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## Copyright (C) 1995-2013 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
## <http://www.gnu.org/licenses/>.
## -*- 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 <gks@maths.uq.oz.au>,
## U of Queensland, Australia, on Nov 19, 1990. Last revision Aug 3,
## 1992.
## Author: Gordon K Smyth <gks@maths.uq.oz.au>,
## Adapted-By: KH <Kurt.Hornik@wu-wien.ac.at>
## 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
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