{"text":"\\section{Introduction}\n\nWehrl proposed \\cite{W} \na hybrid between quantum mechanical and classical entropy\nthat enjoys monotonicity, strong subadditivity and positivity -- physically\ndesirable properties, some of which both kinds of entropy lack \\cite{B,E}. \nThis new entropy is the ordinary Shannon entropy of\nthe probability density provided by the lower\nsymbol of the density matrix. \n\nFor a quantum mechanical system with density matrix $\\rho$, Hilbert space\n$\\H$,\nand a family of normalized\ncoherent states $|z\\rangle$,\nparametrized symbolically by\n$z$ and satisfying\n$\\int dz \\, |z\\rangle\\langle z| = {\\bf 1}$ (resolution of identity),\nthe Wehrl entropy is\n\\begin{equation}\nS_W(\\rho)  = -\\int dz \\, \\langle z|\\rho|z\\rangle \\ln \\langle z|\\rho|z\\rangle . \\label{W0}\n\\end{equation}\nLike quantum mechanical entropy,\n$S_Q = -\\tr \\rho\\ln\\rho $,\nWehrl entropy is always non-negative, in fact\n$S_W > S_Q \\geq 0$.\nIn view of this inequality it is interesting to ask for the minimum\nof $S_W$ and the corresponding minimizing density matrix.\nIt follows from concavity of $-x \\ln x$ that a minimizing\ndensity matrix must be a pure state, \\emph{i.e.}, $\\rho = |\\psi\\rangle\\langle\\psi|$\nfor a normalized vector $|\\psi\\rangle \\in \\H$ \\cite{A}. (Note that\n$S_W(|\\psi\\rangle\\langle\\psi|)$ depends on $|\\psi\\rangle$ and is non-zero,\nunlike the quantum entropy which is of course zero for pure states.)\n\nFor Glauber coherent states  Wehrl conjectured \\cite{W} and Lieb proved\n\\cite{A}\nthat the minimizing state $|\\psi\\rangle$ is again a coherent\nstate. It turns out that all Glauber coherent states\nhave Wehrl entropy one, so Wehrl's conjecture can be written as follows:\n\\begin{thm}[\\rm Lieb] \\label{thmWL}\nThe minimum of $S_W(\\rho)$ for states in $\\H = L^2(\\Re)$\nis one,\n\\begin{equation}\nS_W(|\\psi\\rangle\\langle\\psi|)\n= -\\int dz \\, |\\langle\\psi|z\\rangle|^2 \\ln |\\langle\\psi|z\\rangle|^2 \\geq 1 , \\label{W1}\n\\end{equation}\nwith equality if and only if $|\\psi\\rangle$ is a coherent state.\n\\end{thm}\nTo prove this, Lieb used a clever combination of the sharp\nHausdorff-Young inequality \\cite{Y1,Y3,LL}\nand the sharp Young inequality \\cite{Y2,Y1,Y3,LL}\nto show that\n\\begin{equation}\ns \\int dz \\, |\\langle z|\\psi\\rangle|^{2 s} \\leq 1, \\quad s \\geq 1, \\label{W2}\n\\end{equation}\nagain with equality if and only if $|\\psi\\rangle$ is a coherent state.\nWehrl's conjecture follows from this in the limit $s \\rightarrow 1$\nessentially because\n(\\ref{W1}) is the derivative of (\\ref{W2}) with respect to $s$ at $s=1$.\nAll this easily generalizes to $L^2(\\Re^n)$ \\cite{A,Y4}.\n\nThe lower bound on the Wehrl entropy is related to\nHeisenberg's uncertainty principle \\cite{AH,G} and it has been speculated that\n$S_W$ can be used to measure uncertainty due to both quantum and thermal\nfluctuations \\cite{G}.\n\nIt is very surprising that `heavy artillery' like the sharp constants\nin the mentioned inequalities are needed in Lieb's proof. To elucidate\nthis situation, Lieb suggested \\cite{A} studying the analog of Wehrl's\nconjecture for Bloch coherent states $|\\Omega\\rangle$, where one should\nexpect significant \nsimplification since these are finite dimensional Hilbert spaces.\nHowever, no progress has been made, not even for a single spin, even though\nmany attempts have been made \\cite{B}. Attempts to proceed again \nalong the lines of Lieb's original proof have failed to provide a sharp\ninequality and the direct computation of the entropy and related integrals,\neven numerically, was unsuccesful \\cite{S}.\n\nThe key to the recent progress is a geometric representation of a\nstate of spin~$j$ as $2j$ points on a sphere.\nIn this representation the expression\n$|\\langle\\Omega|\\psi\\rangle|^2$ factorizes into a product\nof $2j$ functions $f_i$ on the sphere,  \nwhich measure the square chordal distance\nfrom the antipode of the point parametrized by\n$\\Omega$ to each of the $2j$ points on the sphere.\nLieb's conjecture,\nin a generalized form analogous to (\\ref{W2}),\nthen looks like the quotient of two H\\\"older inequalities\n\\begin{equation}\n\\frac{|\\!|f_1 \\cdots f_{2j}|\\!|_s}{|\\!|f_1 \\cdots f_{2j}|\\!|_1}\n\\leq \n\\frac{\\prod_{i=1}^{2j}|\\!|f_i|\\!|_{2js}}{\\prod_{i=1}^{2j} |\\!|f_i|\\!|_{2j}},\n\\label{holder}\n\\end{equation}\nwith the one with the higher power winning against the other one.\nWe shall give a group theoretic proof of this inequality\nfor the special case $s \\in \\N$ in theorem~\\ref{natural}.\n\nIn the geometric representation the \nWehrl entropy of spin states finds a direct physical\ninterpretation: It is the classical entropy of a single particle on a sphere\ninteracting via Coulomb potential with $2j$ fixed sources; $s$ plays the\nrole of inverse temperature.\n\nThe entropy integral (\\ref{W0}) can now be done because \n$|\\langle\\Omega|\\psi\\rangle|^2$ factorizes\nand one finds a formula for the Wehrl entropy of any state.\nWhen we evaluate the entropy explicitly for states of spin 1,\n3\/2, and 2 we find surprisingly simple expressions  solely in\nterms of the square chordal distances between the points on the\nsphere that define the given state. \n\nA different, more group theoretic approach seems to point \nto a connection between\nLieb's conjecture and the norm of certain spin $j s$ states with\n$1 \\leq s \\in \\Re$ \\cite{J}.\nSo far, however, this has only been useful for proving the analog\nof inequality (\\ref{W2}) for $s \\in \\N$.\n\nWe find that a proof of Lieb's conjecture for low spins can be reduced\nto some beautiful spherical geometry, \nbut the unreasonable difficulty of a complete proof\nis still a great puzzle; its resolution may very well lead to\ninteresting mathematics and perhaps physics.\n\n\\section{Bloch coherent spin states}\n\nGlauber coherent states \n$|z\\rangle = \\pi^{-\\frac{1}{4}} e^{-(x-q)^2\/2} e^{ipx}$,\nparametrized by $z = (q + i p)\/\\sqrt{2}$ and with measure\n$dz = dp dq\/2\\pi$,\nare usually introduced as\neigenvectors of the annihilation operator\n$a = (\\hat x + i \\hat p)\/\\sqrt{2}$, \n$a|z\\rangle = z |z\\rangle$,\nbut the same states can also be\nobtained by the action of the Heisenberg-Weyl group\n$H_4 = \\{a^\\dagger a, a^\\dagger, a, I\\}$\non the extremal state\n$|0\\rangle = \\pi^{-\\frac{1}{4}} e^{-x^2\/2}$.\nGlauber coherent states are thus elements of the coset\nspace of the Heisenberg-Weyl group\nmodulo the stability subgroup $U(1)\\otimes U(1)$ that leaves the extremal state\ninvariant. (See \\emph{e.g.}\\ \\cite{C1} and references therein.)\nThis construction easily generalizes\nto other groups, in particular to SU(2), where it gives\nthe Bloch coherent spin states \\cite{BC} that we are interested in:\nHere the Hilbert space can be any one of the finite dimensional spin-$j$\nrepresentations $[j] \\equiv \\C^{2j+1}$ of SU(2), \n$j = {1\\over 2}, 1, \\frac{3}{2}, \\ldots$,\nand\nthe extremal state for each $[j]$ is the\nhighest weight vector $|j,j\\rangle$. The stability subgroup is U(1)\nand the coherent states are thus \nelements of the sphere $S_2 = $SU(2)\/U(1);\nthey can be labeled by\n$\\Omega = (\\theta,\\phi)$ and are\nobtained from $|j,j\\rangle$ by rotation:\n\\begin{equation}\n|\\Omega\\rangle_j = \\R_j(\\Omega) |j,j\\rangle. \\label{Om}\n\\end{equation}\nFor spin $j = \\frac{1}{2}$ we find\n\\begin{equation}\n|\\omega\\rangle = p^{\\frac{1}{2}} e^{-i{\\phi\\over 2}} |\\U\\rangle \n           + (1-p)^{\\frac{1}{2}} e^{i{\\phi\\over 2}} |\\D\\rangle ,  \\label{coh}\n\\end{equation}\nwith $p \\equiv \\cos^2\\frac{\\theta}{2}$.\n(Here and in the following $|\\omega\\rangle$ is short for the spin-$\\frac{1}{2}$ coherent\nstate $|\\Omega\\rangle_\\frac{1}{2}$; $\\omega = \\Omega = (\\theta,\\phi)$. \n$|\\U\\rangle \\equiv |\\frac{1}{2},\\frac{1}{2}\\rangle$ and\n$|\\D\\rangle \\equiv |\\frac{1}{2},-\\frac{1}{2}\\rangle$.)\nAn important observation for what follows is that\nthe product of two coherent states for the same $\\Omega$\nis again a coherent state:\n\\begin{eqnarray} \n|\\Omega\\rangle_j \\otimes |\\Omega\\rangle_{j'} \n\t& = & (\\R_j \\otimes \\R_{j'})\\, (|j,j\\rangle \\otimes |j',j'\\rangle)  \\nonumber \\\\\n\t& = & \\R_{j+j'}\\, |j+j',j+j'\\rangle \t           \n\t\\; = \\; |\\Omega\\rangle_{j+j'} .\t     \n\\end{eqnarray}\nCoherent states are in fact the only states for which\nthe product of a spin-$j$ state with a spin-$j'$ state is\na spin-$(j+j')$ state and not a more general element of\n$[j+j'] \\oplus \\ldots \\oplus [\\,|j - j'|\\,]$.\nFrom this key property\nan explicit representation for Bloch coherent states of higher spin \ncan be easily derived:\n\\begin{eqnarray}\n|\\Omega\\rangle_j & = & \\left(|\\omega\\rangle\\right)^{\\otimes 2j} \n             \\; = \\;  \\left(p^{\\frac{1}{2}} e^{-i{\\phi\\over 2}} |\\U\\rangle \n               + (1-p)^{\\frac{1}{2}} e^{i{\\phi\\over 2}} |\\D\\rangle\\right)^{\\otimes 2j} \\nonumber \\\\\n\t     & = & \\sum_{m=-j}^j {2 j \\choose j + m}^{\\frac{1}{2}}\n\t     p^{j+m\\over 2}  (1-p)^{j-m\\over 2} \n\t     e^{-i m {\\phi\\over 2}} |j,m\\rangle.  \\label{Coh}\n\\end{eqnarray}\n(The same expression can also be obtained directly from (\\ref{Om}), see\n\\emph{e.g.}\\ \\cite[chapter 4]{C2}.)\nThe coherent states as given are normalized $\\langle\\Omega|\\Omega\\rangle_j =1$\nand satisfy\n\\begin{equation}\n(2j+1) \\int\\frac{d\\Om}{4\\pi} \\, |\\Omega\\rangle_j\\langle\\Omega|_j = P_j , \\qquad \\mbox{(resolution of\nidentity)} \\label{project}\n\\end{equation}\nwhere $P_j = \\sum |j,m\\rangle\\langle j,m|$ is the projector onto $[j]$.\nIt is not hard to compute the Wehrl entropy for a coherent state\n$|\\Omega'\\rangle$: Since the integral over the sphere is invariant under rotations\nit is enough to consider the coherent state $|j,j\\rangle$; then use\n$|\\langle j,j|\\Omega\\rangle|^2 = |\\langle\\U\\!\\!|\\omega\\rangle|^{2\\cdot 2j} = p^{2j}$\nand $d\\Omega\/4\\pi = -dp\\,d\\phi\/2\\pi$, where \n$p =\\cos^2 \\frac{\\theta}{2}$ as above, to obtain\n\\begin{eqnarray}\nS_W(|\\Omega'\\rangle\\langle\\Omega'|) \n& = & -(2j+1) \\int\\frac{d\\Om}{4\\pi} \\, |\\langle\\Omega|\\Omega'\\rangle|^2 \\ln |\\langle\\Omega|\\Omega'\\rangle|^2 \\nonumber \\\\\n& = & -(2j+1) \\int_0^1 dp \\, p^{2j} \\, 2j \\ln p \\, = \\, \\frac{2j}{2j+1}.\n\\end{eqnarray}\nSimilarly, for later use,\n\\begin{equation}\n(2js+1) \\int \\frac{d\\Om}{4\\pi} |\\langle\\Omega'|\\Omega\\rangle|^{2s} = (2js+1) \\int_0^1 dp \\, p^{2js} = 1.\n\\end{equation}\nAs before the density matrix that minimizes $S_W$\nmust be a pure state $|\\psi\\rangle\\langle\\psi|$. \nThe analog of theorem~\\ref{thmWL} for spin states is:\n\\begin{conj}[\\rm Lieb] \\label{conject1}\nThe minimum of $S_W$ for states in $\\H = \\C^{2j+1}$\nis $2j\/(2j+1)$,\n\\begin{equation}\nS_W(|\\psi\\rangle\\langle\\psi|) \n= -(2j+1) \\int\\frac{d\\Om}{4\\pi} \\, |\\langle\\Omega|\\psi\\rangle|^2 \\ln |\\langle\\Omega|\\psi\\rangle|^2\n\\geq \\frac{2j}{2j+1},\n\\end{equation}\nwith equality if and only if $|\\psi\\rangle$ is a coherent state.\n\\end{conj}\n\n\\noindent\n\\emph{Remark:} For spin 1\/2 this is an identity because\nall spin~1\/2 states are coherent states. The first non-trivial case\nis spin $j=1$.\n\n\\section{Proof of Lieb's conjecture for low spin}\n\nIn this section we shall geometrize the description of spin states, use this\nto solve the entropy integrals\nfor all spin and prove Lieb's conjecture for low spin by actual computation\nof the entropy.\n\n\\begin{lemma}\nStates of spin $j$ are in one to one correspondence to $2j$ points\non the sphere $S_2$: With $2j$ points,\nparametrized by $\\omega_k = (\\theta_k, \\phi_k)$, $k = 1, \\ldots , 2j$,\nwe can associate a state\n\\begin{equation}\n|\\psi\\rangle = c^\\frac{1}{2} P_j (|\\omega_1\\rangle \\otimes \\ldots \\otimes |\\omega_{2j}\\rangle) \\; \\in \\; [j] ,\n\\label{psiprod}\n\\end{equation}\nand every state $|\\psi\\rangle \\in [j]$ is of that form. (The spin-$\\frac{1}{2}$ states\n$|\\omega_k\\rangle$ are given by (\\ref{coh}),\n$c^\\frac{1}{2} \\neq 0$ fixes the\nnormalization of $|\\psi\\rangle$, and $P_j$ is the projector onto spin~$j$.)\n\\label{sphere}\n\\end{lemma}\n\n\\noindent \\emph{Remark:} \nSome or all of the points may coincide.\nCoherent states are exactly\nthose states for which all points on the sphere coincide. \n$c^\\frac{1}{2} \\in \\C$ may contain an (unimportant) phase\nthat we can safely ignore in the following.\nThis representation is unique up to permutation\nof the $|\\omega_k\\rangle$. The $\\omega_k$ may be found by looking at\n$\\langle\\Omega|\\psi\\rangle$ as a function of $\\Omega = (\\theta,\\psi)$:\nthey are the antipodal points to the zeroes of this function. \n\n\\noindent {\\sc Proof}:\nRewrite (\\ref{Coh}) in complex coordinates for $\\theta\\neq 0$\n\\begin{equation}\nz = \\left(\\frac{p}{1-p}\\right)^\\frac{1}{2} e^{i\\phi} = \\cot\\frac{\\theta}{2} e^{i\\phi}\n\\end{equation}\n(stereographic projection)\nand contract it with $|\\psi\\rangle$ to find\n\\begin{equation}\n\\langle\\Omega|\\psi\\rangle = \\frac{e^{-ij\\phi}}{(1+z\\bar z)^j} \n\\sum_{m=-j}^{j_{\\mbox{\\tiny max}}} \n\t   {2 j \\choose j + m}^{\\frac{1}{2}} z^{j+m} \\psi_m , \\label{poly}\n\\end{equation}\nwhere $j_{\\mbox{\\scriptsize max}}$ is the largest value of $m$ for which\n$\\psi_m$ in the expansion\n$|\\psi\\rangle = \\sum\\psi_m |m\\rangle$\nis nonzero. This is a polynomial of degree\n$j+j_{\\mbox{\\scriptsize max}}$ in $z \\in \\C$ and can thus be factorized:\n\\begin{equation}\n\\langle\\Omega|\\psi\\rangle = \n\\frac{ e^{-ij\\phi} \\psi_{j_{\\mbox{\\tiny max}}} }{ (1+z\\bar z)^j } \n\\prod_{k=1}^{j+j_{\\mbox{\\tiny max}}} (z - z_k) . \\label{fact}\n\\end{equation}\nConsider now the spin~$\\frac{1}{2}$ states\n$|\\omega_k\\rangle = (1+z_k \\bar z_k)^{-\\frac{1}{2}}(|\\U\\rangle - z_k |\\D\\rangle)$\nfor $1 \\leq k \\leq j+j_{\\mbox{\\tiny max}}$ and\n$|\\omega_m\\rangle = |\\D\\rangle$ for $j+j_{\\mbox{\\tiny max}} < m \\leq 2j$. According\nto (\\ref{poly}):\n\\begin{equation}\n\\langle\\omega|\\omega_k\\rangle = \\frac{e^{-\\frac{i\\phi}{2}}}{(1+z\\bar z)^\\frac{1}{2} \n(1+ z_k\\bar z_k)^\\frac{1}{2}}(z - z_k) , \n\\qquad \\langle\\omega|\\omega_m\\rangle = \n\\frac{e^{-\\frac{i\\phi}{2}}}{(1+z\\bar z)^\\frac{1}{2}},\n\\end{equation}\nso by comparison with (\\ref{fact}) and with an appropriate constant\n$c$\n\\begin{equation}\n\\langle\\Omega|\\psi\\rangle \n= c^\\frac{1}{2} \\langle\\omega|\\omega_1\\rangle \\cdots \\langle\\omega|\\omega_{2j}\\rangle\n= c^\\frac{1}{2} \\langle\\Omega|\\omega_1\\otimes\\ldots\\otimes\\omega_{2j}\\rangle.\n\\label{fac}\n\\end{equation}\nBy inspection we see that this expression is still valid when\n$\\theta = 0$ and with the help of (\\ref{project}) we can complete the\nproof the lemma.$\\blob$\\\\[1ex]\nWe see that the geometric representation of spin states leads to a\nfactorization of $\\langle\\Omega|\\psi\\rangle|^2$. In this representation we can\nnow do the entropy integrals, essentially because the logarithm becomes a\nsimple sum.\n\n\\begin{thm}   \\label{theorem}\nConsider any state $|\\psi\\rangle$ of spin $j$. According to\nlemma~\\ref{sphere}, it can be written as\n$|\\psi\\rangle = c^\\frac{1}{2} P_j (|\\omega_1\\rangle \\otimes \\ldots \\otimes |\\omega_{2j}\\rangle).$\nLet $\\R_i$ be the rotation that turns $\\omega_i$ to the `north pole',\n$\\R_i|\\omega_i\\rangle = |\\U\\rangle$, let $|\\psi^{(i)}\\rangle = \\R_i|\\psi\\rangle$,\nand let $\\psi_m^{(i)}$ be the coefficient of $|j,m\\rangle$ in the expansion\nof $|\\psi^{(i)}\\rangle$,\nthen the Wehrl entropy is:\n\\begin{equation}\nS_W(|\\psi\\rangle\\langle\\psi|) =\n\\sum_{i=1}^{2j} \\sum_{m=-j}^{j} \\left(\\sum_{n=0}^{j-m}\n\\frac{1}{2j+1-n}\\right) |\\psi_m^{(i)}|^2  -  \\ln c . \\label{formula}\n\\end{equation}\n\\end{thm}\n\n\\noindent \\emph{Remark:}\nThis formula reduces the computation of the Wehrl entropy of any\nspin state to its factorization in the sense of lemma~\\ref{sphere},\nwhich in general requires the solution of a\n$2j$'th order algebraic equation. This may explain why previous \nattempts to do the entropy integrals have failed.\nThe $n=0$ terms in the expression for the entropy\nsum up to $2j\/(2j+1)$, the entropy of a coherent state, \nand Lieb's conjecture can be thus be written\n\\begin{equation}\n\\ln c \\leq \\sum_{i=1}^{2j} \\sum_{m=-j+1}^{j-1} \\left(\\sum_{n=1}^{j-m}\n\\frac{1}{2j+1-n}\\right) |\\psi_m^{(i)}|^2.\n\\end{equation}\nNote that $\\psi^{(i)}_{-j} = 0$ by construction of\n$|\\psi^{(i)}\\rangle$: $\\psi^{(i)}_{-j}$ contains a factor\n$\\langle\\downarrow|\\U\\rangle$.\\\\\nA similar calculation gives\n\\begin{equation}\n\\ln c = 2j + \\int\\frac{d\\Om}{4\\pi} \\, \\ln|\\langle\\Omega|\\psi\\rangle|^2 .\n\\end{equation}\n\n\\noindent\n{\\sc Proof}:\nUsing lemma~\\ref{sphere}, (\\ref{project}),\nthe rotational invariance of the\nmeasure and the inverse Fourier transform in $\\phi$ we find\n\\begin{eqnarray}\n\\lefteqn{S_W(|\\psi\\rangle\\langle\\psi|) \\; = \\;\n{ -(2j+1)}\n\\int\\frac{d\\Om}{4\\pi} |\\langle\\Omega|\\psi\\rangle|^2 \\sum_{i=1}^{2j} \\ln |\\langle\\omega|\\omega_i\\rangle|^2\n- \\ln c} \\nonumber \\\\\n&& = { -(2j+1)} \\sum_{i=1}^{2j} \\int\\frac{d\\Om}{4\\pi}\n|\\langle\\Omega|\\psi^{(i)}\\rangle|^2  \\ln |\\langle\\omega|\\U\\rangle|^2  - \\ln c \\nonumber \\\\\n&& = {\\scriptstyle -(2j+1)} \\sum_{i=1}^{2j}\\sum_{m=-j}^j |\\psi^{(i)}_m|^2\n{\\scriptstyle {2j \\choose j + m}}\n\\int_0^1 dp \\, \n p^{j+m} (1 \\! - \\! p)^{j-m} \\ln p - \\ln c.\n\\end{eqnarray}\nIt is now easy to do the remaining $p$-integral by partial integration\nto proof the theorem.$\\blob$\\\\[1ex]\nLieb's conjecture for low spin can be proved with the help of\nformula (\\ref{formula}). For spin 1\/2 there is\nnothing to prove, since all states  of spin 1\/2 are coherent states.\nThe first nontrivial case is spin 1:\n\n\\begin{cor}[\\rm spin 1]\nConsider an arbitrary state of spin 1. Let\n$\\mu$ be the square of the\nchordal distance between the two points on the sphere of radius~$\\frac{1}{2}$\nthat represent this state. It's Wehrl entropy is given by\n\\begin{equation}\nS_W(\\mu) = \\frac{2}{3} + c\\cdot\\left(\\frac{\\mu}{2} + \\frac{1}{c} \\ln\n\\frac{1}{c}\\right) , \\label{entropy1}\n\\end{equation}\nwith\n\\begin{equation}\n\\frac{1}{c} = 1 - \\frac{\\mu}{2}.\n\\end{equation}\nLieb's conjecture holds for all states of spin 1:\n$S_W(\\mu) \\geq 2\/3 = 2j\/(2j+1)$ with equality for $\\mu = 0$, \\emph{i.e.}\\ for\ncoherent states.\n\\end{cor}\n\n\\noindent {\\sc Proof}:\nBecause of rotational invariance we can assume without loss of generality\nthat the first point is at the `north pole' of the sphere and that\nthe second point is parametrized as $\\omega_2 = (\\tilde\\theta, \\tilde\\phi = 0)$,\nso that\n$\\mu = \\sin^2\\frac{\\tilde\\theta}{2}$ . Up to normalization (and an irrelevant\nphase)\n\\begin{equation}\n|\\tilde\\psi\\rangle = P_{j=1}|\\U\\otimes\\tilde\\omega\\rangle\n\\end{equation}\nis the state of interest. But from (\\ref{coh})\n\\begin{equation}\n|\\U\\otimes\\tilde\\omega\\rangle = (1-\\mu)^\\frac{1}{2}|\\U\\;\\U\\rangle + \\mu^\\frac{1}{2}|\\U\\;\\D\\rangle.\n\\end{equation}\nProjecting onto spin 1 and inserting the normalization constant $c^\\frac{1}{2}$\nwe find\n\\begin{equation}\n|\\psi\\rangle = c^\\frac{1}{2}\\left((1-\\mu)^\\frac{1}{2} |1,1\\rangle + \\mu^\\frac{1}{2}\n\\frac{1}{\\sqrt{2}}|1,0\\rangle\\right). \\label{state}\n\\end{equation}\nThis gives (ignoring a possible phase) \n\\begin{equation}\n1 = \\langle\\psi|\\psi\\rangle = c\\left(1 - \\mu + \\frac{\\mu}{2}\\right) = c\\left(1 -\n\\frac{\\mu}{2}\\right)   \\label{cvalue}\n\\end{equation}\nand so $1\/c = 1 - \\mu\/2$. Now we need to compute the components\nof $|\\psi^{(1)}\\rangle$ and $|\\psi^{(2)}\\rangle$. Note that\n$|\\psi^{(1)}\\rangle = |\\psi\\rangle$ because $\\omega_1$ is already pointing to\nthe `north pole'. To obtain $|\\psi^{(2)}\\rangle$ we need to rotate point 2\nto the `north pole'. We can use the remaining rotational freedom\nto effectively exchange the two points, thereby recovering the original\nstate $|\\psi\\rangle$. The components of \nboth $|\\psi^{(1)}\\rangle$ and $|\\psi^{(2)}\\rangle$ can thus be read off (\\ref{state}):\n\\begin{equation}\n\\psi_1^{(1)} = \\psi_1^{(2)} = c^\\frac{1}{2}(1-\\mu)^\\frac{1}{2} ,\n\\qquad \\psi_0^{(1)} = \\psi_0^{(2)} = c^\\frac{1}{2} \\mu^\\frac{1}{2}\/\\sqrt{2}.\n\\end{equation}\nInserting now $c$, $|\\psi_1^{(1)}|^2 = |\\psi_1^{(2)}|^2 = c(1-\\mu)$,\nand $|\\psi_0^{(1)}|^2 = |\\psi_0^{(2)}|^2 = c \\mu\/2$ into (\\ref{formula})\ngives the stated entropy.\n\nTo prove Lieb's conjecture for states of spin~1 we use\n(\\ref{cvalue}) to show that\nthe second term in (\\ref{entropy1}) is always non-negative and zero only for\n$\\mu = 0$, \\emph{i.e.}\\ for a coherent state. This follows from\n\\begin{equation}\n\\frac{c \\mu}{2} - \\ln c \\geq \\frac{c \\mu}{2} + 1 - c = 0\n\\end{equation}\nwith equality for $c=1$ which is equivalent to $\\mu=0$ .$\\blob$\n\\vspace{2ex}\n\\begin{figure}\n\\begin{center}\n\\unitlength 1.00mm\n\\linethickness{0.4pt}\n\\begin{picture}(30.00,21.00)(10,5)\n\\put(10.00,0.00){\\line(5,1){25.00}}\n\\put(35.00,5.00){\\line(-4,3){20.00}}\n\\put(15.00,20.00){\\line(-1,-4){5.00}}\n\\put(9.00,0.00){\\makebox(0,0)[rc]{$1$}}\n\\put(15.00,21.00){\\makebox(0,0)[cb]{$3$}}\n\\put(36.00,5.00){\\makebox(0,0)[lc]{$2$}}\n\\put(11.00,10.00){\\makebox(0,0)[rc]{$\\mu$}}\n\\put(26.00,13.00){\\makebox(0,0)[lb]{$\\epsilon$}}\n\\put(27.00,2.00){\\makebox(0,0)[ct]{$\\nu$}}\n\\end{picture}\n\\end{center}\n\\caption{Spin~3\/2}\n\\label{three}\n\\end{figure}\n\\begin{cor}[\\rm spin 3\/2]\nConsider an arbitrary state of spin 3\/2. Let $\\epsilon$, $\\mu$, $\\nu$\nbe the squares of the chordal distances between the three points on\nthe sphere of radius $\\frac{1}{2}$ that represent this state (see figure~\\ref{three}). \nIt's Wehrl entropy\nis given by\n\\begin{equation}\nS_W(\\epsilon,\\mu,\\nu) = \\frac{3}{4} + \nc\\cdot\\left(\\frac{\\epsilon+\\mu+\\nu}{3} - \\frac{\\epsilon\\mu + \\epsilon\\nu + \\mu\\nu}{6}\n+\\frac{1}{c} \\ln\\frac{1}{c} \\right)  \\label{spin32}\n\\end{equation}\nwith \n\\begin{equation}\n\\frac{1}{c} = 1 - \\frac{\\epsilon+\\mu+\\nu}{3} .\n\\end{equation}\nLieb's conjecture holds for all states of spin 3\/2:\n$S_W(\\epsilon,\\mu,\\nu) \\geq 3\/4 = 2j\/(2j+1)$ with equality for \n$\\epsilon = \\mu = \\nu = 0$, \\emph{i.e.}\\ for\ncoherent states.\n\\end{cor}\n\n\\noindent {\\sc Proof}: The proof is similar to the spin~1 case, but the\ngeometry and algebra is more involved.\nConsider a sphere of radius ${1\\over 2}$, with points 1, 2, 3 on its surface,\nand two planes through its center; the first plane\ncontaining\npoints 1 and 3, the second plane containing points 2 and 3. The intersection\nangle $\\phi$\nof these two planes satisfies\n\\begin{equation}\n 2\\cos\\phi \\sqrt{\\epsilon\\mu(1-\\epsilon)(1-\\mu)} = \\epsilon + \\mu - \\nu - 2\\epsilon\\mu .\n\\label{phi}\n\\end{equation}\n$\\phi$ is the azimuthal angle of point 2, if point 3 is at the `north pole' of\nthe sphere and point 1 is assigned zero azimuthal angle.\n\nThe states $|\\psi^{(1)}\\rangle$,\n$|\\psi^{(2)}\\rangle$, and $|\\psi^{(3)}\\rangle$ all have one point at the\nnorth pole of the sphere. It is enough to compute the values of\n$|\\psi_m^{(i)}|^2$ for\none $i$, the other values can be found by appropriate permutation of\n$\\epsilon$, $\\mu$, $\\nu$. (Note that we make no restriction on\nthe parameters $0\\leq \\epsilon$, $\\mu$, $\\nu \\leq 1$ other than that they are\nsquare chordal distances between three points on a sphere of\nradius $\\frac{1}{2}$.)\nWe shall start with $i = 3$: Without loss of generality\nthe three points can be parametrized as $\\omega^{(3)}_1 = (\\tilde\\theta,0)$,\n$\\omega^{(3)}_2 = (\\theta,\\phi)$, and $\\omega^{(3)}_3 = (0,0)$\nwith $\\mu = \\sin^2{\\tilde\\theta\\over 2}$ and $\\epsilon = \\sin^2{\\theta\\over 2}$.\nCorresponding spin-$\\frac{1}{2}$ states are\n\\begin{eqnarray}\n|\\omega^{(3)}_1\\rangle & = & (1-\\mu)^\\frac{1}{2}|\\U\\rangle + \\mu^\\frac{1}{2}|\\D\\rangle ,\\label{om1}\\\\\n|\\omega^{(3)}_2\\rangle & = & (1-\\epsilon)^\\frac{1}{2} e^{-i\\phi\\over 2}|\\U\\rangle \n+ \\epsilon^\\frac{1}{2} e^{i\\phi\\over 2}|\\D\\rangle ,  \\label{om2}\\\\\n|\\omega^{(3)}_3\\rangle & = & |\\U\\rangle ,      \\label{om3} \n\\end{eqnarray}\nand up to normalization, the state of interest is\n\\begin{eqnarray}\n|\\tilde\\psi^{(3)}\\rangle \n& = & P_{j=3\/2} |\\omega^{(3)}_1\\otimes\\omega^{(3)}_2\\otimes\\omega^{(3)}_3\\rangle  \\nonumber \\\\\n& = & (1-\\epsilon)^\\frac{1}{2} (1-\\mu)^\\frac{1}{2} e^{-i\\phi\\over 2}\n      |{3\\over 2},{3\\over 2}\\rangle \\nonumber \\\\\n&&    + \\left( (1-\\mu)^\\frac{1}{2} \\epsilon^\\frac{1}{2} e^{i\\phi\\over 2} \n      + \\mu^\\frac{1}{2} (1-\\epsilon)^\\frac{1}{2} e^{-i\\phi\\over 2} \\right) \n      { {1 \\over \\sqrt{3}}} |{3\\over 2},{1\\over 2}\\rangle \\nonumber \\\\\n&&    + \\mu^\\frac{1}{2} \\epsilon^\\frac{1}{2} e^{i\\phi\\over 2} \n      { {1 \\over \\sqrt{3}}} |{3\\over 2},-{1\\over 2}\\rangle .\n\\end{eqnarray}\nThis gives \n\\begin{eqnarray}\n|\\tilde\\psi^{(3)}_{3\\over 2}|^2 & = & (1-\\epsilon)(1-\\mu),\\\\\n|\\tilde\\psi^{(3)}_{1\\over 2}|^2 & = & {1 \\over 3}\\left(\n\\epsilon(1-\\mu) + \\mu(1 - \\epsilon) + 2 \\sqrt{\\epsilon\\mu(1-\\mu)(1-\\epsilon)} \\cos\\phi\\right) \\nonumber \\\\\n& = & {2\\over 3}\\epsilon(1-\\mu) + {2\\over 3}\\mu(1 - \\epsilon) -{\\nu\\over 3}, \\\\\n|\\tilde\\psi^{(3)}_{-{1\\over 2}}|^2 & = & {\\epsilon \\mu\\over 3},\n\\end{eqnarray}\nand\n$|\\tilde\\psi^{(3)}_{-{3\\over 2}}|^2  = 0$. The sum of these expressions\nis\n\\begin{equation}\n{1\\over c} = \\langle\\tilde\\psi|\\tilde\\psi\\rangle =\n1 - {\\epsilon + \\mu + \\nu \\over 3} ,\n\\end{equation}\nwith $0 < 1\/c \\leq 1$.\nThe case $i=1$ is found by exchanging $\\mu \\leftrightarrow \\nu$ (and also\n$3 \\leftrightarrow 1$, $\\phi \\leftrightarrow -\\phi$).\nThe case $i=2$ is found by permuting\n$\\epsilon\\rightarrow\\mu\\rightarrow\\nu\\rightarrow\\epsilon$ (and also $1 \\rightarrow 3\n\\rightarrow 2 \\rightarrow 1$).\nUsing (\\ref{formula}) then gives the stated entropy.\n\nTo complete the proof Lieb's conjecture for all states of \nspin~$3\/2$ we need to show\nthat the second term in (\\ref{spin32}) is always non-negative and zero\nonly for $\\epsilon=\\mu=\\nu=0$.\nFrom the inequality $(1-x)\\ln(1-x) \\geq -x + x^2\/2$ for $0 \\leq x < 1$,\nwe find\n\\begin{equation}\n{1\\over c}\\ln{1\\over c} \\geq -{\\epsilon+\\mu+\\nu\\over 3} + {1\\over 2}\\left(\n{\\epsilon+\\mu+\\nu\\over 3}\\right)^2 ,\n\\end{equation}\nwith equality for $c=1$. Using the inequality between algebraic and geometric\nmean it is not hard to see that\n\\begin{equation}\n\\left({\\epsilon+\\mu+\\nu\\over 3}\\right)^2 \\geq {\\epsilon\\mu + \\nu\\epsilon + \\mu\\nu \\over 3}\n\\end{equation}\nwith equality for $\\epsilon=\\mu=\\nu$. Putting everything together and inserting\nit into (\\ref{spin32}) we have, as desired, $S_W \\geq 3\/4$ with equality\nfor $\\epsilon=\\mu=\\nu=0$, \\emph{i.e.}\\ for coherent states.$\\blob$\n\\vspace{1ex}\n\\begin{figure}\n\\begin{center}\n\\unitlength 1.00mm\n\\linethickness{0.4pt}\n\\begin{picture}(30.00,36.00)(10,7)\n\\put(10.00,15.00){\\line(5,1){25.00}}\n\\put(35.00,20.00){\\line(-4,3){20.00}}\n\\put(15.00,35.00){\\line(-1,-4){5.00}}\n\\put(10.00,15.00){\\line(3,-2){15.00}}\n\\put(25.00,5.00){\\line(2,3){10.00}}\n\\put(9.00,15.00){\\makebox(0,0)[rc]{$1$}}\n\\put(15.00,36.00){\\makebox(0,0)[cb]{$3$}}\n\\put(36.00,20.00){\\makebox(0,0)[lc]{$2$}}\n\\put(25.00,4.00){\\makebox(0,0)[ct]{$4$}}\n\\put(11.00,25.00){\\makebox(0,0)[rc]{$\\mu$}}\n\\put(26.00,28.00){\\makebox(0,0)[lb]{$\\epsilon$}}\n\\put(19.00,24.00){\\makebox(0,0)[lb]{$\\gamma$}}\n\\put(27.00,17.00){\\makebox(0,0)[ct]{$\\nu$}}\n\\put(17.00,9.00){\\makebox(0,0)[rt]{$\\alpha$}}\n\\put(31.00,11.00){\\makebox(0,0)[rt]{$\\beta$}}\n\\put(15.00,35.00){\\line(1,-3){10.00}}\n\\end{picture}\n\\end{center}\n\\caption{Spin~2}\n\\end{figure}\n\\begin{cor}[\\rm spin 2]\nConsider an arbitrary state of spin 2. Let $\\epsilon$, $\\mu$, $\\nu$, $\\alpha$, $\\beta$,\n$\\gamma$\nbe the squares of the chordal distances between the four points on\nthe sphere of radius $\\frac{1}{2}$ that represent this state\n(see figure). It's Wehrl entropy\nis given by\n\\begin{equation}\nS_W(\\epsilon,\\mu,\\nu,\\alpha,\\beta) = \\frac{4}{5} + c \\cdot \\left( \\sigma + \\frac{1}{c}\n\\ln\\frac{1}{c}\\right), \\label{S2}\n\\end{equation}\nwhere\n\\begin{equation}\n\\frac{1}{c} = 1 - \\frac{1}{4}\\sum\\lipic\n+\\frac{1}{12}\\sum\\papic \\label{c2}\n\\end{equation}\nand\n\\begin{equation}\n\\sigma = \\frac{1}{12}\\left(-\\frac{1}{2}\\sum\\trpic\n-\\frac{5}{3}\\sum\\papic-\\sum\\wepic+3\\sum\\lipic\n\\right)\n\\end{equation}\nwith\n\\begin{equation}\n\\sum\\trpic \\equiv \\alpha\\mu\\nu+\\epsilon\\beta\\nu+\\epsilon\\mu\\gamma+\\alpha\\beta\\gamma,\n\\end{equation}\n\\begin{equation}\n\\sum\\papic \\equiv \\alpha\\epsilon+\\beta\\mu+\\gamma\\nu,\n\\qquad\n\\sum\\lipic \\equiv \\alpha+\\beta+\\gamma+\\mu+\\nu+\\epsilon,\n\\end{equation}\n\\begin{equation}\n\\sum\\wepic \\equiv \\alpha\\mu+\\alpha\\nu+\\mu\\nu+\\beta\\epsilon\n+\\beta\\nu+\\epsilon\\nu+\\epsilon\\gamma+\\epsilon\\mu+\\mu\\gamma\n+\\alpha\\beta+\\alpha\\gamma+\\beta\\gamma.\n\\end{equation} \\label{spin2}\n\\end{cor}\n\n\\noindent \\emph{Remark:} The fact that the four points lie on the surface\nof a sphere imposes a complicated constraint on the parameters\n$\\epsilon$, $\\mu$, $\\nu$, $\\alpha$, $\\beta$,\n$\\gamma$. Although we have convincing numerical evidence\nfor Lieb's conjecture for spin~2,\nso far a rigorous proof has been limited to\ncertain symmetric configurations\nlike equilateral triangles with centered fourth point ($\\epsilon=\\mu=\\nu$ and\n$\\alpha=\\beta=\\gamma$), and squares ($\\alpha=\\beta=\\epsilon=\\mu$ and\n$\\gamma=\\nu$). It is not hard to find values of the parameters\nthat give values of $S_W$ below the entropy for coherent states, but they\ndo \\emph{not} correspond to any configuration of points on the sphere,\nso in contrast to spin 1 and spin 3\/2\nthe constraint is now important.\n$S_W$ is concave in each of the parameters $\\epsilon$, $\\mu$, $\\nu$, $\\alpha$, $\\beta$,\n$\\gamma$.\n\n\\noindent {\\sc Proof}: The proof is analogous to the spin~1 and spin~3\/2\ncases but the geometry and algebra are considerably more complicated,\nso we will just give a sketch. Pick four points on the sphere,\nwithout loss of generality parametrized as $\\omega_1^{(3)} =(\\tilde\\theta,0)$,\n$\\omega_2^{(3)} =(\\theta,\\phi)$, $\\omega_3^{(3)} = (0,0)$,\nand $\\omega_4^{(3)} =(\\bar\\theta,\\bar\\phi)$. Corresponding spin $\\frac{1}{2}$ states\nare $|\\omega_1^{(3)}\\rangle$, $|\\omega_2^{(3)}\\rangle$, $|\\omega_3^{(3)}\\rangle$,\nas given in (\\ref{om1}), (\\ref{om2}), (\\ref{om3}), and\n\\begin{equation}\n|\\omega_4^{(3)}\\rangle = (1-\\gamma)^\\frac{1}{2} e^{-i\\bar\\phi \\over 2} |\\U\\rangle\n+ \\gamma^\\frac{1}{2} e^{i\\bar\\phi \\over 2} |\\D\\rangle.\n\\end{equation}\nUp to normalization, the state of interest is\n\\begin{equation}\n|\\tilde\\psi^{(3)}\\rangle =\nP_{j=2} |\\omega_1^{(3)} \\otimes \\omega_2^{(3)} \\otimes\\omega_3^{(3)} \\otimes\\omega_4^{(3)}\\rangle.\n\\end{equation}\nIn the computation of $|\\tilde\\psi^{(3)}_m|^2$ we encounter\nagain the angle $\\phi$, compare (\\ref{phi}),and two new\nangles $\\bar\\phi$ and\n$\\bar\\phi -\\phi$.\nLuckily both can again be expressed as angles between planes that\nintersect the circle's center and we have\n\\begin{eqnarray}\n2 \\cos\\bar\\phi\\sqrt{\\mu\\gamma(1-\\mu)(1-\\gamma)} & = & \n\\mu + \\gamma - \\alpha - 2\\mu\\gamma, \\\\\n2 \\cos(\\bar\\phi-\\phi)\\sqrt{\\epsilon\\gamma(1-\\epsilon)(1-\\gamma)}\n& = & \\gamma + \\epsilon - \\beta - 2\\gamma\\epsilon,\n\\end{eqnarray}\nand find $1\/c = \\sum_m |\\tilde\\psi^{(3)}_m|^2$ as given in (\\ref{c2}).\nBy permuting the parameters $\\epsilon$, $\\mu$, $\\nu$, $\\alpha$, $\\beta$,\n$\\gamma$ appropriately we can derive expressions for the remaining\n$|\\tilde\\psi^{(i)}_m|^2$'s and then compute $S_W$ (\\ref{S2}) with the\nhelp of $(\\ref{formula})$.$\\blob$\n\n\\section{Higher spin}\n\nThe construction outlined in the proof of corollary~\\ref{spin2}\ncan in principle also be applied to states of higher spin, but\nthe expressions pretty quickly become quite unwieldy.\nIt is, however, possible to use theorem~\\ref{theorem} to show that\nthe entropy is extremal for coherent states:\n\n\\begin{cor}[\\rm spin $j$]\nConsider the state of spin $j$ characterized by $2j -1$ coinciding\npoints on the sphere and a $2j$'th point, a small (chordal) distance\n$\\epsilon^\\frac{1}{2}$ away from them. The Wehrl entropy of this small deviation\nfrom a coherent state, up to third order in $\\epsilon$, is\n\\begin{equation}\nS_W(\\epsilon) = {2j\\over 2j+1} + {c \\over 8 j^2} \\epsilon^2 \\quad + {\\cal O}[\\epsilon^4] ,\n\\end{equation}\nwith\n\\begin{equation}\n{1 \\over c} = 1 - {2j - 1 \\over 2j} \\epsilon \\quad \\mbox{(exact)} .\n\\end{equation}\n\\end{cor}\n\nA generalized version of Lieb's conjecture, analogous to (\\ref{W2}), \nis \\cite{A}\n\\begin{conj} \\label{conject2}\nLet $|\\psi\\rangle$ be a normalized state of spin $j$, then\n\\begin{equation}\n(2j s + 1) \\int\\frac{d\\Om}{4\\pi} \\, |\\langle\\Omega|\\psi\\rangle|^{2s} \\leq 1 , \\quad s > 1 ,\n\\label{norms}\n\\end{equation}\nwith equality if and only if $|\\psi\\rangle$ is a coherent state.\n\\end{conj}\n\n\\noindent \\emph{Remark:} \nThis conjecture is equivalent to the ``quotient of two H\\\"older inequalities\"\n(\\ref{holder}).\nThe original conjecture~\\ref{conject1} \nfollows from it in the limit $s \\rightarrow 1$.\nFor $s=1$ we simply get the norm of the spin $j$ state $|\\psi\\rangle$,\n\\begin{equation}\n(2j + 1) \\int\\frac{d\\Om}{4\\pi} \\, |\\langle\\Omega|\\psi\\rangle_j|^2 = | P_j | \\psi\\rangle|^2 ,\n\\label{norm}\n\\end{equation}\nwhere $P_j$ is the projector onto spin $j$.\nWe have numerical evidence for low spin\nthat an analog of conjecture~\\ref{conject2}\nholds in fact for a much larger class of convex functions than\n$x^s$ or $x \\ln x$.\n\n\nFor $s \\in \\N$ there is a surprisingly simple group theoretic argument\nbased on (\\ref{norm}):\n\\begin{thm}\nConjecture~\\ref{conject2} holds for $s \\in \\N$. \\label{natural}\n\\end{thm}\n\n\\noindent \\emph{Remark:} For spin 1 and spin 3\/2  (at $s=2$) this was\nfirst shown by Wolfgang Spitzer by direct computation of the\nintegral.\n\n\\noindent {\\sc Proof}: Let us consider\n$s=2$, $|\\psi\\rangle \\in [j]$ with $||\\psi\\rangle|^2 = 1$,\nrewrite (\\ref{norms}) as follows\nand use (\\ref{norm})\n\\begin{eqnarray}\n\\lefteqn{(2j\\cdot 2 + 1) \\int\\frac{d\\Om}{4\\pi} \\, |\\langle\\Omega|\\psi\\rangle|^{2\\cdot 2}} \\nonumber \\\\\n&& = (2(2j)+ 1) \\int\\frac{d\\Om}{4\\pi} \\, |\\langle\\Omega\\otimes\\Omega|\\psi\\otimes\\psi\\rangle|^2 \n=  |P_{2j} |\\psi\\otimes\\psi\\rangle |^2.\n\\end{eqnarray}\nBut $|\\psi\\rangle\\otimes|\\psi\\rangle \\in [j]\\otimes[j] = [2j]\\oplus[2j-1]\\oplus\\ldots\n\\oplus[0]$, so $|P_{2j} |\\psi\\otimes\\psi\\rangle |^2 < ||\\psi\\otimes\\psi\\rangle |^2 = 1$\nunless $|\\psi\\rangle$ is a coherent state, in which case\n$|\\psi\\rangle\\otimes|\\psi\\rangle \\in [2j]$ and we have equality. The proof for\nall other $s \\in \\N$ is completely analogous.$\\blob$\\\\[1ex]\nIt seems that there should also be a similar group theoretic\nproof for all real, positive $s$ related to (infinite dimensional)\nspin~$js$ representations of su(2) (more precisely: sl(2)). \nThere has been some progress and it is now clear that there will\nnot be an argument as simple as the one given above \\cite{J}.\nCoherent states of the form discussed in \\cite{C3} (for the hydrogen atom)\ncould be of importance here, since they easily generalize to non-integer\n`spin'.\n\nTheorem~\\ref{natural} provides a quick, crude, lower limit on the\nentropy:\n\n\\begin{cor}\nFor states of spin $j$\n\\begin{equation}\nS_W(|\\psi\\rangle\\langle\\psi|) \\geq \\ln{4j+1\\over 2j+1} > 0.\n\\end{equation}\n\\end{cor}\n\n\\noindent {\\sc Proof}: This follows from Jensen's inequality and\nconcavity of $\\ln x$:\n\\begin{eqnarray}\nS_W(|\\psi\\rangle\\langle\\psi|) & = &\n-{\\textstyle (2j+1)}\\int\\frac{d\\Om}{4\\pi}\\, |\\langle\\Omega|\\psi\\rangle|^2 \\ln |\\langle\\Omega|\\psi\\rangle|^2 \\nonumber \\\\\n& \\geq & -\\ln\\left({\\textstyle (2j+1)} \n\\int\\frac{d\\Om}{4\\pi}\\, |\\langle\\Omega|\\psi\\rangle|^{2\\cdot 2}\\right) \\nonumber \\\\\n& \\geq & -\\ln{2j+1 \\over 4j + 1} .\n\\end{eqnarray}\nIn the last step we have used theorem~\\ref{natural}.$\\blob$\\\\[1ex]\nWe hope to have provided enough evidence to convince the\nreader that it is reasonable to expect that\nLieb's conjecture is indeed true for all spin. All cases\nlisted in Lieb's original article, $1\/2$, $1$, $3\/2$, are now\nsettled --\nit would be nice if someone\ncould take care of the remaining ``dot, dot, dot\" \\ldots \n\n\\section*{Acknowledgments}\n\nI would like to thank Elliott Lieb \nfor many discussions, constant support, encouragement, and for reading the\nmanuscript.\nMuch of the early work was done in collaboration with Wolfgang Spitzer.\nTheorem~\\ref{natural} for spin 1 and spin 3\/2 at $s=2$ is due to him and\nhis input was crucial in eliminating many other plausible approaches.\nI would like to thank him for many discussions and excellent team work.\nI would like to thank Branislav Jur\\v co for joint work on the\ngroup theoretic aspects of the problem and stimulating discussions\nabout coherent states.\nIt is a pleasure to thank Rafael Benguria, Almut Burchard, Dirk Hundertmark,\nLarry Thomas, and Pavel Winternitz for many valuable discussions.\nFinancial support by the Max Kade Foundation is gratefully acknowledged.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\n\n\n\n\n\n\n\n\n\n\n\nIn recent years, significant success has been achieved in applying Reinforcement Learning (RL) to different real-world scenarios \\cite{chenTopKOffPolicyCorrection2019, tangDeepValuenetworkBased2019, gauciHorizonFacebookOpen2019}. \nOffline Reinforcement Learning (ORL) \\cite{levineOfflineReinforcementLearning2020} is pioneering a real-world adaptation of RL, focusing on algorithms that can learn a policy from a fixed, previously recorded dataset, without having to interact with the environment during training.\nMany recent community efforts were focused on advanced benchmarks \\cite{fuD4RLDatasetsDeep2021,gulcehreRLUnpluggedSuite2021} and nuanced algorithm comparisons \\cite{brandfonbrenerOfflineRLOffPolicy2021, qinNeoRLRealWorldBenchmark2021}. \n\nHowever, it is widely recognized that even deep online RL is plagued with evaluation issues, such as insufficient account for statistical variability \\cite{agarwalDeepReinforcementLearning2022}, or sensitivity to the choice of hyperparameters \\cite{hendersonDeepReinforcementLearning2018}. The latter problem is one of the reasons why the results of many deep RL methods are usually reported for a narrow range of values.\n\nWhen it comes to deep ORL algorithms, the choice of hyperparameters also plays a major role in the final performance \\cite{wuBehaviorRegularizedOffline2019}. While many in the community note that the whole training and evaluation pipeline needs improvement \\cite{fuBenchmarksDeepOffPolicy2021}, the comparisons of new deep ORL algorithms are still mostly done through online performance reports on the best set of hyperparameters \\cite{kostrikovOfflineReinforcementLearning2021, fujimotoMinimalistApproachOffline2021}.\n\nIn this paper, we argue that the hyperparameter search should not be ignored in the deep offline RL setting, demonstrating that the conclusions about the algorithms change when we control for the number of trained policies deployed online\\footnote{Code is available at \\href{https:\/\/tinkoff-ai.github.io\/eop\/}{tinkoff-ai.github.io\/eop}}. To this end, we introduce the notion of an online budget, i.e. the number of policies deployed online, and suggest to use an entire pipeline similar to the one in \\cite{paineHyperparameterSelectionOffline2020} for reporting results: training, offline selection, and online evaluation, but one where the selection is done by uniform sampling. As we will demonstrate, this decision allows us to re-use the Expected Validations Performance (EVP) \\cite{dodgeShowYourWork2019} technique from the NLP field to get reliable estimates of expected maximum performance under different online budgets from just one round of hyperparameter search.\n\n\n\n\n\n\\begin{figure*}[ht]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\textwidth]{pipeline2.pdf}}\n\\caption{\\textbf{Left}: A widespreaded approach for reporting deep offline RL results, commonly known as online policy selection. \\textbf{Right}: Full deep offline RL evaluation pipeline. We argue for reporting results under the second pipeline with varying sizes of the online evaluation budget $B$. Note that selecting hyperparameters that perform best overall on the online policy selection tasks from the same domain \\cite{fuD4RLDatasetsDeep2021, gulcehreRLUnpluggedSuite2021} can also be put to the right pipeline as a special case.}\n\\label{fig:eval_pipelines}\n\\end{center}\n\\vskip -0.2in\n\\end{figure*}\n\n\\textbf{Our Contributions} Here, we list the main contributions of our work:\n\\begin{itemize}\n    \\item We demonstrate that the preference between deep offline RL algorithms is budget-dependent. We stress that this is more critical for offline settings than for online ones, and that current evaluation methodology does not account for such dependence.\n    \\item We propose to use Expected Validation Performance \\cite{dodgeShowYourWork2019}, a technique actively employed in NLP, for reliable comparison of deep offline RL algorithms under varying online evaluation budgets. To stress the online nature of comparison (in opposition to validation), we refer to it as Expected Online Performance (EOP). This tool can take the both major components of deep offline RL into account: the offline policy selection (OPS) method as well as online evaluation budget. Furthermore, it can be applied without additional computational expenses.\n    \\item Using the proposed tool, we also demonstrate that Behavioral Cloning \\cite{pomerleauEfficientTrainingArtificial1991} is often more favorable under a limited evaluation budget.\n    \\item In addition, EOP can be applied to comparisons of OPS methods. Using EOP, we illustrate that their preference is also budget-dependent.\n\\end{itemize}\n\nIn the end, we also discuss how the proposed solution relates to the recently introduced Active-OPS \\cite{konyushkovaActiveOfflinePolicy2021} and deployment-constrainted RL setup \\cite{matsushimaDeploymentEfficientReinforcementLearning2020}.\n\n\\section{Background}\n\\subsection{Offline RL}\n\nReinforcement learning (\\textit{RL}) is a framework for solving sequential decision-making problems. It is typically formulated as a Markov Decision Process (MDP) over a 5-tuple $(S, A, P, r, \\gamma)$, with action space $A$, state space $S$, transition dynamics $P$, reward function $r$, and discount factor $\\gamma$. The goal of the learning agent is to obtain a policy $\\pi(s, a)$ that maximizes the expected discounted return  $E_{\\pi}[\\sum_{t=0}^{\\infty}\\gamma^{t}r_{t+1}]$ through interaction with the MDP.\n\nIn \\textit{Offline RL}, also known as Batch RL, instead of obtaining data and learning via environment interactions, the agent solely relies on a static dataset $D$ that was collected under some unknown behavioral policy (or a mixture of policies) $\\pi_{\\mu}$. This setting is considered more challenging, as the agent loses its ability for exploration \\cite{levineOfflineReinforcementLearning2020} and is faced with the problem of extrapolation error -- being unable to correct its estimation inaccuracies when selected actions are not present in the training dataset \\cite{fujimotoBenchmarkingBatchDeep2019}.\n\n\\subsection{Offline RL Evaluation}\n\n\\begin{table*}[h]\n\\caption{\\textbf{Best final performance of deep offline RL algorithms if they were evaluated under a different number of policies deployed online} for Hopper-v3 environment. This table highlights that the usage of different online evaluation budgets ($B$) may lead to different conclusions on preference between algorithms. $N$ is the total number of hyperparameters evaluated for a specific algorithm.}\n\\vspace{0.15in}\n\\centering\n\\begin{tabular}{lccccccc|cr}\n\\toprule\nAlgorithm & $B = 1$ & $B = 2$ & $B = 3$ & $B = 4$ & $B = 8$ & $B = 15$ & $B = 30$ & Final & $N$\n\\\\\n\\midrule\n{BC}  &   \\textbf{1794} & \\textbf{2057} & \\textbf{2179} & - & - & - & - & 2179 & 3 \\\\\n{CQL} &  1773 & 1954 & 2072 & \\textbf{2161} & \\textbf{2391} & \\textbf{2603} & \\textbf{2832} & \\textbf{2832} & 30 \\\\\n{PLAS} & 1475 & 1833 & 1996 & 2096 & 2316 & 2507 & - & 2507 & 15 \\\\\n{BCQ} &  1325 & 1605 & 1742 & 1826 & 1986 & - & - & 2062 & 12\\\\\n{CRR}      & 1013 & 1339 & 1477 & 1545 & 1636 & - & - & 1668 & 12 \\\\\n{BREMEN}      & 883 & 1148 & 1318 & 1439 & 1691 & - & - & 1795 & 12 \\\\\n{MOPO}      & 11 & 18 & 24 & 30 & 46 & 63 & 78 & 78 & 30 \\\\\n\\bottomrule\n\\end{tabular}\\\\\n\\label{table:online-budget-dependence-hopper}\n\\end{table*}\n\nTraining and evaluation of deep offline RL algorithms is still in active development, and various authors approach it in different ways by simplifying the genuine offline setting \\cite{gulcehreRLUnpluggedSuite2021}. At the core of the simplification are two primary issues: (1) unlimited amount of online evaluations available, and therefore (2) sidestepping offline policy selection. For example, it is common to report the maximum performance for the best set of hyperparameters (Figure \\ref{fig:eval_pipelines}, Left). Moreover, in many cases, the number of search trials is not made explicit \\cite{kumarConservativeQLearningOffline2020}.\n\nTo eliminate these simplifications, we adhere to a more general setup for training and evaluating offline RL algorithms similar to \\citet{paineHyperparameterSelectionOffline2020} in order to satisfy hard offline constraints (Figure \\ref{fig:eval_pipelines}, Right).\n\nFirst, the dataset $D$ is randomly split trajectory-wise into training $D_{T}$ and validation $D_{V}$ subsets accordingly. Then a sequence of hyperparameter assignments $(h_{1}, h_{2}, ..., h_{N})$ is sampled for running an algorithm of interest, resulting in a sequence of policies $(\\pi_{1}, \\pi_{2}, ..., \\pi_{N})$. Note that at this stage, we do not know how good these policies are.\n\nThen, $B \\leq N$ of policies are arbitrarily chosen for online evaluation, which we refer to as an \\textit{online evaluation budget}. In the most restricted offline RL setting, $B = 1$. However, the generalization to $B > 1$ is justified by the online evaluation budget being conditioned on the relevant decision-making problem and the available resources.\n\nTo choose policies for online evaluation, offline policy selection (OPS) methods can be used. In specific domains, like recommender systems, policies can be picked based on established offline metrics, e.g. Recall, computed on the validation $D_{V}$ dataset \\cite{xinSelfSupervisedReinforcementLearning2020a}. However, such metrics do not always exist, and it is often necessary to rely on general methods \\cite{voloshinEmpiricalStudyOffPolicy2020, fuBenchmarksDeepOffPolicy2021} or proxy tasks \\cite{fuD4RLDatasetsDeep2021, gulcehreRLUnpluggedSuite2021}.\n\n\\section{Online Evaluation Budget Matters}\n\\label{sec:online-eval-matters}\n\nAs can be seen in Figure \\ref{fig:eval_pipelines}, Left, using online policy selection makes the evaluation budget $B$ equivalent to the number of hyperparameter search trials $N$. On the other hand, \\citet{fuD4RLDatasetsDeep2021, gulcehreRLUnpluggedSuite2021} search for the best set of hyperparameters using proxy tasks, but the online evaluation budget on the target task is $B = 1$.\n\nMeanwhile, there is a whole spectrum of values in-between that could be relevant not only for a specific problem, but for a specific context. By context we mean a certain space of resources (computational resources, robotics hardware, time constraints, online testing capacity). Here, a practitioner may work on the same problem, but have a lower or bigger amount of resources available for online evaluation. \n\nTherefore, a natural question to ask when analysing results of deep ORL algorithms is \"Will the conclusions about the algorithms change, if I have a lower or higher online evaluation budget than the one reported in the paper?\". Unfortunately, current evaluation and report methodologies do not provide an answer, and the dependence between varied online evaluation budget and the resulting performance of the algorithms is left unreported. \n\nTo address this issue independently, it is necessary to access detailed experimental results showing which hyperparameters resulted in which performance. While some authors open-source such data \\cite{qinNeoRLRealWorldBenchmark2021}, it is not a common practice to do so.\n\nTo demonstrate that the conclusions about algorithm preference are dependent on the available online evaluation budget, we rely on open-sourced\\footnote{Note that there is a discrepancy in open-sourced and reported results. There is additional data on CRR and BREMEN, but data on MB-PPO is not provided.} results by \\citet{qinNeoRLRealWorldBenchmark2021}. For each algorithm in Table \\ref{table:online-budget-dependence-hopper}, we compute the expected maximum performance under uniform policy selection (i.e. the policies are chosen at random) given a specific online evaluation budget $B$. The final column is what would be reported in the paper, demonstrating that CQL \\cite{kumarConservativeQLearningOffline2020} significantly outperforms its competitors. However, in budgets up to 4, Behavioral Cloning performs the best. Also, note that the preference between CRR \\cite{wangCriticRegularizedRegression2020} and BREMEN \\cite{matsushimaDeploymentEfficientReinforcementLearning2020} is reversed starting from the budget of 8.\n\n\\section{Accounting for the Budget \\raisebox{-0.32ex}{\\includegraphics[scale=0.08]{bank_emoji.png}}}\n\n\\begin{figure*}[!h]\n    \\begin{subfigure}[b]{0.32\\textwidth}\n        \\centering\n        \\centerline{\\includegraphics[width=\\columnwidth]{figures_appendix\/Walker2d-v3_1000_low_uniform.pdf}}\n        \\caption{Walker2d, Low-1000}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.32\\textwidth}\n        \\centering\n        \\centerline{\\includegraphics[width=\\columnwidth]{figures_appendix\/Hopper-v3_1000_medium_uniform.pdf}}\n        \\caption{Hopper, Medium-1000}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.32\\textwidth}\n        \\centering\n        \\centerline{\\includegraphics[width=\\columnwidth]{figures_appendix\/HalfCheetah-v3_1000_high_uniform.pdf}}\n        \\caption{HalfCheetah, High-1000}\n    \\end{subfigure}\n    \\caption{\\textbf{Expected Online Performance graphs under uniform offline policy selection} on \\citet{qinNeoRLRealWorldBenchmark2021} data. The proposed EOP graph clearly demonstrates that preference between algorithms is budget-dependent. Furthermore, it highlights that BC is often more favorable to offline RL algorithms under limited online evaluation budgets. The X-axis denotes the number of policies deployed online, and Y-axis refers to the normalized performance. Note that the number of estimates for a concrete algorithm is upper-bounded by the total number of hyperparameter assignments ($N$) evaluated for this algorithm. Shadowed area represents one standard deviation.}\n   \n    \\label{fig:eop_neorl}\n\\end{figure*}\n\nIn the previous section, we demonstrated multiple model comparisons where authors would have reached a different conclusion if they had used a smaller (or bigger) online evaluation budget. To resolve this issue, we use a tool from the NLP field, Expected Validation Performance (EVP) \\cite{dodgeShowYourWork2019}, that can be adapted for enhancing the quality of experimental reports in a deep ORL setting.\n\n\\subsection{Expected Online Performance}\n\\label{sec:eop_math}\n\nHere, we give a detailed description for EVP, reframed for an offline RL setting, and with the computational budget replaced by an online evaluation budget (typically, $B \\ll N$). We refer to this approach as Expected Online Performance (EOP).\n\nHaving all $N$ policies evaluated online after hyperparameter search, we want an estimate of the expected maximum performance, given that we could deploy only $1 \\leq B \\leq N$ policies out of $N$. \n\\newpage\nThe parameters of interest to us are $\\theta_{1}$,...,$\\theta_{B}$, where \n\\begin{equation} \\label{eq:expected_online_performace}\n    \\theta_{b} := E[max(V(\\pi_{1}),...,V(\\pi_{b}))] = E[V_{(b:b)}]\n\\end{equation}\nfor $1 \\leq b \\leq B$ and $V$ is an random variable (RV) representing the result of online evaluation.\nIn other words, $\\theta_{b}$ is the expected value of the $b^{th}$ order statistic for a sample of size $b$. The $i^{th}$ order statistic $V_{(i:b)}$ is an RV representing the $i^{th}$ smallest value if the RVs were sorted.\n\nOriginally, EVP operates over one stage -- hyperparameter value selection. But in an ORL setting, there is also a second stage -- policy selection (see Figure \\ref{fig:eval_pipelines}, Right). To account for this discrepancy, we note that uniformly sampled hyperparameter values and then uniformly sampled policies result in the probability of a policy being selected for online evaluation proportional to the probability of its hyperparameters being used. Virtually, that makes $\\theta_{b}$ be based on a sample size $b$ drawn independent and identically distributed. \n\nIn this case, the estimator proposed in \\citet{dodgeShowYourWork2019} can be readily applied. The derivation is similar to \\citet{tangShowingYourWork2020}:\n\\begin{equation}\n\\begin{aligned}\n    Pr[V_{(b:b)} < v] & = Pr[V(\\pi_{1}) \\leq v \\wedge ... \\wedge V(\\pi_{b}) \\leq v] \\\\\n    & = \\prod_{i=1}^{b} Pr[V(\\pi_{i}) \\leq v],\n\\end{aligned}\n\\end{equation}\nwhich we denote as $F^{b}(v)$. Then\n\\begin{equation} \\label{eq:expected_online_performace_cdf}\n    \\theta_{b} = E[V_{(b:b)}] = \\int_{-\\inf}^{\\inf} vdF^{b}(v).\n\\end{equation}\nWithout loss of generality, assume $V(\\pi_{1}) \\leq ... \\leq V(\\pi_{N})$. To approximate the Cumulative Distribution Function (CDF), use Empirical Cumulative Distribution Function (ECDF)\n\\begin{equation}\n    \\hat{F}^{b}_{N}(v) = (\\frac{1}{N}\\sum_{i=1}^{N}I[V(\\pi_{i}) \\leq v])^{b}\n\\end{equation}\nTo arrive at the final estimator, replace CDF with an ECDF in Equation \\ref{eq:expected_online_performace_cdf}\n\\begin{equation}\n    \\hat{\\theta}_{b} = \\int_{-\\inf}^{\\inf} vd\\hat{F}^{b}_{N}(v)\n\\end{equation}\nwhich, by definition, evaluates to\n\\begin{equation} \\label{eq:expected_online_performace_uniform_selection}\n    \\hat{\\theta}_{b} = \\frac{1}{N}\\sum_{i=1}^{N}V(\\pi_{i})(\\hat{F}^{b}_{N}(V(\\pi_{i})) - \\hat{F}^{b}_{N}(V(\\pi_{i-1}))\n\\end{equation}\n\nTo summarize, $\\hat{\\theta_{b}}$ corresponds to the estimated expected maximum performance given that (1) hyperparameters were randomly sampled from a pre-defined grid, (2) we could deploy $1 \\leq b \\leq B$ policies out of $N$ for online evaluation, and (3) these $b$ policies were picked by uniform policy selection.\n\nThe major advantage of this estimator is that, if our evaluation methodology satisfies all three conditions described above, then the computation within a single round of hyperparameter search is sufficient to construct a reliable estimate of\nexpected online performance for different values of $b$, without requiring any further experimentation \\cite{dodgeShowYourWork2019}.\nMoreover, for a compact presentation, we can plot a graph over the entire range of values for $b$, demonstrating the dependence between the final performance and online evaluation budget (Figure \\ref{fig:eop_neorl}).\n\nNote, that there are alternative estimators for the quantity of interest \\cite{tangShowingYourWork2020}. However, \\citet{dodgeExpectedValidationPerformance2021} compared different approaches for estimating the expected maximum and found that the employed estimator \\cite{dodgeShowYourWork2019} is favored amongst existing approaches in terms of both MSE criterion and a percent of incorrect conclusions.\n\n\n\\subsection{Target Metric}\n\n\\begin{figure*}[!h]\n    \\begin{subfigure}[b]{0.32\\textwidth}\n        \\centering\n        \\centerline{\\includegraphics[width=\\columnwidth]{figures\/citylearn_999_medium_uniform.pdf}}\n        \\caption{CityLearn, Medium-1000}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.32\\textwidth}\n        \\centering\n        \\centerline{\\includegraphics[width=\\columnwidth]{figures\/finrl_999_high_uniform.pdf}}\n        \\caption{FinRL, High-1000}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.32\\textwidth}\n        \\centering\n        \\centerline{\\includegraphics[width=\\columnwidth]{figures\/industrial_999_medium_uniform.pdf}}\n        \\caption{Industrial Benchmark, Medium-1000}\n    \\end{subfigure}\n    \\caption{\\textbf{Expected Online Performance graphs under uniform offline policy selection.} The proposed EOP graph clearly demonstrates that preference between algorithms is budget dependent. Furthermore, it highlights that BC is often more favorable to offline RL algorithms under limited online evaluation budgets. Shadowed area represents one standard deviation.}\n   \n    \\label{fig:eop_other_domains}\n\\end{figure*}\n\nThe target metric can be represented by any convenient measure used in literature, e.g., absolute policy performance or policy performance normalized by an expert \\cite{fuD4RLDatasetsDeep2021}. In our case studies (Section \\ref{sec:case_studies}), we rely on a modified version of the latter. The main motivation behind this modification is that the original metric is normalized by the value provided by a domain-specific expert \\cite{fuD4RLDatasetsDeep2021}. However, the final results of offline RL algorithms are highly dependent on training data, and expecting to achieve expert performance while training on data from weak policies can be too optimistic. Therefore, we normalize by the performance of the best policy (as there can be multiple) that collected the training data.\n\n\\subsection{Online Evaluation Budget}\n\nThe original EVP makes it possible to use various quantities as an argument to the target metric, e.g. number of hyperparameters enumerated or training time. Similarly, for EOP in the deep ORL setting, several options can be used, such as the number of trajectories, number of timesteps, or number of policies. We suggest using the latter option, and equate the online evaluation budget $B$ with the number of policies deployed for online evaluation. This choice provides researchers with the flexibility of defining their own amount of computation for getting reliable estimates for policy values.\n\n\n\n\n\n\n\\subsection{Beyond Uniform Policy Selection}\n\n\\begin{figure*}[!h]\n    \\begin{subfigure}[b]{0.32\\textwidth}\n        \\centering\n        \\centerline{\\includegraphics[width=\\columnwidth]{figures_appendix\/ops_cql_citylearn_9999_low.pdf}}\n        \\caption{CQL, CityLearn, Low-10000}\n        \\label{fig:eop_ops_a}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.32\\textwidth}\n        \\centering\n        \\centerline{\\includegraphics[width=\\columnwidth]{figures_appendix\/ops_td3+bc_citylearn_9999_low.pdf}}\n        \\caption{TD3+BC, CityLearn, Low-10000}\n        \\label{fig:eop_ops_b}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.32\\textwidth}\n        \\centering\n        \\centerline{\\includegraphics[width=\\columnwidth]{figures_appendix\/ops_cql_finrl_999_low.pdf}}\n        \\caption{CQL, FinRL, Low-1000}\n        \\label{fig:eop_ops_c}\n    \\end{subfigure}\n    \\caption{\\textbf{Inverse of Normalized Regret@K over varied online evaluation budgets}. The suggested method can also be used for comparing offline policy selection methods. A similar pattern emerges -- preference for OPS methods is also budget-dependent. Furthermore, uniform policy selection can be competitive under low budgets, and action difference performs reasonably well across a range of environments, dataset sizes, and policy levels. Shadowed area represents one standard deviation.}\n    \\label{fig:eop_ops}\n\\end{figure*}\n\nIn Section \\ref{sec:eop_math} we outlined an estimator of expected maximum performance under varied online evaluation budgets. Although, an assumption was made that besides randomly sampled hyperparameters, the policies are also selected uniformly. However, comparing deep ORL methods with different OPS methods is also of interest.\n\nSince the policies selected under an arbitrary strategy (e.g. Fitted-Q Evaluation \\cite{leBatchPolicyLearning2019}) are generally not i.i.d, the plug-in estimator derived above would be invalid. However, one can still estimate the parameters of interest using a vanilla average estimator.\n\\begin{equation}\n    \\hat{\\theta_{b}} = \\frac{1}{M}\\sum^{M}_{r=1}max(V^{r}_{1}, ..., V^{r}_{b})\n\\end{equation}\nwhere $M$ is a number of hyperparameter search rounds with offline policy selection, $V^{r}_{i}$ corresponds to an RV representing the result of online evaluation for the $i^{th}$ selected policy in round $r$. Note that this estimator loses the appealing computational side of EVP, and requires multiple runs of hyperparameter search with offline policy selection. This makes it more expensive in terms of computing time.\n\nIn one of our case studies (Section \\ref{sec:case_ops}), we demonstrate how one could use EOP for comparing not only deep ORL algorithms, but OPS methods as well.\n\n\\section{Case Studies}\n\\label{sec:case_studies}\n\nTo further demonstrate the use of the proposed technique and to identify whether online evaluation budget changes the preference between deep ORL algorithms besides the environment analyzed in Section \\ref{sec:online-eval-matters}, we consider several case studies covering a range of decision-making problems: robotics, finances, and energy management. \n\n\\subsection{NeoRL, Robotic Tasks}\n\\label{sec:neorl_robotic_tasks}\n\n\nContinuing the closer look at the results presented in \\citet{qinNeoRLRealWorldBenchmark2021} from Section \\ref{sec:online-eval-matters}, we build Expected Online Performance graphs for other robotics environments (Figure \\ref{fig:eop_neorl}). This once again confirms that the preference is budget-dependent. Moreover, this is consistent across environments, dataset sizes, and policy levels (for a complete set of graphs, check the Appendix). \nThere are many cases when the conclusion changes with a budget. For example, in the Walker-2d environment, CQL is clearly preferred to PLAS \\cite{zhouPLASLatentAction2020a}, but only up to the 9-10 policies available for online evaluation. Another example can be seen in Figure \\ref{fig:eop_neorl}b for the Hopper environment: Behavioral Cloning significantly outperforms its competitiors at low budgets, but loses to PLAS at higher ones. The same holds for the HalfCheetah environment, where CQL starts to prevail at higher budgets.\n\nAs the online budget is upper-bound by the total number of enumerated hyperparameters, it is clear (Figure \\ref{fig:eop_neorl}) that different algorithms were tuned more or less excessively. This results in more optimistic results for one algorithm, and in more pessimistic for another. Consider BCQ \\cite{leBatchPolicyLearning2019} in Figure \\ref{fig:eop_neorl}a. It is tuned up to 13 hyperparameter assignments showing the best result, but as long as a competing algorithm, PLAS, is tuned for 14 and more assignments, it starts to outperform BCQ. An even more vivid example is depicted in Figure \\ref{fig:eop_neorl}c, where MOPO \\cite{yuMOPOModelbasedOffline2020} starts to outperform both BREMEN and CRR at 2x more hyperparameters tested. Note that reporting just one policy value (either using online policy selection or proxy tasks) hides this issue.\n\n\n\\subsection{NeoRL and Other Domains}\n\\label{sec:neorl_other_tasks}\n\nTo validate that our findings hold outside of the open-sourced experimental results provided by \\citet{qinNeoRLRealWorldBenchmark2021}, and to cover a wider range of decision-making problems, we benchmark CQL \\cite{kumarConservativeQLearningOffline2020}, TD3+BC \\cite{fujimotoMinimalistApproachOffline2021}, and BC on the CityLearn \\cite{vazquez-canteliCityLearnV1OpenAI2019}, FinRL \\cite{liuFinRLDeepReinforcement2020}, and Industrial Benchmark \\cite{heinBenchmarkEnvironmentMotivated2017} environments\\footnote{Detailed descriptions of the environments and algorithms used can be found in the Appendix \\ref{appendix:sec:hyperparams}, \\ref{appendix:sec:envs_bases_datasets}.}. In addition, we make sure that the hyperparameter search budgets are equal for all the algorithms to avoid the issue described in the previous section. The hyperparameter grids were deferred to the Appendix \\ref{appendix:sec:hyperparams}. We average mean returns over 100 evaluation trajectories and 3 seeds.\n\nIn Figure \\ref{fig:eop_other_domains}, we see that Behavioral Cloning is quite competitive against both CQL and TD3+BC under limited online evaluation budgets. This akin to the results we observed in robotics environments (Figure \\ref{fig:eop_neorl}), suggesting that BC is often more preferable in restricted settings to deep ORL algorithms.\n\n\n\\subsection{Offline Policy Selection}\n\\label{sec:case_ops}\n\nAs the EOP incorporates offline policy selection, we can also use it to compare how well OPS methods perform against each other.\nTo do so, we use inverse of normalized Regret@K (in our case at $B$) as a target metric. This allows us to answer the following question: \"If we were able to run policies corresponding to $k$ hyperparameter settings in the actual environment and get reliable estimates for their values that way, how far would the best in the set we picked be from the best of all hyperparameter settings considered?\" \\cite{paineHyperparameterSelectionOffline2020}. But instead of reporting one value of $k$ as in \\citet{paineHyperparameterSelectionOffline2020}, we can easily report on all the values of $B$. Moreover, the estimator outlined in Section \\ref{sec:eop_math} can be used for presenting the results of uniform policy selection.\n\nWe do not aim to benchmark and compare the entire myriad of offline policy selection approaches, but rather to demonstrate how one can use the proposed tool for such purposes. To do so, we test several methods on the environments from the previous section, namely $V(s_0)$ using FQE, TD-Error, and Action Difference \\cite{leBatchPolicyLearning2019,hussenotHyperparameterSelectionImitation2021}.\n\nThe results can be found in Figure \\ref{fig:eop_ops}. First, we observe a pattern similar to the one described in Sections \\ref{sec:online-eval-matters}, \\ref{sec:neorl_robotic_tasks}, \\ref{sec:neorl_other_tasks}: \\textit{preference between offline policy selection methods is also budget-dependent}. Therefore, it is not enough to report the result of such methods under just one selected threshold.\nSecond, there is no clear winner among all setups, as even TD-Error may sometimes perform good (Figure \\ref{fig:eop_ops_a}, for more -- check the Appendix). However, Action Difference often performs reasonably well across many dataset sizes and policy levels in Industrial Benchmark and CityLearn environments (Figures \\ref{fig:eop_ops_a}, \\ref{fig:eop_ops_b}). We hypothesize that such a selection method can serve as a post-training conservative regularizator (i.e. picking policies that are more similar to the behavioral ones), but that requires further investigation.\nAnd the last notable observation is that uniform policy selection can be competitive to other considered methods, especially when the online evaluation budget is limited. Sometimes it can even perform the best among the entire set of methods (Figure \\ref{fig:eop_ops_a}).\n\n \n\\section{Related Work}\nTo the best of our knowledge, the closest concept to our work is deployment-constrained RL \\cite{matsushimaDeploymentEfficientReinforcementLearning2020, suMUSBOModelbasedUncertainty2021}. The core idea of this setting is to consider the number of policies deployed online and to reuse the data for iterative training from such deployments. \\citet{matsushimaDeploymentEfficientReinforcementLearning2020,suMUSBOModelbasedUncertainty2021} propose new algorithms that are especially suited for this setting, claiming that they are more deployment-efficient. However, they also relied on an extensive hyperparameter search reporting for the best set of hyperparameters. This hides the actual number of policies evaluated online, while our approach prevents that. An interesting direction for future work would be to adapt the EOP for this iterative setting as well.\n\n\\citet{konyushkovaActiveOfflinePolicy2021} formulated a new problem, which is an extension to OPS, Active Offline Policy Selection (A-OPS). The major difference is to allow for an OPS method to have a feedback loop from newly trained policies deployed online, and re-adjust which policy should be run next. While this is an important step forward, EOP can actually subsume A-OPS as one of the OPS methods, since sequential policy testing is allowed. Furthermore, our paper aims to consolidate all the parts of a deep ORL pipeline, while \\citet{konyushkovaActiveOfflinePolicy2021} focuses on a new problem.\n\n\nRecently, \\citet{agarwalDeepReinforcementLearning2022} scrutinized the evaluation methodology of deep RL algorithms, and advocated for a set of statistical tools to be employed for more reliable comparison. However, \\citet{agarwalDeepReinforcementLearning2022} focuses on reliable evaluation \\textit{after hyperparameter tuning}, while our work highlights its importance in offline deep RL setting, and argues that results should be reported under varied hyperparameter tuning capacity when comparing deep ORL algorithms.\n\n\\citet{brandfonbrenerQuantileFilteredImitation2021} noted the extensive online evaluation budgets used in recent works on deep ORL. To address this issue, comparison between algorithms was made under a small hyperparameter tuning budget ($B=4$). However, evaluating only under a limited budget may not be enough. Our paper demonstrates that the preference between algorithms can be budget-dependent, requiring evaluation under various budgets.\n\nThere is a sizeable body of work on Offline Policy Evaluation \\cite{voloshinEmpiricalStudyOffPolicy2020, fuBenchmarksDeepOffPolicy2021} and, specifically, on Offline Policy Selection \\cite{paineHyperparameterSelectionOffline2020, hussenotHyperparameterSelectionImitation2021, yangOfflinePolicySelection2020}. This work does not aim to compare or benchmark these types of methods, but to provide a procedure for comparing deep ORL algorithms with OPS (and promote the usage of simple uniform selection) for achieving reliable conclusions. In addition, we demonstrate that a similar pattern, online-budget dependence, is relevant for OPS methods as well.\n\nBehavioral Cloning is typically reported in deep ORL papers as a baseline, and many papers claim to beat this baseline \\cite{kumarConservativeQLearningOffline2020, fujimotoMinimalistApproachOffline2021, kumarStabilizingOffPolicyQLearning2019, leBatchPolicyLearning2019, wangCriticRegularizedRegression2020, wuBehaviorRegularizedOffline2019}. However, when considering learning from human demonstrations, \\citet{mandlekarWhatMattersLearning2021} demonstrated the superiority of BC over deep ORL algorithms, especially when recurrence is employed in network architecture. Reinforcing the effectiveness of BC, this work suggests that BC is preferable not only in settings with human demonstrations, but across a diverse range of decision-making problems, given that the online evaluation budget is severely limited.\n\n\n\n\n\n\\section{Closing Remarks}\n\n\n\n\n\n\n\n\n\\textbf{Motivation}:\nWhile a lot of RL community efforts are focused on offline RL datasets' general evaluation, this paper questions the methods' performance under the entire spectrum of their hyperparameters and available resources.\nAs different problems and contexts may allow for different online evaluation budgets, we argue that they can also make different optimal solutions possible within these constraints.\nWe hope that the proposed evaluation technique and our findings will encourage the ORL community to report performance results under different online evaluation budgets.\n\n\n\\textbf{Limitations}:\nAlthough this work emphasizes the importance of several online evaluations, it does not investigate the possible evaluation risks. Many real-world applications have critical corner cases, especially in the autonomous driving, healthcare, and finance domains \\cite{10.1007\/s10664-020-09881-0}. We note that the EOP has limited applicability in such risk-sensitive scenarios due to its focus on maximum performance.\n\n\n\\textbf{Opportunities}:\nEOP proposes a unified methodology for finding the best-performing setup under different online budget constraints. Unlike Deep Learning domains, the deep ORL evaluation is still in active development. The possibility of having a standard performance evaluation report opens avenues for adopting more precise methods for different tasks and contexts or creating online budget-dependent ORL algorithms.\n\n\n\n\n\n\n\\section{Conclusion}\n\nA lot of community effort was recently devoted to developing new algorithms and datasets, while noting that the whole evaluation pipeline is still to be improved upon \\cite{fuD4RLDatasetsDeep2021, gulcehreRLUnpluggedSuite2021}. In this work, we demonstrated (Section \\ref{sec:online-eval-matters}) one of the problems with such pipelines, and proposed a technique named EOP (Section \\ref{sec:eop_math}) to address it when presenting the results of deep ORL algorithms. \nSeveral empirical results were found (Section \\ref{sec:case_studies}): (1) Behavioral Cloning is often more\nfavorable under a limited evaluation budget, (2) Online Policy Selection method preferences are also budget-dependent.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nQR models have become increasingly popular since the seminal work of\n\\cite{koenker1978regression}. In contrast to the mean regression\nmodel, QR belongs to a robust model family, which can give an\noverall assessment of the covariate effects at different quantiles\nof the outcome \\citep{koenker2005quantile}. In particular, we can\nmodel the lower or higher quantiles of the outcome to provide a\nnatural assessment of covariate effects specific for those\nregression quantiles. Unlike conventional models, which only address\nthe conditional mean or the central effects of the covariates, QR\nmodels quantify the entire conditional distribution of the outcome\nvariable. In addition, QR does not impose any distributional\nassumption on the error, except requiring  the error to have a zero\nconditional quantile. The foundations of the methods for independent\ndata are now consolidated, and some statistical methods for\nestimating and drawing inferences about conditional quantiles are\nprovided by most of the available statistical programs (e.g., R,\nSAS, Matlab and Stata). For instance, just to name a few, in the\nwell-known R package \\verb\"quantreg()\" is implemented  a variant of\nthe \\cite{barrodale1977algorithms} simplex (BR) for linear\nprogramming problems described in \\cite{koenker1987algorithm}, where\nthe standard errors are computed by the rank inversion method\n\\citep{koenker2005quantile}. Another method implemented in this\npopular package is Lasso Penalized Quantile Regression (LPQR),\nintroduced by \\cite{tibshirani1996regression}, where a penalty\nparameter is specified to determine how much shrinkage occurs in the\nestimation process. QR can be implemented in a range of different\nways. \\cite{koenker2005quantile} provided an overview of some\ncommonly used quantile regression techniques from a\n\"classical\" framework.\n\n\\cite{kottas2001bayesian} considered\nmedian regression from a Bayesian point of view, which is a special case of quantile regression,\nand discussed non-parametric modeling for the error distribution\nbased on either P\\'{o}lya tree or Dirichlet process priors. Regarding\ngeneral quantile regression, \\cite{yu2001bayesian} proposed a\nBayesian modeling approach by using the ALD,\n\\cite{kottas2009bayesian} developed Bayesian semi-parametric models\nfor quantile regression using Dirichlet process mixtures for the\nerror distribution, {\\cite{geraci07} studied quantile regression\nfor longitudinal data using the ALD.} Recently, \\cite{kozumi2011gibbs} developed\na simple and efficient Gibbs sampling algorithm for fitting the quantile regression model based on a location-scale mixture representation of the ALD.\n\nAn interesting aspect to be considered in statistical modelling is\nthe diagnostic analysis.  This can be carried out by conducting an\ninfluence analysis for detecting influential observations.  One   of\nthe   most technique to detect influential observations is  the\ncase-deletion approach. The famous approach of Cook (1977)  has been\napplied extensively to assess the influence of an observation in\nfitting a statistical model; see \\cite{cook82} and the references\ntherein.   It is difficult to apply this approach directly to the QR\nmodel  because the underlying observed-data likelihood function is\nnot differentiable at zero.  \\cite{zhu2001case} presents an approach\nto perform diagnostic analysis for general statistical models that\nis  based on  the  Q-displacement function. This approach  has  been\napplied successfully to perform influence analysis in several\nregression models, for example, \\cite{xie2007case}  considered  in\nmultivariate  $t$   distribution,\n\\cite{Matos.Lachos.Bala.Labra.2012} obtained case-deletion measures\nfor mixed-effects models following the \\cite{zhu2001case}'s approach\nand   in  \\cite{Zeller.Labra.Lachos.Balakrishnan.2010}  we  can  see\nsome  results about   local influence for mixed-effects models\nobtained by using  the Q-displacement function.\n\nTaking advantage of the likelihood structure imposed by the ALD, the\nhierarchical representation of the ALD,   we  develop here  an\nEM-type algorithm for obtaining the ML estimates at the $p$th level,\nand  by simulation studies our EM algorithm outperformed the\ncompeting  BR and LPQR algorithms, where  the standard error is\nobtained as a by-product. Moreover, we obtain    case-deletion\nmeasures for the QR model. Since QR methods complement and improve\nestablished means regression models, we feel that the assessment of\nrobustness aspects of the parameter estimates in QR is also an\nimportant concern at a given quantile level $p\\in(0, 1)$.\n\n\n\n\\indent The rest of the paper is organized as follows. Section 2\nintroduces the connection between QR and ALD as well as outlining\nthe main results related to ALD.  Section 3 presents  an EM-type\nalgorithm to proceed with ML estimation for the parameters at the\n$p$th level. Moreover,  the observed information matrix  is derived.\nSection \\ref{Sec Diagnostic}  provides a brief sketch  of the case-deletion method\nfor the model with incomplete data, and also develop a methodology pertinent to the ALD. Sections \\ref{sec application}\nand \\ref{sec simulation study} are dedicated to the analysis of real\nand simulated data sets, respectively. Section 6 concludes with a\nshort discussion of issues raised by our study and some possible\ndirections for the future research.\n\n\\section{The quantile regression model} \\label{sec tCR}\nEven though considerable amount of work has been done on  regression  models  and their extensions,  regression  models   by   using asymmetric Laplace distribution    have received little attention in the literature. Only recently, the a   study on quantile regression  model   based  on asymmetric Laplace distribution\n was presented by  Tian et al. (2014) who a derived several interesting\nand attractive properties and   presented   an EM  algorithm. Before\npresenting our derivation, let us recall firstly  the definition of\nthe asymmetric Laplace distribution and   after  this,  we    will\npresent  the  quantile regression model.\n\n\\subsection{Asymmetric Laplace distribution}\nAs discussed in \\cite{yu2001bayesian}, we say that a random variable\nY is distributed as an ALD with location parameter $\\mu$, scale\nparameter $\\sigma>0$ and skewness parameter $p\\in (0,1)$, if its\nprobability density function (pdf) is given by\n\\begin{equation}\\label{pdfAL}\nf(y|\\mu,\\sigma,p)=\\frac{p(1-p)}{\\sigma}\\exp\\Big\n\\{-\\rho_p\\big(\\frac{y-\\mu}{\\sigma}\\big)\\Big\\},\n\\end{equation}\nwhere $\\rho_p(.)$ is the so called check (or loss) function defined\nby $\\rho_p(u)=u(p-\\mathbb{I}\\{u<0\\})$, with   $\\mathbb{I}\\{.\\}$\ndenoting the usual indicator function.   This  distribution is\ndenoted by $ALD(\\mu,\\sigma,p)$. It is easy to see that\n$W=\\rho_p\\big(\\frac{Y-\\mu}{\\sigma}\\big)$ follows an exponential\ndistribution $\\exp(1)$.\n\nA  stochastic representation   is  useful to  obtain  some  properties  of\nthe  distribution, as  for example, the moments, moment generating function (mgf), and estimation algorithm.  For the ALD     \\cite{kotz2001laplace},  \\cite{kozobowski00} and \\cite{zhou13} presented  the  following  stochastic representation:\nLet $U\\sim\n{\\exp}(\\sigma)$ and $Z\\sim N(0,1)$ be two independent random variables. Then,\n$Y\\sim ALD(\\mu,\\sigma,p)$ can be   represented as\n\\begin{equation}\\label{st-ALD}\nY\\buildrel d\\over=\\mu+\\vartheta_p U+\\tau_p\\sqrt{\\sigma U} Z,\n\\end{equation}\nwhere $\\vartheta_p=\\frac{1-2p}{p(1-p)}$ and\n$\\tau^2_p=\\frac{2}{p(1-p)}$,  and $\\buildrel d\\over=$\ndenotes equality in distribution.\n\\indent Figure \\ref{fig:ald}  shows how the skewness of the ALD\nchanges with altering values for $p$. For example, for $p=0.1$\nalmost all the mass of the ALD is situated in the right tail. For $p=0.5$, both tails of the ALD have equal mass and the  distribution then equals the more common double exponential\ndistribution. In contrast to the normal distribution with a\nquadratic term in the exponent, the ALD is linear in the exponent.\nThis results in a more peaked mode for the ALD together with thicker\ntails. On the other hand, the normal distribution has heavier\nshoulders compared to the ALD.\n\\begin{figure}[!htb]\n\\begin{center}\n{\\includegraphics[scale=0.6]{ald.ps\n\\caption{Standard asymmetric Laplace density \\label{fig:ald}}}\n\\end{center}\n\\end{figure}\nFrom (\\ref{st-ALD}),  we   have the hierarchical representation of   the ALD,  see  \\cite{lum12},    given  by\n\\begin{eqnarray}\nY|U=u&\\sim& N(\\mu+\\vartheta_p u,\\tau^2_p\\sigma u ),\\label{hierar1}\\\\\nU&\\sim& exp(\\sigma)\\label{hierar2}.\n\\end{eqnarray}\nThis  representation   will   be   useful   for the  implementation  of  the  EM  algorithm. Moreover,  since   $Y|U=u\\sim N(\\mu+\\vartheta_p u,\\tau^2_p\\sigma u )$, then one  can  derive   easily   the  pdf  of $Y$. That is,  the   pdf in  ( \\ref{pdfAL})   can  be   expressed   as\n\\begin{equation}\\label{pdfALs}\nf(y|\\mu,\\sigma,p)=\\frac{1}{\\sqrt{2\\pi}}\\frac{1}{\\tau_p\\sigma^{\\frac{3}{2}}} \\exp\\Big(\\frac{\\delta(y)}{\\gamma}\\Big)\nA(y),\n\\end{equation}\nwhere $\\delta(y)=\\frac{|y-\\mu|}{\\tau_p\\sqrt{\\sigma}}$, $\\gamma=\\sqrt{\\frac{1}{\\sigma}\\big(2+\\frac{\\vartheta_p^2}{\\tau^2_p}\\big)}=\\frac{\\tau_p}{2\\sqrt{\\sigma}}$ and   $A(y)=2\\Big(\\frac{\\delta(y)}{\\gamma}\\Big)^{1\/2} K_{1\/2}\\big(\\delta(y)\\gamma\\big)$, with $K_{\\nu}(.)$  being  the  modified Bessel function of the third kind.  It   easy  to  see  that\nthat the conditional distribution of $U$, given $Y=y$, is\n$U|(Y=y)\\sim GIG (\\frac{1}{2},\\delta,\\gamma)$. Here,  $GIG(\\nu, a, b)$ denotes the Generalized Inverse\nGaussian (GIG) distribution; see \\cite{barndorff2001non} for more details. The   pdf of GIG   distribution  is   given by\n$$h(u|\\nu,a,b)=\\frac{(b\/a)^{\\nu}}{2K_{\\nu}(ab)}u^{\\nu-1}\\exp\\Big\\{-\\frac{1}{2}\\big(a^2\/{u}+b^2u\\big)\\Big\\},\\,\\,u>0,\\,\\,\\,\\,\\nu\\in \\mathbb{R},\\,\\,a,b>0.$$\nThe moments of $U$  can  be  expressed  as\n$$E[U^k]=\\left(\\frac{a}{b}\\right)^{k}\\frac{K_{\\nu+k}(ab)}{K_{\\nu}(ab)},\\,\\,\\ k\\in \\mathbb{R}.$$\nSome   properties of the Bessel function of  the third kind\n$K_{\\lambda}(u)$    that will  be  useful  for the   developments\nhere  are: (i) $K_{\\nu}(u)=K_{-\\nu}(u)$; (ii)\n$K_{\\nu+1}(u)=\\frac{2\\nu}{u}K_{\\nu}(u)+K_{\\nu-1}(u)$; (iii)  for\nnon-negative integer  $r$,\n$K_{r+1\/2}(u)=\\sqrt{\\frac{\\pi}{2u}}\\exp(-u)\\sum_{k=0}^r\n\\frac{(r+k)!(2u)^{-k}}{(r-k)!k!}$. A  special   case  is  $K_{1\/2}(u)=\n\\sqrt{\\frac{\\pi}{2u}}\\exp(-u)$.\\\\\n\n\n\\subsection{ Linear quantile regression}\n\nLet $y_i$ be a response variable and $\\mathbf{x}_i$ a $k\\times 1$\nvector of covariates for the $i$th observation, and let\n$Q_{y_i}(p|\\mathbf{x}_i)$ be the $p$th $(0 < p < 1)$ quantile regression\nfunction of $y_i$ given $\\mathbf{x}_i$, $i=1,\\ldots,n$ . Suppose that the relationship\nbetween $Q_{y_i}(p|\\mathbf{x}_i)$ and $\\mathbf{x}_i$ can be modeled as $Q_{y_i}(p|\\mathbf{x}_i)\n= \\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p$, where $\\mbox{${\\bm \\beta}$}_p$ is a vector $(k\\times 1)$\nof unknown parameters of interest. Then, we consider the quantile\nregression model given by\n\\begin{equation}\\label{QRmodel}\ny_i = \\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p + \\epsilon_i,\\,\\,\\,i=1,\\ldots,n,\n\\end{equation}\nwhere $\\epsilon_i$ is the error term whose distribution (with density,\nsay, $f_p(.)$) is restricted to have the $p$th quantile equal to\nzero, that is, $\\int^{0}_{-\\infty}f_p(\\epsilon_i)d\\epsilon_i=p$.  The error density $f_p(.)$ is often left unspecified in the classical literature. Thus, quantile regression estimation for\n$\\mbox{${\\bm \\beta}$}_p$ proceeds by minimizing\n\\begin{eqnarray}\\label{lossEq}\n\\widehat{\\mbox{${\\bm \\beta}$}}_p=arg\\,\\, min_{\\mbox{${\\bm \\beta}$}_{p}} \\sum^n_{i=1}\n\\rho_p\\big({y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p}\\big),\n\\end{eqnarray}\nwhere $\\rho_p(.)$ is as in (\\ref{pdfAL}) and $\\widehat{\\mbox{${\\bm \\beta}$}}_p$ is\nthe quantile regression estimate for $\\mbox{${\\bm \\beta}$}_p$ at the $p$th\nquantile. The  special  case  $p=0.5$ corresponds to median\nregression. As the check function is not differentiable at zero,\nwe cannot derive explicit solutions to the minimization problem.\nTherefore, linear programming methods are commonly applied to obtain\nquantile regression estimates for $\\mbox{${\\bm \\beta}$}_p$.  A connection between\nthe minimization of the sum in (\\ref{lossEq}) and the\nmaximum-likelihood theory is provided by the ALD; see \\cite{geraci07}. It is also true that  under\nthe   quantile   regression model,  we   have\n\\begin{equation}\\label{Wi}\nW_i=\\frac{1}{\\sigma}\\rho_p\\big(y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p\\big)\\sim \\exp(1).\n\\end{equation}\nThe  above  result is  useful  to check the model in practice, as\nwill be seen in the Application Section.\n\nNow, suppose $y_1, \\ldots,y_n$ are   independent observations such  as $Y_i \\sim ALD (\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p,\\sigma,p),$\n$i=1,\\ldots,n$. Then,  from (\\ref{pdfALs})  the\nlog--likelihood function for $\\mbox{${ \\bm \\theta}$}=(\\mbox{${\\bm \\beta}$}_p^{\\top},\\sigma)^{\\top} $ can be  expressed  as\n\\begin{equation}\\label{likel}\n\\ell(\\mbox{${ \\bm \\theta}$})=\\sum^n_{i=1} \\ell_i(\\mbox{${ \\bm \\theta}$}),\n\\end{equation}\nwhere\n$\\ell_i(\\mbox{${ \\bm \\theta}$})=c-\\frac{3}{2}\\log{\\sigma}+\\frac{\\vartheta_p}{\\tau_p^2\\sigma}(y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p)+\\log(A_i)$,\nwith  $c$ is a constant does not   depend on $\\mbox{${ \\bm \\theta}$}$ and\n$A_i=2\\big({\\frac{\\delta_i}{\\gamma}}\\big)^{1\/2}K_{1\/2}(\\lambda_i)=\\frac{\\sqrt{2\\pi}}{\\gamma}\\exp(-\\lambda_i),$\nwith $\\delta_i=\\delta(y_i)={|y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p|}\/{\\tau_p\\sqrt{\\sigma}})$   and $\\lambda_i=\\delta_i\\gamma$.\n\nNote that if we consider $\\sigma$ as a nuisance parameter, then the\nmaximization of the likelihood in (\\ref{likel}) with respect to the\nparameter $\\mbox{${\\bm \\beta}$}_p$ is equivalent to the minimization of the\nobjective function in (\\ref{lossEq}). and  hence the relationship\nbetween the check function and ALD can be used to reformulate the QR\nmethod in the likelihood framework.\n\nThe log--likelihood function is not differentiable at zero.\nTherefore,  standard  procedures  the estimation      can not   be\ndeveloped   following  the  usual way. Specifically, the  standard\nerrors  for    the   maximum likelihood    estimates is  not   based\non    the     genuine information matrix. To  overcome   this\nproblem   we   consider the  empirical information matrix as    will\nbe   described in the  next  Subsection.\n\n\n\n\n\n\\subsection{Parameter estimation via the EM algorithm}\nIn this section, we discuss  an  estimation method  for QR  based on\nthe EM algorithm to obtain ML estimates. Also, we consider the\nmethod of moments (MM) estimators,which can be effectively used as\nstarting values in the EM algorithm.\nHere,  we show how to employ the EM  algorithm\nfor ML estimation in QR model under the ALD. From the hierarchical\nrepresentation (\\ref{hierar1})-(\\ref{hierar2}), the QR model in\n(\\ref{QRmodel}) can be presented  as\n\\begin{eqnarray}\nY_i|U_i=u_i&\\sim& N(\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p+\\vartheta_p\\label{repHier1}\nu_i,\\tau_p^2\\sigma u_i),\\\\ U_i&\\sim&\n\\exp(\\sigma),\\,\\,\\,\\,i=1,\\ldots,n,\\label{repHier2}\n\\end{eqnarray}\nwhere $\\vartheta_p$ and $\\tau_p^2$  are as in (\\ref{st-ALD}). This\nhierarchical representation of the QR model is  convenient to  describe\nthe steps  of the   EM algorithm. Let $\\mathbf{y} = (y_1, \\ldots, y_n)$ and  $\\mathbf{u} = (u_1,\\ldots , u_n)$ be\nthe observed data and   the missing data, respectively. Then,   the\ncomplete data log-likelihood function of\n$\\mbox{${ \\bm \\theta}$}=(\\mbox{${\\bm \\beta}$}^{\\top}_p,\\sigma)^{\\top}$, given $(\\mathbf{y},\\mathbf{u})$,\nignoring additive constant terms, is given by\n$\\ell_{c}(\\mbox{${ \\bm \\theta}$}|{\\bf y},\\mathbf{u})=\\sum_{i=1}^{n}\\ell_{c}(\\mbox{${ \\bm \\theta}$}|y_i,u_i)$,\nwhere\n\\begin{eqnarray*}\n\\ell_{c}(\\mbox{${ \\bm \\theta}$}|y_i,u_i) =-\\frac{1}{2} \\log( 2\\pi\\tau_p^2)\n -\\frac{3}{2} \\log(\\sigma)   -\\frac{1}{2}\\log (u_i) -  \\frac{1}{2\\sigma\\tau_p^2}\n{u^{-1}_i}(y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p-\\vartheta_p u_i)^2 -\n\\frac{1}{\\sigma} u_i,\n\\end{eqnarray*}\nfor $i=1,\\ldots,n$.  In what follows the superscript $(k)$ indicates\nthe estimate of the related parameter  at the stage $k$ of the\nalgorithm. The E-step of the EM algorithm requires evaluation of the\nso-called Q-function $Q(\\mbox{${ \\bm \\theta}$}|\\mbox{${ \\bm \\theta}$}^{(k)}) =\n\\textrm{E}_{\\scriptsize \\mbox{${ \\bm \\theta}$}^{(k)}}[\\ell_{c}(\\mbox{${ \\bm \\theta}$}|\\mathbf{y},\\mathbf{u})|{\\bf y}, \\mbox{${ \\bm \\theta}$}^{(k)}]$, where\n$\\textrm{E}_{\\scriptsize \\mbox{${ \\bm \\theta}$}^{(k)}}[.]$ means that the\nexpectation is being effected using $\\mbox{${ \\bm \\theta}$}^{(k)}$ for $\\mbox{${ \\bm \\theta}$}$.\nObserve that the expression of the Q-function is completely\ndetermined by the knowledge of the expectations\n\\begin{eqnarray} \\label{weith}\n{\\cal E}_{ s i}(\\mbox{${ \\bm \\theta}$}^{(k)}) =  \\textrm{E}[U^s_i |y_i, \\mbox{${ \\bm \\theta}$}^{(k)}],\\,\\,\\, s=-1,1,\n\\end{eqnarray}\nthat  are obtained of   properties of   the $GIG(0.5, a, b)$ distribution. Let   $\\mbox{${ \\bm \\xi}$}^{(k)} _{s}=\\big( {\\cal\nE}_{s1}(\\mbox{${ \\bm \\theta}$}^{(k)}), \\ldots, {\\cal E}_{sn}(\\mbox{${ \\bm \\theta}$}^{(k)})\n\\big)^{\\top}$ be the vector  that  contains  all   quantities   defined\nin (\\ref{weith}). Thus, dropping unimportant constants,    the\nQ-function can be written in a synthetic form as\n$Q(\\mbox{${ \\bm \\theta}$}|\\widehat{\\mbox{${ \\bm \\theta}$}})=\\sum_{i=1}^{n}Q_i(\\mbox{${ \\bm \\theta}$}|\\widehat{\\mbox{${ \\bm \\theta}$}})$,\nwhere {\\small{\n\\begin{eqnarray} \\label{eqn qfunction}\nQ_i(\\mbox{${ \\bm \\theta}$}|\\widehat{\\mbox{${ \\bm \\theta}$}}) =\n-\\frac{3}{2}\\log\\sigma-\\frac{1}{2\\sigma\\tau_p^2}\n\\left[ {\\cal\nE}_{-1i}(\\mbox{${ \\bm \\theta}$}^{(k)})(y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p)^2-2(y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}_p)\\vartheta_p+\\frac{1}{4}{\\cal E}_{1\ni}(\\mbox{${ \\bm \\theta}$}^{(k)})\\tau_p^4\\right]. \\, \\, \\, \\, \\,\n\\end{eqnarray}}}\nThis  quite useful expression to implement the M-step, which consists of  maximizing it over $\\mbox{${ \\bm \\theta}$}$. So the  EM algorithm   can be summarized as  follows:\\\\\n\\noindent \\emph{E-step}: Given $\\mbox{${ \\bm \\theta}$}=\\mbox{${ \\bm \\theta}$}^{(k)}$,  compute ${\\cal E}_{si}(\\mbox{${ \\bm \\theta}$}^{(k)})$ through  of the   relation\n\\begin{equation}\\label{deltai}\n{\\cal E}_{ si}(\\mbox{${ \\bm \\theta}$}^{(k)}) =E[U^s_i|y_i,\\mbox{${ \\bm \\theta}$}^{(k)}]=\\left(\\frac{\\delta^{(k)}_i}{\\gamma^{(k)}}\\right)^{s}\\frac{K_{1\/2+s}\\big(\\lambda^{(k)}_i\\big)}{K_{1\/2}\\big(\\lambda^{(k)}_i\\big)}, s=-1,1,\n\\end{equation}\nwhere\n$\\delta^{(k)}_i=\\frac{|y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}^{(k)}_p|}{\\tau_p\\sqrt{\\sigma^{(k)}}}$,  $\\gamma^{(k)}=\\frac{\\tau_p}{2\\sqrt{\\sigma^{(k)}}}$ and\n$\\lambda^{(k)}_i={\\delta^{(k)}_i\\gamma^{(k)}}$;\\\\\n\\noindent{\\em  M-step}: Update ${\\mbox{${ \\bm \\theta}$}}^{(k)}$ by\nmaximizing $Q(\\mbox{${ \\bm \\theta}$}|\\mbox{${ \\bm \\theta}$}^{(k)})$  over $\\mbox{${ \\bm \\theta}$}$,\nwhich leads to the following expressions\n{\\small{\n\\begin{eqnarray*}\n{\\mbox{${\\bm \\beta}$}}^{(k+1)}_p&=&\\left(\\mathbf{X}^{\\top} D(\\mbox{${ \\bm \\xi}$}^{(k)} _{-1})\\mathbf{X} \\right)^{-1}\\mathbf{X}^{\\top}\\big(D(\\mbox{${ \\bm \\xi}$}^{(k)} _{-1})\\mathbf{Y}-\\vartheta_p {\\bf 1}_n\\big),\\, \\, \\\\\n{\\sigma}^{(k+1)}&=&\\frac{1}{3n\\tau^2_p}\\Big[Q(\\mbox{${\\bm \\beta}$}^{(k+1)}, \\mbox{${ \\bm \\xi}$}_{-1}^{(k)})-2{\\bf 1}^{\\top}_n(\\mathbf{Y}-\\mathbf{X}\n\\mbox{${\\bm \\beta}$}^{(k+1)})\\vartheta_p+\\frac{\\tau_p^4}{4}{\\bf\n1}^{\\top}_n\\mbox{${ \\bm \\xi}$}^{(k)} _{1}\\Big],\n\\end{eqnarray*}}}\nwhere $D(\\mathbf{a})$ denotes the diagonal matrix, with the diagonal\nelements given by $\\mathbf{a}=(a_1,\\ldots,a_p)^{\\top}$ and\n$Q(\\mbox{${\\bm \\beta}$}, \\mbox{${ \\bm \\xi}$}_{-1})= (\\mathbf{Y}-\\mathbf{X} \\mbox{${\\bm \\beta}$})^{\\top}D(\\mbox{${ \\bm \\xi}$}_{-1}) (\\mathbf{Y}-\\mathbf{X}\\mbox{${\\bm \\beta}$})$. A similar  expression  for  $\\mbox{${\\bm \\beta}$}^{(k+1)}_p$ is  obtained in  \\cite{tian13}.\nThis process is iterated until some distance involving two\nsuccessive evaluations of the actual log-likelihood $\\ell(\\mbox{${ \\bm \\theta}$})$,\nlike $||\\ell({\\mbox{${ \\bm \\theta}$}}^{(k+1)})-\\ell({\\mbox{${ \\bm \\theta}$}}^{(k)})||$ or\n$||\\ell({\\mbox{${ \\bm \\theta}$}}^{(k+1)})\/\\ell({\\mbox{${ \\bm \\theta}$}}^{(k)})-1||$, is small\nenough. This algorithm is implemented as part of the R package\n\\verb\"ALDqr()\",  which can be downloaded at not cost from the\nrepository CRAN. Furthermore, following  the  results  given in\n\\cite{Yu2005}, the MM estimators for $\\mbox{${\\bm \\beta}$}_p$ and $\\sigma$ are\nsolutions of the following equations:\n\\begin{eqnarray}\\label{ab;m}\n\\widehat{\\mbox{${\\bm \\beta}$}}_{pM}=\\big(\\mathbf{X}^{\\top}\n\\mathbf{X}\\big)^{-1}\\mathbf{X}^{\\top}\\big(\\mathbf{Y}-\\widehat{\\sigma}_M \\vartheta_p{\\bf\n1}_n\\big)\\, \\, \\, \\,{\\rm and} \\, \\,\n\\widehat{\\sigma}_{M}=\\displaystyle  \\frac{1}{n}\\sum_{i=1}^n\n\\rho_p\\big(y_i-\\mathbf{x}_i^{\\top}\\widehat{\\mbox{${\\bm \\beta}$}}_{pM}\\big),\n\\end{eqnarray}\nwhere $\\vartheta_p$  is as (\\ref{st-ALD}). Note that the\nMM estimators do not have explicit closed form and numerical\nprocedures are needed to solve these non-linear equations. They can be used  as  initial values in the iterative procedure for computing the ML estimates  based on the EM-algorithm.\nStandard   errors  for    the   maximum  likelihood    estimates is  based  on\nthe  empirical information matrix, that  according to \\cite{meilijson89} formula, is defined as\n\\begin{eqnarray}\\label{Imatrix }\n\\mathbf{L}(\\mbox{${ \\bm \\theta}$})=\\sum_{j=1}^{n}\\textbf{s}(y_j|\\mbox{${ \\bm \\theta}$})\\textbf{s}^\\top(y_j|\\mbox{${ \\bm \\theta}$})-n^{-1}\\textbf{S}(y_j|\\mbox{${ \\bm \\theta}$})\\textbf{S}^\\top(y_j|\\mbox{${ \\bm \\theta}$}),\n\\end{eqnarray}\nwhere $\\textbf{S}(y_j|\\mbox{${ \\bm \\theta}$})=\\sum_{j=1}^{n}\\textbf{s}(y_j|\\mbox{${ \\bm \\theta}$})$. It is noted from the result of \\cite{louis82} that the individual score can be determined as\n$\\textbf{s}(y_j|\\mbox{${ \\bm \\theta}$}) ={\\partial \\log f(y_j|\\mbox{${ \\bm \\theta}$})}\/{\\partial \\mbox{${ \\bm \\theta}$}} = E\\Big({\\partial \\ell_{c_j}(\\mbox{${ \\bm \\theta}$}|y_j, u_i)}\/{\\partial \\mbox{${ \\bm \\theta}$}} |  y_j,\\mbox{${ \\bm \\theta}$}\\Big)$.  Asymptotic\nconfidence intervals and tests of the parameters at the $p$th\nlevel can be obtained assuming that the ML estimator\n$\\widehat{\\mbox{${ \\bm \\theta}$}}$ has approximately a normal  multivariate distribution. \\\\\n\nFrom  the EM algorithm, we  can see that $ {\\cal E}_{-1 i}(\\mbox{${ \\bm \\theta}$}^{(k)})$ is  inversely proportional to $d_i=|y_i-\\mathbf{x}^{\\top}_i\\mbox{${\\bm \\beta}$}^{(k)}_p|\/\\sigma$. Hence, $u_i(\\mbox{${ \\bm \\theta}$}^{(k)})= {\\cal E}_{-1 i}(\\mbox{${ \\bm \\theta}$}^{(k)})$   can be\ninterpreted as a type of weight for the $i$th case in the estimates\nof $\\mbox{${\\bm \\beta}$}^{(k)}_p$, which   tends to be small for   outlying\nobservations. The behavior of    these weights can be used as tools for\nidentifying outlying observations   as  well as  for  showing that    we   are   considering a robust approach,  as will be seen in Sections 4 and 5.\n\n\\section{Case-deletion measures} \\label{Sec Diagnostic}\nCase-deletion is a classical  approach to study the effects of\ndropping the $i$th case from the data set. Let $\\mathbf{y}_{c}=(\\mathbf{y},\\mathbf{u})$ be  the augmented data set,   and  a quantity\nwith a subscript ``$[i]$''  denotes the original one with the $i$th\nobservation deleted. Thus, The  complete-data log-likelihood function\nbased on  the data with the $i$th case deleted will be denoted by\n$\\ell_{c}(\\mbox{${ \\bm \\theta}$}|\\mathbf{y}_{c[i]})$.  Let\n$\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}=(\\widehat{\\mbox{${\\bm \\beta}$}}^{\\top}_{p[i]},\n\\widehat{\\sigma^2}_{[i]})^{\\top}$ be the  maximizer of the function\n{{ $Q_{[i]}(\\mbox{${ \\bm \\theta}$}|\\widehat{\\mbox{${ \\bm \\theta}$}})=\n\\textrm{E}_{\\scriptsize{\\widehat{\\mbox{${ \\bm \\theta}$}}}}\\left[\\ell_{c}(\\mbox{${ \\bm \\theta}$}|\\mathbf{Y}_{c[i]})|\\mathbf{y}\n\\right] $}},  where $\\widehat{\\mbox{${ \\bm \\theta}$}}=(\\widehat{\\mbox{${\\bm \\beta}$}}^{\\top},\n\\widehat{\\sigma^2})^{\\top}$ is the ML estimate of $\\mbox{${ \\bm \\theta}$}$. To\nassess the influence of the $i$th case on $\\widehat{\\mbox{${ \\bm \\theta}$}}$, we\ncompare the difference between $\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}$  and\n$\\widehat{\\mbox{${ \\bm \\theta}$}}$. If the deletion of a case seriously influences\nthe estimates, more attention needs  to  be paid to that case.\nHence, if $\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}$ is far from  $\\widehat{\\mbox{${ \\bm \\theta}$}}$\nin some sense, then the $i$th case is regarded as influential. As\n$\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}$ is needed for every case, the required\ncomputational  effort can be quite heavy, especially when the sample\nsize  is large. Hence, To calculate the case-deletion estimate $\\widehat{\\mbox{${ \\bm \\theta}$}}^1_{[i]}$ of $\\mbox{${ \\bm \\theta}$}$,  \\citep[see][]{zhu2001case} proposed the following one-step  approximation based on the Q-function,\n\\begin{eqnarray}\\label{theta1}\n\\widehat{\\mbox{${ \\bm \\theta}$}}^1_{[i]}= \\widehat{\\mbox{${ \\bm \\theta}$}}+ \\big\\{\n-\\ddot{Q}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\big\\}^{-1}\n\\dot{Q}_{[i]}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}}),\n\\end{eqnarray}\nwhere\n\\begin{eqnarray} \\label{eqn Hessian Matrix and Grad}\n\\ddot{Q}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})=\\displaystyle\\frac{\\partial^2\nQ(\\mbox{${ \\bm \\theta}$}|\\widehat{\\mbox{${ \\bm \\theta}$}})}{\\partial\\mbox{\\mbox{${ \\bm \\theta}$}}\\partial{\\mbox{${ \\bm \\theta}$}}^{\\top}}\n\\big\\vert_{\\mbox{${ \\bm \\theta}$}=\\widehat{\\mbox{${ \\bm \\theta}$}}}   \\,\\,\\,\\, \\textrm{and}\n\\,\\,\\,\\,\n  \\dot{Q}_{[i]}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})=\\displaystyle\\frac{\\partial{{Q}_{[i]}(\\mbox{${ \\bm \\theta}$}|\\widehat{\\mbox{${ \\bm \\theta}$}})}}{\\partial{\\mbox{${ \\bm \\theta}$}}}\\big\\vert_{\\mbox{${ \\bm \\theta}$}=\\widehat{\\mbox{${ \\bm \\theta}$}}},\n\\end{eqnarray}\nare  the Hessian matrix  and the gradient vector evaluated at\n$\\widehat{\\mbox{${ \\bm \\theta}$}}$, respectively. The Hessian matrix\nis an essential element in the method  developed by \\cite{zhu2001case} to obtain the measures for\ncase-deletion diagnosis. For    developing  the case-deletion measures, we  have   to   obtain   the  elements in   (\\ref{theta1}), $\\dot{Q}_{[i]}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})$ and  $\\ddot{Q}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})$.  These formulas  can be obtained quite easily from (\\ref{eqn qfunction}):\n\n\\begin{enumerate}\n\\item[1.]  The  components  of $\\dot{Q}_{[i]}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})$ are\n{\\small{\n\\begin{eqnarray*}\n\\dot{Q}_{[i]\\mbox{${\\bm \\beta}$}}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})&=& \\frac{  \\partial{Q_{[i]}({\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})}}{\\partial{\\mbox{${\\bm \\beta}$}}}\\big\\vert_{\\mbox{${ \\bm \\theta}$}=\\widehat{\\mbox{${ \\bm \\theta}$}}} ={{\\frac{1}{\\widehat{\\sigma}}}} E_{1[i]}\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n\\,\\, \\,\n\\dot{Q}_{[i]\\sigma}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})=\n\\frac{\\partial{Q_{[i]}({\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})}}{\n\\partial{\\sigma}}\\big\\vert_{\\mbox{${ \\bm \\theta}$}=\\widehat{\\mbox{${ \\bm \\theta}$}}}=-\\frac{1}{2\\widehat{\\sigma^2}}\nE_{2[i]},    \\label{sigma}\n\\end{eqnarray*}\nwhere\n\\begin{eqnarray}\n E_{1[i]} & = & \\frac{1}{\\tau_{p}^{2}} \\sum_{j\\neq i} \\left[{\\cal E}_{-1j}(\\widehat{\\mbox{${ \\bm \\theta}$}}^{(k)})(y_{j}-\\mathbf{x}_{j}^{\\top}\\widehat{\\mbox{${\\bm \\beta}$}})\\mathbf{x}_{j}-\\mathbf{x}_{j}\\vartheta_{p} \\right] \\,\\,\\,\\, \\textrm{ and} \\label{eqn E1i}  \\\\\nE_{2[i]} & = & \\sum_{j\\neq i} \\left[3\n\\widehat{\\sigma}-\\frac{1}{\\tau_{p}^{2}}  {\\cal\nE}_{-1j}(\\widehat{\\mbox{${ \\bm \\theta}$}}^{(k)})(y_{j}-\\mathbf{x}_{j}^{\\top}\\widehat{\\mbox{${\\bm \\beta}$}_p})^{2}-2(y_{j}-\\mathbf{x}_{j}^{\\top}\\widehat{\\mbox{${\\bm \\beta}$}_p})\\vartheta_{p}\n+ \\frac{1}{4}{\\cal\nE}_{1j}(\\widehat{\\mbox{${ \\bm \\theta}$}}^{(k)})\\tau_{p}^{4}\\right].\n\\label{eqn E2i}\n\\end{eqnarray}}}\n\\item[2.]  The elements   of  the  second order partial derivatives of $Q(\\mbox{${ \\bm \\theta}$}|\\widehat{\\mbox{${ \\bm \\theta}$}})$ evaluated at $\\widehat{\\mbox{${ \\bm \\theta}$}}$ are\n{{\n\\begin{eqnarray*}\n\\ddot{Q}_{\\beta}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})& = &  -\\frac{1}{\\widehat{\\sigma}\\tau_p^{2}} \\mathbf{X}^{\\top}D\\big(\\mbox{${ \\bm \\xi}$} _{-1}^{(k)}\\big)\\mathbf{X},\\nonumber\\\\\n\\ddot{Q}_{\\sigma}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\}&\n=& \\frac{3}{4\\widehat{\\sigma^2}} - \\frac{1}{2\\widehat{\\sigma^3}\\tau^2_p}\\Big[Q\\big(\\mbox{${\\bm \\beta}$},\\mbox{${ \\bm \\xi}$}^{(k)} _{-1}\\big)-2{\\bf 1}^{\\top}_n(\\mathbf{Y}-\\mathbf{X}\n\\mbox{${\\bm \\beta}$})\\vartheta_p+\\frac{\\tau_p^4}{4}{\\bf\n1}^{\\top}_n\\mbox{${ \\bm \\xi}$}^{(k)} _{1}\\Big]\n\\end{eqnarray*}}}\nand  $\\ddot{Q}_{\\beta \\sigma}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\}\n= {\\bf 0}$.\n\\end{enumerate}\nIn   the   following result,  we will  obtain the\none-step approximation of\n$\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}=(\\widehat{\\mbox{${\\bm \\beta}$}}^{\\top}_{p[i]},\n\\widehat{\\sigma}_{[i]})^{\\top}$, $i=1,\\ldots,n$ based on\n(\\ref{theta1}), viz., the relationships between  the parameter\nestimates for the full data set and the data with the $i$th case\ndeleted.\n\\begin{theorem} \\label{the;1}\nFor the QR model defined in (\\ref{repHier1}) and (\\ref{repHier2}), the\nrelationships between the parameter estimates for full data set and\nthe data with the $i$th case deleted are  as follows:\n\\begin{eqnarray*}\\label{aprox1}\n\\widehat{\\mbox{${\\bm \\beta}$}}^1_{p[i]}&=& \\widehat{\\mbox{${\\bm \\beta}$}}_p+  \\tau_p^{2} \\big(\\mathbf{X}^{\\top}D\\big(\\widehat{\\mbox{${ \\bm \\xi}$}}_{-1}\\big)\\mathbf{X}\\big)^{-1} \\textbf{E}_{1[i]}\\,\\,\\, \\, \\,{\\rm and }\\,\\, \\, \\, \\,\n\\widehat{\\sigma^2}^1_{[i]}= \\widehat{\\sigma^2} - \\frac{1}{2\\widehat{\\sigma^2}}\\Big(\\ddot{Q}_{\\sigma}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\Big)^{-1} E_{2[i]},\n\\end{eqnarray*}\nwhere  $\\textbf{E}_{1[i]}$ and   $E_{2[i]}$ are as in (\\ref{eqn E1i}) and  (\\ref{eqn E2i}), respectively.\n\\end{theorem}\nTo asses the influence of the $i$th case on the ML estimate\n$\\widehat{\\mbox{${ \\bm \\theta}$}}$, we  compare $\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}$\nand $\\widehat{\\mbox{${ \\bm \\theta}$}}$ based on  metrics,  proposed by \\cite{zhu2001case},  for measuring the\ndistance between $\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}$ and  $\\widehat{\\mbox{${ \\bm \\theta}$}}$.  For  that, we consider here the following;\n\n\\begin{enumerate}\n\\item  {\\it Generalized Cook distance}:\n\\begin{equation}\\label{GCD}\nGD_i=(\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}-\\widehat{\\mbox{${ \\bm \\theta}$}})^{\\top}\\big\\{ -\\ddot{\nQ}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\big\\}(\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}-\\widehat{\\mbox{${ \\bm \\theta}$}}),\n\\quad i=1,\\ldots,n.\n\\end{equation}\nUpon  substituting  (\\ref{theta1}) into  (\\ref{GCD}), we obtain the\napproximation\n$$GD^1_i=\\dot{Q}_{[i]}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})^{\\top}\\big\\{-\\ddot{Q}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\big\\}^{-1}\n\\dot{Q}_{[i]}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}}), \\quad i=1,\\ldots,n.$$\nAs $\\ddot{Q}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})$ is  a  diagonal matrix, one  can    obtain   easily a  type of  Generalized Cook distance for  parameters $\\mbox{${\\bm \\beta}$}$ and $\\sigma$, respectively,  as  follows\n$$GD^1_i(\\mbox{${\\bm \\beta}$})=\\dot{Q}_{[i]\\mbox{${\\bm \\beta}$}}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})^{\\top}\\big\\{-\\ddot{Q}_{\\beta}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\big\\}^{-1}\n\\dot{Q}_{[i]\\mbox{${\\bm \\beta}$}}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}}), \\quad i=1,\\ldots,n.$$\n$$GD^1_i(\\sigma)=\\dot{Q}_{[i]\\sigma}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})^{\\top}\\big\\{-\\ddot{Q}_{\\sigma}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\big\\}^{-1}\n\\dot{Q}_{[i]\\sigma}(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}}), \\quad i=1,\\ldots,n.$$\n\n\\item  {\\it  Q-distance}:  This  measure of the influence of the $i$th case is based  on  the\n$Q$-distance function, similar to the likelihood distance $LD_i$\n\\citep{cook82},  defined as\n\\begin{equation}\\label{QD}\nQD_i=2\\big\\{Q(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})-Q(\\widehat{\\mbox{${ \\bm \\theta}$}}_{[i]}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\big\\}.\n\\end{equation}\nWe can calculate an approximation of the  likelihood displacement\n$QD_i$ by  substituting  (\\ref{theta1}) into  (\\ref{QD}), resulting\nin  the following approximation $QD^{1}_i$ of $QD_i$:\n\\begin{equation*}\\label{QD1}\nQD^1_i=2\\big\\{Q(\\widehat{\\mbox{${ \\bm \\theta}$}}|\\widehat{\\mbox{${ \\bm \\theta}$}})-Q(\\widehat{\\mbox{${ \\bm \\theta}$}}^1_{[i]}|\\widehat{\\mbox{${ \\bm \\theta}$}})\\big\\}.\n\\end{equation*}\n\\end{enumerate}\n\n\\section{Application}\\label{sec application}\n\nWe illustrate the proposed methods by applying them to the\nAustralian Institute of Sport (AIS) data, analyzed by Cook and\nWeisberg (1994) in a normal regression setting. The data set\nconsists of several variables measured in $n=202$ athletes (102\nmales and 100 females). Here, we focus on body mass index (BMI),\nwhich is assumed to be explained by  lean body mass (LBM) and gender\n(SEX). Thus, we consider the following QR model:\n$$BMI_i=\\beta_0+\\beta_1 LBM_i+\\beta_2 SEX_i+\\epsilon_i,\\,\\,\\,\\,\\,i=1,\\ldots,202,$$\nwhere $\\epsilon_i$ is a zero $p$ quantile. This model can be fitted\nin the R software by using the package \\verb\"quantreg()\", where one\ncan arbitrarily use the BR or the LPQR algorithms. In order to\ncompare with our proposed EM algorithm, we carry out quantile\nregression at three different quantiles, namely $p= \\{0.1, 0.5,\n0.9\\}$ by using the ALD distribution as described in Section 2. The\nML estimates and associated standard errors were obtained by using\nthe EM algorithm and the observed information matrix described in\nSubsections 2.3, respectively. Table \\ref{table.application}\ncompares the results of our EM, BR and the LPQR estimates under the\nthree selected quantiles. The standard error of the LPQR estimates\nare not provided in the R package \\verb\"quantreg()\" and are not\nshown in Table \\ref{table.application}. From this table we can see\nthat estimates under the three methods only exhibit slight\ndifferences, as expected. However, the standard errors of our EM\nestimates are smaller than those via the BR algorithm. This suggests\nthat the EM algorithm seems to produce more accurate estimates of\nthe regression parameters at the $p$th level.\n\\begin{table}[ht!]\n\\centering\n\\caption{AIS data. Results of the parameter estimation via EM, Barrodale and Roberts (BR) and  Lasso Penalized Quantile Regression (LPQR) algorithms for three selected quantiles.}\n{\\small{\n\\begin{tabular}{ccccccc}\n\n       \\hline\\hline\n    \\multicolumn{2}{c}{} & \\multicolumn{2}{c}{EM} &  \\multicolumn{2}{c}{BR}   &  \\multicolumn{1}{c}{LPQR} \\\\\n\n        $p$ & Parameter   & MLE        &     SE   & Estimative &  SE    &Estimative \\\\\n        \\hline\n        0.1 & $\\beta_{0}$ & 9.3913     &   0.7196 & 9.3915     & 1.2631 & 9.8573     \\\\\n            & $\\beta_{1}$ & 0.1705     &   0.0091 & 0.1705     & 0.0160 & 0.1647     \\\\\n            & $\\beta_{2}$ & 0.8312     &   0.2729 & 0.8209     & 0.4432 & 0.6684     \\\\\n            & $\\sigma$    & 0.2617     &   0.0252 & 1.0991     & ------ & 1.0959    \\\\\n        \\hline\n        0.5 & $\\beta_{0}$ & 7.6480     &   0.8717 & 7.6480     & 1.1120 & 7.6480    \\\\\n            & $\\beta_{1}$ & 0.2160     &   0.0116 & 0.2160     & 0.0159 & 0.2160    \\\\\n            & $\\beta_{2}$ & 2.2499     &   0.3009 & 2.2226     & 0.4032 & 2.2226    \\\\\n            & $\\sigma$    & 0.6894     &   0.0590 & 0.6894     & ------ & 0.6894    \\\\\n        \\hline\n        0.9 & $\\beta_{0}$ & 5.8000     &   0.5887 & 5.8000     & 1.6461 & 6.0292     \\\\\n            & $\\beta_{1}$ & 0.2700     &   0.0084 & 0.2700     & 0.0256 & 0.2678     \\\\\n            & $\\beta_{2}$ & 3.9596     &   0.1937 & 3.9658     & 0.6203 & 3.8271     \\\\\n            & $\\sigma$    & 0.3391     &   0.0258 & 1.2677     & ------ & 1.2767     \\\\\n      \\hline\\hline\n\n\n\\end{tabular}\n}}\n  \\label{table.application}%\n\\end{table}%\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.7]{perfilThetaAIS2.eps}~\n\\caption{AIS data: ML estimates  and $95\\%$ confidence intervals for\nvarious values of $p$. \\label{fig:2b}}\n\\end{center}\n\\end{figure}\nTo obtain a more complete picture of the effects, a series of QR\nmodels over the grid $p=\\{0.1, 0.15,\\ldots, 0.95\\}$ is estimated.\nFigure  \\ref{fig:2b} gives a graphical summary of this analysis. The\nshaded area depicts the $95\\%$ confidence interval from all the\nparameters. From Figure \\ref{fig:2b} we can observe some interesting\nevidences which cannot be detected by mean regression. For example,\nthe effect of the two variables (LBM and gender) become stronger for\nthe higher conditional quantiles,  indicating that the BMI are\npositively correlated with the quantiles. The robustness of the\nmedian regression $(p=0.5)$ can be assessed by considering the\ninfluence of a single outlying observation on the EM estimate of\n$\\mbox{${ \\bm \\theta}$}$. In particular, we can assess how much the EM estimate of\n$\\mbox{${ \\bm \\theta}$}$ is influenced by a change of $\\delta$ units in a single\nobservation $y_{i}$. Replacing $y_{i}$ by\n$y_{i}(\\delta)=y_{i}+\\delta sd(\\mathbf{y})$, where $sd(.)$ denotes\nthe standard deviation. Let $\\widehat{\\beta}_{j}(\\delta)$ be the EM\nestimates of $\\beta_j$ after contamination, $j=1,2,3$. We are\nparticularly interested in the relative changes\n$|(\\widehat{\\beta}_{j}(\\delta)-\\widehat{\\beta}_{j})\/\\widehat{\\beta}_{j}|$.\nIn this study we contaminated the observation corresponding to\nindividual $\\left\\{\\#146\\right\\}$ and for $\\delta$ between 0 and 10.\nFigure \\ref{fig:change} displays  the results of the relative\nchanges of the estimates for different values of $\\delta$. As\nexpected, the estimates from the median regression model are less\naffected by variations on $\\delta$ than those of the mean\nregression. Moreover,  Figure \\ref{fig:2c} shows  the Q-Q plot and envelopes  for mean and\nmedian regression,  which  are  obtained based on the distribution of\n$W_i$,  given in (\\ref{Wi}), that  follows  $\\exp(1)$ distribution. The lines in these figures represent the 5th\npercentile, the mean and the $95$th percentile of $100$ simulated\npoints for each observation. These figures clearly show that the\nmedian regression\ndistribution provides a better-fit  than the standard mean regression to the AIS data set.\\\\\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.35]{meanVsmedianbeta0.ps}~\\includegraphics[scale=0.35]{meanVsmedianbeta1.ps}~\\includegraphics[scale=0.35]{meanVsmedianbeta2.ps}\n\\caption{Percentage of change in the estimation of $\\beta_0$,\n$\\beta_1$ and $\\beta_2$ in comparison with the true value, for\nmedian $(p=0.5)$ and mean regression, for different contaminations\n$\\delta$. \\label{fig:change}}\n\\end{center}\n\\end{figure}\n\\begin{figure}[!tb]\n\\begin{center}\n\\includegraphics[scale=0.55]{envelopesAIS.ps}~\n\\caption{AIS data: Q--Q plots and simulated envelopes for mean and\nmedian regression.\\label{fig:2c}}\n\\end{center}\n\\end{figure}\nAs discussed at  the  end  of  Section 2.3 the estimated distance\n$\\widehat{d}_i=|y_i-\\mathbf{x}^{\\top}_i\\widehat{\\mbox{${\\bm \\beta}$}}_p|\/\\widehat{\\sigma}$\ncan be used efficiently as a  measure  to identify possible outlying\nobservations. Figure \\ref{fig:mahal}(left panel) displays the index\nplot of the distance $d_i$ for the median regression model\n$(p=0.5)$. We see from this figure that observations {\\#75, \\#162,\n\\#178 and \\#179} appear as possible outliers. From the EM-algorithm,\nthe estimated weights $u_i(\\widehat{\\mbox{${ \\bm \\theta}$}})={\\cal E}_{\nsi}(\\widehat{\\mbox{${ \\bm \\theta}$}})$ for these observations are the smallest ones\n(see right panel in Figure \\ref{fig:mahal}), confirming the\nrobustness aspects of the maximum likelihood estimates against\noutlying observations of the QR models. Thus, larger $d_i$ implies a\nsmaller $u_i(\\widehat{\\mbox{${ \\bm \\theta}$}})$, and the estimation of $\\mbox{${ \\bm \\theta}$}$\ntends to give smaller weight to outlying observations in the sense\nof the distance $d_i$.\n\nFigure \\ref{fig:1b} shows the estimated quartiles of two levels of\ngender at each LBM point from our EM algorithm along with the\nestimates obtained via mean regression. From this figure we can see\nclear attenuation in $\\beta_1$ due to the use of the median\nregression related to the mean regression. It is possible to observe\nin this figure some atypical individuals that could have an\ninfluence on the ML estimates for different values of quantiles.  In\nthis figure, the individuals $\\#75, ~\\#130, ~\\#140 ~\\#162, ~\\#160$\nand $\\#178$ were marked since they were detected as potentially\ninfluential.\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.47]{di.ps}~\\includegraphics[scale=0.5]{weights_di.ps}\n\\caption{AIS data: Index plot of the distance  $d_i$ and the\nestimated weights $u_i$.\\label{fig:mahal}}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.5]{Female.ps}~\\includegraphics[scale=.5]{Man.ps}\n\\caption{AIS data: Fitted regression lines for the three selected\nquantiles along with the mean regression line. The influential\nobservations are numbered. \\label{fig:1b}}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\centering \\hspace {1cm}\\centering \\\\\n\\includegraphics[scale=0.35]{gdi01.ps}~\\includegraphics[scale=0.35]{gdi05.ps}~\\includegraphics[scale=0.35]{gdi09.ps}\\\\\n\\includegraphics[scale=0.35]{QDip01.ps}~\\includegraphics[scale=0.35]{QDip05.ps}~\\includegraphics[scale=0.35]{QDip09.ps}\\\\\n\\caption{Index plot of (first row) approximate likelihood distance\n$GD^1_i$. (second row). Index plot of approximate likelihood\ndisplacement $QD^1_i$. The influential observations are\nnumbered.\\label{pert3} }\n\\end{center}\n\\end{figure}\n\n\n\nIn order to identify influential observations at different quantiles\nwhen some observation is eliminated, we can generate graphs of the\ngeneralized Cook distance $GD^{l}_i$, as explained in Section\n\\ref{Sec Diagnostic}. A high value for $GD^{l}_{i}$ indicates that\nthe $i$th observation has a high impact on the maximum likelihood\nestimate of the parameters. Following \\cite{Barros}, we can use\n$2(p+1)\/n$ as benchmark for the $GD^{l}_{i}$ at different quantiles.\nFigure \\ref{pert3} (first row) presents the index plots of\n$GD^{l}_i$. We note from this figure that, only observation $\\#140$\nappears as influential in the ML estimates at $p=0.1$ and\nobservations $\\#75, \\#178$ as influential at $p=0.5$, whereas\nobservations $\\#75,\\#162, \\#178$ and $\\#179$ appear as influential\nin the ML estimates at $p=0.9$. Figure \\ref{pert3} (second row)\npresents the index plots of $QD^1_i$. From this figure, it can be\nnoted that observations $\\#76,\\#130, \\#140$ appear to be influential\nat $p=0.1$, whereas observations $\\#75,\\#162$ and $\\#178$ seem to be\n influential in the ML estimates at $p=0.1$, and in addition\nobservation  $\\#179$ appears to be influential at $p=0.9$.\n\n\n\\section{Simulation studies} \\label{sec simulation study}\nIn this section, the results from two simulation studies are\npresented to illustrate the performance of the proposed method.\n\n\\subsection{Robustness of the EM estimates (Simulation study 1)}\n\nWe conducted a simulation study to assess the performance of the\nproposed EM algorithm, by mimicking the setting of the AIS data by\ntaking the sample size $n = 202$. We simulated data from the model\n\n\\begin{equation}\ny_{i}=\\beta_{1} + \\beta_{2}x_{i2} + \\beta_{3}x_{i3} + \\epsilon_{i},\n\\,\\,\\,\\,\\,\\,\\,\\, i=1,\\ldots,202, \\label{simulation_1}\n\\end{equation}\nwhere the $x_{ij}'s$ are simulated from a uniform distribution\n(U(0,1)) and the errors $\\epsilon_{ij}$ are simulated from four\ndifferent distributions: $(i)$ the standard normal distribution\n$N(0,1)$, $(ii)$ a Student-t distribution with three degrees of\nfreedom, $t_{3}(0,1)$, $(iii)$ a heteroscedastic normal\ndistribution, $(1+x_{i2})N(0,1)$ and, $(iv)$ a bimodal mixture\ndistribution $0.6t_3(-20,1)+0.4t_3(15,1)$. The true values of the\nregression parameters were taken as $\\beta_1=\\beta_2=\\beta_3=1$. In\nthis way, we had four settings and for each setting we generated\n$10000$ data sets.\n\n\nOnce the simulated data were generated, we fit a QR model, with $p=\n0.1,\\, 0.5$ and $0.9$, under Barrodale and Roberts (BR), Lasso\n(Lasso) and EM algorithms by using the \"quantreg()\" package and our\n\\verb\"ALDqr()\" package, from the R language, respectively. For the\nfour scenarios, we computed the  bias and the square root of the\nmean square error (RMSE), for each parameter over the $M=10,000$\nreplicas. They are defined as:\n\\begin{eqnarray}\nBias(\\gamma) &=& \\overline{\\widehat{\\gamma}}-\\gamma\\label{bias} \\,\\, {\\rm and}\\, \\, \\, \\,\nRMSE(\\gamma) = \\sqrt{SE(\\gamma)^2 + Bias(\\gamma)^2}\\label{EQM}\n\\end{eqnarray}\nwhere $\\overline{\\widehat{\\gamma}} =\n\\frac{1}{M}\\sum_{i=1}^{M}\\widehat{\\gamma}_i$ and  $SE(\\gamma)^2 =\n{\\frac{1}{M-1}\\sum_{i=1}^{M}\\lp\\widehat{\\gamma}_i -\n\\overline{\\widehat{\\gamma}}\\rp^2},$ with $\\gamma =\n\\beta_1,\\beta_2,\\beta_3$ or $\\sigma$, $\\widehat{\\gamma}_i$ is the\nestimate of $\\gamma$ obtained in replica $i$ and $\\gamma$ is the\ntrue value. Table \\ref{table.simul1} reports the simulation results for $p =\n0.1,\\, 0.5$ and $0.9$. We observe that the EM yields lower biases\nand RMSE than the other two estimation methods under all the\ndistributional scenarios. This finding suggests that the EM would\nproduce better results than other alternative methods typically used\nin the literature of QR models.\n\\begin{table}[htbp!]\n  \\centering\n  \\caption{Simulation study. Bias and root mean-squared\n  error (RMSE)  of $\\mbox{${\\bm \\beta}$}$ under different error distributions. The estimates under Barrodale and Roberts (BR)\nand Lasso  (Lasso) algorithms were obtained by the \"quantreg()\"\npackage from the R language.} {\\footnotesize{\n\\begin{tabular}{lccccccc}\n\n        \\hline\\hline\n    \\multicolumn{2}{c}{} & \\multicolumn{2}{c}{$\\beta_{1}$}  & \\multicolumn{2}{c}{$\\beta_{2}$} & \\multicolumn{2}{c}{$\\beta_{3}$} \\\\\n\n     \\hline\n    Method                     & $p$ &    Bias &  RMSE  & Bias & RMSE & Bias & RMSE  \\\\\n        \\hline\n    $\\epsilon \\sim N(0,1)$ & \\multicolumn{7}{c}{} \\\\\n        BR  & 0.1 & -1.2639 & 1.3444 &   0.0076 &  0.5961 &-0.0030 &  0.5934  \\\\\n                               & 0.5 &  0.0064 & 0.3376 &  -0.0048 &  0.4390 &-0.0051 &  0.4453  \\\\\n                               & 0.9 &  1.2640 & 1.3460 &   0.0030 &  0.6051 & 0.0069 &  0.6039 \\\\\n\n               LPQR                  & 0.1 & -0.9664 & 1.0464 &  -0.3072 &  0.6165 &-0.3110 &  0.6187  \\\\\n                               & 0.5 &  0.1474 & 0.3628 &  -0.1463 &  0.4534 &-0.1462 &  0.4576  \\\\\n                               & 0.9 &  1.5901 & 1.6460 &  -0.3164 &  0.6173 &-0.3076 &  0.6179  \\\\\n\n        EM                     & 0.1 & -1.2551 & 1.3362 &  -0.0055 &  0.5964 &-0.0090 &  0.6020  \\\\\n                               & 0.5 &  0.0040 & 0.3286 &  -0.0050 &  0.4332 &-0.0031 &  0.4363  \\\\\n                               & 0.9 &  1.2694 & 1.3484 &  -0.0071 &  0.6019 &-0.0120 &  0.5955  \\\\\n                                                    \\hline\n    $\\epsilon \\sim t_{3}(0,1)$  & \\multicolumn{7}{c}{} \\\\\n        BR  & 0.1 & -1.2446 & 1.3364 &  -0.0290 &  0.6274 &-0.0313 &  0.6259 \\\\\n                               & 0.5 &  0.1049 & 0.4870 &   0.1213 &  0.6714 & 0.1123 &  0.6708  \\\\\n                               & 0.9 &  2.3618 & 2.8408 &   1.0056 &  2.4928 & 0.9459 &  2.4332 \\\\\n\n                LPQR                  & 0.1 & -0.9315 & 1.0219 &  -0.3478 &  0.6422 &-0.3412 &  0.6354  \\\\\n                               & 0.5 &  0.3007 & 0.5410 &  -0.0928 &  0.6310 &-0.0831 &  0.6237  \\\\\n                               & 0.9 &  3.0443 & 3.2880 &   0.1911 &  1.6375 & 0.2231 &  1.6601 \\\\\n\n        EM                     & 0.1 & -1.2287 & 1.3213 &  -0.0402 &  0.6209 &-0.0374 &  0.6265  \\\\\n                               & 0.5 &  0.0965 & 0.4866 &   0.1352 &  0.6789 & 0.1304 &  0.6758  \\\\\n                               & 0.9 &  2.3781 & 2.8459 &   0.9464 &  2.4082 & 0.9264 &  2.4167  \\\\\n                                                    \\hline\n     $\\epsilon \\sim (1+x_{2})N(0,1)$  & \\multicolumn{7}{c}{} \\\\\n        BR  & 0.1 & -1.2869 & 1.4256 &   0.0130 &  0.8706 &-1.2554 &  1.5381  \\\\\n                               & 0.5 & -0.0051 & 0.4468 &   0.0049 &  0.6336 & 0.0061 &  0.6509 \\\\\n                               & 0.9 &  1.2868 & 1.4259 &   0.0018 &  0.8686 & 1.2307 &  1.5256 \\\\\n\n                LPQR                  & 0.1 & -1.1393 & 1.2272 &  -0.3694 &  0.7773 &-1.1450 &  1.2756  \\\\\n                               & 0.5 &  0.1834 & 0.4520 &  -0.1906 &  0.6193 &-0.1963 &  0.6304  \\\\\n                               & 0.9 &  1.6972 & 1.7933 &  -0.3621 &  0.7925 & 0.7494 &  1.1587  \\\\\n\n        EM                     & 0.1 & -1.2772 & 1.4140 &   0.0051 &  0.8646 &-1.2341 &  1.5195 \\\\\n                               & 0.5 &  0.0954 & 0.4892 &   0.1289 &  0.6724 & 0.1316 &  0.6694 \\\\\n                               & 0.9 &  1.2599 & 1.3987 &   0.0076 &  0.8723 & 1.2488 &  1.5315  \\\\\n                                            \\hline\n       $\\epsilon \\sim 0.6t_3(-20,1)+0.4t_3(15,1)$  & \\multicolumn{7}{c}{} \\\\\n        BR  & 0.1 & -1.2350 & 1.3268 &  -0.0395 &  0.6160 &-0.0396 &  0.6192  \\\\\n                               & 0.5 &  0.1029 & 0.4896 &   0.1214 &  0.6780 & 0.1212 &  0.6741 \\\\\n                               & 0.9 &  2.3857 & 2.8737 &   0.9657 &  2.4574 & 0.9558 &  2.4585 \\\\\n\n                LPQR                  & 0.1 & -0.9664 & 1.0464 &  -0.3072 &  0.6165 &-0.3110 &  0.6187  \\\\\n                               & 0.5 &  0.1474 & 0.3628 &  -0.1463 &  0.4534 &-0.1462 &  0.4576  \\\\\n                               & 0.9 &  1.5901 & 1.6460 &  -0.3164 &  0.6173 &-0.3076 &  0.6179  \\\\\n\n        EM                     & 0.1 & -0.9327 & 1.0201 &  -0.3491 &  0.6433 &-0.3355 &  0.6372 \\\\\n                               & 0.5 &  0.2880 & 0.5343 &  -0.0745 &  0.6216 &-0.0717 &  0.6159 \\\\\n                               & 0.9 &  3.0624 & 3.3102 &   0.1702 &  1.6627 & 0.2221 &  1.6575  \\\\\n                               \\hline\\hline\n\n\\end{tabular}\n}}\n\n  \\label{table.simul1}%\n\\end{table}%\n\n\n\\subsection{Asymptotic properties (Simulation study 2)} \\label{sec simulation study 2}\n\nWe also conducted a simulation study to evaluate the finite-sample\nperformance of the parameter estimates. We generated artificial\nsamples from the regression model (\\ref{simulation_1}) with\n$\\beta_1=\\beta_2=\\beta_3=1$ and $x_{ij}\\sim U(0,1)$. We chose\nseveral distributions for the random term $\\epsilon_i$ a little\ndifferent than the simulation study 1, say, $(i)$ normal\ndistribution $N(0,2)$ (N1), $(ii)$ a Student-t distribution\n$t_{3}(0,2)$ (T1), $(iii)$ a heteroscedastic normal distribution,\n$(1+x_{i2})N(0,2)$ (N2) and, $(iv)$ a bimodal mixture distribution\n$0.6t_3(-20,2)+0.4t_3(15,2)$ (T2). Finally, the sample sizes were\nfixed at $n = 50, 100, 150, 200, 300,$ $400, 500, 700$ and $800$.\n\nFor each combination of parameters and sample sizes, $10000$ samples\nwere generated under the four different situations of error\ndistributions (N1, T1, N2, T2). Therefore, 36 different simulation\nruns are performed. Once all the data were simulated, we fit the QR\nmodel with $p=0.5$ and the  bias (\\ref{bias}) and the square root of\nthe mean square error (\\ref{EQM}) were recorded. The results  are\nshown in Figure \\ref{fig:77a}. We can see a pattern of convergence\nto zero of the bias and MSE when $n$ increases. As a general rule,\nwe can say that bias and MSE tend to approach to zero when the\nsample size increases, indicating that the estimates based on the\nproposed EM-type algorithm do provide good asymptotic properties.\nThis same pattern of convergence to zero is repeated considering\ndifferent levels of the quantile $p$.\n\n\n\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[scale=0.35]{bias1_beta1.ps}~\\includegraphics[scale=0.35]{RMSE1_beta1.ps}\\\\\n\\includegraphics[scale=0.35]{bias2_beta2.ps}~\\includegraphics[scale=0.35]{RMSE1_beta2.ps}\\\\\n\\includegraphics[scale=0.35]{bias3_beta3.ps}~\\includegraphics[scale=0.35]{RMSE1_beta3.ps}\\\\\n\\caption{Simulation study 2. Average bias (first column) and average\nMSE (second column) of the estimates of $\\beta_1$,$\\beta_2$,\n$\\beta_3$ with $p=0.5$ (median regression), where $N1=N(0,2)$,\n$T1=t_3(0,2)$, $N2=(1+x_2)N(0,2)$ and\n$T2=0.6t_3(-20,2)+0.4t_3(15,2)$ .\\label{fig:77a}}\n\\end{center}\n\\end{figure}\n\n\\section{Conclusion}\nWe have studied a likelihood-based approach to the estimation of the\nQR based on the asymmetric Laplace distribution (ALD). By utilizing\nthe relationship between the QR check function and the ALD, we cast\nthe QR problem into the usual likelihood framework. The mixture\nrepresentation  of the ALD allows us to express a QR model as a\nnormal regression model, facilitating the implementation of an  EM\nalgorithm, which naturally provides the ML estimates of the model\nparameters with the observed information matrix as a by product. The\nEM algorithm  was implemented as part of the R package \\textit{ALDqr()}. We hope that by making the code of our method\navailable, we will lower the barrier for other researchers to use\nthe EM algorithm  in their studies of quantile regression. Further,\nwe presented diagnostic analysis in QR models, which was based on\nthe case-deletion technique suggested by \\cite{zhu2001case} and\n\\cite{ZhuLee2001},  which are the counterparts for missing data\nmodels of the well-known ones proposed by \\cite{cook77} and\n\\cite{cook86}. The simulation studies  demonstrated the superiority\nof the proposed methods to the existing methods, implemented in the\npackage \\verb\"quantreg()\". We applied our methods to a real data set\n(freely downloadable from R) in order to illustrate how the\nprocedures can be used to identify outliers and to obtain robust ML\nparameter estimates. From these results, it is encouraging that the\nuse of ALD offers a better alternative in the analysis of QR models.\n\nFinally, the  proposed  methods can  be extended to a more general\nframework, such as, censored (Tobit) regression models, measurement\nerror models, nonlinear regression models, stochastic volatility\nmodels, etc and should yield satisfactory results at the expense of\nadditional complexity in implementation. An in-depth investigation\nof such extensions is beyond the scope of the present paper, but\nthese are interesting topics for further research.\n\n\n\n\\section*{Acknowledgements}\nThe research of V. H. Lachos was supported by Grant 305054\/2011-2\nfrom Conselho Nacional de Desenvolvimento Cient\\'{\\i}fico e Tecnol\\'{o}gico\n(CNPq-Brazil) and by Grant 2014\/02938-9 from  Funda\\c c\\~ao de\nAmparo \\`{a} Pesquisa do Estado de S\\~ao Paulo (FAPESP-Brazil).\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\\label{1}\n\nThe low dimensional microscopic dynamics of heat conduction has\nbeen an attractive question since early year last century. Much\nmore attention has been payed to this problem in the last two\ndecades due to the dramatic achievement in the application of\nminiaturized devices \\cite{Forsman,Taillefer,Tighe,Hone,Kim,Zhang}\nwhich can be described by 1D or 2D models. More and more numerical\ncalculations are focused on the minimal requirements for a\ndynamical model where Fourier's law holds or not\n\\cite{Lepri_1,Casati,Posch,Kaburaki,Lepri_2,Alonso_1,Moasterio,Li_2,Li_3,Alonso_2}.\nA convergent heat conductivity was shown in ding-a-ling model\n\\cite{Casati,Posch} which is chaotic. The studies on Lorentz gas\nmodel \\cite{Alonso_1,Moasterio} (the circular scatters are\nperiodically placed in the channel) of which the Lyapunov exponent\nis nonzero gave a finite heat conductivity which fulfils the\nFourier law explicitly. Hence, chaos was ever regarded as an\nindispensable factor to normal heat conduction. Whereas, the FPU\nmodel \\cite{Kaburaki,Lepri_2} indicated that the chaotic behavior\nis not sufficient to arrive at normal heat conduction. Recently, a\nseries of billiard gas models \\cite{Li_2,Li_3,Alonso_1,Alonso_2}\nwere devoted to explore the normal heat conduction of quasi 1D\nchannels with zero Lyapunov exponent. However, the role of chaos\nin heat conduction has not been well understood. Additionally, the\nexponential stability and instability frequently coexist in the\nscatters of real system. Thus the model with various degree of\nchaos deserves further investigation from the microscopic point of\nview, and it will be also interesting to explore non-equilibrium\nstationary states and to determine the steady temperature field.\n\nIn this paper, we focus on the quasi one-dimensional gas model\nwhich is closer to the real system. The scatters in our model are\nthe isosceles right triangle with a segment of circle substituting\nfor the right angle. In this case the edges of scatters are the\ncombination of line and a quarter of circle. Such a channel is of\nchaos, which indicates exponential instability of microscopic\ndynamics. Our paper is organized as follows. In section \\ref{2},\nwe introduce the model and investigate the degree of chaos for\nvarious channels with different arc-radius and channel height. In\nsection \\ref{3}, we study the heat transport behavior and the\ncorresponding diffusive behavior by changing the radius of top arc\nand channel height. In section \\ref{4}, we investigate the\nnon-equilibrium stationary state and determine the steady\ntemperature field numerically. We also analyze the dependence on\ndiffusion exponent $\\beta$ and system size $N$ of temperature\nprofile theoretically. In section \\ref{5}, we discuss the relation\nbetween our work and others and summarize our main conclusions.\n\n\n\\section{the model}\\label{2}\n\nWe consider a billiard gas channel with two parallel walls and a\nseries of scatters. The channel consists of $N$ replicate cells of\nlength $l$ and height $h$, and each cell is placed with two\nscatters as shown in Fig. \\ref{fig:systematic}. The scatter's\ngeometry is an isosceles right triangle of hypotenuse $a$ whose\nvertex angle is replaced by a segment of circle with radius $R$\nwhich is tangential to the two sides of the triangle. At the two\nends of the channel are two heat baths with temperature $T_{L}$\nand $T_{R}$. Noninteracting particles coming from these heat baths\nare scattered by the walls and the straight lines as well as the\narcs of the scatters in the channel.\n\n\\begin{figure}[h]\n\\includegraphics[width=0.46\\textwidth]{fig_1}%\n\\caption{\\label{fig:systematic} The channel with $N$ replicate\ncells. Here, $l=2.2$, $a=1.2$, $h$ changes from $1.0$ to $0.27$,\nand $R$ from $0$ to $0.848528$ to ensure a quarter of circle\nalways.}\n\\end{figure}\n\nFor such a channel, the degree of chaos can be characterized\nqualitatively by Poincare surface of section (SOS) \\cite{Vega}.\nSuppose we take out one unit cell from the channel and close the\ntwo ends by straight walls. Then the problem becomes a billiard\nproblem. A particle moves within the cell and makes elastic\ncollision with the mirror-like boundary. We investigate the\nsurfaces of section ($s$, $v_\\tau$) under different initial\nconditions. $s$ is the length along the billiard boundary from the\ncollision point to the reference point. $v_\\tau$ is the tangential\ncomponent of velocity with respect to the boundary at that point.\nThe filling behavior of phase space shown in Fig. \\ref{fig:sos}\nindicates the degree of chaos. In case I, it's non chaotic. The\nsurface of section is regular and periodic as shown in Fig.\n\\ref{fig:sos}(a). As the radius $R$ is increased from (a) to (e),\nthe motion becomes more complex and the map becomes dense with\npoints except some regular islands. In Fig. \\ref{fig:sos}(f), the\nregular parts disappear which indicates strong chaos.\n\n\\begin{figure}[h]\n\\includegraphics[1.5cm,1cm][8cm,12cm]{fig_2}\n\\caption{\\label{fig:sos} Poincare surface-of-section of the\nbilliard problem. The billiard starts with an incident angle $0.8$\nand unit velocity. (a) $R=0$, $h=1.0$; (b) $R=0.001$, $h=1.0$; (c)\n$R=0.015$, $h=1.0$; (d) $R=0.1$, $h=1.0$; (e) $R=0.848528$,\n$h=1.0$; (f) $R=0.848528$, $h=0.27$. }\n\\end{figure}\n\n\n\\section{the heat transport and diffusion behavior}\\label{3}\n\nTo study the heat conduction of the model, the heat flux is\ninvestigated firstly. In calculating the heat flux, we follow Ref.\n\\cite{Alonso_1}. For simplicity, the particles from the two heat\nbaths are supposed to have definite velocities $\\sqrt{2T_{L}}$ and\n$\\sqrt{2T_{R}}$ respectively \\cite{Li_2}. We consider one particle\ncolliding with a heat bath during a period of simulating time. The\nenergy exchange $(\\Delta$E$)_{j}$ at the $j$th collision with the\nheat bath is defined as\n\\begin{equation}\n(\\Delta E)_{j}=E_{h}-E_{p},\n\\end{equation}\nwhere $E_{h}$ denotes for the energy of particle taken from the\nbath and $E_{p}$ for that carried in the channel. For $M$\ncollisions between the particle and the bath wall during the\nsimulation time $t$, the heat flux is given by\n\\begin{equation}\nJ_{1} (N)=\\frac{\\sum_{j=1}^M(\\Delta E)_{j}}{t}.\n \\label{eq:flux}\n\\end{equation}\n\nAs there is one heat carrier in each cell and the channel has $N$\nreplicas, there are $N$ particles in the whole channel. Summing\nover the heat flux of $N$ heat carriers, we have $J_{N} (N)=N\nJ_{1} (N)$. Meanwhile, the Fourier's law reads\n\\begin{equation}\nJ_{N} (N)=-\\kappa\\frac{d T}{d x}=\\kappa\\frac{T_{L}-T_{R}}{Nl},\n\\label{eq:fourier}\n\\end{equation}\nwhere $\\kappa$ refers to the heat conductivity which is determined\nby Eqs. (\\ref{eq:flux}) and (\\ref{eq:fourier}),\n\\begin{equation}\n\\kappa\\thicksim N^{2}J_{1} (N).\n\\end{equation}\n\nWe consider various cases by changing the radius $R$ of the top\narc of the scatters to investigate their effects on heat\nconduction. The heat flux of a single particle versus system size\nshown in Fig. \\ref{fig:heatflow} are four typical cases. Namely,\ncase I: the $ {}_{\\blacksquare}$ studies for $R=0,\\,h=1.0$; case\nI\\!I: the $\\circ$ for $R=0.001,\\,h=1.0$; case I\\!I\\!I: the\n$\\vartriangle$ for $R=0.848528,\\,h=1.0$, and case I\\!V: the\n$\\triangledown$ for $R=0.848528,\\,h=0.27$, respectively. The total\ncell numbers are chosen as $N=20$, $40$, $80$, $160$, $320$, $640$\nand $1280$ respectively. After a sufficient long period of\nsimulation time, the heat flux approaches to a constant value.\nClearly, the value of heat flux decreases with increasing $R$ for\nthe same size. Remarkably, there is $20$ times difference of heat\nflux between case I\\!I\\!I and case I\\!V, which indicates that\nsmaller height suppresses the heat flux greatly. Thus, it appears\nthat the value of heat flux can be adjusted in this way in\ndesigning heat-control devices. Furthermore, our calculations show\nthat the heat flux dependence on $N$ exhibits faint non-linearity\nalthough the curve looks linear for all cases except case I\\!V in\nthe log-log scale .\n\n\\begin{figure}[h]\n\\includegraphics[2cm,1cm][8cm,6cm]{fig_3}\n\\caption{\\label{fig:heatflow} The heat flux of a single particle\nversus system size($N=20$, $40$, $80$, $160$, $320$, $640$ and\n$1280$) with the divergence exponent of heat conductivity\n$\\alpha=0.721$, $0.526$, $0.101$, $0.009$ for four typical cases\nrespectively (left panel). The ratio of heat flux\n$J_{1}(N)\/J_{1}(2N)$ for different system sizes (right panel). }\n\\end{figure}\n\nIn order to observe the deviation from the line, which arises from\nthe finite-size effect, we calculate the ratio of heat flux versus\nsystem size for various radius. The data for the aforementioned\nfour cases are plotted in the right panel of Fig.\n\\ref{fig:heatflow}, from which one can see that both the\nincreasing of system size and of the arc radius bring the ratio an\nupward tendency to the value $4$ which ensures the Fourier law. In\ncase I where $R=0$, there is only a slight increase for the ratio\naround $2.4$. Whereas, the ratio rises drastically along with the\nincreasing of systems size even if the radius $R$ is merely\n$0.001$ (the case I\\!I). When $R=0.848528$ (the case I\\!I\\!I), the\nscatters become a full segment of quarter circle. The ratio also\nrises drastically at first and gradually after $N>160$ in this\ncase. In both cases I\\!I and I\\!I\\!I, it seemly approaches to\ndistinct asymptotic values which are all different from that for\nnormal conduction. This implies that smaller degree of chaos is\ninsufficient to bring about a normal heat conduction although the\nincreasingly chaotic degree makes the divergent exponent of heat\nconduction smaller. In case I\\!V, we maintain the scatters at\nradius $R=0.848528$ and reduce the height $h$ from $1.0$ to\n$0.27$. In this strongly chaotic case, the ratio fluctuates around\nthe value $4$ (dotted line) which means that Fourier law is\nobeyed.\n\nIt is known that the normal heat conduction happens when\n$\\alpha=0$, which indicates the heat conductivity is independent\nof system size, and the anomalous heat conduction corresponds to\nthe case of $\\alpha>0$. The heat conductivity $\\kappa$ we\ncalculated can be given by $\\kappa\\thicksim N^{\\alpha}$ with\n$\\alpha \\gtrsim 0$ despite the heat-flux ratio has a different\nincrease in asymptotic value for all cases (except case I\\!V).\n\nWe calculate $\\alpha$ at the range of system sizes $N$ from $20$\nto $1280$ by averaging over many realizations for various radius\n$R$ at fixed channel height $h=1.0$, and the plot of the\ndependence of $\\alpha$ on $R$ is shown in Fig. \\ref{fig:alpha}(a).\nOne can see that $\\alpha$ descends from $0.721$ through $0.526$ to\n$0.092$ if $R$ increases from $0$ to $0.848528$ for a fixed height\n$h=1.0$. Clearly, the $\\alpha$ descends rapidly for small radius\n({\\it e.g. } $R=0.001$ in case I\\!I) and slowly for larger ones. This\nillustrates that the appearance of arc on the top of the scatter\nsuppresses the divergent exponent $\\alpha$ drastically. If the\nchannel height $h$ for fixed $R=0.848528$ is changed from $1.0$ to\n$0.27$, the $\\alpha$ is found to diminish to $0.009$. Therefore\nthe $\\kappa$ appears to be independent of system size and the\nFourier law holds in this case.\n\n\\begin{figure}[h]\n\\includegraphics[width=0.52\\textwidth]{fig_4}\n\\caption{\\label{fig:alpha} (color on line) (a) Conductivity\ndivergence exponent $\\alpha$ versus circular radius $R$. The\n${}_{^\\blacksquare}$, refers to the magnitude of $\\alpha$ for\n$h=1.0$, $R=0,\\,0.001,\\,0.05,\\,0.1,\\,0.2,\\,0.4,\\,0.6,$ and\n$0.848528$; the $\\blacktriangle$ for $h=0.5$ and $R=0.848528$; the\n$\\bigstar$ for $h=0.27$ and $R=0.848528$ has the value of $0.009$.\n(b) The relation between $\\beta$ and $\\alpha$, where the circle is\nthe numerical result, the red line is of $\\alpha=2-2\/\\beta$\n\\cite{Li_4} and the dashed line is the result of Ref.\n\\cite{Denisov}. (c) Log-log plot of mean square displacement\n$\\langle x(t)^{2}\\rangle$ versus time $t$. The curves from top to\nbottom on the right correspond to cases I, I\\!I, I\\!I\\!I and I\\!V\nrespectively. The ensemble has $10^5$ particles starting from the\ncenter of the channel at time $t=0$ where $x=0$ with the unit\nvelocity and random direction.}\n\\end{figure}\n\nSince the characteristic of heat transport is found being closely\nrelated to the diffusion\nbehavior\\cite{Alonso_2,Li_2,Li_3,Li_4,Li_5,Denisov}, we\ninvestigate the diffusion property for the above cases\nsubsequently. For a particle starting at the origin at time $t=0$\nand diffusing along $x$ direction, the mean square displacement\n$\\langle(x(t)-x(0))^{2}\\rangle$ characterizes its diffusion\nbehavior. For normal diffusion, the Einstein relation of\n$\\langle(x(t)-x(0))^{2}\\rangle = Dt$ holds, where $D$ is diffusion\ncoefficient. If the mean square displacement does not grow\nlinearly in time, {\\it {i.e.}}, $\\langle(x(t)-x(0))^{2}\\rangle =\nDt^{\\beta}$, we refer to anomalous diffusion. Recently, the\nconnection between anomalous diffusion and corresponding heat\nconduction in 1D system was discussed hotly\n\\cite{Li_4,Li_5,Denisov}. We plot the mean square displacement\nversus time $t$ in Fig. \\ref{fig:alpha}(c) for the aforementioned\nfour cases. Note that $10^{5}$ particles were put at the center of\nthe channel where $x=0$ with unit velocity and random direction in\nthe simulations. The top solid line and the bottom short-dash-dot\nline are precisely straight in the whole simulation period\n($t=10^5$), which correspond to the case I (non-chaotic) and case\nI\\!V (strong chaotic), respectively.  We obtain $\\beta=1.628$ for\nthe case I which corresponds to $\\alpha=0.721$, and $\\beta=1.001$\nto $\\alpha=0.009$ for the case I\\!V. Beyond these two cases do the\ncurves keep asymptotically linear at large time $t$ with diffusion\nexponent $\\beta$ between the values of above two cases. The best\nfits of the slope give $\\beta=1.357$ which corresponds to\n$\\alpha=0.526$ for case I\\!I and $\\beta=1.050$ to $\\alpha=0.101$\nfor case I\\!I\\!I, respectively. The relation between divergent\nexponent $\\alpha$ and diffusion exponent $\\beta$ fits the relation\nof $\\alpha=2-2\/\\beta$ proposed by Li and Wang in Ref. \\cite{Li_4},\nas is plotted in Fig. \\ref{fig:alpha}(b). Whereas Denisov {\\it et\nal.} presented another connection of $\\alpha$ with $\\beta$ on the\nbasis of the L$\\acute{e}$vy walk model \\cite{Denisov}. More\ndetails about the origin of the discrepancy between above two\nrelations can be found in Ref. \\cite{Li_5}.\n\n\\begin{figure}[h]\n\\includegraphics[width=0.54\\textwidth]{fig_5}\n\\caption{\\label{fig:MFP} (color on line) (a), (b), (c) and (d) The\nPDFs of the flight distance $|\\delta x|$ between two consecutive\ncollisions for above four cases I, I\\!I, I\\!I\\!I and I\\!V\nrespectively. $N$ represents the system size. Note that the\nlog-log scale is used in (b) and the inset of (d). In the latter\ncase (d), Gaussian distribution (red dashed-line) is in comparison\nto the numerical PDF. (e) The typical trajectory with periodicity\nin case of $R=0$ , $h=1.0$.}\n\\end{figure}\n\nAs different diffusion behaviors are likely related to the\ntrajectory characteristics of the particle propagation, we\ninvestigate the PDF $\\psi(|\\delta x|)$ of the flight distance\n$|\\delta x|$ in $x$-direction between two consecutive collisions\nwith the scatters. After a long time for adequate collisions in\nthe channel, the PDF for aforementioned fore cases, shown in Fig.\n\\ref{fig:MFP}(a) to (d) respectively, take on completely different\nforms for different cases. In case I, the discrete values of\nprobability indicate that the trajectories are abundant of\nperiodicity, which is almost alike for larger system size. The\nmaximum value of PDF appears when $|\\delta x|=0.447$, and the\ntypical trajectory is plotted in Fig. \\ref{fig:MFP}(e) which shows\nexplicitly that the parallel passage makes the periodical\ntrajectory possible, and the particles are easier to propagate\nalong the channel with fewer collisions. It is superdiffusion in\nthis case. In case I\\!I, only smaller system size has the explicit\nperiodicity. With the system size growing, the periodicity is\ndestroyed by the collisions with the segment of circle time and\ntime. The PDF gets smoother in this case. In case I\\!I\\!I, the\nperiodicity happens only for large flight distance $|\\delta x|$\nwith very small number of families. Note that the maximum of PDF\nis corresponding to the value $|\\delta x|$ of $2.2$ which is just\nthe length of a cell, and the PDF decays in power law. In this\ncase it requires more collisions and takes more time for the\nparticles to escape a certain region. Thus the propagation is\nsuppressed but is still of superdiffusion. The normal diffusion\ntakes place when the particles are scattered by sufficiently large\ndensity of hyperbolic scatters (case I\\!V). Consequently, the\nstrong chaos presents the trajectory of heat carriers with more\naperiodicity. The PDF takes on its characteristic form which has a\nGaussian tail as shown in the inset of Fig. \\ref{fig:MFP}(d).\n\nThus, the propagation modes are responsible for the diffusion\nbehavior. The abundance of aperiodicity of trajectory is the\ncharacteristic of chaotic channel and may also play an crucial\nrole in the normal diffusion. In other words, if the trajectory in\na certain system emerges the aperiodicity due to some other\nmechanisms, such as in polygonal billiard gas model\n\\cite{Alonso_2}, the normal diffusion behaviors may happen.\n\n\n\\section{The calculation of temperature field }\\label{4}\n\nWe calculate the temperature field following the approach proposed\nin Ref. \\cite{Alonso_1}. The temperature of $i$th cell is defined\nby averaging the kinetic energy over all visits into the cell\n\\begin{equation}\nT_{i}=<E_{i}>=\\frac{\\displaystyle\\sum_{j=1}^{m}t_{j}E_{ij}}{\\displaystyle\\sum_{j=1}^{m}t_{j}},\n\\end{equation}\nwhere $t_{j}$ denotes for the time spent within the cell in the\n$j$th visit, and $m$ for the total number of visit. For\nsufficiently large $m$ we expect a steady temperature profile, and\nthis is indeed verified in our calculations for totally $10^{10}$\nvisits. The temperature profiles we obtained are plotted in Fig.\n\\ref{fig:temp}. It is worthwhile to point out that the steady\ntemperature profiles between non-chaotic and chaotic system are\nquite different in thermodynamics limit, which is due to the\ndifferent diffusion behaviors as shown in Fig. \\ref{fig:alpha}(c).\nAs case I is non-chaotic and has uniform diffusion exponent\n$\\beta$, the temperature profiles keep almost the similar shape\nfor different system sizes. At the two ends of channel there are\nlarge temperature jumps which play an important role in the\nFourier transport and dynamics of the system \\cite{Aoki}. These\njumps arise from the boundary heat resistance which usually\nappears when there is a heat flux across the interface of the two\nadjacent materials. In case I\\!I and I\\!I\\!I which are chaotic and\nhave asymptotically decreasing $\\beta$ versus time $t$, there also\nexists the boundary heat resistance. Unlike in case I, the\ntemperature jump here is smaller and diminishes when the system\nsize grows. For larger size is there almost no temperature jump\nwhich corresponds to a nearly linear temperature profile. Both the\nlarger system size $N$ and the arc radius lead to the increase of\nchaos degree which is responsible for the decrease of diffusion\nexponent $\\beta$ ($\\geqslant 1$). In case I\\!V, which is strong\nchaotic, the temperature profiles are almost linear for various\nsystem sizes we considered, corresponding to the normal heat\nconduction.\n\n\\begin{figure}[h]\n\\includegraphics[width=0.48\\textwidth]{fig_6}\n\\caption{\\label{fig:temp} (color on line) Numerical results of\ntemperature profiles for $T_{L}=1.0$, $T_{R}=0.9$ and sizes\n$N=20$(solid), $40$(dash), $80$(dot), $160$(dash dot), $320$(dash\ndot dot), $640$(short dash) and $1280$(short dot), respectively.\nThe four panels refer to (I)$R=0,\\,h=1.0$; (I\\!I)\n$R=0.001,\\,h=1.0$; (I\\!I\\!I) $R=0.848528,\\,h=1.0$, and (I\\!V)\n$R=0.848528,\\,h=0.27$. The red lines correspond to the best fits\nfor the numerical temperature profile at $N=1280$ with Eq.\n(\\ref{eq:temp}), giving the analytical values $\\beta$ with $1.65$,\n$1.34$, $1.06$ and $1.00$ for above four cases respectively. }\n\\end{figure}\n\nWe estimate the temperature profiles from the average point of\nview. Considering the incident particles from the left heat bath\n(where $x=0$) propagating along the x-axis to the right end, we\nsuppose that a reflecting boundary is placed at the origin of the\nx-axis and an absorbing one at the other end. When\n$2\\geqslant\\beta\\geqslant 1$, we assume that the mean density\n$n_{L}(x)$ of the particles at site $x$ in the steady state is\nproportional to $(1-x)^{\\gamma}$ with\n$\\gamma=(2\/\\beta-1)\\beta^{3\/2}$, where we set $x=i\/N$. Under this\nassumption, we have $n_{L}(x)\\sim 1-x$ \\cite{Alonso_1} when\n$\\beta=1$ (normal diffusion) and $n_{L}(x)\\sim const$ when\n$\\beta=2$ (ballistic diffusion). The conservation of particle\nnumber requires\n\\begin{equation}\nn_{L}(x)\\sim \\frac{(1-x)^{\\gamma}}{D_{L}},\n\\end{equation}\nwhere $D_{L}$ are the diffusion coefficient. Likewise, for a\nparticle propagating from right to left,we have\n\\begin{equation}\nn_{R}(x)\\sim \\frac{x^{\\gamma}}{D_{R}},\n\\end{equation}\n\nWe assume $D_{L}=T_{L}^{\\beta\/2}$ and $D_{R}=T_{R}^{\\beta\/2}$.\nThus, if $2\\geqslant\\beta\\geqslant 1$, the temperature is given by\n\\begin{eqnarray}\n&&T(x)=\\frac{T_{L}n_{L}(x)+T_{R}n_{R}(x)}{n_{L}(x)+n_{R}(x)}\\nonumber\\\\\n&&\n=\\frac{T_{L}T_{R}^{\\beta\/2}(1-x)^{\\gamma}+T_{L}^{\\beta\/2}T_{R}x^{\\gamma}}\n{T_{L}^{\\beta\/2}x^{\\gamma}+T_{R}^{\\beta\/2}(1-x)^{\\gamma}}.\n\\label{eq:temp}\n\\end{eqnarray}\nwhere $\\gamma=(2\/\\beta-1)\\beta^{3\/2}$.\n\n\\begin{figure}[h]\n\\includegraphics[width=0.4\\textwidth]{fig_7}\n\\caption{\\label{fig:comp}(color on line) Numerical results of\ntemperature profile in comparison to the analytical results. The\nnumerical temperature profiles with $R=0.848528, N=40$; $R=0.4,\nN=40$; $R=0.2, N=40$; $R=0.1, N=80$; $R=0.05, N=160$ at $h=1.0$\nand $R=0.848528, N=20$ at $h=0.5$ almost share the same shape. The\nred line is the plot of Eq.(\\ref{eq:temp}) with $\\beta=1.25$. }\n\\end{figure}\n\nAs shown in Figs. \\ref{fig:temp} and \\ref{fig:comp}, the\nanalytical results (in red lines) are in good agreement with the\nnumerical ones for all the cases except those at the two ends of\nthe channel for superdifusion cases. These deviations are likely\ndue to the different boundary conditions we used. Furthermore, the\nvalue of $\\beta$, obtained by the best fits for the numerical\ntemperature profile at $N=1280$ with Eq. (\\ref{eq:temp}), agrees\nwith the simulating result greatly for aforementioned four cases.\nOne can see clearly from Eq.(\\ref{eq:temp}) that temperature\nprofiles are closely related to the diffusion exponent $\\beta$,\nnamely, the case with smaller diffusion exponent tends to have\nsmaller temperature jump. Accordingly, it is not unexpected that\ndifferent chaotic cases may share the same temperature profile if\nthey have the identical diffusion exponent $\\beta$. As shown in\nFig. \\ref{fig:comp}, the case with smaller diffusion exponent\nrequires smaller system size for achieving the same temperature\nprofile. Moreover, our calculations show that the results of\nEq.(\\ref{eq:temp}) are consistent with the numerical ones even in\nlarger temperature gradient. Thus, the temperature profile is\nmostly dependent on the diffusion behavior which is remarkably\naffected by the finite-size effect for chaotic cases of\nsuperdiffusion.\n\n\n\\section{discussion and conclusion}\\label{5}\n\nIn summary, we have investigated the role that the chaos plays in\nthe heat conduction by billiard gas channel. We have demonstrated\nthat the degree of dynamical chaos is enhanced by increasing the\narc radius or the system size for chaotic channel, and the mass\nand heat transport behavior is significantly related to the degree\nof dynamical chaos of a channel. The stronger the chaos is, the\ncloser to normal transport behaviors the model seems to be.\nFurthermore, our numerical results of two exponents $\\alpha$ and\n$\\beta$ for both non-chaotic and chaotic cases when\n$\\beta\\geqslant 1$ satisfies the formula $\\alpha=2-2\/\\beta$\n\\cite{Li_4}. We also discussed the microscopic dynamics by the PDF\nof flight distance in $x$-direction. It seems that aperiodicity of\ntrajectory plays an important role in diffusion behavior. Finally,\nour results showed that the temperature jumps at both ends of the\nchannel depend mostly on the diffusion property for both\nnon-chaotic and chaotic channels, and the finite-size effect is\nmore crucial for chaotic ones.\n\nAs is known that the billiard gas model is applicable for\ncapturing the underlying dynamics of particles without\ninteraction. It is therefore worthwhile to discuss the relation\nbetween our work and others in this field.\n\nAlonso {\\it et al. } \\cite{Alonso_1} investigated 1D Lorentz gas model\nfull of periodically distributed half circular scatters. By\ndefining the heat conductivity and temperature field as\nstatistical average over time on the hypothesis of local thermal\nequilibrium, the Fourier law holds in this case, and a linear\ngradient is given for quite small temperature difference . Our\nwork starts from the same approach but different scatter geometry\nis taken into account. Thus it is not surprising that our work has\nsome overlap with theirs in spirit. However, we pay much more\nattention to the role played by different degree of dynamical\nchaos in heat conduction. As a result, our intensive calculations\nextended the results in  Ref. \\cite{Alonso_1} and concluded that\nonly sufficient strong chaos results in the normal diffusion,\nthus, the normal heat transport.\n\nLi {\\it et al. } \\cite{Li_2} presented the dependence of heat conductivity\non system size and the temperature profile in channel with zero\nLyapunov exponent where the right triangle scatters are\nperiodically distributed. In this case, the exponent stability\nleads to abnormal transport behavior. Clearly, their result is the\nnon-chaotic limit of our model.\n\n\n\\section*{Acknowledgement}\n\nWe would like to thank B. Li for providing Ref. \\cite{Li_5} prior\nto publication and helpful discussion. The work is supported by\nNSFC No.10225419 and 90103022.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{}\n\n\nWe present an analysis of the average spectral properties of ~12,000 SDSS quasars as a function of accretion disc inclination, as measured from the equivalent width of the [O III] 5007\\AA\\ line. The use of this indicator on a large sample of quasars from the SDSS DR7 has proven the presence of orientation effects on the features of UV\/optical spectra, confirming the presence of outflows in the NLR gas and that the geometry of the BLR is disc-like. Relying on the goodness of this indicator, we are now using it to investigate other bands\/components of AGN. Specifically, the study of the UV\/optical\/IR SED of the same sample provides information on the obscuring \"torus\". The SED shows a decrease of the IR fraction moving from face-on to edge-on sources, in agreement with models where the torus is co-axial with the accretion disc. Moreover, the fact we are able to observe the broad emission lines also in sources in an edge-on position, suggests that the torus is rather clumpy than smooth as in the Unified Model. The behaviour of the SED as a function of EW[OIII] is in agreement with the predictions of the clumpy torus models as well.\n\n\n\n\n\\tiny\n \\fontsize{8}{11}\\helveticabold { \\section{Keywords:} galaxies: active, galaxies: nuclei, galaxies: Seyfert, quasars: emission lines, quasars: general}\n\\end{abstract}\n\n\\section{Introduction}\n\nThe fact we are not able to spatially resolve the inner regions of Active Galactic Nuclei (AGN), combined with their axisymmetric geometry \\citep{AntonucciMiller1985, Antonucci1993}, can make it difficult to interpret their emissions.\nThe Unified Model predicts the orientation to be one of the main drivers of the diversification in quasars spectra. For this reason, an indicator of the inclination of the source with respect the line of sight of the observer is essential to get further in studying these objects.\n\nDespite several quasars properties have been found to provide information on the inclination of the inner nucleus \\citep{WillsBrowne1986,WillsBrotherton1995,Boroson2011,Decarli2011,VanGorkom2015}, we still lack an univocal measurement of this quantity. This problem is even harder when dealing with not-jetted objects, the most among quasars \\citep[$>90\\%$,][]{Padovani2011}, for which we can not rely on the presence of the strongly collimated radio-jets, directed perpendicularly to the accretion disc.\n\nIn order to give a more accurate description of the components surrounding the central engine and to understand where are the boundaries between one and another, we need to know which components are being intercepted by our line of sight.\n\nAssuming that some of these inner components are characterised by a spherical geometry can often simplify the scenario, while at the same time misleading us. We use the emission lines coming from the Broad Line Region (BLR) to give an estimate of the mass of the central Super Massive Black Hole (SMBH), but in doing that we do not take properly into consideration the geometry of the BLR, i.e we use an average \\emph{virial factor} $f$ to account for the uknown in the geometry and kinematics of the emitting region and we overlook the effects of orientation on the emission lines \\citep{JarvisMcLure2006,Shen2013}. These measurements can then be used in turn to examine the relations between the SMBH and their host galaxies - one of the few tools available to understand the connection between structures on such different spatial scales - their uncertainties affecting these studies \\citep[e.g. ][]{ShenKelly2010}.\nIf the BLR is characterised by a non-spherical geometry we are sistematically underestimating the BH masses in all the sources in which the velocity we intercept, i.e. the line width we measure, is only a fraction of the intrinsic velocity of the emitting gas orbiting around the SMBH.\nThe inclination of the source with respect to the line of sight is therefore crucial to both the understanding of how the nuclei work and how they affect the formation and evolution of galaxies in the Universe.\nIn this proceeding we show recent results on the optical spectra and we present a preliminary result on the Spectral Energy Distribution of quasars, that we obtained using the EW[OIII] as an orientation indicator.\n\n\n\\section{Orientation effects on emission features}\n\n\\subsection{Optical spectra}\nBased on the properties of the [OIII] $5007$\\AA~ line - negligibly contaminated by non-AGN processes and coming from the Narrow Line Region (NLR), whose dimensions ensure the isotropy of the emissions \\citep{Mulchaey1994, Heckman2004} - and on the strong anisotropy of the continuum emitted by the optically-thick\/geometrically-thin accretion disc \\citep{ShakuraSunyaev1973}, we proposed the equivalent width (EW) of the [OIII] line, the ratio between the two luminosities, as an indicator of quasars orientation \\citep{Risaliti2011,Bisogni2017}. \n\nIn \\cite{Risaliti2011} we examined the distribution of the observed EW[OIII] in a large sample of quasars from the SDSS DR5 ($\\sim 6000$) and verified the presence of an orientation effect: the distribution shows a power law tail at the high EW[OIII] values that can not be ascribable to the intrinsic differences in the NLR among different objects, i.e. the intrinsic EW[OIII] distribution, the one we would observe if all the sources were seen in a face-on position.\nThe observed EW[OIII] distribution is a convolution of the intrinsic properties of the NLR emissions in the different objects, such as the ionising continuum and the covering factor of the clouds, and the effects due to their inclination angle. \n\nIn \\cite{Bisogni2017} we selected a larger sample of objects from the SDSS DR7 ($\\sim 12000$), this time with the aim of looking for evidences of orientation effects in the optical spectra.\nWe split our sample in six bins of EW[OIII], each one corresponding to an inclination angle range. Within each bin, the spectra were stacked in order to produce a master spectrum.\nWe then analysed both the broad and the narrow emission lines as a function of EW[OIII], i.e of the inclination angle, finding orientation effects on both of them.\nFig. \\ref{fig1} shows the presence of orientation effects on the broad component of H$\\beta$: the width of the broad line, either represented by the line dispersion $\\sigma$, the Full Width Half Maximum (FWHM) or the Inter-Percentile Velocity width (IPV), increases steadily when we move from low to high EW[OIII]. We found the same result for the other broad lines examined (H$\\alpha$ and MgII, see \\cite{Bisogni2017} for more details). This behaviour is what is expected if the BLR geometry is disc-like and we are moving from sources in a face-on position to sources in an edge-on position.\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=18cm]{FWHM_sigma_IPV_Hb.pdf}\n\\end{center}\n\\caption{Full Width Half Maximum, Inter-Percentile Velocity width and dispersion $\\sigma$ as a function of the EW[OIII] for the H$\\beta$ line. All these quantities, describing the rotational velocity of the gas orbiting around the central SMBH, increase moving from low to high EW[OIII] as expected if the BLR is disc-shaped and we are moving from face-on to edge-on positions.}\\label{fig1}\n\\end{figure}\n\n\n\n\nAs for the narrow emission lines, we examined the [OIII] $\\lambda 5007$\\AA, the most prominent among them in the optical spectrum. This line is known to be contaminated by emissions coming from non-virialized gas, i.e. not orbiting around the central SMBH, but outflowing perpendicularly to the accretion disc \\citep{Heckman1984, Boroson2005}. \nIf the EW[OIII] is a good indicator of the inclination of the source, we should see the blue component of the line, emitted by outflowing gas, decreasing both in intensity and in velocity shift with respect to the nominal wavelength of the emission moving to high EW[OIII] values. Going from face-on to edge-on position in fact we are not intercepting anymore the outflow perpendicular to the accretion disc. This behaviour is found in the [OIII] line profile (Fig. \\ref{fig2}).\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=18cm]{profile+vel_OIII.pdf}\n\\end{center}\n\\caption{Left panel: [OIII] $5007$\\AA~ line profile as a function of EW[OIII]. The blue component of the line decreases moving from low to high EW[OIII]. Right panel: velocity shift of the blue component of the [OIII] line with respect to the velocity of the [OII] $\\lambda 3727$ line, accounting for the systemic velocity of the host galaxy. The shift is decreasing (in modulus) when moving from low to high EW[OIII].  Both the trends are expected if we are moving from face-on to edge-on positions, where the outflow velocity component of the gas is not intercepted anymore by the line of sight.}\\label{fig2}\n\\end{figure}\n\nWe want to stress that within each EW[OIII] bin, therefore within each inclination angle range, the population of quasars is characterised by different SMBH masses, luminosities and accretion rates. These properties are considered among the main drivers of the variance in quasars spectra \\citep{Marziani2003,Zamanov2002,ShenHo2014}. In our study however, as it is designed, even if the effects produced by these properties are present, they are diluited in the stacked representative spectra. \nAs a confirmation that the orientation, even if not the only driver, plays a major role in the variance of quasars spectra, we found a clear trend of the Eigenvector 1, i.e. the anticorrelation between the FeII and [OIII] emissions intensity that \\cite{BorosonGreen1992} identify as the main responsible for quasars spectral variance, with the EW[OIII].\nSpecifically, when we move from low to high EW[OIII], i.e. from low to high inclination angles, we see the [OIII] intensity increasing, while FeII emission becomes less and less intense. This can be explained in terms of orientation: the BLR shares the same anisotropy of the accretion disc and therefore the intensity of its emissions, in this case FeII, is decreased by a factor $\\cos \\theta$, i.e. decreases when moving to edge-on sources. On the other hand the [OIII] line appears as more evident in edge on positions because the luminosity of the continuum emitted by the accretion disc is decreased by the factor $\\cos \\theta$.\n\n\n\\subsection{Infrared emissions}\n\n\nThe observed EW[OIII] distribution has implications for the obscuring component as well.\n\nThe torus is depicted in the Unified Model as a smooth and toroidal structure that can reach $\\sim 1- 10$ pc in size \\citep{Burtscher2013}. If this is true then, there is a maximum inclination angle beyond which we are not able anymore to intercept the emissions coming from the very inner components, such as the continuum emitted by the accretion disc and the broad lines emitted by the BLR.\nIn this case, however, the observed EW[OIII] distribution would drop when the line of sight is starting to intercept the torus. This is not what we observe: the power law keeps going very steadily to the highest EW[OIII] values.\nMoreover, we are intercepting broad emission lines in positions corresponding to high inclination angles.\nBoth these facts are not compatible with the torus being a smooth structure and rather suggest a clumpy structure.\n\nTo test the indicator and exploit its potential, we are now interested in investigating the infrared emissions.\n \nWe then collect photometric data for the same sample in the UV, optical and IR band to study the Spectral Energy Distributions (SED) of the sources.\n\n\\section{Sample and data analysis}\nFor the sample of $\\sim 12000$ objects we selected from the SDSS DR7 the following photometric data are available:\n\n\\begin{itemize}\n\\item[-] Far Ultra Violet and Near Ultra Violet bands from \\emph{Galaxy Evolution Explorer} (\\emph{GALEX}) DR5 \\citep{Bianchi2011}.\n\\item[-] \\emph{ugriz} SDSS photometric data from \\cite{Shen2011}.\n\\item[-] The J, H and K bands from the \\emph{Two Micron All-Sky Survey} (2MASS) \\citep{Skrutskie2006}.\n\\item[-] The $3.4$, $4.6$, $12$ and $22\\,\\mu$m photometric data from the \\emph{Wide-field Infrared urvey Explorer} (\\emph{WISE}) \\citep{Wright2010}.\n\\end{itemize}\n\nWe first correct all the magnitudes for Galactic extinction using the maps from the \\cite{Schlegel1998}. Then, for each EW[OIII] bin, we use the same approach as for the optical spectroscopic data: we rest-frame the data according to the sources redshift and then we perform a stacking of the interpolated SED in order to produce a master SED, on which we can examine the effects produced by the orientation.\nBefore stacking them, we normalise each SED by dividing for the value of $\\nu L_{\\nu}$ at $\\lambda=15 \\, \\mu$m, a reference wavelength in the mid-infrared spectral range, the band in which the torus emits. In doing that, we are normalizing the individual SED for the intrinsic differences of the torus and of other components in the different objects, such as the size and covering factor of the obscuring region, its distance from the central engine and the properties of the ionising continuum, whose emission is being reprocessed by the torus. This makes us able to compare the average behaviour of the obscuring structure at different inclinations with respect to the line of sight of the observer. The final SED are shown in Fig. \\ref{fig3}.\n\n\\section{Results and discussion: implications for the obscuring torus}\n\nIn the optical stacks corresponding to the highest inclination angles (high EW[OIII]) we are able to detect emissions from the BLR.\nThis evidence implies three possible scenarios: the absence of the torus, a torus that is mis-aligned with respect the plane of the accretion disc (and of the BLR) and a clumpy torus.\nThe first scenario is ruled out by the fact that the IR emission is clearly visible in the SED of the sample (Fig. \\ref{fig3}).\nAs for the second one, we see the IR emission in the stacked SED decreasing progressively as a function of the indicator, defined through the anisotropic properties of the emission coming from the accretion disc itself. If torus and accretion disc were not co-axial, we would not see such an orderly behaviour.\n\nThe only scenario we are left with is therefore a clumpy torus, leading to a differention between type 1 and type 2 AGN due only to the photon escaping probability \\citep{Elitzur2008}.\nDue to the selection we performed (i.e. we selected blue objects, and verified that the continuum in our stacked optical spectra was not experiencing any reddening, see \\cite{Bisogni2017}), when we are looking at sources with a high EW[OIII], i.e. with a high inclination angle, we are dealing with type 1 sources in which the BLR is intercepted through the dusty clouds of the torus.\n\n\n\\begin{figure}[h!]\n\\begin{minipage}[b]{.5\\linewidth}\n\\centering\\includegraphics[scale=0.6]{SED_stacks.pdf\n\\end{minipage}%\n\\begin{minipage}[b]{.5\\linewidth}\n\\centering\\includegraphics[scale=0.575]{SED_zoom_torus_stacks.pdf\n\\end{minipage}\n\\caption{(Left panel) Spectral Energy Distributions for the six EW[OIII] bins, corresponding to different inclination angle ranges. The master SED for a EW[OIII] bin was realized as follows: the photometric data for each source from the \\emph{GALEX}, SDSS, 2MASS and \\emph{WISE} surveys were corrected for Galactic reddening, rest-framed, interpolated on a common grid and normalized to the $\\nu L_{\\nu}$ value corresponding to the reference wavelength ($15\\mu$m); for every channel in the grid we then selected the median value. (Right panel) Total flux for the six EW[OIII] bins in the spectral range in which the emission coming from the torus is predominant. SED corresponding to low EW[OIII] values are characterised by a shallower decrease in the emission at the shorter IR wavelengths with respect to the longer ones, while in the case of high EW[OIII] the decrease is steeper. This behaviour is in agreement with the clumpy torus models (see text for details).}\\label{fig3}\n\\end{figure}\n\nWe can compare our results with the clumpy models in literature \\citep{Nenkova2008a, Nenkova2008b} that examine the infrared emission of the torus as a function of the inclination angle with respect to the observer.\nIf the torus is a clumpy structure, what we expect is that the IR emission at shorter wavelengths decreases progressively more than the ones at longer wavelengths when we are reaching edge-on position, due to the combination of an increasing number of clouds intercepted by the line of sight and of a higher absorption at the shorter than at the longer wavelengths \\citep{Nenkova2008b}.\n\nThe behaviour of the stacked SED as a function of EW[OIII] confirms this scenario. At low EW[OIII] (low inclination angle) we are able to intercept the IR emissions coming from the inner clouds of the torus, that are directly illuminated by the ionising continuum, while at high EW[OIII] (high inclination angle) the IR emission coming from the inner clouds is shielded and we can detect it only after it is absorbed by the clouds in the outskirts of the torus. This produces the decrease in the flux at the shorter wavelengths, that becomes progressively more important for stacks corresponding to higher inclinations.\n\nAs a final verification that our results are not biased by any characteristics of the sample, we made the following checks:\n1.) since our sample contains non-jetted as well as jetted quasars, the most extreme among them (blazars) could contaminate the part of the SED pertaining to the torus emission. We verified that the sub-sample composed by non-jetted quasars only gives the same result as the complete sample.\\\\\n2.) $\\sim 50\\%$ of the objects in our sample has a redshift $z>0.47$. This is the critical value beyond which the normalisation flux at $15\\,\\mu$m is retrieved through an extrapolation rather than an interpolation of the SED.\nWe verified that the analysis on the $z<0.47$ and $z>0.47$ sub-samples does not give different results. The only differences in the $z<0.47$ ($z>0.47$) sub-sample we recognise with respect to the complete sample SED are: a lower (higher) luminosity in Big Blue Bump (accretion disc), due to the fact that our sample is on average more luminous at higher redshifts, and a higher (lower) emission in the optical\/NIR band, due to a higher contribution from the host galaxy for sources at lower redshift. We conclude that the extrapolation of the $15\\,\\mu$m flux in $z>0.47$ sources does not affect our results.\n\n\n\\section{Conclusions}\n\nIn this proceeding we summarise the results of the analysis on the optical spectra and we present the preliminary results of the analysis on the infrared emission of $12000$ sources of the SDSS DR7 as a function of the EW[OIII], a new orientation indicator.\nWe find that:\n\\begin{itemize}\n\\item[-] the BLR shares the same geometry of the accretion disc; we are intercepting the intrinsic velocity of the orbiting gas only when we are looking at sources in edge-on positions. If not properly taken into account, the orientation effects affecting the broad emissions lead to an underestimation of the SMBH virial masses in every position but the edge-on ones.\n\\item[-] the presence of outflowing gas in the NLR is clearly seen in the profile of the [OIII] $\\lambda 5007$\\AA~ as a function of the inclination angle. The blue component decreases both in intensity and in the shift with respect to the reference wavelength moving from face-on to edge-on positions.\n\\item[-] the preliminary analysis of the SED reveals a stronger decrease in the IR emission corresponding to the shorter wavelengths with respect to the longer ones when moving from low to high EW[OIII] values, as expected in the theoretical models for clumpy tori when moving from low to high inclination angles.\n\\end{itemize}\n\nFurther analysis is needed in order to investigate properly the emission coming from the torus. Starting from these first results, we are in the process of performing a SED fitting for each source in the sample with \\emph{AGNfitter} \\citep{CalistroRivera2016}. We will then be able to repeat the analysis on the representative SED, this time having also information on the single components contributing to the total emission.\n\n\nWe will also investigate the sources in our sample for which multiple observations are available (e.g. Stripe82, new BOSS spectra) in order to look for evidences of \\emph{changing look} behaviours as a function of the EW[OIII]. If our interpretation of the data implying a clumpy structure for the torus is correct, we expect to see some of the sources that were included in our sample as Type 1 AGN changing to Type 2 objects at a different epoch. This behaviour is expected more frequently for sources with a high EW[OIII], where the orientation effect is dominant, but it is not excluded even for sources with low EW[OIII] values.\n\n\n\\section*{Conflict of Interest Statement}\n\nThe authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.\n\n\n\\section*{Funding}\nSupport for this work was provided by the National Aeronautics and Space Administration through Chandra Award Number AR7-18013 X issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. E.L. is supported by a European Union COFUND\/Durham Junior Research Fellowship (under EU grant agreement no. 609412).\n\n\n\\bibliographystyle{frontiersinSCNS_ENG_HUMS}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}