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+## Copyright (C) 1995-2012 Kurt Hornik
+##
+## This file is part of Octave.
+##
+## Octave is free software; you can redistribute it and/or modify it
+## under the terms of the GNU General Public License as published by
+## the Free Software Foundation; either version 3 of the License, or (at
+## your option) any later version.
+##
+## Octave is distributed in the hope that it will be useful, but
+## WITHOUT ANY WARRANTY; without even the implied warranty of
+## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+## General Public License for more details.
+##
+## You should have received a copy of the GNU General Public License
+## along with Octave; see the file COPYING. If not, see
+## .
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {[@var{theta}, @var{beta}, @var{dev}, @var{dl}, @var{d2l}, @var{p}] =} logistic_regression (@var{y}, @var{x}, @var{print}, @var{theta}, @var{beta})
+## Perform ordinal logistic regression.
+##
+## Suppose @var{y} takes values in @var{k} ordered categories, and let
+## @code{gamma_i (@var{x})} be the cumulative probability that @var{y}
+## falls in one of the first @var{i} categories given the covariate
+## @var{x}. Then
+##
+## @example
+## [theta, beta] = logistic_regression (y, x)
+## @end example
+##
+## @noindent
+## fits the model
+##
+## @example
+## logit (gamma_i (x)) = theta_i - beta' * x, i = 1 @dots{} k-1
+## @end example
+##
+## The number of ordinal categories, @var{k}, is taken to be the number
+## of distinct values of @code{round (@var{y})}. If @var{k} equals 2,
+## @var{y} is binary and the model is ordinary logistic regression. The
+## matrix @var{x} is assumed to have full column rank.
+##
+## Given @var{y} only, @code{theta = logistic_regression (y)}
+## fits the model with baseline logit odds only.
+##
+## The full form is
+##
+## @example
+## @group
+## [theta, beta, dev, dl, d2l, gamma]
+## = logistic_regression (y, x, print, theta, beta)
+## @end group
+## @end example
+##
+## @noindent
+## in which all output arguments and all input arguments except @var{y}
+## are optional.
+##
+## Setting @var{print} to 1 requests summary information about the fitted
+## model to be displayed. Setting @var{print} to 2 requests information
+## about convergence at each iteration. Other values request no
+## information to be displayed. The input arguments @var{theta} and
+## @var{beta} give initial estimates for @var{theta} and @var{beta}.
+##
+## The returned value @var{dev} holds minus twice the log-likelihood.
+##
+## The returned values @var{dl} and @var{d2l} are the vector of first
+## and the matrix of second derivatives of the log-likelihood with
+## respect to @var{theta} and @var{beta}.
+##
+## @var{p} holds estimates for the conditional distribution of @var{y}
+## given @var{x}.
+## @end deftypefn
+
+## Original for MATLAB written by Gordon K Smyth ,
+## U of Queensland, Australia, on Nov 19, 1990. Last revision Aug 3,
+## 1992.
+
+## Author: Gordon K Smyth ,
+## Adapted-By: KH
+## Description: Ordinal logistic regression
+
+## Uses the auxiliary functions logistic_regression_derivatives and
+## logistic_regression_likelihood.
+
+function [theta, beta, dev, dl, d2l, p] = logistic_regression (y, x, print, theta, beta)
+
+ ## check input
+ y = round (vec (y));
+ [my, ny] = size (y);
+ if (nargin < 2)
+ x = zeros (my, 0);
+ endif;
+ [mx, nx] = size (x);
+ if (mx != my)
+ error ("logistic_regression: X and Y must have the same number of observations");
+ endif
+
+ ## initial calculations
+ x = -x;
+ tol = 1e-6; incr = 10; decr = 2;
+ ymin = min (y); ymax = max (y); yrange = ymax - ymin;
+ z = (y * ones (1, yrange)) == ((y * 0 + 1) * (ymin : (ymax - 1)));
+ z1 = (y * ones (1, yrange)) == ((y * 0 + 1) * ((ymin + 1) : ymax));
+ z = z(:, any (z));
+ z1 = z1 (:, any(z1));
+ [mz, nz] = size (z);
+
+ ## starting values
+ if (nargin < 3)
+ print = 0;
+ endif;
+ if (nargin < 4)
+ beta = zeros (nx, 1);
+ endif;
+ if (nargin < 5)
+ g = cumsum (sum (z))' ./ my;
+ theta = log (g ./ (1 - g));
+ endif;
+ tb = [theta; beta];
+
+ ## likelihood and derivatives at starting values
+ [g, g1, p, dev] = logistic_regression_likelihood (y, x, tb, z, z1);
+ [dl, d2l] = logistic_regression_derivatives (x, z, z1, g, g1, p);
+ epsilon = std (vec (d2l)) / 1000;
+
+ ## maximize likelihood using Levenberg modified Newton's method
+ iter = 0;
+ while (abs (dl' * (d2l \ dl) / length (dl)) > tol)
+ iter = iter + 1;
+ tbold = tb;
+ devold = dev;
+ tb = tbold - d2l \ dl;
+ [g, g1, p, dev] = logistic_regression_likelihood (y, x, tb, z, z1);
+ if ((dev - devold) / (dl' * (tb - tbold)) < 0)
+ epsilon = epsilon / decr;
+ else
+ while ((dev - devold) / (dl' * (tb - tbold)) > 0)
+ epsilon = epsilon * incr;
+ if (epsilon > 1e+15)
+ error ("logistic_regression: epsilon too large");
+ endif
+ tb = tbold - (d2l - epsilon * eye (size (d2l))) \ dl;
+ [g, g1, p, dev] = logistic_regression_likelihood (y, x, tb, z, z1);
+ disp ("epsilon"); disp (epsilon);
+ endwhile
+ endif
+ [dl, d2l] = logistic_regression_derivatives (x, z, z1, g, g1, p);
+ if (print == 2)
+ disp ("Iteration"); disp (iter);
+ disp ("Deviance"); disp (dev);
+ disp ("First derivative"); disp (dl');
+ disp ("Eigenvalues of second derivative"); disp (eig (d2l)');
+ endif
+ endwhile
+
+ ## tidy up output
+
+ theta = tb (1 : nz, 1);
+ beta = tb ((nz + 1) : (nz + nx), 1);
+
+ if (print >= 1)
+ printf ("\n");
+ printf ("Logistic Regression Results:\n");
+ printf ("\n");
+ printf ("Number of Iterations: %d\n", iter);
+ printf ("Deviance: %f\n", dev);
+ printf ("Parameter Estimates:\n");
+ printf (" Theta S.E.\n");
+ se = sqrt (diag (inv (-d2l)));
+ for i = 1 : nz
+ printf (" %8.4f %8.4f\n", tb (i), se (i));
+ endfor
+ if (nx > 0)
+ printf (" Beta S.E.\n");
+ for i = (nz + 1) : (nz + nx)
+ printf (" %8.4f %8.4f\n", tb (i), se (i));
+ endfor
+ endif
+ endif
+
+ if (nargout == 6)
+ if (nx > 0)
+ e = ((x * beta) * ones (1, nz)) + ((y * 0 + 1) * theta');
+ else
+ e = (y * 0 + 1) * theta';
+ endif
+ gamma = diff ([(y * 0), (exp (e) ./ (1 + exp (e))), (y * 0 + 1)]')';
+ endif
+
+endfunction