1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
|
## Copyright (C) 1995, 1996, 1997 Kurt Hornik
##
## This program 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 2, or (at your option)
## any later version.
##
## This program 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 this file. If not, write to the Free Software Foundation,
## 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
## Performs ordinal logistic regression.
##
## Suppose Y takes values in k ordered categories, and let gamma_i (x)
## be the cumulative probability that Y falls in one of the first i
## categories given the covariate x. Then
## [theta, beta] =
## logistic_regression (y, x)
## fits the model
## logit (gamma_i (x)) = theta_i - beta' * x, i = 1, ..., k-1.
## The number of ordinal categories, k, is taken to be the number of
## distinct values of round (y) . If k equals 2, y is binary and the
## model is ordinary logistic regression. X is assumed to have full
## column rank.
##
## theta = logistic_regression (y)
## fits the model with baseline logit odds only.
##
## The full form is
## [theta, beta, dev, dl, d2l, gamma] =
## logistic_regression (y, x, print, theta, beta)
## in which all output arguments and all input arguments except y are
## optional.
##
## print = 1 requests summary information about the fitted model to be
## displayed; print = 2 requests information about convergence at each
## iteration. Other values request no information to be displayed. The
## input arguments `theta' and `beta' give initial estimates for theta
## and beta.
##
## `dev' holds minus twice the log-likelihood.
##
## `dl' and `d2l' are the vector of first and the matrix of second
## derivatives of the log-likelihood with respect to theta and beta.
##
## `p' holds estimates for the conditional distribution of Y given x.
## 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@ci.tuwien.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 ("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 ("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
|
