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1 ## Copyright (C) 1995, 1996, 1997 Kurt Hornik |
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2 ## |
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3 ## This file is part of Octave. |
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4 ## |
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5 ## Octave is free software; you can redistribute it and/or modify it |
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6 ## under the terms of the GNU General Public License as published by |
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7 ## the Free Software Foundation; either version 2, or (at your option) |
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8 ## any later version. |
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9 ## |
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10 ## Octave is distributed in the hope that it will be useful, but |
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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13 ## General Public License for more details. |
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14 ## |
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15 ## You should have received a copy of the GNU General Public License |
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16 ## along with Octave; see the file COPYING. If not, write to the Free |
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17 ## Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |
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18 ## 02110-1301, USA. |
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19 |
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20 ## -*- texinfo -*- |
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21 ## @deftypefn {Functio 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}) |
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22 ## Perform ordinal logistic regression. |
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23 ## |
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24 ## Suppose @var{y} takes values in @var{k} ordered categories, and let |
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25 ## @code{gamma_i (@var{x})} be the cumulative probability that @var{y} |
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26 ## falls in one of the first @var{i} categories given the covariate |
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27 ## @var{x}. Then |
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28 ## |
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29 ## @example |
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30 ## [theta, beta] = logistic_regression (y, x) |
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31 ## @end example |
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32 ## |
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33 ## @noindent |
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34 ## fits the model |
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35 ## |
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36 ## @example |
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37 ## logit (gamma_i (x)) = theta_i - beta' * x, i = 1, ..., k-1 |
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38 ## @end example |
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39 ## |
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40 ## The number of ordinal categories, @var{k}, is taken to be the number |
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41 ## of distinct values of @code{round (@var{y})}. If @var{k} equals 2, |
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42 ## @var{y} is binary and the model is ordinary logistic regression. The |
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43 ## matrix @var{x} is assumed to have full column rank. |
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44 ## |
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45 ## Given @var{y} only, @code{theta = logistic_regression (y)} |
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46 ## fits the model with baseline logit odds only. |
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47 ## |
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48 ## The full form is |
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49 ## |
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50 ## @example |
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51 ## [theta, beta, dev, dl, d2l, gamma] |
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52 ## = logistic_regression (y, x, print, theta, beta) |
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53 ## @end example |
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54 ## |
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55 ## @noindent |
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56 ## in which all output arguments and all input arguments except @var{y} |
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57 ## are optional. |
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58 ## |
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59 ## Stting @var{print} to 1 requests summary information about the fitted |
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60 ## model to be displayed. Setting @var{print} to 2 requests information |
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61 ## about convergence at each iteration. Other values request no |
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62 ## information to be displayed. The input arguments @var{theta} and |
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63 ## @var{beta} give initial estimates for @var{theta} and @var{beta}. |
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64 ## |
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65 ## The returned value @var{dev} holds minus twice the log-likelihood. |
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66 ## |
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67 ## The returned values @var{dl} and @var{d2l} are the vector of first |
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68 ## and the matrix of second derivatives of the log-likelihood with |
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69 ## respect to @var{theta} and @var{beta}. |
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70 ## |
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71 ## @var{p} holds estimates for the conditional distribution of @var{y} |
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72 ## given @var{x}. |
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73 ## @end deftypefn |
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74 |
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75 ## Original for MATLAB written by Gordon K Smyth <gks@maths.uq.oz.au>, |
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76 ## U of Queensland, Australia, on Nov 19, 1990. Last revision Aug 3, |
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77 ## 1992. |
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78 |
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79 ## Author: Gordon K Smyth <gks@maths.uq.oz.au>, |
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80 ## Adapted-By: KH <Kurt.Hornik@wu-wien.ac.at> |
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81 ## Description: Ordinal logistic regression |
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82 |
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83 ## Uses the auxiliary functions logistic_regression_derivatives and |
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84 ## logistic_regression_likelihood. |
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85 |
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86 function [theta, beta, dev, dl, d2l, p] ... |
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87 = logistic_regression (y, x, print, theta, beta) |
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88 |
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89 ## check input |
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90 y = round (vec (y)); |
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91 [my, ny] = size (y); |
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92 if (nargin < 2) |
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93 x = zeros (my, 0); |
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94 endif; |
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95 [mx, nx] = size (x); |
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96 if (mx != my) |
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97 error ("x and y must have the same number of observations"); |
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98 endif |
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99 |
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100 ## initial calculations |
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101 x = -x; |
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102 tol = 1e-6; incr = 10; decr = 2; |
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103 ymin = min (y); ymax = max (y); yrange = ymax - ymin; |
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104 z = (y * ones (1, yrange)) == ((y * 0 + 1) * (ymin : (ymax - 1))); |
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105 z1 = (y * ones (1, yrange)) == ((y * 0 + 1) * ((ymin + 1) : ymax)); |
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106 z = z(:, any (z)); |
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107 z1 = z1 (:, any(z1)); |
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108 [mz, nz] = size (z); |
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109 |
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110 ## starting values |
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111 if (nargin < 3) |
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112 print = 0; |
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113 endif; |
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114 if (nargin < 4) |
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115 beta = zeros (nx, 1); |
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116 endif; |
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117 if (nargin < 5) |
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118 g = cumsum (sum (z))' ./ my; |
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119 theta = log (g ./ (1 - g)); |
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120 endif; |
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121 tb = [theta; beta]; |
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122 |
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123 ## likelihood and derivatives at starting values |
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124 [g, g1, p, dev] = logistic_regression_likelihood (y, x, tb, z, z1); |
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125 [dl, d2l] = logistic_regression_derivatives (x, z, z1, g, g1, p); |
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126 epsilon = std (vec (d2l)) / 1000; |
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127 |
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128 ## maximize likelihood using Levenberg modified Newton's method |
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129 iter = 0; |
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130 while (abs (dl' * (d2l \ dl) / length (dl)) > tol) |
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131 iter = iter + 1; |
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132 tbold = tb; |
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133 devold = dev; |
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134 tb = tbold - d2l \ dl; |
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135 [g, g1, p, dev] = logistic_regression_likelihood (y, x, tb, z, z1); |
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136 if ((dev - devold) / (dl' * (tb - tbold)) < 0) |
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137 epsilon = epsilon / decr; |
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138 else |
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139 while ((dev - devold) / (dl' * (tb - tbold)) > 0) |
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140 epsilon = epsilon * incr; |
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141 if (epsilon > 1e+15) |
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142 error ("epsilon too large"); |
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143 endif |
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144 tb = tbold - (d2l - epsilon * eye (size (d2l))) \ dl; |
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145 [g, g1, p, dev] = logistic_regression_likelihood (y, x, tb, z, z1); |
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146 disp ("epsilon"); disp (epsilon); |
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147 endwhile |
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148 endif |
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149 [dl, d2l] = logistic_regression_derivatives (x, z, z1, g, g1, p); |
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150 if (print == 2) |
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151 disp ("Iteration"); disp (iter); |
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152 disp ("Deviance"); disp (dev); |
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153 disp ("First derivative"); disp (dl'); |
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154 disp ("Eigenvalues of second derivative"); disp (eig (d2l)'); |
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155 endif |
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156 endwhile |
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157 |
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158 ## tidy up output |
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159 |
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160 theta = tb (1 : nz, 1); |
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161 beta = tb ((nz + 1) : (nz + nx), 1); |
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162 |
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163 if (print >= 1) |
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164 printf ("\n"); |
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165 printf ("Logistic Regression Results:\n"); |
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166 printf ("\n"); |
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167 printf ("Number of Iterations: %d\n", iter); |
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168 printf ("Deviance: %f\n", dev); |
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169 printf ("Parameter Estimates:\n"); |
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170 printf (" Theta S.E.\n"); |
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171 se = sqrt (diag (inv (-d2l))); |
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172 for i = 1 : nz |
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173 printf (" %8.4f %8.4f\n", tb (i), se (i)); |
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174 endfor |
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175 if (nx > 0) |
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176 printf (" Beta S.E.\n"); |
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177 for i = (nz + 1) : (nz + nx) |
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178 printf (" %8.4f %8.4f\n", tb (i), se (i)); |
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179 endfor |
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180 endif |
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181 endif |
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182 |
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183 if (nargout == 6) |
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184 if (nx > 0) |
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185 e = ((x * beta) * ones (1, nz)) + ((y * 0 + 1) * theta'); |
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186 else |
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187 e = (y * 0 + 1) * theta'; |
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188 endif |
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189 gamma = diff ([(y * 0), (exp (e) ./ (1 + exp (e))), (y * 0 + 1)]')'; |
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190 endif |
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191 |
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192 endfunction |
