilon}}~g_L(\\mathbf{u}) + H_L(\\mathbf{u}) + \\frac{1}{2}\\sum_{j=1}^n\\mathbf{w}_j\\mathbf{u}_j^2,\n\\end{align*}\nthen appropriately threshold the solution to recover the solution to \\eqref{eq:regularized_support_knapsack} for all possible values of $\\mu\\in\\mathbb{R}_{\\geq\\epsilon}$. Because $\\psi_j$ is finite and strictly increasing on all of $\\mathbb{R}$, we can simply select $\\epsilon = 0$.\n\nGiven the solutions to the regularized problem $A^\\mu$ specified by Theorem \\ref{thm:bach_convex_submod}, we select the set $A^\\mu$ with smallest $\\mu\\in\\mathbb{R}$ such that the constraint $W(\\mathbf{x}) \\leq B$ is satisfied. We can then use the result of Theorem \\ref{thm:main_result} to compute the minimizer in the original optimization problem over $\\mathbb{R}^n_{\\geq 0}$.\n\n\\subsection{Continuous Budget Constraints}\nAs shown above, the Lov\\`asz extension lets us handle problems with discrete budget constraints, so a natural next step is to consider continuous budget constraints. In particular, these are continuous functions $W:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$, such that:\n\\begin{align*}\nW(\\mathbf{x}) &= \\sum_{i=1}^nW_i(\\mathbf{x}_i),\n\\end{align*}\nwith each $W_i:\\mathbb{R}_{\\geq 0}\\rightarrow\\mathbb{R}$ a strictly increasing function. With this particular $W$, the regularized optimization problem \\eqref{eq:regularized_case} with Lagrange multiplier $\\mu\\in\\mathbb{R}_{\\geq 0}$ becomes:\n\\begin{align*}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}}~f(\\mathbf{x}) + g(\\supp{\\mathbf{x}}) + \\mu\\sum_{i=1}^nW_i(\\mathbf{x}_i).\n\\end{align*}\nTo recover the problem form \\eqref{eq:family_submodular} specified by Theorem \\ref{thm:bach_convex_submod}, we further assume that $f:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ is separable, i.e., $f(\\mathbf{x}) = \\sum_{i=1}^nf_i(\\mathbf{x}_i)$. In this case, the relaxed optimization problem \\eqref{eq:lattice_opt_prob_relax} is:\n\\begin{align}\n\\minimize{A\\in 2^{[n]}}~g(A) + \\sum_{i\\in A} H_i(\\mu),\\label{eq:cont_budget_regularized}\n\\end{align}\nwhere we defined $H_i : \\mathbb{R}_{> 0}\\rightarrow\\mathbb{R}$ as the function:\n\\begin{align}\nH_i(\\mu) &= \\underset{\\mathbf{z}\\geq 0}{\\min}~f_i(\\mathbf{z}) + \\mu W_i(\\mathbf{z}),\\quad i = 1,2,...,n,\\label{eq:scalar_H}\n\\end{align}\nand assumed (without loss of generality) that $W_i(0) = f_i(0) = 0$.\n\nTo apply Theorem \\ref{thm:bach_convex_submod}, we need $H_i:\\mathbb{R}_{> 0}\\rightarrow\\mathbb{R}$ to be strictly increasing on its domain. We verify this property in the following proposition, whose proof we detail in Appendix \\ref{apdx:continuous_constraints}.\n\n\\begin{proposition}\nThe function $H_i:\\mathbb{R}_{\\geq 0}\\rightarrow\\mathbb{R}_{\\leq 0}$ defined in \\eqref{eq:scalar_H} is monotone in $\\mu$ for all $i=1,2...,n$. It is strictly increasing for all $\\mu\\in[0,c]$, where $c\\in\\mathbb{R}_{\\geq 0}$ is the smallest constant such that $H_i(c) = 0$. In addition, $H_j$ is constant and zero on the interval $[c,\\infty[$.\n\\end{proposition}\n\nBecause the only point at which $H_i$ is not strictly increasing occurs when its value is exactly zero (implying that allowing the element $\\mathbf{x}_i$ to be nonzero provides no decrease in continuous cost), the desired result from Theorem \\ref{thm:bach_convex_submod} still holds with only a minor modification, the details of which we also defer to Appendix \\ref{apdx:continuous_constraints}.\n\nIt then follows from Theorem \\ref{thm:bach_convex_submod} that again, by solving the single convex optimization problem:\n\\begin{align}\n\\minimize{\\mathbf{u}\\in\\mathbb{R}^n_{\\geq 0}}~g_L(\\mathbf{u}) + \\sum_{i=1}^n\\int_\\epsilon^{\\epsilon+\\mathbf{u}_i}H_i(\\mu)d\\mu,\n\\end{align}\nwe can recover the solution to a family of regularized optimization problems \\eqref{eq:cont_budget_regularized}. As before, we select the set $A^\\mu$ with the largest $\\mu\\in\\mathbb{R}_{\\geq \\epsilon}$ such that the budget constraint $W(\\mathbf{x})\\leq B$ is satisfied.\n\n\\section{Robust Optimization}\nGiven our ability to efficiently solve the joint continuous and discrete optimization problem in \\eqref{eq:cont_discrete_opt_problem}, and even some of its constrained variants, we may seek problem settings where they arise as a subproblem. This situation often occurs in \\emph{robust optimization}, where we seek to solve an optimization problem while remaining resilient to worst-case problem instances.\n\n\\subsection{Motivating Example from Multiple Domain Learning}\nRecent work in \\cite{qian2019robust} highlighted the concept of \\emph{multiple domain learning}, where a single machine learning model is trained on sets of data from $K$ different domains. By training against worst-case distributions of the data in these domains, they show that the resulting machine learning model often achieves lower generalization and worst-case testing errors.\n\nIn particular, let the training data for a learning model be $S = \\{S_1,S_2,...,S_K\\}$ with $S_i$ the data from domain $i$. We also let $f_i : W\\rightarrow\\mathbb{R}$ for $i=1,2,...,K$ be the empirical risk of the model on the data from each domain $i$, given parameters in some convex subset $W\\subseteq\\mathbb{R}^n$. The proposed robust optimization problem is then:\n\\begin{align*}\n\\minimize{\\mathbf{w}\\in W}~\\underset{\\mathbf{p}\\in C}{\\max}~\\sum_{i=1}^K\\mathbf{p}_if_i(\\mathbf{w}),\n\\end{align*}\nwith $C = \\{\\mathbf{p}\\in \\mathbb{R}^K_{\\geq 0}\\mid \\sum_{i=1}^K\\mathbf{p}_i \\leq 1\\}$, the simplex. If we additionally reward the use of data from domain $i$ (or equivalently, penalize the worst-case distribution of data for including domain $i$), then we form the robust continuous and discrete optimization problem:\n\\begin{align*}\n\\minimize{\\mathbf{w}\\in W}~\\underset{\\mathbf{p}\\in C}{\\max}~\\sum_{i=1}^K\\mathbf{p}_if_i(\\mathbf{w}) - g(\\supp{\\mathbf{p}}),\n\\end{align*}\nwith $g: 2^K\\rightarrow\\mathbb{R}$ a monotone submodular set function. By considering a penalty on the set of nonzero entries of the worst-case distribution, we encode some prioritization of which domains are more or less relevant to us in our application. Then by Theorem \\ref{thm:bach_convex_submod}, we can solve the inner maximization problem (with an appropriate change of signs) by adding a Lagrange multiplier $\\mu$ and solving a related convex problem.\n\n\\subsection{General Results}\nMore generally, robust optimization problems can often be expressed as a min-max saddle point optimization problem of a function $q:\\mathcal{X}\\times\\mathcal{Y}\\rightarrow\\mathbb{R}$:\n\\begin{align}\n\\maximize{\\mathbf{x}\\in\\mathcal{X}}\\underset{\\mathbf{y}\\in \\mathcal{Y}}{\\min}~q(\\mathbf{x},\\mathbf{y}). \\label{eq:saddle_point_prob}\n\\end{align}\nThis problem is interpreted as maximizing the function $q(\\mathbf{x},\\mathbf{y})$ with respect to our available parameters $\\mathbf{x}\\in\\mathcal{X}\\subseteq \\mathbb{R}^n$, under the worst case choice of additional problem parameters $\\mathbf{y}\\in \\mathcal{Y}\\subseteq\\mathbb{R}^m_{\\geq 0}$ \\cite{ben2009robust}.\n\nGiven some appropriate structure for the function $q$, the min-max problem \\eqref{eq:saddle_point_prob} is surprisingly tractable. If we define $Q:\\mathcal{X}\\rightarrow\\mathbb{R}$ as:\n\\begin{align*}\nQ(\\mathbf{x}) &= \\underset{\\mathbf{y}\\in \\mathcal{Y}}{\\min}~q(\\mathbf{x},\\mathbf{y}),\n\\end{align*}\nwe can express the saddle-point problem \\eqref{eq:saddle_point_prob} as:\n\\begin{align}\n\\maximize{\\mathbf{x}\\in\\mathcal{X}}~Q(\\mathbf{x}).\\label{eq:Q_opt}\n\\end{align}\nIf the function $q(\\mathbf{x},\\mathbf{y})$ is concave in $\\mathbf{x}$ for any fixed $\\mathbf{y}\\in \\mathcal{Y}$, then the function $Q$ is concave \\cite{borwein2000convex}. Moreover, we can compute a subgradient of $Q$ at any $\\mathbf{x}\\in\\mathcal{X}$ as:\n\\begin{gather*}\n\\nabla_{\\mathbf{x}}Q(\\mathbf{x}_0) = \\nabla_\\mathbf{x} q(\\mathbf{x}_0,\\mathbf{y}^*), \\\\\n\\mathbf{y}^*\\in\\textrm{arg}\\min{\\mathbf{y}\\in\\mathcal{Y}}~q(\\mathbf{x}_0,\\mathbf{y}).\n\\end{gather*}\nIn other words, efficiently solving the minimization problem defining $Q$ for an $\\mathbf{x}_0\\in\\mathcal{X}$ also gives a subgradient of $Q$. Because $Q$ is concave in $\\mathbf{x}$, even a straightforward algorithm such as projected subgradient ascent in the problem \\eqref{eq:Q_opt} will converge to a global optimum.\n\nIn this work, we showed that minimization problems in the form of \\eqref{eq:cont_discrete_opt_problem} with functions satisfying Assumptions 1-3 can be solved efficiently. Suppose then, that the function $q:\\mathcal{X}\\times\\mathcal{Y}$ is of the form:\n\\begin{align*}\nq(\\mathbf{x},\\mathbf{y}) &= f(\\mathbf{x},\\mathbf{y}) + g(\\eta(\\mathbf{y}))\n\\end{align*}\nwith $f:\\mathcal{X}\\times \\mathcal{Y}\\rightarrow\\mathbb{R}$ concave in $\\mathbf{x}$ for any fixed $\\mathbf{y}$ and also convex and submodular on $\\mathcal{Y}\\subseteq \\mathbb{R}^n_{\\geq 0}$ in $\\mathbf{y}$ for any fixed $\\mathbf{x}$. If $\\eta:\\mathcal{Y}\\rightarrow\\mathcal{L}$ satisfies Assumption 3, $g:\\mathcal{L}\\rightarrow\\mathbb{R}$ is monotone and submodular, and we assume the set of $\\mathbf{y}\\in\\mathcal{Y}$ such that $\\eta(\\mathbf{y})\\leq \\mathbf{l}$ is a convex subset for any $\\mathbf{l}\\in\\mathcal{L}$, then the robust optimization problem \\eqref{eq:saddle_point_prob} becomes:\n\\begin{align}\n\\maximize{\\mathbf{x}\\in\\mathbb{R}^n}~\\underset{\\mathbf{y}\\in \\lattwo}{\\min}~f(\\mathbf{x},\\mathbf{y}) + g(\\eta(\\mathbf{y})).\\label{eq:max_min_prob}\n\\end{align}\n\nFor a given $\\mathbf{x}_0\\in\\mathbb{R}^n$, we view the selection of $\\mathbf{y}\\in \\mathcal{Y}$ as a worst-case, or ``adversarial'' choice of parameters for the function $f$. The penalty on $\\eta(\\mathbf{y})$ then suggests that by simply considering some structured sets of adversarial parameters $\\eta(\\mathbf{y})$ while selecting the optimal $\\mathbf{x}$, we gain some associated benefit. The submodularity of $g$ then implies that as we consider more potential variations in $\\mathbf{y}$ (i.e., larger $\\supp{\\mathbf{y}}$) the ``gain'' we receive for considering more variations is smaller.\n\nIn addition, $Q$ becomes:\n\\begin{align*}\nQ(\\mathbf{x}) &= \\underset{\\mathbf{y}\\in \\lattwo}{\\min}~f(\\mathbf{x},\\mathbf{y}) + g(\\eta(\\mathbf{y})),\n\\end{align*}\nwhich is still the minimum of a family of concave functions, and therefore amenable to subgradient ascent methods as discussed above. A subgradient of $Q$ can easily be computed as:\n\\begin{gather*}\n\\nabla_{\\mathbf{x}} Q(\\mathbf{x}_0) = \\nabla_{\\mathbf{x}}q(\\mathbf{x}_0,\\mathbf{y}^*) = \\nabla_\\mathbf{x} f(\\mathbf{x}_0,\\mathbf{y}^*), \\\\\n\\mathbf{y}^* \\in \\textrm{arg}\\min{\\mathbf{y}\\in\\lattwo}~f(\\mathbf{x}_0,\\mathbf{y}) + g(\\eta(\\mathbf{y})).\n\\end{gather*}\nWe collect these ideas into the following theorem.\n\n\\begin{theorem}\\label{thm:robust_opt}\nConsider the robust optimization problem \\eqref{eq:max_min_prob}. Assume $f:\\latone\\times\\lattwo\\rightarrow\\mathbb{R}$ is concave in $\\mathbf{x}\\in\\latone$ for any fixed $\\mathbf{y}\\in\\lattwo$, and also convex and submodular in $\\mathbf{y}\\in\\lattwo$ for any fixed $\\mathbf{x}\\in\\latone$. Let $\\eta:\\lattwo\\rightarrow\\mathcal{L}$ satisfy Assumption 3, $g:\\mathcal{L}\\rightarrow\\mathbb{R}$ be a monotone submodular function and assume that for a given $\\mathbf{\\ell}\\in\\mathcal{L}$, the set of $\\mathbf{y}\\in\\lattwo$ such that $\\eta(\\mathbf{y})\\leqtwo \\mathbf{\\ell}$ is a convex subset of $\\lattwo$. For any $\\epsilon \\in\\mathbb{R}_{>0}$, let $T\\in\\mathbb{Z}_{>0}$ be of order $O(\\frac{1}{\\epsilon^2})$, meaning as $T$ tends to infinity, there exists a constant $M\\in\\mathbb{R}_{>0}$ such that $T \\leq \\frac{M}{\\epsilon^2}$. Then $T$ iterations of projected subgradient ascent using step lengths $\\eta_i = \\frac{1}{\\sqrt{T}}$ produces, in polynomial time, iterates $\\mathbf{x}^{(i)}\\in\\latone$ for $i = 1,2,...,T$ such that $\\frac{1}{T}\\sum_{i=1}^TQ(\\mathbf{x}^{(i)}) \\leq Q(\\mathbf{x}^*) + \\epsilon$.\n\\end{theorem}\nThe computational complexity of this approach may be high, as projected subgradient ascent can be slow in practice. However, each sub-problem instance involves a mixed continuous and discrete optimization problem, so this complexity is warranted.\n\n\\section{Examples and Computational Evaluation}\nIn this section, we illustrate the proposed theoretical results on several numerical examples involving optimization on the lattices $\\mathbb{R}^n_{\\geq 0}$ and $2^{[n]}$. We compare against two state-of-the-art techniques: a direct application of the continuous submodular function minimization algorithms outlined in \\cite{bach2019submodular}, and the projected subgradient descent method proposed in \\cite{eloptimal}. \n\nThe algorithms for continuous submodular function minimization rely on discretizing the domain $\\mathbb{R}^n_{\\geq 0}$ into $k$ discrete points in each dimension, so we consider the domain $[0,1]^n\\subseteq\\mathbb{R}^n_{\\geq 0}$ and set the discretization level to $k = 51$ unless otherwise specified. In our implementation, we use the Pairwise Frank-Wolfe algorithm to solve the discretized optimization problem, with all relevant results plotted in blue and labeled \\textit{Cont Submodular}.\n\nThe projected subgradient method is known to provide approximation guarantees even in the non-submodular case \\cite{eloptimal}, but as shown in Section \\ref{sec:polytime}, amounts to a specific choice of algorithms in our theory. To implement this approach, we use IBM's CPLEX 12.8 constrained quadratic program solver in MATLAB to evaluate the function $H$ (as expressed in \\eqref{eq:H_definition}) and use Polyak's rule for updating the step size. The relevant results are plotted in red, and labeled \\textit{Projected (Sub)Gradient} in figures.\n\nOur approach is agnostic to the choice of convex optimization and submodular set function minimization routines, so we also use CPLEX to evaluate $H$. However, we instead use the minimum-norm point algorithm from \\cite{fujishige2011submodular} as implemented in MATLAB by \\cite{krause2010sfo}, which has fast performance in practice, coupled with the semi-gradient based lattice pruning strategy proposed in \\cite{iyer2013fast}. Our results are plotted in black, and labeled \\textit{Min Norm + CPLEX} in figures.\n\nAll three methods are presented with identical cost functions to minimize, and are run until either convergence to suboptimality below $10^{-4}$ or a maximum of 100 iterations. The experiments were all run on a mid-2012 Macbook Pro with a 2.9 GHz Intel Core i7 processor and 16GB of 1600 MHz DDR3 RAM, and reported running times are an average of five runs on randomized problem instances.\n\n\\subsection{Regularized Sparse Regression}\nWe first examine a regularized sparse regression problem, similar in spirit to \\eqref{eq:compressed_sensing}. Consider some $\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}$, $\\mathbf{D}\\in\\mathbb{R}^{m\\times n}$, $\\mathbf{b}\\in\\mathbb{R}^m$, and define the function $f:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ as:\n\\begin{align}\nf(\\mathbf{x}) &= \\Vert \\mathbf{Dx}-\\mathbf{b}\\Vert_2^2.\\label{eq:ls_mr_f}\n\\end{align}\nThen define the monotone submodular set function $g:2^{[n]}\\rightarrow\\mathbb{R}$ as:\n\\begin{align}\ng(A) &= \\begin{cases}\n\\lambda\\left[(n-1) + \\max(A) - \\min(A) + \\vert A\\vert\\right], & A\\neq \\emptyset, \\\\\n0 & A = \\emptyset,\n\\end{cases}\\label{eq:ls_mr_g}\n\\end{align}\nwith $\\lambda\\in\\mathbb{R}_{\\geq 0}$, and $\\max(A)$ and $\\min(A)$ denoting the largest and smallest index element, respectively, in the set of indices $A$. This choice of $g$ in the sparse regression problem \\eqref{eq:lattice_opt_problem} places a high penalty on large sets of nonzero entries in the vector $\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}$ that are far apart in index.\n\nWe generate a series of random problem instances with $m = n$ satisfying the assumption of submodularity on $\\mathbb{R}^n_{\\geq 0}$ and also the convexity condition of Corollary \\ref{cor:polytime}. Let $\\mathrm{chol}:\\mathbb{R}^{n\\times n}\\rightarrow\\mathbb{R}^{n\\times n}$ denote a Cholesky decomposition of a positive semidefinite matrix, and construct the matrix $\\mathbf{D}$ in \\eqref{eq:ls_mr_f} as:\n\\begin{align*}\n\\mathbf{D} &= \\mathrm{chol}\\left(\\mathbf{C} + \\mathbf{C}^T + n\\mathbf{I}\\right), \\quad \\mathbf{C}_{ij}\\sim\\mathrm{unif}(-1,0),\\4all i,j=1,2,...,n.\n\\end{align*}\nThis construction guarantees that the function $f$ in \\eqref{eq:ls_mr_f} is both convex and submodular on $\\mathbb{R}^n_{\\geq 0}$, satisfying the conditions for Corollary \\ref{cor:polytime}. For the parameter $\\mathbf{b}\\in\\mathbb{R}^m$, we use the signal in the top plot of Fig. \\ref{fig:LS_figs}, and we set the regularization strength to $\\lambda = 0.05$ so that both the functions $f$ and $g$ play nontrivial roles in the objective function.\n\n\nWe plot the results from each algorithm in Fig. \\ref{fig:LS_figs}. Because the minimizer of the optimization problem is a representation of $\\mathbf{y}$ using structured sparse columns of $\\mathbf{D}$, we show the the reconstructed vector $\\mathbf{Dx}$ in the second, third, and fourth plots of Fig. \\ref{fig:LS_figs}. Because there is no reliance on discretization, both the projected subgradient descent and minimum-norm point algorithms produce a much smoother result, as expected.\n\nIn the bottom left plot of Fig. \\ref{fig:LS_figs}, we show the cost achieved over iterations of each algorithm. The minimum-norm point converges rapidly to the globally optimal cost, while the projected subgradient descent method takes longer to achieve the same cost. However, the discretization error associated with the continuous submodular function minimization approach prevents it from ever achieving the true optimal cost.\n\nFinally, over a small window of problem sizes, we show the running times of each algorithm in the bottom right plot of Fig. \\ref{fig:LS_figs}. Interestingly, our approach presents a compromise between the slow optimality of the projected subgradient descent method and the fast but inexact continuous submodular function minimization algorithm. \n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}{0.95\\linewidth}\n\t\\includegraphics[width=\\linewidth]{.\/new_pics\/ls_mr_combined_signals.eps}\n\t\\vspace{-15mm}\n\\end{subfigure}\\\\\n\\begin{subfigure}{0.95\\linewidth}\n\t\\includegraphics[width=\\linewidth]{.\/new_pics\/ls_mr_costs_and_runtimes.eps}\n\\end{subfigure}\n\\end{center}\n\\caption{Results from the sparse regression problem simulations. The reconstructed signal representations using columns of $\\mathbf{D}$ created by each algorithm are shown in the second, third, and fourth plot. Note the solutions produced by projected subgradient and the minimum-norm point algorithm are identical. We plot the cost function value over each algorithm's iterations in the bottom left, while in the bottom right we compare the running times of the algorithms over a small window of problem dimensions.}\n\\label{fig:LS_figs}\n\\vspace{-3mm}\n\\end{figure}\n\n\n\\subsection{Signal Denoising}\nWe next study a simple denoising example, where we consider a signal $\\mathbf{x}\\in\\mathbb{R}^n$, which is corrupted by some additive disturbance $\\mathbf{w}\\in\\mathbb{R}^n$, with $\\mathbf{w}\\sim \\mathcal{N}(0,0.1\\mathbf{I})$. We would like to recover the signal $\\mathbf{x}$ from the noisy measurements $\\mathbf{y} = \\mathbf{x} + \\mathbf{w}$, under the assumption that the true signal $\\mathbf{x}$ is smooth (meaning variations between adjacent entries ought to be small), and that the meaningful content arrived in a small number of contiguous windows of entries.\n\nWe can express the desire to match the noisy signal $\\mathbf{y}$ with a smooth one with the convex and submodular function $f:\\mathbb{R}^n\\rightarrow\\mathbb{R}$ defined as:\n\\begin{align}\nf(\\mathbf{x}) &= \\frac{1}{2}\\Vert \\mathbf{x} - \\mathbf{y}\\Vert + \\mu \\sum_{i=1}^{n-1}\\left(\\mathbf{x}_i-\\mathbf{x}_{i+1}\\right)^2.\\label{eq:DN_f}\n\\end{align}\nThe first term promotes matching the slightly corrupted signal, while the quadratic penalty on adjacent entries of $\\mathbf{x}\\in\\mathbb{R}^n$ promotes smoothness.\n\nSimilarly, we can express the knowledge of a small and contiguous set of nonzero entries in the vector $\\mathbf{x}$ with the monotone submodular set function $g:2^{[n]}\\rightarrow\\mathbb{R}$ defined by:\n\\begin{align}\ng(A) &= \\lambda\\left(\\vert A\\vert + \\mathrm{\\# int}(A)\\right),\\label{eq:DN_g}\n\\end{align}\nwhere $\\lambda\\in\\mathbb{R}_{\\geq 0}$, and the function $\\mathrm{\\# int}(A)$ counts the number of sets of contiguous indices in the set $A$. This set function is smallest on subsets with a small number of entries that are adjacent in index.\n\nFor experiments, we use the signal $\\mathbf{x}\\in\\mathbb{R}^n$ shown in the top plot of Fig. \\ref{fig:DN_figs}, with the noise-corrupted measurements $\\mathbf{x} + \\mathbf{w} = \\mathbf{y}\\in\\mathbb{R}^n$ with an example shown in dotted orange. We then let $\\mu = 0.8$ in \\eqref{eq:DN_f} and $\\lambda = 0.05$ in \\eqref{eq:DN_g} so that the overall problem's cost function has nontrivial contributions from both the smoothness-promoting function and the sparsity-inducing regularizer. In this case, for the continuous submodular algorithm we discretize the compact set $[-1,1]^n\\subseteq\\mathbb{R}^n$ into $k=101$ distinct values per index.\n\nWe show the resulting denoised signals in the second, third, and fourth plots in Fig. \\ref{fig:DN_figs}, with the running time comparison over a small window of problem dimensions in the bottom right. The discretization of the domain in the continuous submodular function minimization approach produces artifacts in the resulting signal, whereas the result of the projected subgradient and minimum-norm point algorithms are smoother with smaller sets of nonzero entries. We see once more that for most problem sizes, our proposed minimum-norm point algorithm poses a compromise between speed and accuracy, providing guaranteed global optimality without the high running time of projected subgradient descent.\n\nWe also compare the objective value achieved during the iterations of each algorithm for a single instance in the bottom left plot of Fig. \\ref{fig:DN_figs} with $n=100$. Again, the minimum-norm point algorithm converges rapidly to the minimum alongside the projected subgradient method, while the continuous submodular function minimization approach's discretization error prevents it from ever achieving global optimality.\n\n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}{0.95\\linewidth}\n\t\\includegraphics[width=\\linewidth]{.\/new_pics\/smooth_int_combined_signals.eps}\n\t\\vspace{-15mm}\n\\end{subfigure}\\\\\n\\begin{subfigure}{0.95\\linewidth}\n\t\\includegraphics[width=\\linewidth]{.\/new_pics\/smooth_int_costs_runtimes.eps}\n\\end{subfigure}\n\\end{center}\n\\caption{Results of the denoising problem simulations. The true signal and its noisy counterpart are shown in the top plot. The second, third, and fourth plots show the denoised signals produced by each of the three algorithms. Note that the results from the minimum-norm point algorithm and the projected subgradient descent method are identical. The bottom left plot shows the objective value across iterations for $n=100$, and bottom right shows the running times of each algorithm for a window of problem dimensions.}\n\\label{fig:DN_figs}\n\\vspace{-3mm}\n\\end{figure}\n\n\\subsection{Discretization Error Dependence}\nIn this section, we explore the relationship between the continuous submodular function minimization algorithm's discretization error and its running time. To this end, we ran instances of the sparse regression example with the modified range function penalty, using a discretization resolution in each dimension ranging from $k=50$ to $k=400$.\n\nThe minimum cost achieved at each discretization level $k$ is shown in the left plot of Fig. \\ref{fig:k_comp}. Similarly, the associated running times of the algorithm are shown in the right-hand plot of Fig. \\ref{fig:k_comp}. Interestingly, near the value of $k = 250$, the achieved cost becomes effectively optimal, but the running time increases by an order of magnitude.\n\n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}{.4\\textwidth}\n\t\\includegraphics[width=1.0\\linewidth]{pics\/k_vs_optimality_mr}\n\t\\vspace{-6mm}\n\t\\label{fig:k_vs_optimality}\n\\end{subfigure}%\n\\begin{subfigure}{.4\\textwidth}\n\t\\includegraphics[width=1.0\\linewidth]{pics\/k_vs_runtime_mr}\n\t\\vspace{-6mm}\n\t\\label{fig:k_vs_runtime}\n\\end{subfigure}\n\\end{center}\n\\caption{Results highlighting the role of the discretization resolution $k$ on the continuous submodular algorithm's optimality (left) and running times (right) in an instance of the sparse regression problem with $n=100$.}\n\\label{fig:k_comp}\n\\vspace{-3mm}\n\\end{figure}\n\nTo give a coarse estimate on the origin of higher running times for projected subgradient descent and the minimum-norm point algorithms, we note that the computational cost of each iteration is dominated by the cost of computing the Lov\\`asz extension of $H$. This computation has time complexity $O(n\\log n + n EO)$, where $EO$ is the complexity of evaluating $H$. If $H$ is evaluated through convex optimization, many generic interior-point methods have time complexity that is approximately $EO = O(n^3)$. Therefore, each iteration of the minimum-norm point algorithm and the projected subgradient descent algorithm might have complexity on the order of $O(n\\log n + n^4)$.\n\nOne possible remedy to this issue would be to use a more specialized algorithm for solving the constrained convex optimization problem evaluating $H$, thus reducing the complexity of $EO$. Another easy alternative would be to stop early, producing an approximate evaluation of $H$. Projected subgradient descent still has some approximation guarantees in this regime \\cite{eloptimal}, but our agnostic combination may be more brittle to these variations.\n\n\\section{Conclusions}\nIn this work, we showed that model-fitting problems with structure-promoting regularizers could be expressed as optimization problems defined over two connected lattices. Using submodularity theory, we derived conditions on these functions and their domains under which we can directly solve these problems exactly and efficiently. We focused on continuous and Boolean lattices, and derived conditions under which an agnostic combination of submodular set function minimization and convex optimization algorithms can compute the exact solution in polynomial time.\n\nWe then extended this theory to handle optimization problems with simple continuous or discrete budget constraints on the model parameters. We did this by naively adding the constraint to the cost with a Lagrange multiplier, but then used submodular function theory to solve for all possible Lagrange multiplier values with a single convex optimization problem. Finally, we also highlighted robust or adversarial optimization scenarios, where our exact solutions could provide subgradients to be used in globally convergent ascent methods.\n\nOne promising next direction of research would be examining if a slightly weaker satisfaction of the assumptions on the functions $f$ and $g$ would result in only a slightly weaker guarantee, with the same algorithm-agnostic approach. Moreover, the assumptions and conditions outlined in this work are merely sufficient. Future work might examine if these conditions are necessary as well.\n\n\\begin{comment}\n\\subsection{System Observability}\nConsider a linear dynamical system:\n\\begin{align*}\n\\dot{\\mathbf{x}} &= \\mathbf{Ax} \\\\\n\\mathbf{y} &= \\mathbf{Cx}\n\\end{align*}\nwith $\\mathbf{x}\\in\\mathbb{R}^n$, $\\mathbf{A}\\in\\mathbb{R}^{n\\times n}$, $\\mathbf{y}\\in\\mathbb{R}^m$, and $\\mathbf{C}\\in\\mathbb{R}^{m\\times n}$. The system's Observability Gramian $\\mathcal{W}_O \\in \\mathbb{R}^{n\\times n}$ is defined as:\n\\begin{align*}\n\\mathcal{W}_O &= \\int_0^\\infty e^{\\mathbf{A}^T\\tau}\\mathbf{C}^T\\mathbf{C}e^{\\mathbf{A}\\tau}d\\tau.\n\\end{align*}\nThe Observability Gramian is a positive semidefinite matrix that characterizes the ``observable space,'' which can be thought of as the portion of the state space where estimating the state $\\mathbf{x}$ through outputs $\\mathbf{y}$ is possible.\n\nIf we let the output $\\mathbf{y}$ be ``filtered'' through some matrix $\\mathbf{S}\\in\\mathbb{R}^{m\\times m}$ such that $\\mathbf{y} = \\mathbf{SCx}$, then the Observability Gramian becomes:\n\\begin{align*}\n\\mathcal{W}_O&=\\int_0^\\infty e^{\\mathbf{A}^T\\tau}\\mathbf{C}^T\\mathbf{S}^T\\mathbf{S}\\mathbf{C}e^{\\mathbf{A}\\tau}d\\tau.\n\\end{align*}\nWhen determining the best filtering matrix $\\mathbf{S}$ for the system, one objective would be to maximize the ``average observable space'' of the system, which would mean maximizing the quantity:\n\\begin{align*}\n\\tr{\\mathcal{W}_O} &= \\tr{\\int_0^\\infty e^{\\mathbf{A}^T\\tau}\\mathbf{C}^T\\mathbf{S}^T\\mathbf{S}\\mathbf{C}e^{\\mathbf{A}\\tau}d\\tau}.\n\\end{align*}\nBecause the matrix $\\mathbf{S}$ appears in a quadratic fashion here, optimization problems determining $S$ directly as a function of its entries is difficult. We instead parameterize $\\mathbf{S}^T\\mathbf{S}$ with a conic combination of $q$ symmetric positive semidefinite matrices $\\mathbf{E}_i\\in\\mathbb{R}^{m\\times m}$, weighted by some $\\mathbf{p}\\in\\mathbb{R}^p_{\\geq 0}$:\n\\begin{align*}\n\\mathbf{S}^T\\mathbf{S} &= \\sum_{i=1}^q\\mathbf{p}_i\\mathbf{E}_i.\n\\end{align*}\nAfter determining the desired weights $\\mathbf{p}$, we can extract $\\mathbf{S}$ through the Cholesky decomposition. Note however, that because we are using a specific choice of basis $\\mathbf{E}_i$ for our matrix $\\mathbf{S}^T\\mathbf{S}$, we can only select the best $\\mathbf{S}^T\\mathbf{S}$ (and therefore $\\mathbf{S}$) of those that can be written as a nonnegative weighted sum of our selected matrices $\\mathbf{E}_i$, which is not necessarily the entire space of possible matrices.\n\nInterestingly, the trace of the Observability Gramian is a linear function in $\\mathbf{p}$, which we then express as:\n\\begin{align*}\n\\tr{\\mathcal{W}_O}(\\mathbf{p}) &= \\mathbf{w}^T\\mathbf{p}.\n\\end{align*}\nwith $\\mathbf{w}\\in\\mathbb{R}^q$.\n\nIn addition to maximizing the observable space, we may want to reduce the complexity of the output $\\mathbf{y}$ by preferring that it depend on only a few elements of the state variable $\\mathbf{x}$. Conveniently, when the filtering matrix $\\mathbf{S}$ does not depend on an index $i$ of the state, the resulting matrix $\\mathbf{S}^T\\mathbf{S}$ has all zeros in both row and column $i$.\n\nGiven a choice of weightings $\\mathbf{p}$, we can compute the corresponding number of nonzero rows (or columns) in the matrix $\\sum_{i=1}^q\\mathbf{p}_i\\mathbf{E}_i$ and assign an associated penalty. Doing this creates a set function $g:2^q\\rightarrow\\mathbb{R}$, and as $g$ is the sum of indicator functions of (potentially overlapping) groups, it is naturally a monotone submodular function.\n\nIn order to avoid selecting unbounded amounts of particular bases, we can place a budget, or an $\\ell_2$-norm ball constraint on the vector $\\mathbf{p}$, and we are left with the final optimization problem:\n\\begin{align*}\n\\begin{array}{cc}\n\\minimize{\\mathbf{p}\\in\\mathbb{R}^q_{\\geq 0}}&-\\mathbf{w}^T\\mathbf{p} + g(\\supp{\\mathbf{p}}) \\\\\n\\text{subject to} & \\Vert\\mathbf{p}\\Vert_2^2 \\leq 1.\n\\end{array}\n\\end{align*}\nBecause the function $\\tr{\\mathcal{W}_O}:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ is separable, and the $\\ell_2$ norm constraint is defined by a strictly increasing and separable function, we can directly apply Theorem \\ref{thm:bach_convex_submod_ii} and solve the regularized form of this problem for all regularization strengths.\n\\end{comment}\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\ph{Autonomous Exploration and SubT} Robotic exploration and the advancement of autonomy offer new ways to explore potentially dangerous and hard-to-access underground environments. Multi-agent systems have matured in controlled and structured environments like warehouses, factories, and laboratories, while current robotic challenges seek to advance these technologies for search and rescue scenarios, planetary prospecting, and subsurface exploration \\cite{asada2019robocup, Hambuchen2017NASA_Rob_Challenge, Link2021_ESA_ESRIC}. Motivated by the search for life on other planets, NASA JPL's team CoSTAR \\cite{agha2021nebula} took part in the Defense Advanced Research Projects Agency's (DARPA) Subterranean Challenge (SubT) seeking to advance robotic multi-agent systems and their technology readiness for potential future missions. If brought to other planets (e.g. Mars), subsurface missions could bring new insights into their geologic past as well as on their potential for supporting life in the environmentally protected undergrounds \\cite{Titus2021}. In contrast to traditional exploration missions where a team of operators and scientists controls one rover, SubT introduced the challenging requirement that only \\textit{a single human supervisor} can directly interface with the deployed multi-agent team in real-time and when a communication link is established. SubT is divided into three, one year development pushes with major field testing demonstrations. This work focuses on the advancements in our supervisor autonomy and game-inspired user interface that were developed under the restrictions of a worldwide pandemic and deployed during the SubT final competition comprising two preliminary missions (P1 and P2) and the final prize run (F). \n\n\n\\begin{figure}[t]\n\\centering\n \\includegraphics[width=0.45\\textwidth]{figures\/robot_team-min.png}\n \\caption{Team CoSTAR's Mission Control user interface (A). (B) a subset of CoSTAR's ground robots showing four customized Boston Dynamic's Spot and Clearpath Husky powered by JPL's autonomy platform NeBULA. Typically a deployment of 4 to 6 ground vehicles was targeted during SubT, but the number of agents is extendable (e.g., see A with 11 robots).}\n \\label{fig:robot_team}\n\\end{figure}\n\n\\ph{Human-Robot Collaboration} Achieving man-computer symbiosis \\cite{Licklider1960ManComputerSymbiosis} has been a long-time goal of the community to promote a close coupling of human and machine capabilities and ultimately inspire the evolving field of human-robot interaction \\cite{chen2021human}. \nThis work improves collaborative human multi-robot exploration and search performance fusing our extended autonomy assistant Copilot \\cite{Kaufmann2020copilotMike} that uses automated planning techniques with a game-inspired interface design for effective robot deployment, operations, and single operator supervision to create a more symbiotic interaction. \n\nWe present key design choices that are breaking away\nfrom common robot interfacing strategies that were deployed in similar challenge\ncontexts \\cite{kohlbrecher2015human,cerberus} and used interfaces based on the\nRobot Operating System's (ROS) visualization tool RViz. Further, we leverage\nhuman-robot interdependencies to inform the design and development of supervised\nautonomy and interaction paradigms to achieve our set interaction objectives.\nThe latest results from the SubT competition ``Finals'' are compared to a\nbaseline from previous competition runs, namely the ``Urban Circuit'', which\ndeployed earlier interface and system implementations and interaction paradigms\nthat we improve with our combined game-inspired interface and enhanced\nsupervisory autonomy.\n\n\nIn \\Cref{sec:related_work}, we look at related work from human-robot interaction\nand user interface design. \\Cref{sec:problem_requirements_objectives} outlines\nthe SubT requirements and interaction objectives for our multi-agent exploration\nscenario.\nWe then describe our supervisory autonomy Copilot and its latest implementation\nin \\Cref{sec:supervised_autonomy}, while \\Cref{sec:interface} outlines the user\ninterface components and human-robot interaction capabilities. Finally, we\npresent results from our development pushes in \\Cref{sec:results} and close with\nconclusions and an outlook on future work in\n\\Cref{sec:conclusions_future_work}.\n\n\n\\section{Related Work}\n\\label{sec:related_work}\n\\ph{Human-Robot Interaction and Interface Design} More than sixty years after\nthe introduction of man-computer symbiosis by\n\\textit{Licklider}~\\cite{Licklider1960ManComputerSymbiosis}, \\textit{Chen and\n Barnes}~\\cite{chen2021human} conclude that the boundaries of long-term\nhuman-robot symbiosis are still to be pushed by interdisciplinary\ncollaborations. \\textit{Szafir and Szafir}~\\cite{Szafir2021HRI_DATA_VIZ} have\nidentified best practices in the field of data visualization as a key driver to\nadvance both HRI and data visualization. Complex visualizations and renderings\nhave become achievable with off-the-shelf hardware, which allows the integration\nof visualization principles such as sensemaking~\\cite{Szafir2021HRI_DATA_VIZ}\nthat helps a human digest information. In human-space systems \\textit{Rahmani\n et al.}~\\cite{rahmani2019space} identified that interface technologies are\ncurrently in development, but their technology readiness levels are not very\nmature. Multiple design methods have been introduced in the literature, for\ninstance, Coactive Design~\\cite{coactive_design} which is a structured\napproach to analyze human and robot requirements and was used in the context of\nthe 2015 DARPA Virtual Robotics Challenge that aimed at advancing disaster\nresponse capabilities. \\textit{Roundtree et\n al.}~\\cite{Roundtree2019VizDesignCollectiveTeams} found that abstract\ninterface designs that visualize collective status over single agent\ninformation could increase performance; however such designs depend on the task\nat hand, team size and mission goals~\\cite{chen2021human}. A common testing\nstrategy in computer game development is\nPlaytesting~\\cite{Wallner2019Playtesting},\nwhich is comparable to simulation and field testing in the multi-robot domain.\nThe game-inspired development technique RITE, which was introduced in the\ncontext of interface development for the computer game Age of\nEmpires~\\cite{medlock2002usingRITE_AgeOfEmpires}, was used and adapted for fast\ndevelopment sprints. Additionally, we drew inspiration from real-time strategy games like Age of Empires,\nwhich guided the design of the 3D portion of the interface.\n\n\n\\ph{Robot Challenge Interfaces} During 2013's DARPA Robotics Challenge, team\nViGIR leveraged ROS to control a humanoid robot. The team decided to implement\ntheir interfaces using RViz and built an Operation Control Center consisting of\nat least six screens. Robot challenges are found to typically influence\nhuman-robot interaction design and interfaces~\\cite{Szafir2021HRI_DATA_VIZ} and\nfor DARPA's SubT teams, the common design practice was based on RViz and ROS\nplugins (\\cite{csiro,cornellSubTJFR,scherer2021resilient,norlab,cerberus}). Even\nour team started off using RViz as a quick way to prototype\ninterfaces~\\cite{Otsu2020IEEEAerospace} and used it as the main way to interact with\nthe robot agents due to its tight integration with ROS and ability to access\nrobot data for debugging purposes. We shifted away from this approach for the final competition, and the resulting HRI modalities and supervisory interface are presented in this work.\n\n\n\\section{Background and Objectives}\n\\label{sec:problem_requirements_objectives}\n\\ph{Challenge Requirements} The overall SubT goals are two common problems faced\nby real-world multi-agent systems: first, the autonomous exploration of unknown\nenvironments, and second, the search for objects of interest hidden within. While\nexploration and search provide a need for specific capabilities, DARPA further\nintroduced a set of guidelines and rules to motivate higher levels of autonomy\nfor the deployed systems:\n\\begin{inparaenum}[(i)]\n \\item only a single human operator is allowed to interact, supervise, and interface with the robots;\n \\item each mission is bound by a fixed \\textit{setup time} limit of 30 minutes and an \\textit{exploration time} limit between 30 and 60 minutes;\n \\item a pit crew of four (Finals) or nine (Urban Circuit) can support the supervisor by setting up hardware in a designated area without access to wireless data streams, robot control, or interface;\n \\item there is a limited number of attempts to submit discovered objects of interest;\n \\item the final challenge environment comprises tunnel, urban, and cave terrains to be explored. \n\\end{inparaenum} \n\n\\ph{Objectives} Deploying and operating large teams of robots like Team CoSTAR's robot fleet, shown in \\Cref{fig:robot_team}B, are complex real-world problems. Addressing this set of problems creates the need for a resource-efficient and robust human and multi-agent system to i) not overwhelm the single human supervisor, ii) meet the timing requirements, and iii) increase the performance of both exploration and search tasks. \n\n\nTo tackle this challenge and develop a system that can deploy reliably even beyond the SubT challenge, we embed the following interaction objectives into our system design:\n\\begin{inparaenum}[(1)]\n \\item Reducing overhead and human workload (e.g., from application switching and manual task execution)\n \\item Creating and maintaining situational awareness\n \\item Managing large teams of robots (from setup, deployment to exploration) while allowing for a flexible configuration\n \\item Accessing critical information in a single unified interface\n \\item Maintaining an enjoyable performance that can visualize the complete robot team\n \\item Collaborating with autonomy and trusting automation.\n\\end{inparaenum}\n\n\n\n\n\\section{Supervised Autonomy}\n\\label{sec:supervised_autonomy}\n\\subsection{Copilot}\n\\ph{Motivation} After SubT's ``Urban Circuit'', the allowed personnel in the\ncompetition staging area was reduced from ten to five team members which\nincludes the main supervisor. This required a shift in how robots were\nstrategically and physically handled (minimum 2 people are needed to lift and\nstage a single robot). Task coordination was done by a pit crew member directing\nthe operator and influencing their actions while following static paper\n\\textit{checklist procedures}. Developing and deploying a computerized assistant\nthat could take over this role was soon desired.\n\n\\ph{Original Implementation} A first version of Copilot, ``an autonomous\nassistant for human-in-the-loop multi-robot operations'' was introduced in\n\\cite{Kaufmann2020copilotMike}. This early Copilot was only tested in realistic\ncave simulations or during preparatory missions with one deployed robot. Copilot supports a single human supervisor in monitoring robot teams, aids with strategic task planning, scheduling, and execution, and communicates high-level commands between agents and a human supervisor if a communication link exists. The autonomy assistant aims at keeping workload acceptable while maintaining high situational awareness that allows rapid responses in case system failures are observed.\n \n\\ph{Task Interaction} Copilot takes over the decision-making processes regarding planning and scheduling, which reduces the need to memorize tasks and task sequences or the need to delegate a team member to take over such checklist-like tasks. Some tasks were implemented with higher autonomy levels and automatically executed limited actions, but most required the human to start the task, manually execute parts of it, and confirm that the task had been completed successfully or unsuccessfully while monitoring the system. \nOn one hand, it reduced the need to remember tasks; on the other hand, more interactions with the newly introduced system were needed.\n\n\\ph{Scalability Limitations} Due to computational limitations, a full mission\nsimulation could not be achieved with more than three robots at reduced\nreal-time and not more than two in real-time. However, upon tightly integrating\nCopilot with multiple real robot platforms, we noticed that the current concept\nof operations didn't scale well when adding more robots to a mission. We learned\nthat task execution on the real hardware requires different timing and\nintroduces many sources for machine and human errors (e.g., if cables are loose,\nsensors don't power up, or unknown unknowns occur).\n\n\\ph{Visualization Limitations} In robotics interfaces, scheduling, and timeline\nviews are often presented in a robot- or task-centric way, focusing on who or which\nagent is scheduled for a certain task and when, respectively\n\\cite{BaeRossiDavidoff2020VizAnalytics}. The main task-centric approach that was\nused in early Copilot tests showed a vertical list view with a scrollable\ntimeline. This timeline showed the four tasks closest in time on top. As the\nnumber of tasks scaled linearly with the number of deployed robots this list\nview became inefficient --- especially when tasks had to be deferred and worked\non in a non-sequential order.\n\n\\subsection{Improved Copilot}\n\nThe identified shortcomings motivated a redesigning and rethinking of Copilot's back-end and front-end to reduce and not just shift workload; thus, we implemented higher levels of automation.\n\n\\ph{Architecture Changes} \\Cref{fig:copilot_architecture} provides a simplified\noverview of Copilot's updated task management architecture. A multi-robot task\nauto-generator and verifiable task executor have been added to the system, and\nthe underlying planner has been replaced. All modules access a centralized task\ndatabase which stores pending, active, successful, or failed mission tasks for\nsetup, deployment, and during exploration.\n\n\\begin{figure}[t]\n \\centering\n \\def\\columnwidth{\\columnwidth}\n \\import{figures\/}{architecture.pdf_tex}\n\\caption{Copilot's task management architecture. Auto-generator, Planner, and Executor have been added or updated and access a centralized task database which stores pending, active, successful, or failed tasks.}\n\\label{fig:copilot_architecture}\n\\end{figure}\n\n\n\\ph{Task Dependency Graph} A robot mission can be fairly complex, even when\nlooking at the deployment of a single robot. In\n\\Cref{fig:task_dependencies} such a single robot mission is shown as a directed\ngraph indicating the temporal\nconstraints and execution dependencies with arcs between the nodes that represent a\npre-defined set of mission tasks. Each task is defined by its duration, earliest\nstart time, latest end time, and its dependency relations with other tasks.\n\n\\begin{figure}[!tb]\n \\centering\n \\def\\columnwidth{\\columnwidth}\n \\import{figures\/}{s1_tasks.pdf_tex}\n\\caption{Pre-defined Copilot tasks for a single robot mission indicating task dependencies. The number of tasks scales linearly with the number of deployed robots. Spot1 related tasks are depicted in blue and operator tasks in orange. A superscript O or P at the beginning of a task indicate that the operator or pit crew has to manually fulfill some pre-condition. A superscript at the end indicates that a human sign-off is implemented before proceeding with the next task. For instance ``Power on robot platform'' requires a physical push of the robot platform startup button.}\n\\label{fig:task_dependencies}\n\\end{figure}\n\nTo deploy multiple robots without the need to hard-coding all possible agent\ncombinations and graphs, we use a scalable auto generator. The preceding\nsuperscript O in the graph (see \\Cref{fig:task_dependencies}) indicates\nthat human inputs or actions are required for the task. In the case of the\n\\texttt{Launch base software} task, this means that the operator has to initiate\nthe software launch as a pre-condition and is prompted to select the robots that\nthey would like to deploy for the upcoming mission. Similarly, superscripts at\nthe end of a task indicate that human action is needed before the next task can\nbegin. Tasks without either have been fully automated for nominal cases in this\nnewer Copilot version.\n\n\\ph{Task Planning and Scheduling} The aforementioned task dependency graph for\nthe selected robots forms the input for Copilot's task planner and is stored in\nthe MongoDB task database. The generation of a task plan for setting up, deploying, and\nassisting the operator during exploration is framed as an automated temporal\nplanning problem. In the first version of Copilot, we formulated such problem as\na Simple Temporal Network (STN), encoded as a linear program. In the improved\nversion of Copilot, deployed in the final events of SubT, we moved to a PDDL\ntemporal planning formulation to allow 1) flexibility on task representation\nwith respect to state constraints, resources, and planning, and 2) use the body\nof planners available in the literature. Herein we integrated the OPTIC planner\n\\cite{benton-etat-2012-OPTIC}, a PDDL temporal planner that handles time window\nspecification (timed initial literals), and discrete and continuous resources.\n\nTo perform planning, OPTIC uses both a PDDL domain file and a problem\nfile. The domain file has been designed to represent tasks (modelled as\noperators) and its dependencies (preconditions). The problem file is generated\nprior to calling the planner, and it is built based on the current state of\nmission and tasks execution. For example, if a task is ongoing, the PPDL file\nwould represent the task as ongoing and add constraints to ensure it continues\nthe execution to meet the necessary constraints. As a notional example\nof the scale of the planning problem, a mission with four robots would have\napproximately 60 tasks to be scheduled during setup and deployment. Planning is\nperformed at a predefined cadence (e.g., every 1.5 seconds), but it also follows\nan event-based approach when task execution is late, or the human-in-the-loop\nchanges their strategy --- this helps mitigate execution uncertainty. The\ngenerated plan is parsed and stored in a Task Database (for logging and\nvisualization across the system); each task is then dispatched for execution.\n\nIf a plan is not found by OPTIC due to temporal constraint violations (e.g.,\ndelays in task execution), Copilot will attempt to increasingly relax some of\nthe key temporal constraints, such as the latest end time of certain activities\n(e.g., allowing setup tasks to end a few minutes after the setup time,\noverlapping with the beginning of the exploration time window). In critical\nscenarios, Copilot would notify the operator of a schedule relaxation to allow\nfor further strategy changes.\n\n\\ph{Task Verification and Execution} A verifiable and generic task framework is\nintroduced to Copilot, allowing for quick implementations and standardized task\nautomation. Each task follows a strict precondition, execution, and\npost-condition template. Condition checks and execution can be triggered across\nagents, including the base station at which the human can oversee all automated\nprocesses at a high level in the new Copilot interface, which is described in\n\\Cref{sec:interface}. The task template execution covers both fully automated\ntasks and semi-automated tasks where an operators confirmation is required (e.g.\ndeploying a robot into a cave requires a Go\/No-go decision from the supervisor ---\ndeploying itself is an automated process). If a task fails during execution or\npost-condition checking, Copilot will try to resolve the issue by retrying tasks\nseveral times and allowing for more execution time. Failed tasks will be reported\nto the supervisor, who can choose to debug the issue at hand or trigger another\nautomated retry. Retries and resets are possible at all levels, and completed\ntasks can be reset during an active mission in case a robot platform has to be\nrebooted.\n\n\n\n\\section{Game-Inspired Interface}\n\\label{sec:interface}\n\t\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{figures\/ui_overview.png}\n\\caption{An overview of the major UI components. (A) The Robot and associated Copilot task cards. (B) The split-screen 3D visualization view with view controls, WiFi signal strength overlay, and an artifact card showing on the map. (C) The artifact drawer. (D) The robot health systems component.}\n\\label{fig:ui_overview}\n\\end{figure*}\n\n\\ph{Game Inspiration} Inspiration for multi-agent interaction and interface\ndesign is partially drawn from real-time strategy games such as Age of Empires,\nStarCraft, and {Command \\& Conquer}. When played competitively, these games\nrequire a high sense of micro and macro-management of units and their\nenvironment and the ability to efficiently switch between these two ways of\nmanaging a team. Micromanagement involves short-term strategy and\ndecision-making, where individual units may require critical attention to win a\nbattle, overcome an obstacle, or navigate to the next point of interest, while\nmacromanagement refers to longer-term strategizing that involves resource\ngathering, unit production over time, and overall exploration and control of the\nmap~\\cite{rtsCombatStrategy}. Parallels can be applied to the management of a\nrobot team in the SubT competition. Even autonomous robots can benefit from or\nrequire human intervention and commanding, especially if critical attention\ntowards failing subsystems is needed. Supervised multi-agent control draws from\nthe human's situational awareness regarding the environment and robot states to\neffectively coordinate multi-agent behaviors, successfully locate artifacts, and\nscore points.\n\n\\ph{Mission Phases} The user interface is designed to be adaptable to the\noverall mission and two major phases of an individual robot's competition run in\nparticular:\n1) setup and deployment, and 2) mission execution with its exploration and\nsearch components. Across these phases, the visibility and abstraction of\ninformation need to be flexible to facilitate focus on the anticipated operator\ninteractions. In deployment, the user interface uses the Copilot-generated tasks and status information to guide the sole operator through the multitude of individual tasks while allowing them to maintain their situational awareness, manage the entire robot team, and coordinate with the pit crew.\n\n\\ph{Three Column Layout} The Mission Control interface is organized into\ndifferent view components. \\Cref{fig:ui_overview} shows the main split-screen\nwith three columns aiming at creating reliable locations for the operator to\nlook at when needing to accomplish functionally distinct tasks (A). The aim here\nis to reduce the amount of visual scanning, application switching, and to parse\nrobot needs on an individual or team level swiftly. Individual robot information\npertinent to monitoring health systems is available on the left, planned and\nactively re-scheduled Copilot tasks for individual robots are placed alongside\neach agent in the middle, and a 3D interactive visualization of the robots in\ntheir environment is anchored to the right. During mission execution, the\nprimary goal of the user interface is to keep the operator situationally aware\nof a multitude of individual robot health systems and data sensed from the\nsurrounding environment while presenting the most important information and thus\nreducing their cognitive workload. In \\Cref{fig:robot_team} the 3D visualization\nis expanded, and robot sensor and status information is minimized to select\nmission-critical information.\n \n\n\\ph{Health Systems and Robot Status} In order to effectively survey the status\nof any individual robot in the team, visibility into over 30 unique sensors and\nstatuses needed to be surfaced to the operator per robot. This required\nidentifying which indicators were critical to display at all times, which could\nbe hidden within a sub-view, which were good candidates to be combined and\nabstracted, and which would be prioritized across either the deployment (split)\nor mission execution (split and expanded visualization) modes of the user\ninterface. In addition to sensors visible at an individual level, an additional\nview was created to organize sensors compactly across the team, providing easy\nvisual scanning for the operator during macro-management and deployment, as\nshown in \\Cref{fig:ui_overview}D. An abstraction of robot behaviors (e.g.,\nexploring, dropping a communications node) and mobility states presents an\noverall status of each robot to the operator by color and a high-level\ndescription. This status is prioritized based on criticality to ensure the\noperator's attention will be requested for the most important issue at any given\ntime.\n\nPreviously, Copilot tasks resided in an entirely separate module of the\ninterface with limited screen estate, requiring the operator to move other\nrelated and necessary sensor and status information out of physical view. A\nreorganization where Copilot tasks are paired alongside their respective robots\nis utilized to reduce context loss and pair necessary information to complete\nthe tasks together, as shown in \\Cref{fig:ui_overview}A. Over time during the\ndevelopment roll-out, this pairing of health, sensor, and status indicators\nalongside Copilot tasks facilitated a level of trust from the operator where\nfocus on a particular robot was not necessary unless a critical task requiring\noperator intervention appeared.\n\n\\ph{3D Visualization View} A 3D interactive visualization leveraging React Three\nFiber (a React-based renderer for three.js) was created within the UI with the\naim of achieving a significant reduction in operator task and application\nswitching. Prior to this version of the interface, the operator was required to\nswitch between a web browser to view robot health systems and status information\nand RViz (a visualizer for ROS) to view the robots within the 3D environment and\ncommand them. In the split view of the UI, the operator can have the full\ncontext of robot sensors and status information along with any outstanding\nCopilot tasks. When in the expanded visualization view, the layout shares\nsimilarities with layouts of traditional Real-Time Strategy (RTS) games, where\ncontent is functionally organized from the corners of the view and leave the\ncenter-most screen real estate where the operator will primarily interact with\nrobots and information unobstructedly. From this view, the operator can take\non any of the following tasks: surveying the mapped environment and robot\npositions for locations to scout, locating, and submitting object or signal\nartifacts, directing or course-correcting robot autonomy with manual navigation\ncommands, viewing signal strength of the communications backbone within the\nenvironment, and assigning robots to drop communication nodes manually. The\nvisualization allows the operator to navigate the 3D environment through\npanning, zooming, and filtering points of interest categories. To effectively\nmanage the switching between micro and macro-level interactions, a single-click\nshortcut was implemented on each robot status card for the operator to quickly\nfocus on any robot that requires attention. An additional shortcut is\nprovided to zoom back out to an overview of the map.\n\t\nImprovements over traditional RTS commanding controls were also made to minimize\nthe amount of mouse control and coordination necessary. Instead of requiring to\nselect or drag a bounding box prior to commanding a robot, the operator could\nsimply interact with the visualized information roadmap (IRM) --- a breadcrumb\ntrail used for safely navigating the environment constructed by the team of\nrobots~\\cite{plgrim} --- and assign any robot with a high priority navigation point or\ncommunications node drop location through a context menu, regardless of whether the\nparticular robots are currently in view or not.\n\t\nTo help with artifact management, the locations of detected artifacts are\nvisualized and interactivity is added to allow the operator to quickly hover\ninto a thumbnail and click to navigate to the dedicated Artifact Drawer\n\\Cref{fig:ui_overview}C for deeper analysis and submission. Additional\ninteractions are, for example, manually adding and manipulating detected artifact\nlocations within the 3D space, by dragging its location across a plane for\nfine-tuning if a submission location was deemed incorrect and needed adjustment.\n\t\nWhile in the expanded visualization view, compressed versions of the robot\nstatus modules are shown horizontally in the bottom left of the view with the\nmission status indicator made more prominent and placed above each module. These\noverall status indicators were given visual priority to ensure grabbing the\noperator's attention. For instance, the indicator would flash red when a robot\nhad fallen over, was low on battery, or required assistance. The operator could\nimmediately click the respective robot module and be oriented over it for\nmicromanagement.\n\n\\ph{Artifact Drawer} Artifact submission was a critical part of SubT that also has many real-world parallels, for instance, in search and rescue. Especially under time constraints, it is necessary to quickly identify artifacts of interest in the environment, whether these be human survivors or other objects of interest. Detecting and localizing artifacts automatically is done using a state-of-the-art image processing pipeline \\cite{terry2020object}, but no AI system is infallible, especially in unknown environments, so having a system for an operator to manually review artifacts efficiently was critical considering mission time and submission attempts.\n\nIn the old system \\cite{terry2020object}, a manual artifact review system did exist, but it was built with a focus on only basic functionality and a high reliance on initially accurate artifact detections. Each artifact report took roughly 90 seconds to review. In redesigning this component, we wanted to focus on improving the review process from an ease of use perspective and decrease the time spent to confidently review an artifact report down to 15 seconds. Beyond simply making the system more intuitive for the operator, this actually had a major functional benefit from a trustability standpoint in that it allowed us to decrease the confidence threshold for flagging artifact detections and have the operator go through and verify nearly 6 times more potential artifact reports while not increasing total time spent. \n \nTo better design the new system for speed, it was important to understand which areas of the old one were slowing the process down the most. Testing the old system in simulation and operator feedback revealed that the artifact review process needed too many clicks. Then, time had to be spent zooming in on and reviewing images and checking with RViz separately to verify that artifact coordinates were correct. No visual aid was given if corrections were necessary, and coordinates had to be updated by manually entering them for each axis in ${\\rm I\\!R^3}$. Borrowing from game interface design, integrating the 3D visualization view directly into the web UI removed the need for application switching, and drag controls were added to adjust locations providing correctly scaled coordinate updates from the 3D environment. A minified list that provides an overview of all artifact reports by confidence levels, plus maximizing the screen real estate of a single selected artifact helped increase efficiency. Finally, adding keyboard shortcuts as commonly used in gaming made meeting our target goal of 15 seconds possible. \n\n\n\n\\section{Results}\n\\label{sec:results}\n\nOver the course of the last challenge year, we conducted a limited series of field tests in three testing locations, including the abandoned tunnels at the Los Angeles Subway Terminal building, the Lava Bed National Monument in Northern California, and the Kentucky Underground lime-stone cave for which we applied our rapid development and testing strategy. We experimented with different robot configurations and in different stages of readiness as our system's capabilities matured. We deployed up to 11 vehicles simultaneously during these tests stressing the overall system (including Copilot and all the UI elements) and learning about its technical limitations like bandwidth and computing resources which will be presented in upcoming work.\n\nWe deployed the presented game-inspired user interface and supervised autonomy system during the SubT challenge using four to six ground robots nominally. While we could have exceeded the number of six robots using the newly designed interface and autonomy, six became the preferred number of agents to explore large-scale environments while allowing reliable communication links that would not exceed bandwidth limitations when robots disseminated information from autonomously explored out-of-comms areas. This allowed meeting the set interaction objectives, especially maintaining an enjoyable performance that can visualize the complete robot team while contributing to a lower workload due to fewer deployed agents.\n\nIn what follows, we analyzed screen recordings and log files (approved by Caltech IRB protocol number 19-0461 and Polytechnique Montreal project CER-2122-50-D) collected during the SubT final competition. We extracted time-to-task information, robot deployment times, mouse locations, and application usage from runs P1, P2, and F that consist of a setup-time and mission phase of 30+30 and 30+60 minutes, respectively. Robots were only allowed to leave the setup area and enter the course when the mission time began. Readying the team of robots and not bleeding into the mission time was a crucial effort to maximize available mission and exploration time. The results are compared to an earlier state of the system that did not implement Copilot and used different interfaces, namely the SubT ``Urban Circuit'' similar to \\cite{Otsu2020IEEEAerospace}. During the ``Urban Circuit'' task, coordination was done by humans only.\n\n\\ph{Robot Deployment} \\Cref{fig:robot_deployment} shows the robot deployment times that were achieved by deploying Copilot and compares them to the baseline. We can see that during run P1, we achieved sending one robot in less than 60 seconds each, deploying a total of 6 ground vehicles in 5 minutes and 31 seconds. In runs P2 and F, we achieved staying below the one minute mark for the first three robots. Deploying the robots without Copilot and the new interface in the `Urban Circuit'' runs A1, B1, and B2 took more than 5.5 min per robot on average, thus significantly reducing the time available for exploration and consequently reducing ground coverage and information gain regarding the search task. \n\n\\begin{figure}[t]\n\\centering\n \\includegraphics[width=0.5\\textwidth]{figures\/robot_deployment.png}\n \\caption{Robot deployment times per game run measured upon entering the course. The black dotted line ($\\sim$1min\/r) indicates the team's internal goal for robot deployment and represents a deployment of one robot per minute. F backup marks insertion points of 2 robots that were not part of the initial deployment strategy but were added ad-hoc to compensate for robot failures during run F.}\n \\label{fig:robot_deployment}\n\\end{figure}\n\n\\ph{Application Usage} The new interface resulted in a shift in application usage and reduced switching between different applications and computers with a second set of peripherals, as RViz was running on a second device during the ``Urban Circuit''. \\Cref{fig:application_usage} presents the relative usage of applications for six SubT runs. Designing a unified interface resulted in a shift in application usage that reduced the use of RViz significantly. While more than 50\\% of time was spent on RViz during the ``Urban Circuit'' runs, we were able to unify user interactions and situation awareness in a single Mission Control interface. Only run F uses RViz for some time as a debugging tool that gave access to the robot's cost maps depicting the perceived risks around them. This information was not visualized by the new interface, but presents valuable key information in case of unexpected and off-nominal operations.\n\n\\ph{UI Feature Usage} With the main Mission Control interface being the main\ninteraction point for human supervisory control, we then look at the feature\nusage within the interface itself. \\Cref{fig:radar_plot} shows the relative\ninteraction times with the split-screen view, the 3D full-screen console view,\nthe sensor health overview, artifact submission drawer, and the BPMN modal that\ngives a detailed overview of a robot's inner state machine (which was relied\nupon during the ``Urban Circuit''). We see that, especially during runs P2 and F,\nlarge amounts of time were spent on the artifact drawer and thus performing the\nsearch task analyzing the artifact reports that were generated by the\nmulti-agent system. To gain situational awareness and potentially interact with\nthe robot team, the human supervisor primarily relied on the split-screen view of\nthe Mission Control app that is shown in the background of \\Cref{fig:heatmap}\noverlaid by a heat map that indicates the most active areas derived from mouse\ncursor positions sampled at 1.5 Hz. In this analysis, an area is deemed inactive\nif the mouse has been stationary for more than ten seconds. \\emph{Huang et\n al.}~\\cite{andwhite2012user} found that the median difference between human\ngaze and mouse position during an active task is 77 pixels with a standard\ndeviation of 33.9 pixels at 96 dpi screen resolution. A Gaussian kernel with\n$\\mu=98$ and $\\sigma=43$ adjusting for 122 dpi is used to derive our heat maps.\n\\Cref{fig:heatmap} indicates that the robot cards, Copilot tasks and the 3D view\nwere all crucial tools while overseeing the robotic system and performing the\nexploration and search tasks.\n \n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{figures\/application_usage.png}\n \\caption{Application usage (foreground application) for six SubT mission runs in percent. A1, B1, and B2 represent the usage before the redesign that integrated 3D visualization and interactions for P1, P2, and F in a single Mission Control application using only one computer and screen. Note that node manager and terminal usage are underrepresented in runs A1, B1, and B2 because the initial setup phase of up to 10 minutes was not recorded for these runs due to different logging procedures.}\n \\label{fig:application_usage}\n\\end{figure}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.23\\textwidth]{figures\/radar_ui_sub_category_usage_prelim1.png}\n \\includegraphics[width=0.23\\textwidth]{figures\/radar_ui_sub_category_usage_prelim2.png}\n \\includegraphics[width=0.27\\textwidth]{figures\/radar_ui_sub_category_usage_final_prize.png}\n \\caption{Analysis of the redesigned user interface interaction by view component in percent for runs P1, P2, and F.}\n \\label{fig:radar_plot}\n\\end{figure}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{figures\/heatmap_prelim2_costar_mission_control.png}\n \\caption{Activity heat map showing the x and y positions of cursor interactions (and indirectly gaze) overlaid on the Mission Control Split-Screen view exemplary for game run P2. The view consists of robot cards, a column for Copilot tasks, and the split-screen 3D view. A brighter heat map indicates higher interaction times in this area. Stationary cursors for more than 10 seconds are classified as inactive.}\n \\label{fig:heatmap}\n\\end{figure}\n\n\n\n\n\\section{Conclusions and Future Work}\n\\label{sec:conclusions_future_work}\nIn this work we \n\\begin{inparaenum}[(i)]\n \\item create a game-inspired user interface for multi-agent robot missions\n \\item integrate an automated planner for task planning and scheduling,\n \\item add a verifiable task framework for increased reliability, and\n \\item present results on how the overall system performed over the course of\n several real-world deployments, including the DARPA SubT Challenge final.\n\\end{inparaenum}\nIn future work, we plan to deploy our interface and Copilot during scientific\nexploration missions to autonomously map and identify geological features and\nassess exploration strategies in lava tubes. This will lead to further\nvalidation of the subsystems and a structured assessment of a supervisor's\nworkload outside the realm of the SubT challenge with experts and potentially\nnon-expert users. Ultimately, we would like to assess operator workload from\nwearable sensors in real-time and consider such constraints in Copilot's task\nplanning. \n\n\n\n\\section*{Acknowledgment}\n\\footnotesize{The work is partially supported by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004), and Defense Advanced Research Projects Agency (DARPA). This work was conducted in collaboration with the Making Innovative Space Technologies Laboratory (MIST Lab) at Polytechnique Montreal. The first author would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for their generous support in the form of a Vanier Canada Graduate Scholarship. Thank you to all members of Team CoSTAR for their valuable discussions and support.}\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{#1}\\setcounter{equation}{0}}\n\n\\newcommand{\\req}[1]{$(\\ref{#1})$}\n\\newcommand{\\dfn}{\\stackrel{\\triangle}{=}}\n\\newcommand{\\bydef}{\\stackrel{\\bigtriangleup}{=}}\n\\newcommand{\\limsupn}{\\limsup_{n \\rightarrow \\infty}}\n\\newcommand{\\liminfn}{\\liminf_{n \\rightarrow \\infty}}\n\\newcommand{\\val}{{\\rm v }}\n\\newcommand{\\co}{{\\rm conv }}\n\\newcommand{\\sgn}{{\\rm sign }}\n\\newcommand{\\ri}{{\\rm ri }}\n\n\\newcommand{\\vect}[1]{{\\boldsymbol #1 }}\n\n\\newcommand{\\td}[1]{\\tilde{#1}}\n\\newcommand{\\ceil}[1]{\\left\\lceil #1 \\right\\rceil}\n\\newcommand{\\floor}[1]{\\left\\lfloor #1 \\right\\rfloor}\n\\newcommand{\\inprod}[2]{\\langle #1 , #2 \\rangle }\n\\newcommand{\\card}[1]{\\left\\lvert #1 \\right\\rvert}\n\\newcommand{\\bc}{\\begin{center}}\n\\newcommand{\\ec}{\\end{center}}\n\\newcommand{\\bz}{\\vect{z}}\n\\newcommand{\\vv}[1]{\\boldsymbol #1}\n\\newcommand{\\by}{\\vect{y}}\n\\newcommand{\\bY}{\\vect{Y}}\n\\newcommand{\\refs}[1]{$(\\ref{#1})$}\n\\newcommand{\\refds}[2]{({#1}.\\ref{#2})}\n\\newcommand{\\bX}{\\vect{X}}\n\\newcommand{\\bx}{\\vect{x}}\n\\newcommand{\\bq}{\\vect{q}}\n\\newcommand{\\bZ}{\\boldsymbol Z}\n\\newcommand{\\R}{\\mathbb R}\n\\newcommand{\\N}{\\mathbb N}\n\\newcommand{\\Z}{\\mathbb Z}\n\\newcommand{\\m}[1]{\\mathcal{#1}}\n\\newcommand{\\ul}[1]{\\underline{#1}}\n\\newcommand{\\ol}[1]{\\overline{#1}}\n\n\\newcommand{\\be}{\\begin{equation}}\n\\newcommand{\\ee}{\\end{equation}}\n\\newcommand{\\beaa}{\\begin{eqnarray*}}\n\\newcommand{\\eeaa}{\\end{eqnarray*}}\n\\newcommand{\\ben}{\\begin{enumerate}}\n\\newcommand{\\een}{\\end{enumerate}}\n\\newcommand{\\db}{\\hspace*{\\fill}{\\zapf o}}\n\\newcommand{\\cpn}[1]{\\caption{\\myfont #1}}\n\\newcommand{\\bpn}{\\begin{proposition}\\twlsf}\n\\newcommand{\\epn}{\\db\\end{proposition}}\n\\newcommand{\\bdm}{\\begin{displaymath}}\n\\newcommand{\\edm}{\\end{displaymath}}\n\\newcommand{\\ba}{\\begin{array}}\n\\newcommand{\\ea}{\\end{array}}\n\n\\newcommand{\\supp}{\\rm{supp}}\n\\newcommand{\\op}[1]{\\mbox{op}_{#1}}\n\n\n\\newcommand{\\st}{\\mathop{\\rm s.t.}}\n\\newcommand{\\OR}{\\mathop{\\,\\,\\,{\\rm or}\\,\\,\\,}}\n\\newcommand{\\conv}{\\mathop{\\rm conv}}\n\\newcommand{\\aff}{\\mathop{\\rm aff}}\n\\newcommand{\\MP}{{\\cal P}}\n\\newcommand{\\dd}{\\mathrm{d}}\n\\newcommand{\\mbal}{\\mb{\\alpha}}\n\\newcommand{\\mbbt}{\\mb{\\beta}}\n\\newcommand{\\mbga}{\\mb{\\gamma}}\n\\newcommand{\\mbth}{\\mb{\\theta}}\n\\newcommand{\\mbmu}{\\mb{\\mu}}\n\n\\newcommand{\\br}{\\mb{b}_R}\n\\newcommand{\\argmin}{\\mathop{\\rm argmin}}\n\\newcommand{\\argmax}{\\mathop{\\rm argmax}}\n\n\n\\newtheorem{assumption}{Assumption}\n\\newtheorem{definition}{Definition}\n\\newtheorem{lemma}{Lemma}\n\\newtheorem{proposition}{Proposition}\n\\newtheorem{corollary}{Corollary}\n\\newtheorem{remark}{Remark}\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{claim}{Claim}\n\\newtheorem{example}{Example}\n\n\\newcommand{\\boldone}{\\mbox{\\bf 1}}\n\\newcommand{\\Ga}{\\Gamma}\n\\newcommand{\\GA}{\\Gamma}\n\\newcommand{\\La}{\\Lambda}\n\\newcommand{\\LA}{\\Lambda}\n\\newcommand{\\la}{\\lambda}\n\\newcommand{\\eps}{\\epsilon}\n\\newcommand{\\Om}{\\Omega}\n\n\\newcommand{\\Rd}{\\reals^d}\n\\newcommand{\\DU}{\\Delta(\\cU)}\n\\newcommand{\\lip}{\\langle}\n\\newcommand{\\rip}{\\rangle}\n\\newcommand{\\rome}{\\mbox{\\rm e}}\n\n\\newcommand{\\limsupt}{\\limsup_{t \\rightarrow \\infty}}\n\\newcommand{\\liminft}{\\liminf_{t \\rightarrow \\infty}}\n\\newcommand{\\E}{\\mathbb{E}}\n\\newcommand{\\U}{\\mathbb{U}}\n\\newcommand{\\rP}{\\mathbb{P}}\n\\newcommand{\\norm}[1]{\\left\\lVert#1\\right\\rVert}\n\\newcommand{\\tnorm}[1]{\\lVert\\mkern-2mu |#1|\\mkern-2mu\\rVert}\n\\title{Robust Newsvendor Games with Ambiguity in Demand Distributions}\n\\author{Xuan Vinh Doan\\thanks{DIMAP and ORMS Group, Warwick Business School, University of Warwick, Coventry, CV4 7AL, United Kingdom, xuan.doan@wbs.ac.uk.} \\and Tri-Dung Nguyen\\thanks{Mathematical Sciences and Business School, University of Southampton, Southampton, SO17 1BJ, United Kingdom, T.D.Nguyen@soton.ac.uk.}}\n\n\n\\date{January 2016}\n\n\n\\begin{document}\n\\maketitle\n\n\n\\begin{abstract}\nWe investigate newsvendor games whose payoff function is uncertain due to ambiguity in demand distributions. We discuss the concept of stability under uncertainty and introduce solution concepts for robust cooperative games which could be applied to these newsvendor games. Properties and numerical schemes for finding core solutions of robust newsvendor games are presented.\n\\end{abstract}\nKeywords: Cooperative games; uncertain payoffs; newsvendor games; robust optimization; stability.\n\n\\section{Introduction}\n\\label{sec:intro}\nA joint venture is usually an effective approach for individual players in the market to share costs, reduce risk, and increase the total joint revenue or profit. For example, individual retailers can decide whether to order inventories together and share the profit from selling ordered products later. As a part of the joint venture formation, all players should agree on how to share the joint profit or payoff before the cooperation is established. Cooperative game theory provides a mathematical framework for addressing this problem, which is modeled as \\emph{newsvendor centralization games} (or newsvendor games for short). The model of these games has been introduced in Hartman \\cite{hartman1994cooperative}. Formally, consider the set ${\\cal N}$ of $N$ retailers and let $\\td{d}_i\\in\\mathbb{R}_+$ be the random demand for retailer $i$, $i\\in{\\cal N}$. In the setting of newsvendor games, we assume that the unit ordering cost $c$ and the unit selling price $p$ are the same for all retailers, $0x_i(u)$ for all $i\\in{\\cal S}$.\n\\end{definition}\nIn other words, a sub-coalition has the incentive to stay in the grand coalition if no matter what decision it takes, there will be at least one player in the sub-coalition is better off by staying in the grand coalition in any realization of uncertain parameters. Applying this definition for individual players, we can say that a player has the incentive to break away if there exists an action $\\alpha\\in{\\cal A}(\\{i\\})$ which guarantees that he\/she will be better off by leaving the grand coalition for at least one realization of $u\\in{\\cal U}$, i.e., $v_u(\\alpha,\\{i\\})>x_i(u)$. Note that the idea of defining these realization-based concepts is similar to those in Bitran \\cite{bitran1980linear}, which are used to define necessary and sufficient solution concepts for multi-objective optimization. We are now ready to define main solution concepts for cooperative games with uncertain characteristic functions $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ using the proposed payoff distribution scheme and the above definition of break away incentive.\n\nA decision $(a,\\bz)$ is \\emph{individually rational} if there is no individual player who has the incentive to break away. We can then define the concept of imputations.\n\\begin{definition}\n\\label{def:rimputation\nAn imputation of the cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ is an individually rational decision $(a,\\bz)$ whose payoff distribution scheme $\\bz$ is efficient.\n\\end{definition}\nSimilar to the deterministic setting, let $\\mbox{impu}({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ denote the set of all imputations of the cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$.\n\nWe now define the concept of stability. A decision $(a,\\bz)$ is \\emph{stable} if there is no sub-coalition ${\\cal S}\\subsetneq{\\cal N}$ that has the incentive to break away and we have the following definition of the cores of cooperative games $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$.\n\\begin{definition}\n\\label{def:score}\nThe core of the cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ is the set of all stable decisions with efficient payoff distribution schemes and is denoted by $\\emph{core}({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$.\n\\end{definition}\n\nThe above definition of stability indicates that all sub-coalitions of a stable grand coalition necessarily do not have the incentive to break away no matter how $u\\in {\\cal U}$ is realized. \nThis follows the ``immunized-against-uncertainty'' principle of robust optimization (see Ben-Tal et al. \\cite{ben2009robust} and references therein); therefore, we call cooperative games $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ with the above definition of break away incentive \\emph{robust cooperative games}.\n\nWe are now ready to characterize the existence of imputations and core decisions, i.e., decisions belong to the core, of robust cooperative games $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$. We start by making the following assumptions.\n\n\\begin{assumption}\n\\label{as:pos}\n\\indent\n\\begin{itemize}\n\\item[(i)] For any player $i\\in{\\cal N}$, there exists an action such that his\/her payoff is always non-negative, i.e., there exists $a \\in {\\cal A}(\\{i\\})$ such that $v_u(a,\\{i\\})\\geq 0$ for all $u\\in{\\cal U}$.\n\\item[(ii)] The payoff of the grand coalition $\\cal N$ is always positive, i.e., $v_u(a,{\\cal N})>0$ for all $a\\in{\\cal A}({\\cal N})$ and $u\\in{\\cal U}$.\n\\end{itemize}\n\\end{assumption}\nSimilar to Timmer et al. \\cite{timmer2005convexity}, these assumptions emphasize the fact that we are focusing on profit games with possible nonnegative payoff for each individual sub-coalition and it is indeed worth considering the grand coalition given that its profit is always positive. In other words, Assumption~\\ref{as:pos}(ii) implies that we should only consider the set of actions ${\\cal A}({\\cal N})$ which create positive profits under any circumstances for the grand coalition. For robust newsvendor games that we are going to discuss in Section~\\ref{stoc.newsvendor}, these assumptions are easily satisfied in general.\n\n\n\n\n\nWe now state the following result on existence conditions for imputations of robust cooperative games.\n\\begin{theorem}\n\\label{prop:impucond}\nGiven a robust cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$, an imputation exists if and only if there exists an action $a\\in{\\cal A}({\\cal N})$ such that\n\\be\n\\label{eq:impucond}\n\\sum_{i\\in{\\cal N}} v_{\\max}(a,\\{i\\}) \\leq 1,\n\\ee\nwhere $\\displaystyle v_{\\max}(a,\\{i\\}) = \\max_{u\\in{\\cal U}}\\left\\{\\frac{\\displaystyle\\max_{\\alpha_i\\in{\\cal A}(\\{i\\})}v_u(\\alpha_i,\\{i\\})}{v_u(a,{\\cal N})}\\right\\}$.\n\\end{theorem}\nIn order to prove the theorem, we need the following lemma.\n\\begin{lemma}\n\\label{prop:individual:rationality:cond}\nGiven a robust cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ and a decision $(a,\\bz)$, a player $ i \\in {\\cal N}$ has no incentive to break away if and only if\n$\\displaystyle z_i \\geq v_{\\max}(a,\\{i\\})$.\n\\end{lemma}\n\n\\begin{pf}\nWe have: $x_i(u)=v_u(a,{\\cal N})\\cdot z_i$ for all $i\\in{\\cal N}$ and $u\\in{\\cal U}$. Player $i\\in{\\cal N}$ does not have the incentive to break away if and only if\n$$\nv_u(\\alpha_i,\\{i\\})\\leq v_u(a,{\\cal N})\\cdot z_i,\\quad\\,\\forall\\,\\alpha_i\\in{\\cal A}(\\{i\\}),~u\\in{\\cal U}.\n$$\nUnder Assumption \\ref{as:pos}(ii), this holds if and only if $\\displaystyle z_i\\geq\\max_{\\alpha_i\\in{\\cal A}(\\{i\\})}\\frac{v_u(\\alpha_i,\\{i\\})}{v_u(a,{\\cal N})}$ for all $u\\in{\\cal U}$, i.e.,\n$$\nz_i\\geq \\max_{u\\in{\\cal U}}\\left\\{\\frac{\\displaystyle\\max_{\\alpha_i\\in{\\cal A}(\\{i\\})}v_u(\\alpha_i,\\{i\\})}{v_u(a,{\\cal N})}\\right\\}=v_{\\max}(a,\\{i\\}).\n$$\n\\end{pf}\n\nWe are now ready to prove Theorem \\ref{prop:impucond}.\n\n\\begin{pf}\nSuppose an imputation $(a,\\bz)$ exists. According to the definition of imputations, there is no player $i\\in{\\cal N}$ who has the incentive to break away. Thus, according to Lemma~\\ref{prop:individual:rationality:cond}, we have:\n$$\nz_i\\geq v_{\\max}(a,\\{i\\}),\\quad\\,\\forall\\,i\\in{\\cal N}.\n$$\nNow, $(a,\\bz)$ is a decision of the grand coalition; therefore, $\\displaystyle\\sum_{i\\in{\\cal N}}z_i=1$. Summing over all $i\\in{\\cal N}$ the above inequality, we then achieve condition \\refs{eq:impucond}.\n\nNow, suppose condition \\refs{eq:impucond} holds. Let $\\displaystyle\\eps = 1-\\sum_{i\\in{\\cal N}} v_{\\max}(a,\\{i\\}) \\geq 0$ and define\n$$\nz_i=v_{\\max}(a,\\{i\\})+\\frac{\\eps}{N},\\quad\\,\\forall\\,i\\in{\\cal N}.\n$$\nWe will show that $(a,\\bz)$ is an imputation. Clearly, $\\bz$ is efficient, i.e., $\\displaystyle\\sum_{i\\in{\\cal N}}z_i=1$ given the definition of $z_i$ and $\\eps$. Now we have: for all $u\\in{\\cal U}$, $\\displaystyle z_i\\geq v_{\\max}(a,\\{i\\}) \\geq \\frac{\\displaystyle\\max_{\\alpha_i\\in{\\cal A}(\\{i\\})}v_u(\\alpha_i,\\{i\\})}{v_u(a,{\\cal N})}$ for all $i\\in{\\cal N}$ since $\\eps\\geq 0$. Thus, under Assumption \\ref{as:pos}(ii),\n$$\nv_u(a,{\\cal N})\\cdot z_i\\geq \\max_{\\alpha_i\\in{\\cal A}(\\{i\\})}v_u(\\alpha_i,\\{i\\}),\\quad\\,\\forall\\,i\\in{\\cal N},\\,u\\in{\\cal U}.\n$$\nIt shows that for all $u\\in{\\cal U}$,\n$$\nx_i(u)\\geq v_u(\\alpha_i,\\{i\\}),\\quad \\forall\\,i\\in{\\cal N},\\,\\alpha_i\\in{\\cal A}(\\{i\\}).\n$$\nThus, there is no player $i$ who has the incentive to break away; that is $(a,\\bz)$ is individually rational, which implies that $(a,\\bz)$ is an imputation.\n\\end{pf}\n\nSimilar to the deterministic and stochastic cooperative games, the existence of core decisions is related to the concept of balancedness. A map $\\mu:2^{{\\cal N}}\\setminus\\emptyset\\rightarrow[0,+\\infty)$ is called a \\emph{balanced} map if $\\displaystyle\\sum_{{\\cal S}\\subsetneq{\\cal N}}\\mu({\\cal S})\\cdot\\mb{e}_{{\\cal S}}=\\mb{e}_{{\\cal N}}$, where, for all ${\\cal S}\\subseteq{\\cal N}$, $\\mb{e}_{{\\cal S}}\\in\\{0,1\\}^N$ with $\\left(e_{{\\cal S}}\\right)_i=1$ if and only if $i\\in{\\cal S}$. We are now ready to define the balanced robust cooperative game.\n\\begin{definition}\nA robust cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ is called balanced if there exists an action $a\\in{\\cal A}({\\cal N})$ such that for all balanced map $\\mu$,\n\\be\n\\label{eq:balanced}\n\\sum_{{\\cal S}\\subsetneq{\\cal N}}\\mu({\\cal S})v_{\\max}(a,{\\cal S})\\leq 1,\n\\ee\nwhere $\\displaystyle v_{\\max}(a,{\\cal S})=\\max_{u\\in{\\cal U}}\\left\\{\\frac{\\displaystyle\\max_{\\alpha_{{\\cal S}}\\in{\\cal A}({\\cal S})}v_u(\\alpha_{{\\cal S}},{\\cal S})}{v_u(a,{\\cal N})}\\right\\}$.\n\\end{definition}\nThis definition of balanced robust cooperative games matches the definition of balanced deterministic games when $\\card{{\\cal U}}=\\card{{\\cal A}({\\cal S})}=1$ for all ${\\cal S}\\subseteq{\\cal N}$ with the balancedness condition $\\displaystyle\\sum_{{\\cal S}\\subsetneq{\\cal N}}\\mu({\\cal S})v({\\cal S})\\leq v({\\cal N})$. We can now state the following theorem regarding the existence of core decisions.\n\n\\begin{theorem}\n\\label{thm:corecond}\nA robust cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ has a non-empty core if and only if it is balanced.\n\\end{theorem}\nIn order to prove Theorem~\\ref{thm:corecond}, we need the following lemma.\n\\begin{lemma}\n\\label{lem:breakcond}\nGiven a robust cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ and an imputation $(a,\\bz)$, a coalition ${\\cal S}\\subsetneq{\\cal N}$ has the incentive to break away if and only if\n\\be\n\\label{eq:breakcond}\n\\sum_{i\\in{\\cal S}}z_i< v_{\\max}(a,{\\cal S}).\n\\ee\n\\end{lemma}\n\n\\begin{pf}\nGiven an imputation $(a,\\bz)$, a coalition ${\\cal S}$ has the incentive to break away if there exists an efficient decision $(\\hat{a},\\hat{\\bz})$\nand a realization $u\\in{\\cal U}$ such that for all $i\\in{\\cal S}$,\n$$\nv_u(\\hat{a},{\\cal S})\\cdot \\hat{z}_i>v_u(a,{\\cal N})\\cdot z_i.\n$$\nSince $(\\hat{a},\\hat{\\bz})$ is efficient, we have: $\\displaystyle\\sum_{i\\in{\\cal S}}\\hat{z}_i=1$. Summing over all $i\\in{\\cal S}$ the inequality above, we then obtain the following statement:\n$$\n\\exists\\,u\\in{\\cal U}\\,:\\,v_u(a,{\\cal N})\\cdot\\sum_{i\\in{\\cal S}} z_i0$ for all $u\\in{\\cal U}$ under Assumption \\ref{as:pos}(ii). Thus, if a coalition ${\\cal S}$ has the incentive to break away, then\n$$\n\\sum_{i\\in{\\cal S}}z_i<\\max_{u\\in{\\cal U}}\\left\\{\\frac{\\displaystyle\\max_{\\alpha_{{\\cal S}}\\in{\\cal A}({\\cal S})}v_u(\\alpha_{{\\cal S}},{\\cal S})}{v_u(a,{\\cal N})}\\right\\}=v_{\\max}(a,{\\cal S}).\n$$\n\nNow, suppose $\\displaystyle\\sum_{i\\in{\\cal S}}z_iv_u(a,{\\cal N})\\cdot z_i.\n$$\nLet $(\\hat{a},\\hat{u})\\in\\displaystyle\\arg\\max_{\\alpha_{{\\cal S}}\\in{\\cal A}({\\cal S})}\\max_{u\\in{\\cal U}}\\left\\{\\frac{v_u(\\alpha_{\\cal S},{\\cal S})}{v_u(a,{\\cal N})}\\right\\}$. Clearly, we have:\n$$\n\\frac{v_{\\hat{u}}(\\hat{a},{\\cal S})}{v_{\\hat{u}}(a,{\\cal N})}=\\max_{\\alpha_{{\\cal S}}\\in{\\cal A}({\\cal S})}\\max_{u\\in{\\cal U}}\\left\\{\\frac{v_u(\\alpha_{{\\cal S}},{\\cal S})}{v_u(a,{\\cal N})}\\right\\}=\\max_{u\\in{\\cal U}}\\left\\{\\frac{\\displaystyle\\max_{\\alpha_{\\cal S}\\in{\\cal A}({\\cal S})}v_u(\\alpha_{\\cal S},{\\cal S})}{v_u(a,{\\cal N})}\\right\\}=v_{\\max}(a,{\\cal S}).\n$$\nFrom Lemma~\\ref{prop:individual:rationality:cond}, we have $\\displaystyle z_i \\geq v_{\\max}(a,\\{i\\})$ since $(a,\\bz)$ is an imputation. We also have, by Assumption~\\ref{as:pos}(i), $\\displaystyle v_{\\max}(a,\\{i\\}) \\geq 0$. Thus, $\\displaystyle v_{\\max}(a,{\\cal S}) > \\displaystyle\\sum_{i\\in{\\cal S}}z_i \\geq 0$ and hence $v_{\\hat{u}}(\\hat{a},{\\cal S})\\neq 0$.\n\nLet $\\displaystyle\\eps=\\frac{1}{\\card{{\\cal S}}}\\left(v_{\\max}(a,{\\cal S})-\\sum_{i\\in{\\cal S}}z_i\\right)>0$ and define $\\displaystyle\\hat{z}_i=\\frac{z_i+\\eps}{v_{\\max}(a,{\\cal S})}$ for all $i\\in{\\cal S}$. Clearly, $\\displaystyle\\sum_{i\\in{\\cal S}}\\hat{z}_i=1$ given the definition of $\\eps$. In addition, for all $i\\in{\\cal S}$, we have:\n$$\nv_{\\hat{u}}(\\hat{a},{\\cal S})\\cdot\\hat{z}_i=v_{\\hat{u}}(a,{\\cal N})\\cdot z_i+v_{\\hat{u}}(a,{\\cal N})\\cdot \\eps>v_{\\hat{u}}(a,{\\cal N})\\cdot z_i.\n$$\nThus, $(\\hat{a},\\hat{\\bz})$ is an efficient decision and the inequality above shows that coalition $\\cal S$ indeed has the incentive to break away.\n\\end{pf}\n\nWe are now ready to prove Theorem \\ref{thm:corecond}.\n\n\\begin{pf}\nSuppose there exists a core decision $(a,\\bz)$. Clearly, $(a,\\bz)$ is an imputation. Applying Lemma \\ref{lem:breakcond}, we can show that $\\bz$ is a feasible (and optimal) solution of the following linear program:\n\\be\n\\label{eq:primal}\n\\ba{rl}\n\\displaystyle Z_P=\\min_{\\bx} & \\displaystyle\\sum_{i\\in{\\cal N}}0\\cdot x_i\\\\\n\\st & \\displaystyle\\sum_{i\\in{\\cal S}}x_i\\geq v_{\\max}(a,{\\cal S}),\\quad \\forall\\,{\\cal S}\\subsetneq{\\cal N},\\\\\n& \\displaystyle\\sum_{i\\in{\\cal N}}x_i=1.\n\\ea\n\\ee\nThe dual problem is written as follows:\n\\be\n\\label{eq:dual}\n\\ba{rll}\nZ_D=\\max & \\displaystyle\\sum_{{\\cal S}\\subsetneq{\\cal N}}y_{{\\cal S}}v_{\\max}(a,{\\cal S})-p\\\\\n\\st & \\displaystyle\\sum_{{\\cal S}:i\\in{\\cal S}}y_{{\\cal S}}-p=0, & \\forall\\,i\\in{\\cal N},\\\\\n& y_{{\\cal S}}\\geq 0, & \\forall\\,{\\cal S}\\subsetneq{\\cal N}.\n\\ea\n\\ee\nApplying strong duality, we have: $Z_D=Z_P=0$, which means, for all feasible solution $\\left(\\left\\{y_{{\\cal S}}\\right\\}_{{\\cal S}\\subsetneq{\\cal N}},p\\right)$,\n$$\n\\sum_{{\\cal S}\\subsetneq{\\cal N}}y_{{\\cal S}}v_{\\max}(a,{\\cal S})\\leq p.\n$$\nNow consider any balanced map $\\mu$, it is clear that $\\left(\\left\\{\\mu({\\cal S})\\right\\}_{{\\cal S}\\subsetneq{\\cal N}},1\\right)$ is a feasible solution of the dual problem. Thus we have:\n$$\n\\sum_{{\\cal S}\\subsetneq{\\cal N}}\\mu({\\cal S})v_{\\max}(a,{\\cal S})\\leq 1.\n$$\nIt shows that the robust cooperative game is balanced.\n\nNow, suppose the robust cooperative game is balanced. There exists an action $a\\in{\\cal A}({\\cal N})$ such that for all balanced map $\\mu$,\n$$\n\\sum_{{\\cal S}\\subsetneq{\\cal N}}\\mu({\\cal S})v_{\\max}(a,{\\cal S})\\leq 1.\n$$\nWe are going to show that $Z_D=0$. The dual problem is indeed feasible. $(\\mb{0},0)$ is a feasible solution and it implies that $Z_D\\geq 0$. Let us consider a feasible solution $\\left(\\left\\{y_{{\\cal S}}\\right\\}_{{\\cal S}\\subsetneq{\\cal N}},p\\right)$. Since $y_{{\\cal S}}\\geq 0$ for all ${\\cal S}\\subsetneq{\\cal N}$, we have: $p\\geq 0$. If $p=0$, it is easy to show that $y_{{\\cal S}}=0$ for all ${\\cal S}\\subsetneq{\\cal N}$. If $p>0$, let $\\mu({\\cal S})=y_{\\cal S}\/p$ for all ${\\cal S}\\subsetneq{\\cal N}$. The map $\\mu$ is indeed a balanced map, thus we have:\n$$\n\\sum_{{\\cal S}\\subsetneq{\\cal N}}\\left(y_{{\\cal S}}\/p\\right)v_{\\max}(a,{\\cal S})\\leq 1\\,\\Leftrightarrow\\,\\sum_{{\\cal S}\\subsetneq{\\cal N}}y_{{\\cal S}}v_{\\max}(a,{\\cal S})-p\\leq 0.\n$$\nThus for all feasible solution $\\left(\\left\\{y_{{\\cal S}}\\right\\}_{{\\cal S}\\subsetneq{\\cal N}},p\\right)$, $\\displaystyle \\sum_{{\\cal S}\\subsetneq{\\cal N}}y_{{\\cal S}}v_{\\max}(a,{\\cal S})-p\\leq 0$, which means $Z_D\\leq 0$ and hence $Z_D=0$. According linear strong duality, the primal problem is feasible (and with the obvious optimal objective $Z_P=0$). Thus there exists at least a feasible solution $\\bz$ of the primal problem. The decision $(a,\\bz)$ is an imputation and indeed a core decision given the conditions obtained from the constraints of the primal problem. Thus the robust cooperative game has a non-empty core.\n\\end{pf}\n\nTheorem \\refs{thm:corecond} establishes the relationship between the game balancedness and the existence of core solutions. Computationally, it implies that we can attempt to solve the following optimization problem to check the existence of core solutions:\n\\be\n\\label{eq:rleastcore}\n\\ba{rl}\n\\displaystyle s({\\cal N},{\\cal A},{\\cal V}({\\cal U}))=\\min_{\\bx,\\eps,a} & \\eps\\\\\n\\st & \\displaystyle\\sum_{i\\in{\\cal S}}x_i\\geq v_{\\max}(a,{\\cal S})-\\eps,\\quad \\forall\\,{\\cal S}\\subsetneq{\\cal N},\\\\\n& \\displaystyle\\sum_{i\\in{\\cal N}}x_i=1,\\\\\n& a\\in{\\cal A}({\\cal N}).\n\\ea\n\\ee\nIf $s({\\cal N},{\\cal A},{\\cal V}({\\cal U}))\\leq 0$, then the core of the robust cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ is non-empty. Note that this optimization problem is no longer a linear program like Problem \\refs{eq:leastcore} or \\refs{eq:stability} given the fact that the action of the grand coalition is now a decision variable. We will investigate further the computational aspect of finding a core solution for a specific robust cooperative game, the \\emph{robust newsvendor game}, which is going to be discussed next. Even though the framework of robust cooperative games developed in this section is indeed suitable for the newsvendor games with distributional ambiguity which we are interested in, we would like to emphasize that it is plausible to develop other frameworks of cooperative games with uncertain characteristic functions using different payoff distribution schemes and different preference relations to suit other applications better.\n\n\n\\section{Robust Newsvendor Games} \\label{stoc.newsvendor}\nIn this section, we consider newsvendor games with ambiguity in demand distributions in the framework of robust cooperative games, which we call \\emph{robust newsvendor games}.\n\\begin{comment}Individual retailers usually collect historical demands independently before they join any coalition and the current assumption of a known joint distribution can be considered quite strong. In order to make the problem is more realistic, we only assume that some (multivariate) marginal distributions are known. For example, it is more reasonable to assume the knowledge of the joint demand distribution of a subset of retailers that are located close to each other and hence likely serve customers from a same area.\n\\end{comment}\nAs discussed in the previous section, the uncertainty comes from the fact that the joint demand distribution is unknown. We assume that only some (multivariate) marginal distributions of the joint demand are known. More concretely, consider a partition of ${\\cal N}$ with $R$ subsets ${\\cal N}_1,\\ldots,{\\cal N}_R$ such that\n$$\n{\\cal N}=\\bigcup_{r=1}^R{\\cal N}_r\\quad\\mbox{and}\\quad{\\cal N}_r\\cap{\\cal N}_s=\\emptyset\\quad\\mbox{for all }r\\neq s.\n$$\nGiven a vector $\\mb{d}\\in\\mathbb{R}^n$, let $\\mb{d}_r\\in\\mathbb{R}^{N_r}$ denote the sub-vector formed with the elements in the $r$th subset ${\\cal N}_r$ where $N_r=\\card{{\\cal N}_r}$ is the size of the subset. We assume that probability measures $P_r$ of random vectors $\\td{\\mb{d}}_r$ are known for all $r=1,\\ldots,R$. Let $\\mathcal{P}(P_1,\\ldots,P_R)$ denote the set of joint probability measures of the random vector $\\td{\\mb{d}}$ consistent with the prescribed probability measures of the random vectors $\\td{\\mb{d}}_r$ for all $r=1,\\ldots,R$, which acts as the uncertainty set ${\\cal U}$ in the general framework of robust cooperative games. Note that $\\mathcal{P}(P_1,\\ldots,P_R)$ is always non-empty since the independent measure among the sub-vectors is a feasible distribution. The set of joint distributions with fixed marginal distributions $\\mathcal{P}(P_1,\\ldots,P_R)$ is referred to as the Fr\\'echet class of distributions (see R\\\"uschendorf \\cite{ruschen91b}). It has been used to evaluate bounds on the cumulative distribution function of a sum of random variables with an application in risk management (Embrechts and Puccetti \\cite{ep06b}). Doan and Natarajan \\cite{doan12} developed a robust optimization model using $\\mathcal{P}(P_1,\\ldots,P_R)$ with an application in project management. We now investigate this Fr\\'echet class of distributions in the context of robust newsvendor games.\n\nGiven a subset ${\\cal S}\\subseteq{\\cal N}$, we define ${\\cal S}_r={\\cal S}\\cap{\\cal N}_r$ for all $r=1,\\dots,R$. Clearly, if all retailers $i$, $i\\in{\\cal S}$, join together, we know the non-overlapping marginal distributions of the joint demand vector with respect to the partition $({\\cal S}_1,\\ldots,{\\cal S}_R)$ of ${\\cal S}$. If ${\\cal S}\\subseteq{\\cal N}_r$ for some $r$, the joint distribution of $\\td{d}_i$, $i\\in{\\cal S}$, is completely known. In this case, the action the coalition should take, i.e., the decision on the ordering quantity, is well-defined as in the deterministic setting. The action set of coalition ${\\cal S}$ can be simply defined as ${\\cal Y}({\\cal S})=\\{y^*({\\cal S})\\}$, where $y^*({\\cal S})$ is the $(p-c)\/p$-quantile of the known distribution of $\\tilde{d}({\\cal S})$ as defined in \\refs{eq:optquantbar}. Note that ${\\cal Y}({\\cal S})$ acts as the action set ${\\cal A}({\\cal S})$ in the general framework of robust cooperative games. In general, the distribution $P({\\cal S})$ is unknown and coalition ${\\cal S}$ can choose its action regarding the ordering quantity from a general action set ${\\cal Y}({\\cal S})$.\nTo keep it simple, we shall let ${\\cal Y}({\\cal S})=\\mathbb{R}_+$ given the fact that the ordering quantities are non-negative for all ${\\cal S}\\subseteq{\\cal N}$.\n As mentioned earlier, if ${\\cal S}\\subseteq{\\cal N}_r$ for some $r$, we can restrict ${\\cal Y}({\\cal S})=\\{y^*({\\cal S})\\}$, where $y^*({\\cal S})$ is the $(p-c)\/p$-quantile of the known distribution of $\\tilde{d}({\\cal S})$ as defined in \\refs{eq:optquantbar}.\n\nGiven an ordering decision $y\\in{\\cal Y}({\\cal S})$, the \\emph{uncertain} payoff, i.e., total expected profit, of coalition $\\cal S$ is $v_P(y,{\\cal S})=\\mathbb{E}_{P({\\cal S})}\\left[p\\min\\{\\td{d}({\\cal S}),y\\}-cy\\right]$ for $P\\in{\\cal P}(P_1,\\ldots,P_R)$ as in \\refs{eq:quantbar}, where $P({\\cal S})$ is the corresponding marginal joint distribution of $\\td{d}_i$, $i\\in{\\cal S}$, derived from $P$. For each individual retailer $i$, ${\\cal Y}(\\{i\\})=y_i^*$ and $v_P(y_i^*,\\{i\\})\\geq v_P(0,\\{i\\})=0$ for all $P\\in{\\cal P}(P_1,\\ldots,P_R)$, which implies Assumption~\\ref{as:pos}(i) is automatically satisfied.\n\nFinally, for the grand coalition, in order to satisfy Assumption~\\ref{as:pos}(ii), we let\n\\be\n\\label{eq:yn}\n{\\cal Y}({\\cal N})=\\left\\{y\\in\\mathbb{R}_+\\,:\\,v_P(y,{\\cal N})>0,\\,\\forall\\,P\\in{\\cal P}(P_1,\\ldots,P_R) \\right\\}.\n\\ee\nWe shall provide a simple condition with which the action set of the grand coalition is non-empty. Let $d_{\\min}({\\cal S})$ be the minimum value that the random demand $\\td{d}({\\cal S})$ can achieve, the following lemma sets out a sufficient condition for ${\\cal Y}({\\cal N})$ to be empty.\n\n\\begin{lemma}\n\\label{lem:nonempty}\nIf $\\displaystyle\\max_{r=1,\\ldots,R}d_{\\min}({\\cal N}_r)>0$, then ${\\cal Y}({\\cal N})\\neq \\emptyset$.\n\\end{lemma}\n\n\\begin{pf}\nWe have: $\\displaystyle\\td{d}({\\cal N})=\\sum_{r=1}^R\\td{d}({\\cal N}_r)$. Thus, if $\\displaystyle\\max_{r=1,\\ldots,R}d_{\\min}({\\cal N}_r)>0$, then:\n$$\nd_{\\min}({\\cal N})\\geq\\sum_{r=1}^Rd_{\\min}({\\cal N}_r)> 0.\n$$\nFor $y\\in[0,d_{\\min}(N)]$, $v_P(y,{\\cal N})=(p-c)y$, which is strictly increasing for any $P\\in{\\cal P}(P_1,\\ldots,P_R)$ given the fact that $p>c$. In addition, for $y\\geq d_{\\max}({\\cal N})$, $v_P(y,{\\cal N})=-cy+\\displaystyle p\\sum_{r=1}^R\\mathbb{E}_{P_r}\\left[\\td{d}({\\cal N}_r)\\right]$, which is strictly decreasing for any $P\\in{\\cal P}(P_1,\\ldots,P_R)$ given the fact that $c>0$.\n\nNow consider the function $\\displaystyle\\bar{v}(y,{\\cal N})=\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)}v_P(y,{\\cal N})$. It is clear that $\\bar{v}(\\cdot,{\\cal N})$ is again strictly increasing in $[0,d_{\\min}({\\cal N})]$ and strictly decreasing in $[d_{\\max}({\\cal N}),+\\infty)$. Thus we have:\n$$\n\\arg\\max_{y\\geq 0}\\bar{v}(y,{\\cal N})\\in[d_{\\min}({\\cal N}),d_{\\max}({\\cal N})],\n$$\nand\n$$\n\\max_{y\\geq 0}\\bar{v}(y,{\\cal N})\\geq \\bar{v}(d_{\\min}({\\cal N}),{\\cal N})=(p-c)d_{\\min}({\\cal N})>0.\n$$\nThus there exists $y\\geq 0$ such that $\\bar{v}(y,{\\cal N})>0$, or equivalently, $v_P(y,{\\cal N})>0$ for all $P\\in{\\cal P}(P_1,\\ldots,P_R)$. It shows that ${\\cal Y}({\\cal N})\\neq\\emptyset$.\n\\end{pf}\n\nThe condition in Lemma \\ref{lem:nonempty} simply requires that one of the (marginal) distributions $P_r$, $r=1,\\ldots,R$ has the support set of solely non-zero demand vectors, which can be considered as a reasonable assumption in reality.\n In the rest of the paper, we shall make that assumption to ensure ${\\cal Y}({\\cal N})\\neq\\emptyset$, or equivalently, that Assumption~\\ref{as:pos}(ii) is satisfied.\nWe are now ready to consider the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$ and investigate the existence of its imputations and core solutions.\n\\subsection{Existence of Imputations and Core Solutions}\n\\label{ssec:existence}\nThe uncertain characteristic function $v_P(y,{\\cal S})$ can be written as\n\\be\n\\label{eq:char}\nv_{P}(y,{\\cal S})=(p-c)y-p\\,\\mathbb{E}_{P({\\cal S})}\\left[\\left(y-\\td{d}({\\cal S})\\right)^+\\right],\n\\ee\nfor all ${\\cal S}\\subseteq{\\cal N}$, $y\\in{\\cal Y}({\\cal S})$, and $P\\in{\\cal P}(P_1,\\ldots,P_R)$. The deterministic newsvendor games always have imputations since the corresponding characteristic function is super-additive. The following theorem claims the existence of imputations of the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$.\n\\begin{theorem}\n\\label{thm:rimpu}\nThe robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$ always has an imputation.\n\\end{theorem}\n\nIn order to prove Theorem \\ref{thm:rimpu}, we first study a particular action that each coalition can take, the worst-case optimal ordering quantity $y^*_{wc}({\\cal S})$:\n\\be\n\\label{eq:optquant2}\ny^*_{wc}({\\cal S})\\in\\arg\\max_{y\\geq 0}\\left\\{(p-c)y-p\\max_{P\\in{\\cal P}(P_1,\\ldots,P_R)}\\mathbb{E}_{P}\\left[\\left(y-\\td{{d}}({\\cal S})\\right)^+\\right]\\right\\}.\n\\ee\nBasically, this ordering quantity is optimal under the worst-case scenario with respect to the joint demand distribution $P$. Let us also define $v_{wc}({\\cal S})$ as the maximum worst-case expected profit for coalition ${\\cal S}$, i.e.\n\\be\nv_{wc}({\\cal S}) = \\max_{y\\geq 0}\\left\\{(p-c)y-p\\max_{P\\in{\\cal P}(P_1,\\ldots,P_R)}\\mathbb{E}_{P}\\left[\\left(y-\\td{{d}}({\\cal S})\\right)^+\\right]\\right\\}.\n\\ee\n\nWhen ${\\cal S}\\subseteq{\\cal N}_r$ for some $r$, clearly, $y^*_{wc}({\\cal S})=y^*({\\cal S})$, the $(p-c)\/p$-quantile of the known distribution of $\\td{d}({\\cal S})$, and $v_{wc}({\\cal S})=\\bar{v}({\\cal S})$ as defined in \\refs{eq:optval}. The following lemma shows how to calculate the worst-case optimal ordering quantities $y^*_{wc}({\\cal S})$ and the worst-case expected profit $v_{wc}({\\cal S})$ for an arbitrary ${\\cal S}\\subseteq{\\cal N}$.\n\\begin{lemma}\n\\label{lem:optquant}\nFor an arbitrary ${\\cal S}\\subseteq{\\cal N}$, the worst-case optimal ordering quantity $y^*_{wc}({\\cal S})$ defined in \\refs{eq:optquant2} can be calculated as follows:\n\\be\n\\label{eq:optquantform1}\ny^*_{wc}({\\cal S})=\\sum_{r=1}^R{y}^*({\\cal S}_r),\n\\ee\nwhere ${\\cal S}_r={\\cal S}\\cap{\\cal N}_r$ for all $r=1,\\ldots,R$, ${y}^*({\\cal S}_r)$ is the $(p-c)\/p$-quantile of the known distribution of $\\td{d}({\\cal S}_r)$, and ${y}^*(\\emptyset)=0$. In addition, $v_{wc}({\\cal S}) = \\displaystyle \\sum_{r=1}^R v_{wc}({\\cal S}_r) = \\displaystyle \\sum_{r=1}^R \\bar{v}({\\cal S}_r)$.\n\\end{lemma}\n\n\\begin{pf}\nConsider the optimization problem in \\refs{eq:optquant2}. For $y\\leq d_{\\min}({\\cal S})=\\min\\{\\td{d}({\\cal S})\\}$, we can write $(p-c)y-p\\,\\mathbb{E}_{P}\\left[\\left(y-\\td{{d}}({\\cal S})\\right)^+\\right]=(p-c)y$ for any distribution $P$. Since $p-c>0$, it is an increasing function in $y$ in $(-\\infty;d_{\\min}({\\cal S})]$. Since $d_{\\min}({\\cal S})\\geq 0$, we can then remove the non-negative constraint $y\\geq 0$ from \\refs{eq:optquant2} when calculating $y^*_{wc}({\\cal S})$. Now, consider the inner optimization problem of \\refs{eq:optquant2}. This is an instance of the distributionally robust optimization problem studied in Doan and Natarajan \\cite{doan12}. Without loss of generality, we can assume that ${\\cal S}_r\\neq\\emptyset$ for all $r=1,\\ldots,R$ knowing that ${y}^*(\\emptyset)=0$. Applying Proposition 1(ii) from \\cite{doan12}, we obtain the following reformulation:\n$$\n\\ba{rl}\n\\displaystyle\\max_{P\\in{\\cal P}(P_1,\\ldots,P_R)}\\mathbb{E}_{P}\\left[\\left(y-\\td{{d}}({\\cal S})\\right)^+\\right]=\\min_{\\mbs{x}} &\\displaystyle\\sum_{r=1}^R\\mathbb{E}_{P_r}\\left[\\left(x_r-\\td{d}({\\cal S}_r)\\right)^+\\right]\\\\\n\\st &\\displaystyle\\sum_{r=1}^Rx_r=y.\n\\ea\n$$\nThus, in order to find $y^*_{wc}({\\cal S})$, we can solve the following optimization problem\n$$\n\\ba{rl}\n\\displaystyle\\max_{y,\\mbs{x}}&\\displaystyle (p-c)y-p\\sum_{r=1}^R\\mathbb{E}_{P_r}\\left[\\left(x_r-\\td{d}({\\cal S}_r)\\right)^+\\right]\\\\\n\\st & \\displaystyle\\sum_{r=1}^Rx_r=y.\\\\\n\\ea\n$$\nThe optimal ordering quantity $y^*_{wc}({\\cal S})$ can then be calculated as $\\displaystyle y^*_{wc}({\\cal S})=\\sum_{r=1}^Rx_r^*$, where $\\bx^*$ is the optimal solution of the following separable optimization problem:\n$$\n\\max_{\\mbs{x}}\\sum_{r=1}^R\\left((p-c)x_r-p\\sum_{r=1}^R\\mathbb{E}_{P_r}\\left[\\left(x_r-\\td{d}({\\cal S}_r)\\right)^+\\right]\\right),$$\nor equivalently,\n$$\n\\sum_{r=1}^R\\max_{x_r}\\left((p-c)x_r-p\\mathbb{E}_{P_r}\\left[\\left(x_r-\\td{d}({\\cal S}_r)\\right)^+\\right]\\right).\n$$\nFor each sub-problem, $x_r^*$ is the $(p-c)\/p$-quantile of the distribution of $\\td{d}({\\cal S}_r)$, which means $x^*_r={y}^*({\\cal S}_r)$ for all $r=1,\\ldots, R$, according to \\refs{eq:optquantbar}. Thus we have $\\displaystyle y^*_{wc}({\\cal S})=\\sum_{r=1}^R{y}^*({\\cal S}_r)$. This also leads to $v_{wc}({\\cal S}) = \\displaystyle \\sum_{r=1}^R v_{wc}({\\cal S}_r) = \\displaystyle \\sum_{r=1}^R \\bar{v}({\\cal S}_r)$. Here, the joint demand distribution for players in ${\\cal S}_r$ is known and hence the worst-case expected profit ${v}_{wc}(S_r)$ is exactly the same with the deterministic expected profit $\\bar{v}({\\cal S}_r)$ shown in Formulation~\\refs{eq:optval}.\n\\end{pf}\n\n We are now ready to use this lemma to prove Theorem \\ref{thm:rimpu}.\n\n\\begin{pf}\nAccording to Theorem \\ref{prop:impucond}, the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$ has imputations if and only if there exists $y\\in{\\cal Y}({\\cal N})$ such that\n$$\n\\sum_{i\\in{\\cal N}}v_{\\max}(y,\\{i\\})\\leq 1,\n$$\nwhere $\\displaystyle v_{\\max}(y,\\{i\\})=\\max_{P\\in{\\cal P}(P_1,\\ldots,P_R)}\\left\\{\\frac{\\displaystyle\\max_{y_i\\in{\\cal Y}(\\{i\\})}v_P(y_i,\\{i\\})}{v_P(y,{\\cal N})}\\right\\}$. For each individual retailer $i$, the demand distribution is known; therefore ${\\cal Y}(\\{i\\})=\\{y_i^*\\}$, where $y_i^*$ is the $(p-c)\/p$-quantile of the distribution function of $\\td{d}_i$. Thus\n$$\n\\max_{y_i\\in{\\cal Y}(\\{i\\})}v_P(y_i,\\{i\\})=v_i(y^*)=\\bar{v}_i\\geq 0,\\quad\\forall\\,P\\in{\\cal P}(P_1,\\ldots,P_R).\n$$\nWe then have $\\displaystyle v_{\\max}(y,\\{i\\})=\\bar{v}_i\\cdot\\left(\\displaystyle\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)}v_P(y,{\\cal N})\\right)^{-1}$ and the existence condition of imputation can be written as follows:\n$$\n\\exists\\,y\\in{\\cal Y}({\\cal N}):\\,\\sum_{i\\in{\\cal N}}\\bar{v}_i\\leq \\displaystyle\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)}v_P(y,{\\cal N}),\n$$\ngiven the fact that $v_P(y,{\\cal N})>0$ for all $y\\in{\\cal Y}({\\cal N})$ and $P\\in{\\cal P}(P_1,\\ldots,P_R)$. Equivalently, the existence condition is\n$$\n\\sum_{i\\in{\\cal N}}\\bar{v}_i\\leq \\displaystyle\\max_{y\\in{\\cal Y}({\\cal N})}\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)}v_P(y,{\\cal N})=\\max_{y\\geq 0}\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)}v_P(y,{\\cal N}).\n$$\nThe optimization problem $\\displaystyle\\max_{y\\geq 0}\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)}v_P(y,{\\cal N})$ is the same as Problem \\refs{eq:optquant2}. Let us consider the worst-case optimal ordering quantity $y^*_{wc}({\\cal N})$, the existence condition can then be written as follows:\n$$\n\\sum_{i\\in{\\cal N}}\\bar{v}_i\\leq v_P(y^*_{wc}({\\cal N}),{\\cal N}),\\quad\\forall\\,P\\in{\\cal P}(P_1,\\ldots,P_R).\n$$\nApplying Lemma \\ref{lem:optquant} for ${\\cal S}={\\cal N}$, we have $\\displaystyle y^*_{wc}({\\cal N})=\\sum_{i=1}^R{y}^*({\\cal N}_r)$. Using the fact that $(x+y)^+\\leq x^++y^+$, we have, for all $P\\in{\\cal P}(P_1,\\ldots,P_R)$,\n$$\n\\ba{rl}\nv_P(y^*_{wc}({\\cal N}),{\\cal N}) &=\\,(p-c)y^*_{wc}({\\cal N})-p\\,\\mathbb{E}_P\\left[\\left(y^*_{wc}({\\cal N})-\\td{d}({\\cal N})\\right)^+\\right]\\\\\n&\\geq\\,\\displaystyle (p-c)\\sum_{i=1}^R{y}^*({\\cal N}_r)-p\\sum_{r=1}^R\\mathbb{E}_{P_r}\\left[\\left({y}^*({\\cal N}_r)-\\td{d}({\\cal N}_r)\\right)^+\\right]\\\\\n&=\\,\\displaystyle\\sum_{r=1}^R\\left\\{(p-c){y}^*({\\cal N}_r)-p\\,\\mathbb{E}_{P_r}\\left[\\left({y}^*({\\cal N}_r)-\\td{d}({\\cal N}_r)\\right)^+\\right]\\right\\}\\\\\n&=\\,\\displaystyle\\sum_{r=1}^R\\bar{v}({\\cal N}_r),\n\\ea\n$$\nwhere $\\bar{v}({\\cal N}_r)$ is the optimal total expected profit of coalition ${\\cal N}_r$ and is computed using \\refs{eq:optval} for all $r=1,\\ldots,R$ since $P_1,\\ldots,P_R$ are completely known.\n\\begin{comment}\nWe have $\\bar{v}({\\cal N}_r)\\geq 0$ for all $r=1,\\ldots,R$ and under Assumption \\ref{as:mpos}, there exists $r$ such that $\\bar{v}({\\cal N}_r)>0$. Thus, we have\n$$\nv_P({\\cal N})\\geq\\sum_{r=1}^R\\bar{v}({\\cal N}_r)>0.\n$$\n\\end{comment}\n\nNow consider the deterministic newsvendor game for coalition ${\\cal N}_r$ with the complete knowledge of the joint distribution $P_r$. There exists at least one imputation for this cooperative game; thus, we have\n$$\n\\sum_{i\\in{\\cal N}_r}\\bar{v}_i\\leq\\bar{v}({\\cal N}_r),\\quad\\forall\\,r=1,\\ldots,R.\n$$\nUsing the fact that ${\\cal N}=\\bigcup_{r=1}^R{\\cal N}_r$ and ${\\cal N}_r\\cap{\\cal N}_s = \\emptyset$ for all $r\\neq s$, we have\n$$\nv_P({\\cal N})\\geq\\sum_{r=1}^R\\bar{v}({\\cal N}_r)\\geq\\sum_{r=1}^R\\sum_{i\\in{\\cal N}_r}\\bar{v}_i=\\sum_{i\\in{\\cal N}}\\bar{v}_i,\\quad\\forall\\,P\\in{\\cal P}(P_1,\\ldots,P_R).\n$$\nThus, the existence condition is satisfied and the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$ always has an imputation.\n\\end{pf}\n\nWe focus on properties of core solutions of the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$ next.\n\\begin{theorem}\n\\label{thm:rcore2}\nIf $(y,\\mb{z})$ is a core solution of the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$, then the following statements hold:\n\\begin{itemize}\n\\item[(a)] The ordering quantity $y$ is the worst-case optimal ordering quantity, i.e., $y = y^*_{wc}({\\cal N})$.\n\\item[(b)] $v_{wc}({\\cal N})\\cdot\\mb{z}({\\cal N}_r)$ is a core solution of the (deterministic) newsvendor game $({\\cal N}_r,\\bar{v})$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{pf}\nLet us consider a core solution $(y,\\mb{z})$ of the robust newsvendor game {\\small $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$}. For each coalition ${\\cal N}_r$, $r =1,\\ldots,R$, we have\n\n\\begin{eqnarray}\nv_{\\max}(y,{\\cal N}_r)&=&\\max_{P\\in{\\cal P}(P_1,\\ldots,P_R)} \\left\\{\\frac{\\displaystyle\\max_{y_r \\in{\\cal Y}({\\cal N}_r)}v_P(y_r,{\\cal N}_r)}{v_P(y,{\\cal N})}\\right\\} \\nonumber\\\\\n&=& \\frac{\\displaystyle\\max_{y_r \\in{\\cal Y}({\\cal N}_r)}v_{P_r}(y_r,{\\cal N}_r)}{\\displaystyle \\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(y,{\\cal N})} \\nonumber\\\\\n&=& \\frac{\\displaystyle v_{wc}({\\cal N}_r)}{\\displaystyle \\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(y,{\\cal N})} = \\frac{\\displaystyle \\bar{v}({\\cal N}_r)}{\\displaystyle \\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(y,{\\cal N})}. \\label{eq:thm:rcore1}\n\\end{eqnarray}\nThis is due to the fact that $P_r$ is known with certainty and $v_{wc}({\\cal N}_r)\\geq 0$. According to Lemma~\\ref{lem:breakcond} and Equality \\refs{eq:thm:rcore1}, for $(y,\\mb{z})$ to be a core solution, we need to have\n\\begin{eqnarray}\n\\displaystyle \\sum_{i\\in{\\cal N}_r}z_i \\geq v_{\\max}(y,{\\cal N}_r) = \\frac{\\displaystyle \\bar{v}({\\cal N}_r)}{\\displaystyle \\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(y,{\\cal N})}. \\label{eq:thm:rcore2}\n\\end{eqnarray}\nSumming this over $r=1,\\ldots,R,$ we obtain\n\\begin{eqnarray}\n\\displaystyle 1 &=& \\sum_{i\\in {\\cal N}} z_i = \\sum_{r=1}^R~\\sum_{i\\in{\\cal N}_r}z_i \\geq \\frac{\\displaystyle \\sum_{r=1}^R \\bar{v}({\\cal N}_r)}{\\displaystyle \\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(y,{\\cal N})}.\\label{eq:thm:rcore3}\n\\end{eqnarray}\nGiven Assumption~\\ref{as:pos}(ii), $\\displaystyle \\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(a,{\\cal N})>0$. Hence, we can rewrite Inequality~\\refs{eq:thm:rcore3} as\n\\begin{eqnarray*}\n\\displaystyle \\sum_{r=1}^R \\bar{v}({\\cal N}_r) \\leq \\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(y,{\\cal N}).\n\\end{eqnarray*}\nThis leads to\n\\begin{eqnarray}\n\\displaystyle \\sum_{r=1}^R \\bar{v}({\\cal N}_r) &\\leq& \\max_{y \\geq 0}~\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(y,{\\cal N}) \\label{eq:thm:rcore4} \\\\\n&\\equiv& v_{wc}({\\cal N}) \\nonumber\\\\\n&=& \\sum_{r=1}^R v_{wc}({\\cal N}_r) = \\sum_{r=1}^R \\bar{v}({\\cal N}_r), \\nonumber\n\\end{eqnarray}\nwhere the last equality comes from Lemma~\\ref{lem:optquant}. Thus, all the inequalities in the chain need to be tight. It implies that the ordering quantity $y$ is the worst-case optimal ordering quantity, $y=y_{wc}^*({\\cal N})$ for \\refs{eq:thm:rcore4} to be tight. We also obtain $\\displaystyle \\sum_{i \\in N_r} z_i = \\displaystyle \\frac{\\bar{v}({\\cal N}_r)}{v_{wc}({\\cal N})}$ for \\refs{eq:thm:rcore2} to be tight. In addition, for all coalition ${\\cal S}_r \\subset {\\cal N}_r$, by using the similar argument in deriving Equality~\\refs{eq:thm:rcore1}, we have\n\\begin{eqnarray*}\n\\displaystyle \\sum_{i \\in S_r} z_i \\geq v_{\\max}(y,{\\cal S}_r) = \\displaystyle \\frac{\\bar{v}(S_r)}{v_{wc}({\\cal N})},\n\\end{eqnarray*}\nwhich means $v_{wc}({\\cal N})\\cdot\\bz({\\cal N}_r)$ is a core solution of the deterministic newsvendor game $({\\cal N}_r,\\bar{v})$ for $r=1,\\ldots,R$.\n\\end{pf}\n\nTheorem~\\ref{thm:rcore2} shows that in order to check whether a particular decision $(y,\\bz)$ is a core solution of the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$, we only need to consider $y=y^*_{wc}({\\cal N})$. The optimization problem \\refs{eq:rleastcore} for checking the existence of core solutions can be reduced to the following linear program:\n\\be\n\\label{eq:rnleastcore}\n\\ba{rl}\n\\displaystyle \\sigma(y^*_{wc}({\\cal N}))=\\min_{\\bx,\\eps} & \\eps\\\\\n\\st & \\displaystyle\\sum_{i\\in{\\cal S}}x_i\\geq v_{\\max}(y^*_{wc}({\\cal N}),{\\cal S})-\\eps,\\quad{\\cal S}\\subsetneq{\\cal N},\\\\\n& \\displaystyle\\sum_{i\\in{\\cal N}}x_i=1.\n\\ea\n\\ee\n\n\nUnlike the deterministic newsvendor games, the robust newsvendor games do not always have core solution for $N\\geq 3$. The following example shows a simple robust newsvendor game with $N=3$ whose core is empty.\n\n\\begin{example}\n\\label{ex:nocore}\nLet us consider the partition of ${\\cal N}=\\{1,2,3\\}$ with $R=2$, ${\\cal N}_1=\\{1,2\\}$ and ${\\cal N}_3=\\{3\\}$. The probability distribution $P_1$ of $(\\td{d}_1,\\td{d}_2)$ is characterized by the uniform marginal distribution of $\\td{d}_1$, $\\td{d}_1 \\sim U(0,D)$, for some $D>0$, and the relationship $\\td{d}_2 = D-\\td{d}_1$. The probability distribution $P_2$ of $\\td{d}_3$ is another uniform distribution, $\\td{d}_3 \\sim U(0,D)$. Clearly, ${\\cal Y}({\\cal N})\\neq\\emptyset$ given the fact that $d_{\\min}({\\cal N}_1)=D>0$.\n\nAccording Theorem \\ref{thm:rcore2}, if the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,P_2)))$ has a core solution $(y,\\bz)$, then $y=y^*_{wc}({\\cal N})$. Given $P_1$ and $P_2$, we are able to compute and bound some values of $v_{\\max}(y,{\\cal S})$ where ${\\cal S}\\subset{\\cal N}$, as follows:\n$$\nv_{\\max}(y,\\{1,2\\})=\\frac{2p}{3p-c}\\leq v_{\\max}(y,\\{2,3\\})=v_{\\max}(y,\\{1,3\\}).\n$$\nFor clarity of the exposition, we leave the detailed computation of these values in the Appendix. Now, since $(y,\\bz)$ is a core solution, $\\displaystyle\\sum_{i\\in{\\cal S}}z_i\\geq v_{\\max}(y,{\\cal S})$ for ${\\cal S}\\subset{\\cal N}$ and $z_1+z_2+z_3=1$. Thus\n\\begin{eqnarray*}\n2 &=& (z_1+z_2)+(z_2+z_3)+(z_3+z_1)\\\\\n&\\geq& v_{max}(y,\\{1,2\\}) + v_{max}(y,\\{2,3\\})+v_{max}(y,\\{1,3\\})\\\\\n&\\geq& 6p\/(3p-c) > 2,\n\\end{eqnarray*}\nwhich is a contradiction or this robust newsvendor game does not have a core solution.\n\\end{example}\n\nWe now focus on how to solve Problem \\refs{eq:rnleastcore} to check the existence of core solutions of robust newsvendor games. If the core is empty, i.e., $\\sigma(y^*_{wc}({\\cal N}))>0$, \\emph{least core solutions} for the robust newsvendor game can be found by solving the general problem \\refs{eq:rleastcore}, which will be discussed in the next section.\n\\subsection{Core and Least Core Computation}\\label{robust_core_computation}\n\\begin{comment}\nIn order to solve the problem \\refs{eq:rnleastcore}, we need to be able to compute $v_{\\max}(y^*_{wc}({\\cal N}),{\\cal S})$ for each coalition $\\cal S\\subsetneq{\\cal N}$. If ${\\cal S}\\subseteq {\\cal N}_r$ for some $r$, we have shown that $\\displaystyle v_{\\max}(y^*_{wc}({\\cal N}),{\\cal S})=\\frac{\\bar{v}(S)}{v_{wc}({\\cal N})}$.\n\\end{comment}\nBoth Problems \\refs{eq:rleastcore} and \\refs{eq:rnleastcore} involve the function $v_{\\max}$. We first show how to compute $v_{\\max}(y,{\\cal S})$ for an arbitrary ordering quantity $y\\in{\\cal Y}({\\cal N})$ and an arbitrary ${\\cal S}\\subsetneq{\\cal N}$ with ${\\cal Y}({\\cal S})=\\mathbb{R}_+$. (Note that if ${\\cal S}\\subseteq {\\cal N}_r$ for some $r$, $r=1,\\ldots,R$, we then have ${\\cal Y}({\\cal S})=\\{y^*({\\cal S})\\}$ and $v_{\\max}(y,{\\cal S})$ can be computed by simply solving a single optimization problem, $\\displaystyle\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)}v_P(y,{\\cal N})$.) We have:\n\\be\n\\label{eq:gvmax}\nv_{\\max}(y,{\\cal S})=\\max_{\\gamma \\in \\R^+}~ \\max_{P\\in{{\\cal P}(P_1,\\ldots,P_R)}} \\frac{(p-c)\\gamma-p\\,\\mathbb{E}_P\\left[\\left(\\gamma-\\td{d}({\\cal S})\\right)^+\\right]}\n{(p-c)y-p\\,\\mathbb{E}_P\\left[\\left(y-\\td{d}({\\cal N})\\right)^+\\right]}.\n\\ee\nFor newsvendor games, it is reasonable to assume that demands follow discrete non-negative distributions, which could be constructed from historical sales or market analysis. More specifically, let each distribution $P_r$ of $\\td{\\mb{d}}_r$ be represented as a discrete non-negative distribution with $K_r$ values, $\\mb{d}^k_r$ of probability $p^k_r$, for $k = 1,\\ldots,K_r$, $r=1,\\ldots,R$. Thus, each probability distribution $P$ in $\\mathcal{P}(P_1,\\ldots,P_R)$ is a discrete distribution with a support of $\\displaystyle K =\\prod_{r=1}^RK_r$ values $\\mb{d}_k$ and each has an unknown probability of $q_k$, $k = 1,\\ldots,K$. For $P$ to be consistent with $P_1,\\ldots,P_R$, the following constraints on $\\mb{q}$ must hold:\n\\be\n\\label{eq:qdef}\n\\left\\{\\displaystyle \\mb{q} \\geq 0,~~\\sum_{k=1}^Kq_k=1,~~\\sum_{k=1}^K\\mathbb{I}\\{\\mb{d}_{k,r}=\\mb{d}_r^l\\}q_k = p_r^l,~~ r=1,\\ldots,R,\\,l=1,\\ldots,K_r\\right\\}.\n\\ee\nGiven the one-to-one mapping between $P$ and $\\mb{q}$, we abuse the notations and write both $v_P(y,{\\cal S})$ and $v_{\\vect{q}}(y,{\\cal S})$ interchangeably when the context is clear for $\\mb{q}$ to be the corresponding representation of $P$.\n\nProblem \\refs{eq:gvmax} can be reformulated as\n\\begin{equation}\n\\label{eq:vmax}\n\\begin{array}{rl}\nv_{\\max}(y,{\\cal S})=\\displaystyle \\max_{\\gamma, \\vect{q}} & \\displaystyle\\frac{(p-c)\\gamma - p\\,\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]q_k}{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k}\\\\\n\\st & \\displaystyle\\sum_{k=1}^K\\mathbb{I}\\{\\mb{d}_{k,r}=\\mb{d}_r^l\\}q_k = p_r^l,\\quad\\forall\\,r=1,\\ldots,R,\\,l=1,\\ldots,K_r,\\\\\n& \\displaystyle\\sum_{k=1}^Kq_k=1,\\\\\n& \\gamma,\\mb{q}\\geq 0.\n\\end{array}\n\\end{equation}\nFor each fixed $\\gamma$, we can apply the standard method for transforming a linear fractional optimization problem into a linear program (see, for example, Cambini et al. \\cite{cambini05}). To this end, let us introduce new decision variables\n$$\\displaystyle \\theta = \\frac{1}{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k},$$\n and\n$$\\displaystyle \\psi_k = \\frac{q_k}{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k},\\quad k=1,\\ldots,K.$$\nUnder Assumption~\\ref{as:pos}(ii), we have $\\theta > 0$. The objective function then becomes $$\\displaystyle (p-c)\\gamma\\cdot \\theta - p\\,\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]\\psi_k,$$\nand Problem~\\refs{eq:vmax} can be reformulated as\n\n\\begin{equation}\n\\label{eq:mpriLP}\n\\begin{array}{rl}\nv_{\\max}(y,{\\cal S})=\\displaystyle \\max_{\\gamma, \\theta, \\vect{\\psi}} & \\displaystyle (p-c)\\gamma\\cdot \\theta - p\\,\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]\\psi_k\\\\\n\\st & \\displaystyle\\sum_{k=1}^K\\mathbb{I}\\{\\mb{d}_{k,r}=\\mb{d}_r^l\\}\\psi_k - p_r^l\\cdot\\theta = 0,\\quad\\forall\\,r=1,\\ldots,R,\\,l=1,\\ldots,K_r,\\\\\n& \\displaystyle\\sum_{k=1}^K \\psi_k-\\theta = 0,\\\\\n& \\displaystyle (p-c)y\\cdot \\theta - p\\,\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right] \\psi_k = 1,\\\\\n& \\gamma,\\theta, \\mb{\\psi} \\geq 0,\n\\end{array}\n\\end{equation}\nwhere the first two constraints in \\refs{eq:mpriLP} are derived directly from the first two constraints in \\refs{eq:vmax} by multiplying both sides of those with $\\theta$. The third constraint in \\refs{eq:mpriLP} is derived by the definitions of $\\theta$ and $\\mb{\\psi}$. Finally, we have replaced the constraint $\\theta > 0$ by $\\theta \\geq 0$ without loss of generality since $\\theta = 0$ is not a feasible solution (otherwise, $\\mb{\\psi}$ must be equal to zero from the second constraint and that violates the third constraint.)\n\nProblem \\refs{eq:mpriLP} is a bilinear optimization problem, which is generally not easy to solve. We show, however, in the following proposition that one of the distinct values of $\\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}$, which are known, is an optimal value of $\\gamma$. Given a fixed $\\gamma$, the resulting bilinear optimization problem is reduced to a linear program. It implies that we can solve Problem \\refs{eq:mpriLP} by solving at most $K$ linear programs with $\\gamma$ set to each and every distinct value of $\\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}$.\n\n\\begin{proposition}\n\\label{prop:linearequiv}\nThere exists an optimal solution $(\\gamma^*,\\theta^*,\\mb{\\psi}^*)$ of Problem \\refs{eq:mpriLP} such that $$\\gamma^*\\in\\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}.$$\n\\end{proposition}\n\n\\begin{pf}\nGiven an arbitrary value of $\\gamma$, Problem \\refs{eq:mpriLP} is reduced to a linear program for $\\theta$ and $\\mb{\\psi}$ over a fixed feasible set $\\cal F$ defined by the set of constraints in \\refs{eq:mpriLP}, i.e., $\\gamma$ only affects the objective function. Under Assumption~\\ref{as:pos}(ii), $\\theta$ and $\\mb{\\psi}$ are non-negative and bounded, which means $\\cal F$ is bounded and Problem \\refs{eq:mpriLP} can be written as follows:\n$$\nv_{\\max}(y,{\\cal S}) = \\max_{\\gamma\\geq 0}\\left(\\max_{s=1,\\ldots,S}\\left\\{(p-c)\\gamma\\cdot \\theta^s - p\\,\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]\\psi_k^s\\right\\}\\right),\n$$\nwhere $\\left\\{(\\theta^s,\\mb{\\psi}^s)\\right\\}_{s=1,\\ldots,S}$ is the set of extreme points of $\\cal F$. Equivalently, we have:\n$$\nv_{\\max}(y,{\\cal S}) = \\max_{s=1,\\ldots,S}\\left\\{\\max_{\\gamma\\geq 0}\\left((p-c)\\gamma\\cdot \\theta^s - p\\,\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]\\psi_k^s\\right)\\right\\}.\n$$\n\nFor an arbitrary solution $(\\theta^s,\\mb{\\psi}^s)$, $s=1,\\ldots,S$, it is easy to show that function $\\displaystyle f(\\gamma;\\theta^s,\\mb{\\psi}^s)=(p-c)\\gamma\\cdot \\theta^s - p\\,\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]\\psi_k^s$ is a \\emph{concave} piece-wise linear function with intersection points as distinct values of the set $\\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}$. Since $\\theta^s>0$ as shown previously, $f(\\cdot;\\theta^s,\\mb{\\psi}^s)$ tends to $-\\infty$ when $\\gamma$ tends to $+\\infty$. It means there is at least an optimal solution for the problem $\\displaystyle\\max_{\\gamma\\geq 0}f(\\gamma;\\theta^s,\\mb{\\psi}^s)$ which belongs to the set $\\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}$. Thus we have:\n$$\nv_{\\max}(y,{\\cal S}) = \\max_{s=1,\\ldots,S}\\left\\{\\max_{l=1,\\ldots,K}\\left\\{(p-c)d_l({\\cal S})\\cdot \\theta^s - p\\,\\sum_{k=1}^K\\left[\\left(d_l({\\cal S})-d_k({\\cal S})\\right)^+\\right]\\psi_k^s\\right\\}\\right\\},\n$$\nwhich shows that there exists an optimal solution $(\\gamma^*,\\theta^*,\\mb{\\psi}^*)$ of Problem \\refs{eq:mpriLP} such that\n $$\\gamma^*\\in\\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}.$$\n\n\n\\end{pf}\n\nProposition \\ref{prop:linearequiv} shows us how to compute $v_{\\max}(y,{\\cal S})$ by solving at most $K$ linear programs. We can use this approach to compute $v_{\\max}(y^*_{wc}({\\cal N}),{\\cal S})$ as inputs of the linear program \\refs{eq:rnleastcore}, which is then solved to check the existence of core solutions of the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$. If the core is empty, we need to consider the general problem \\refs{eq:rleastcore} applying to the robust newsvendor game, whose optimal solutions can be considered as its least core solutions. Let us consider the following problem, which is similar to \\refs{eq:rnleastcore}, for an arbitrary $y \\in {\\cal Y}({\\cal N})$:\n\\be\n\\label{eq:ly}\n\\ba{rl}\n\\displaystyle \\sigma(y)=\\min_{\\bx,\\eps} & \\eps\\\\\n\\st & \\displaystyle\\sum_{i\\in{\\cal S}}x_i\\geq v_{\\max}(y,{\\cal S})-\\eps,\\quad{\\cal S}\\subsetneq{\\cal N},\\\\\n& \\displaystyle\\sum_{i\\in{\\cal N}}x_i=1.\n\\ea\n\\ee\n\\begin{comment}\n\\begin{equation}\n\\label{eq:ly}\n\\begin{array}{rl}\n\\displaystyle \\Upsilon(y) = \\min_{\\mb{x},\\epsilon} & \\epsilon\\\\\ns.t. & \\mb{x}({\\cal S}) + \\epsilon \\geq v_{max}(y,{\\cal S}), \\forall {\\cal S} \\subsetneq {\\cal N},\\\\\n& \\mb{e}^T \\mb{x} = 1.\n\\end{array}\n\\end{equation}\n\\end{comment}\nClearly, $\\displaystyle s({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))=\\min_{y \\in {\\cal Y}({\\cal N})} \\sigma(y)$, which is a reformulation of the least core problem \\refs{eq:rleastcore}. We will show that $\\sigma(y)$ is a convex function in the following proposition.\n\\begin{comment}\nWe can utilize the convexity property of $\\Upsilon(y)$ in the numerical computation of a $y^*$ with the smallest least core value. Specifically, we can search for the (one dimensional) $y$ that is a local optimal solution and conclude that this is also the global optimal solution.\n\\end{comment}\n\\begin{proposition}\n\\label{prop:ly_vmax_properties}\nThe following statements hold:\n\\begin{itemize}\n\\item[(a)] For each coalition ${\\cal S}\\subsetneq {\\cal N}$, $\\displaystyle v_{max}(y,{\\cal S})$ is a convex function of $y$ on ${\\cal Y}({\\cal N})$.\n\\item[(b)] $\\sigma(y)$ is a convex function of $y$ on ${\\cal Y}({\\cal N})$.\n\\end{itemize}\n\\end{proposition}\n\nIn order to prove the proposition, we need the following lemma.\n\n\\begin{lemma}\n\\label{lemma:vmax_properties}\nThe inverse function $\\displaystyle \\frac{1}{v_P(y,{\\cal N})}$ is a convex function of $y$ on ${\\cal Y}({\\cal N})$ for all $P\\in{\\cal P}(P_1,\\ldots,P_R)$.\n\\end{lemma}\n\n\\begin{pf}\nLet $\\mb{q}$ be the corresponding probability vector of a joint distribution $P\\in {\\cal P}(P_1,\\ldots,P_R)$. We have:\n$$\nv_P(y,{\\cal N})\\equiv v_{\\vect{q}}(y,{\\cal N})=(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k,\n$$\nwhich is a piecewise linear concave function of $y$ with at most $(K+1)$ linear pieces positioning within the intervals induced by the sorted sequence of $\\{d_1({\\cal N}),d_2({\\cal N})\\ldots d_K({\\cal N})\\}$. We can therefore rewrite $\\displaystyle v_P(y,{\\cal N})=\\min_{k=1,\\ldots,K+1}\\{a_ky+b_k\\}$ where $a_k,b_k$ are appropriate linear coefficients that can be derived from $p$, $c$, $\\mb{d}({\\cal N})$ and $\\mb{q}$. Since $v_P(y,{\\cal N})>0$ for all $y\\in{\\cal Y}({\\cal N})$, we have: $a_ky+b_k>0$ for all $k=1,\\ldots,K+1$, and $y\\in{\\cal Y}({\\cal N})$. We then have:\n$$\n\\frac{1}{v_P(y,{\\cal N})}=\\max_{k=1,\\ldots,K+1}\\frac{1}{a_ky+b_k},\n$$\nwhich is the maximum of convex inverse linear functions on ${\\cal Y}({\\cal N})$ and hence is also a convex function in ${\\cal Y}({\\cal N})$.\n\\begin{comment}\nLet $d^{(1)}({\\cal N})\\leq d^{(2)}({\\cal N})\\leq \\ldots \\leq d^{(K)}({\\cal N})$ be the corresponding sorted sequence of $\\{d_1({\\cal N}),d_2({\\cal N})\\ldots d_K({\\cal N})\\}$. We also denote $q^{(j)}$ as the corresponding probability of the realization of the joint demand at $d^{(j)}({\\cal N})$ for each $j = 1,\\ldots,K$.\n\nLet us denote $a_k = (\\rho-c - \\rho \\sum_{j=1}^k q^{(j)})$ and $b_k = \\rho \\sum_{j=1}^k q^{(j)} d^{(k)}({\\cal N})$ for $k= 0,\\ldots,K+1$. We have\n$$\n\\displaystyle v_P(y,{\\cal N}) =\n\\begin{cases}\n\\displaystyle a_0 y + b_0, &\\mbox{if } y \\leq d^{(1)}({\\cal N}) \\\\\n\\displaystyle a_k y+b_k, &\\mbox{if } d^{(k)}({\\cal N}) \\leq y \\leq d^{(k+1)}({\\cal N})\\\\\n\\displaystyle a_K y+b_K, &\\mbox{if } y \\geq d^{(K)}({\\cal N}).\n\\end{cases}\n$$\nLet us define $\\displaystyle g_k(y) = \\frac{v_P(\\gamma,{\\cal S})}{a_k y+b_k},~k=0,\\ldots,K+1,$ which are inverse linear functions and are convex. Then $\\displaystyle \\frac{v_P(\\gamma,{\\cal S})}{v_P(y,{\\cal N})}$ composes of these $K+1$ convex pieces. In addition, by the definition of $(a_k,b_k),~k=0,\\ldots,K$, we can show that\n$$ \\displaystyle v_P(y,{\\cal N}) = \\max_{j=0,\\ldots,K} g_j(y),$$\nwhich is the maximum of convex functions and hence is also a convex function.\n\\end{comment}\n\\end{pf}\n\nWe are now ready to prove the Proposition~\\ref{prop:ly_vmax_properties}.\n\n\\begin{pf}\n(a) If ${\\cal S}\\subseteq{\\cal N}_r$ for some $r$, $r=1,\\ldots,R$, we have: ${\\cal Y}({\\cal S})=\\{y^*({\\cal S})\\}$ and\n$$\n\\ba{rl}\nv_{\\max}(y,{\\cal S})&=\\displaystyle\\max_{P\\in{{\\cal P}(P_1,\\ldots,P_R)}} \\frac{\\bar{v}({\\cal S})}\n{(p-c)y-p\\,\\mathbb{E}_P\\left[\\left(y-\\td{d}({\\cal N})\\right)^+\\right]}\\\\\n& =\\displaystyle\\frac{\\bar{v}({\\cal S})}{\\min_{\\vect{q}\\in{\\cal Q}}\\left\\{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k\\right\\}}\\equiv \\displaystyle\\frac{\\bar{v}({\\cal S})}{\\min_{\\vect{q}\\in{\\cal Q}} v_{\\vect{q}}(y,{\\cal N})},\n\\ea\n$$\nwhere ${\\cal Q}={\\cal Q}(P_1,\\ldots,P_R)$ is the feasible set of the probability vector $\\mb{q}$ as described in \\refs{eq:qdef}. The second equality follows from the fact that $\\bar{v}({\\cal S})\\geq 0$ and $v_P(y,{\\cal N}) > 0$. Since ${\\cal Q}$ is a bounded polytope; there exists an optimal solution $\\mb{q}^*\\in{\\cal Q}^*$, where ${\\cal Q}^*$ is the set of extreme points of ${\\cal Q}$. Thus we have:\n$$\nv_{\\max}(y,{\\cal S})=\\max_{\\vect{q}\\in{\\cal Q}^*}\\frac{\\bar{v}({\\cal S})}{v_{\\vect{q}}(y,{\\cal N})}.\n$$\nSince $\\bar{v}({\\cal S})\\geq 0$ and since $\\displaystyle \\frac{1}{v_{\\vect{q}}(y,{\\cal N})}$ is convex for each $\\mb{q}$ according to Lemma \\ref{lemma:vmax_properties}, we have: $v_{\\max}(y,{\\cal S})$ is convex.\n\nNow consider an arbitrary ${\\cal S}\\subsetneq{\\cal N}$ with ${\\cal Y}({\\cal S})=\\mathbb{R}_+$. We have,\n$$\nv_{\\max}(y,{\\cal S})=\\max_{P\\in{{\\cal P}(P_1,\\ldots,P_R)}} \\frac{\\displaystyle\\max_{\\gamma \\in {\\cal Y}({\\cal S})}v_P(\\gamma,{\\cal S})}\n{v_P(y,{\\cal N})}.\n$$\nWe have: $v_P(y,{\\cal N})>0$ for all $P\\in{\\cal P}(P_1,\\ldots,P_R)$ and $y\\in{\\cal Y}({\\cal N})$. In addition, ${\\cal Y}({\\cal S})=\\mathbb{R}_+$, thus there always exists $\\gamma\\in{\\cal Y}({\\cal S})$ small enough such that $v_P(\\gamma,{\\cal S})\\geq 0$ for any $P$. We then have: $v_{\\max}(y,{\\cal S})\\geq 0$ for all $y\\in{\\cal Y}({\\cal N})$. We can rewrite the formulation of $v_{\\max}(y,{\\cal S})$ as follows:\n$$\nv_{\\max}(y,{\\cal S})=\\max_{\\gamma \\in \\R^+}~ \\max_{\\vect{q}\\in{{\\cal Q}}} \\frac{(p-c)\\gamma - p\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]q_k}\n{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k}.\n$$\n\nProposition~\\ref{prop:linearequiv} shows that we can restrict the domain of $\\gamma$ to the discrete set $\\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}$, that is,\n$$\nv_{\\max}(y,{\\cal S})=\\max_{\\gamma \\in \\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}}~ \\max_{\\vect{q}\\in{{\\cal Q}}} \\frac{(p-c)\\gamma - p\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]q_k}\n{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k}.\n$$\n\\begin{comment}\nThus, Problem~\\refs{eq:mpriLP} can be reformulated as\n\n\\begin{equation}\n\\label{eq:vmax2}\n\\begin{array}{rl}\nv_{\\max}(y,{\\cal S})=\\displaystyle \\max_{\\gamma \\in \\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}} ~~ \\max_{\\vect{q} \\in Q} & h(y,{\\cal S},\\gamma,\\mb{q}),\n\\end{array}\n\\end{equation}\nwhere $\\displaystyle h(y,{\\cal S},\\gamma,\\mb{q}) = \\displaystyle\\frac{(p-c)\\gamma - p\\,\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]q_k}{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k}$.\n\\end{comment}\nFor a fixed $\\gamma$, the inner problem is a linear fractional optimization problem over the bounded polyhedron $\\cal Q$ and hence there exists an optimal solution $\\mb{q}^*$ given that $v_P(y,{\\cal N})>0$ for all $y\\in{\\cal Y}({\\cal N})$ and $P\\in{\\cal P}(P_1,\\ldots,P_R)$. Let us consider the level sets ${\\cal L}_{\\alpha}$ of the linear fractional objective function, which are hyperplanes. For the optimal objective value $\\alpha^*$, we have: ${\\cal L}_{\\alpha^*}\\cap{\\cal Q}\\neq\\emptyset$. We claim that ${\\cal L}_{\\alpha^*}\\cap{\\cal Q}^*\\neq\\emptyset$, where ${\\cal Q}^*$ is the set of extreme points of $\\cal Q$. Since $\\alpha^*$ is the optimal objective value, $\\cal Q$ belongs to a half-space defined by ${\\cal L}_{\\alpha^*}$. Suppose, on contradiction, that ${\\cal L}_{\\alpha^*}\\cap{\\cal Q}^*=\\emptyset$. Due to convexity and since the entire ${\\cal Q}^*$ belongs to the same half-space defined by ${\\cal L}_{\\alpha^*}$, we have: ${\\cal L}_{\\alpha^*}\\cap{\\cal Q}=\\emptyset$ (contradiction).\n\nWith ${\\cal L}_{\\alpha^*}\\cap{\\cal Q}^*\\neq\\emptyset$, we can now compute $v_{\\max}(y,{\\cal S})$ as follows:\n$$\nv_{\\max}(y,{\\cal S})=\\max_{\\gamma \\in \\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}}~ \\max_{\\vect{q}\\in{{\\cal Q}^*}} \\frac{(p-c)\\gamma - p\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]q_k}\n{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k}.\n$$\nSince $v_{\\max}(y,{\\cal S})\\geq 0$, we can focus on the set ${\\cal H} \\in \\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\} \\times {\\cal Q}^*$ of $(\\gamma,\\mb{q})$ such that $(p-c)\\gamma - p\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]q_k\\geq 0$ when computing $v_{\\max}(y,{\\cal S})$, that is,\n$$\nv_{\\max}(y,{\\cal S})=\\max_{(\\gamma,\\vect{q}) \\in {\\cal H}} \\frac{(p-c)\\gamma - p\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]q_k}\n{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k}.\n$$\nApplying Lemma \\refs{lemma:vmax_properties}, clearly, $v_{\\max}(y,{\\cal S})$ is the maximum of convex functions, which means it is also a convex function.\n\\begin{comment}\nLet $\\mb{q}^{(1)},\\ldots,\\mb{q}^{(M)}$ be the corresponding extreme points. Since level sets of the linear fractional objective function are hyperplanes, the optimal level set that passes through $\\mb{q}^* \\in Q$ must also pass through at least an extreme point of $Q$. This is because otherwise all these extreme points must belongs to the same half-plane produced by the optimal level set hyperplane, but not on that hyperplane, which means the entire polyhedron $Q$ does not intersect with the hyperplane and this contradicts with $\\mb{q}^* \\in Q$.\n\nThus, the optimality of Problem~\\refs{eq:vmax2} can be attained at one of the extreme points $\\mb{q}^{(1)},\\ldots,\\mb{q}^{(M)}$. Problem~\\refs{eq:vmax2} can then be reformulated as\n\\begin{equation}\n\\label{eq:vmax3}\n\\begin{array}{rl}\nv_{\\max}(y,{\\cal S})=\\displaystyle \\max_{\\gamma \\in \\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}} ~~ \\max_{\\vect{q} \\in \\{\\mb{q}^{(1)},\\ldots,\\mb{q}^{(M)}\\}} & h(y,{\\cal S},\\gamma,\\mb{q}),\n\\end{array}\n\\end{equation}\nwhich is the maximum of a number of functions $h(y,{\\cal S},d_l({\\cal S}),\\mb{q}^{(m)}),~l=1,\\ldots,K,~m=1,\\ldots,M$. Since each of these function is convex over $y$ by Lemma~\\ref{lemma:vmax_properties}, we have $v_{\\max}(y,{\\cal S})$ is also a convex function of $y$.\n\\end{comment}\n\n(b) To prove the convexity of $\\sigma(y)$, we show that, for any $\\{y_1, y_2, y_3\\} \\in {\\cal Y}({\\cal N})$ such that there exists $\\alpha \\in [0,1]$ with $y_2 = \\alpha y_1+ (1-\\alpha) y_3$, then $\\sigma(y_2) \\leq \\alpha \\sigma(y_1) + (1-\\alpha) \\sigma(y_3)$.\n\nLet $(\\mb{x}_1,\\epsilon_1)$ and $(\\mb{x}_3,\\epsilon_3)$ be the optimal solutions of ~\\refs{eq:ly} when $y = y_1$ and $ y=y_3$, respectively. Let us define $(\\mb{x}_2,\\epsilon_2) = \\alpha (\\mb{x}_1,\\epsilon_1)+(1-\\alpha)(\\mb{x}_3,\\epsilon_3)$. It is easy to verify that $\\mb{e}^T \\mb{x}_2 = 1$. In addition, for all $ {\\cal S} \\subsetneq {\\cal N}$, we have\n\\begin{eqnarray}\n\\mb{x}_2({\\cal S}) + \\epsilon_2 &=& \\alpha (\\mb{x}_1({\\cal S})+\\epsilon_1)+(1-\\alpha)(\\mb{x}_3({\\cal S})+\\epsilon_3) \\label{eq:ly1}\\\\\n &\\geq& \\alpha v_{max}(y_1,{\\cal S})+(1-\\alpha) v_{max}(y_3,{\\cal S}) \\label{eq:ly2}\\\\\n &\\geq& v_{max}( \\alpha y_1+ (1-\\alpha) y_3,{\\cal S}) \\label{eq:ly3}\\\\\n &=& v_{max}( y_2,{\\cal S}) \\label{eq:ly4},\n\\end{eqnarray}\nwhere \\refs{eq:ly1} comes directly from the construction of $(\\mb{x}_2,\\epsilon_2)$; \\refs{eq:ly2} comes from the feasibility of $(\\mb{x}_1,\\epsilon_1)$ and $(\\mb{x}_3,\\epsilon_3)$; \\refs{eq:ly3} comes from the convexity of $v_{max}(y,{\\cal S})$ as shown in part (a). Finally, \\refs{eq:ly4} comes directly from the definition of $y_2$.\n\nThis shows that $(\\mb{x}_2,\\epsilon_2)$ is a feasible solution of ~\\refs{eq:ly} when $y = y_2$. Therefore,\n$$\\sigma(y_2) \\leq \\epsilon_2= \\alpha \\epsilon_1 + (1-\\alpha) \\epsilon_3= \\alpha \\sigma(y_1) + (1-\\alpha) \\sigma(y_3),$$\ni.e., $\\sigma(y)$ is a convex function.\n\\end{pf}\n\n\n\n\nProposition \\ref{prop:ly_vmax_properties} shows that the least core problem \\refs{eq:rleastcore} for our robust newsvendor game is a convex optimization problem in terms of $y$ and we could apply simple one-dimensional search algorithms to find the optimal solution. The next section provides some numerical results on the properties and computation of the core (and least core) solutions of robust newsvendor games.\n\\subsection{Numerical Results}\nWe consider the following experimental setting. We are given a set of retailers ${\\cal N}$ and a partition ${\\cal N}_1,\\ldots,{\\cal N}_R$. In addition, for each $r= 1,\\ldots, R$, we are given the discrete historical joint demand distribution $P_r$ for the subset of retailers ${\\cal N}_r$ but not the joint demand distribution of all retailers. Discussions in Section~\\ref{robust_core_computation} allow us to compute a robust core (or least core) solution by solving \\refs{eq:rnleastcore} (and \\refs{eq:rleastcore} if necessary) under the framework of robust newsvendor games, which consists of the allocation scheme $\\bz_{rob}$ and an order quantity $y_{rob}$ for the grand coalition.\n\nIn order to evaluate the performance of the robust solution $(y_{rob},\\bz_{rob})$, we are going to compare it with the solution derived from the deterministic newsvendor game under the assumption that all multivariate marginal distributions $P_r$, $r=1,\\ldots,R$, are independent of each other, that is, $\\displaystyle \\mathbb{P}(\\tilde{\\mb{d}} = (\\mb{d}_1^{l_1},\\ldots,\\mb{d}_r^{l_R})) = \\prod_{r=1}^R \\mathbb{P}(\\tilde{\\mb{d}}_r = \\mb{d}_r^{l_r})=\\prod_{r=1}^R p_r^{l_r}$. This could be considered as a common assumption on the joint distribution given its marginal distributions. Clearly, the resulting joint distribution $P_I$ belongs to $\\mathcal{P}(P_1,\\ldots,P_R)$. Given this distribution $P_I$, we can compute the allocation scheme $\\bz_{det}=\\bx\/\\bar{v}({\\cal N})$, where $\\bx$ is a core solution of the deterministic newsvendor game with respect to $P_I$. In addition, the optimal order quantity $y_{det}$ for the grand coalition in this deterministic newsvendor game is used to form the solution $(y_{det},\\bz_{det})$, which will be compared with the robust solution $(y_{rob},\\bz_{rob})$.\n\nWe shall compare the performance of these two solutions with respect to joint distributions which belong to $\\mathcal{P}(P_1,\\ldots,P_R)$. Given a distribution $P\\in \\mathcal{P}(P_1,\\ldots,P_R)$, we compute the maximum normalized dissatisfaction or worst normalized excess value for each solution. For $(y_{rob},\\bz_{rob})$, the excess value is computed as\n\\be\n\\label{eq:excess}\n\\eps_{rob}^P=\\max_{{\\cal S}\\subsetneq{\\cal N}}\\left\\{\\left(\\frac{\\displaystyle\\max_{\\gamma\\in{\\cal Y}({\\cal S})}v_P(\\gamma,{\\cal S})}{v_P(y_{rob},{\\cal N})}-z_{rob}({\\cal S})\\right)^+\\right\\}.\n\\ee\nThe excess value $\\eps_{det}^P$ can be defined in the same fashion for $(y_{det},\\bz_{det})$. We follow the stress test approach proposed by Dupa\\v cov\\' a \\cite{dupacova06} with the contaminated distributions $P_{\\lambda} = \\lambda P_I + (1-\\lambda)P_{ext}$ for $\\lambda\\in[0,1]$, where $P_{ext}$ are extremal distributions, i.e., those distributions which are likely to be the ones with which $v_{\\max}(y,{\\cal S})$ are computed. Similar approach has been discussed in Bertsimas et al. \\cite{bertsimas10} to test the quality of some stochastic optimization solutions.\n\\begin{comment}\nThe joint demand distributions can be represented by a probability vector $\\mb{q}$ which satisfies the set of linear constraints included in Model~\\ref{eq:vmax}. We denote the corresponding polytop as $Q$. Here, we notice that the probability vector $\\mb{q}_I$ that corresponds to $P_I$ lies in the interior of $Q$. Obviously if we choose $\\mb{q}=\\mb{q}_I$, then we would expect the deterministic strategy to work well as it has used the truth joint distribution. To provide the overall assessment of ROBUST and INDEPT, we will randomize $q$ within $Q$. For each random cost vector $\\mb{c}$ of the same size with $\\mb{q}$, if we solve the problem $\\{ \\max \\mb{c}^T \\mb{q} ~:~ \\mb{q} \\in Q\\}$ using a simplex method, we would obtain an extreme point $\\mb{q}_e \\in Q$. We also vary $\\alpha \\in [0,1]$ to produce a new probability vector $\\mb{q} = \\alpha \\mb{q}_e + (1-\\alpha)\\mb{q}_I$. Here, $\\alpha = 0$ means the chosen distribution is $P_I$ while $\\alpha = 1$ means the joint distribution is an extreme point of $Q$. By varying $\\alpha$ and by randomizing $\\mb{c}$, we can compare the performance between ROBUST and INDEPT as the joint distribution changes.\n\\end{comment}\n\nWe now consider a numerical example with $n=10$ and $R=2$, with the sizes of subsets, $\\card{{\\cal N}_1} = 4$ and $\\card{{\\cal N}_2} = 6$, respectively. We construct multivariate marginal distributions $P_1$ and $P_2$ from a randomly generated discrete joint distribution with the support set of each individual retailer's demand set to $[1,10]$. Other parameters include $p=1.5$ and $c= 1$. All the numerical results are tested on a PC with 2.67 gigahertz CPU, 12 gigabyte RAM, and a 64-bit Windows 7 operating system. We use MATLAB 8.0 for coding and IBM CPLEX Studio Academic version 12.5 for solving LPs problems under default settings. In order to compute a robust core (or least core) solution of this game, we would need to compute $v_{\\max}(y,{\\cal S})$ for each of $2^n=1024$ coalitions. This is accomplished by solving a number of linear programs as presented in Proposition~\\ref{prop:linearequiv}. On average, the total time it took to compute a single value $v_{\\max}(y,{\\cal S})$ is approximately $12$ seconds under this setting. Problem \\refs{eq:rnleastcore} can then be directly solved whereas \\refs{eq:rleastcore} is solved with a simple one-dimensional search algorithm whose main subroutine depends on the solution of \\refs{eq:rnleastcore} for different values of $y$.\n\nIn this numerical example, for a given value of $\\lambda$, we simply generate $100$ random extremal distributions $P_{ext}$ by solving the linear program $\\{ \\max \\mb{c}^T \\mb{q} ~:~ \\mb{q} \\in {\\cal Q}\\}$ with random cost vectors $\\mb{c}$. We also include extremal distributions produced while calculating $v_{max}$. The excess values $\\eps_{rob}^P$ and $\\eps_{det}^P$ are computed for all contaminated distributions $P_\\lambda$. Figure \\ref{fig:robust_vs_deterministic} provides comparisons on three statistics of $\\eps_{rob}^P$ and $\\eps_{det}^P$ for each fixed value of $\\lambda$: the maximum, the minimum, and the average.\n\\begin{figure}[htp]\n \\begin{center}\n\n\\includegraphics[width=0.7\\textwidth]{contaminated}\n\\end{center}\n\\caption{Comparison between $\\eps_{rob}^P$ and $\\eps_{det}^P$ for contaminated distributions with different $\\lambda$.}\n\\label{fig:robust_vs_deterministic}\n\\end{figure}\nThe maximum values of both $\\eps_{rob}^P$ and $\\eps_{det}^P$ increase when $\\lambda$ increases. For $\\lambda>0.5$, the robust solution $(y_{rob},\\bz_{rob})$ yields smaller excess values in the worst case as compared to those of $(y_{det},\\bz_{det})$ for contaminated distributions. It shows that the robust solution hedges against the worst case as expected even though on average, the solution $(y_{det},\\bz_{det})$ is slightly better in terms of worst excess values. It is worth noting that in the best case, both solutions are core solutions with no dissatisfaction even for the case of $\\lambda=1$.\n\n\n\nWe run the experiment again for $M=20$ different instances. Figure \\ref{fig:robust_vs_deterministic1} shows the statistics of $\\eps_{rob}^P$ and $\\eps_{det}^P$ when $\\lambda=1$ for all of these instances. The results again show that the robust solution $(y_{rob},\\bz_{rob})$ consistently outperforms $(y_{det},\\bz_{det})$ for all these instances in the worst case and is slightly worse on average.\n\\begin{figure}[htp]\n \\begin{center}\n\n\\includegraphics[width=0.7\\textwidth]{instances}\n\\end{center}\n\\caption{Comparison between $\\eps_{rob}^P$ and $\\eps_{det}^P$ for different instances with $\\lambda=1$.}\n\\label{fig:robust_vs_deterministic1}\n\\end{figure}\nFinally, we run the experiment again for four different settings with respect to sizes of the two subsets, $(1,9)$, $(2,8)$, $(3,7)$, and $(5,5)$, in addition to the original setting of $(4,6)$, with $M=20$ instances for each setting. Figure \\ref{fig:robust_vs_deterministic2} shows the box plots for the maximum values of $\\eps_{rob}^P$ and $\\eps_{det}^P$ when $\\lambda=1$. The results again show that the robust solution $(y_{rob},\\bz_{rob})$ outperforms $(y_{det},\\bz_{det})$ in the worst case.\n\\begin{figure}[htp]\n \\begin{center}\n\n\\includegraphics[width=0.7\\textwidth]{structures}\n\\end{center}\n\\caption{Comparison between maximum $\\eps_{rob}^P$ and $\\eps_{det}^P$ for different subset structures with $\\lambda=1$.}\n\\label{fig:robust_vs_deterministic2}\n\\end{figure}\n\\section{Conclusion}\nIn this paper, we develop a framework for newsvendor games with ambiguity in demand distributions, which we call robust cooperative games. We discuss solution concepts of robust cooperative games and study them in the context of newsvendor games with ambiguity in demand distributions when only marginal distributions are known. Some numerical results are provided, which show the robust core solutions hedge against the worst cases as expected. It is possible to develop other frameworks for cooperative games with uncertain characteristic functions by using different payoff distribution schemes and preference relations, which could be applied to other applications.\n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}