/usr/share/octave/packages/statistics-1.3.0/copulacdf.m is in octave-statistics 1.3.0-4.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 | ## Copyright (C) 2008 Arno Onken <asnelt@asnelt.org>
##
## 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 3 of the License, 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 program; if not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {@var{p} =} copulacdf (@var{family}, @var{x}, @var{theta})
## @deftypefnx {Function File} {} copulacdf ('t', @var{x}, @var{theta}, @var{nu})
## Compute the cumulative distribution function of a copula family.
##
## @subheading Arguments
##
## @itemize @bullet
## @item
## @var{family} is the copula family name. Currently, @var{family} can
## be @code{'Gaussian'} for the Gaussian family, @code{'t'} for the
## Student's t family, @code{'Clayton'} for the Clayton family,
## @code{'Gumbel'} for the Gumbel-Hougaard family, @code{'Frank'} for
## the Frank family, @code{'AMH'} for the Ali-Mikhail-Haq family, or
## @code{'FGM'} for the Farlie-Gumbel-Morgenstern family.
##
## @item
## @var{x} is the support where each row corresponds to an observation.
##
## @item
## @var{theta} is the parameter of the copula. For the Gaussian and
## Student's t copula, @var{theta} must be a correlation matrix. For
## bivariate copulas @var{theta} can also be a correlation coefficient.
## For the Clayton family, the Gumbel-Hougaard family, the Frank family,
## and the Ali-Mikhail-Haq family, @var{theta} must be a vector with the
## same number of elements as observations in @var{x} or be scalar. For
## the Farlie-Gumbel-Morgenstern family, @var{theta} must be a matrix of
## coefficients for the Farlie-Gumbel-Morgenstern polynomial where each
## row corresponds to one set of coefficients for an observation in
## @var{x}. A single row is expanded. The coefficients are in binary
## order.
##
## @item
## @var{nu} is the degrees of freedom for the Student's t family.
## @var{nu} must be a vector with the same number of elements as
## observations in @var{x} or be scalar.
## @end itemize
##
## @subheading Return values
##
## @itemize @bullet
## @item
## @var{p} is the cumulative distribution of the copula at each row of
## @var{x} and corresponding parameter @var{theta}.
## @end itemize
##
## @subheading Examples
##
## @example
## @group
## x = [0.2:0.2:0.6; 0.2:0.2:0.6];
## theta = [1; 2];
## p = copulacdf ("Clayton", x, theta)
## @end group
##
## @group
## x = [0.2:0.2:0.6; 0.2:0.1:0.4];
## theta = [0.2, 0.1, 0.1, 0.05];
## p = copulacdf ("FGM", x, theta)
## @end group
## @end example
##
## @subheading References
##
## @enumerate
## @item
## Roger B. Nelsen. @cite{An Introduction to Copulas}. Springer,
## New York, second edition, 2006.
## @end enumerate
## @end deftypefn
## Author: Arno Onken <asnelt@asnelt.org>
## Description: CDF of a copula family
function p = copulacdf (family, x, theta, nu)
# Check arguments
if (nargin != 3 && (nargin != 4 || ! strcmpi (family, "t")))
print_usage ();
endif
if (! ischar (family))
error ("copulacdf: family must be one of 'Gaussian', 't', 'Clayton', 'Gumbel', 'Frank', 'AMH', and 'FGM'");
endif
if (! isempty (x) && ! ismatrix (x))
error ("copulacdf: x must be a numeric matrix");
endif
[n, d] = size (x);
lower_family = lower (family);
# Check family and copula parameters
switch (lower_family)
case {"gaussian", "t"}
# Family with a covariance matrix
if (d == 2 && isscalar (theta))
# Expand a scalar to a correlation matrix
theta = [1, theta; theta, 1];
endif
if (any (size (theta) != [d, d]) || any (diag (theta) != 1) || any (any (theta != theta')) || min (eig (theta)) <= 0)
error ("copulacdf: theta must be a correlation matrix");
endif
if (nargin == 4)
# Student's t family
if (! isscalar (nu) && (! isvector (nu) || length (nu) != n))
error ("copulacdf: nu must be a vector with the same number of rows as x or be scalar");
endif
nu = nu(:);
endif
case {"clayton", "gumbel", "frank", "amh"}
# Archimedian one parameter family
if (! isvector (theta) || (! isscalar (theta) && length (theta) != n))
error ("copulacdf: theta must be a vector with the same number of rows as x or be scalar");
endif
theta = theta(:);
if (n > 1 && isscalar (theta))
theta = repmat (theta, n, 1);
endif
case {"fgm"}
# Exponential number of parameters
if (! ismatrix (theta) || size (theta, 2) != (2 .^ d - d - 1) || (size (theta, 1) != 1 && size (theta, 1) != n))
error ("copulacdf: theta must be a row vector of length 2^d-d-1 or a matrix of size n x (2^d-d-1)");
endif
if (n > 1 && size (theta, 1) == 1)
theta = repmat (theta, n, 1);
endif
otherwise
error ("copulacdf: unknown copula family '%s'", family);
endswitch
if (n == 0)
# Input is empty
p = zeros (0, 1);
else
# Truncate input to unit hypercube
x(x < 0) = 0;
x(x > 1) = 1;
# Compute the cumulative distribution function according to family
switch (lower_family)
case {"gaussian"}
# The Gaussian family
p = mvncdf (norminv (x), zeros (1, d), theta);
# No parameter bounds check
k = [];
case {"t"}
# The Student's t family
p = mvtcdf (tinv (x, nu), theta, nu);
# No parameter bounds check
k = [];
case {"clayton"}
# The Clayton family
p = exp (-log (max (sum (x .^ (repmat (-theta, 1, d)), 2) - d + 1, 0)) ./ theta);
# Product copula at columns where theta == 0
k = find (theta == 0);
if (any (k))
p(k) = prod (x(k, :), 2);
endif
# Check bounds
if (d > 2)
k = find (! (theta >= 0) | ! (theta < inf));
else
k = find (! (theta >= -1) | ! (theta < inf));
endif
case {"gumbel"}
# The Gumbel-Hougaard family
p = exp (-(sum ((-log (x)) .^ repmat (theta, 1, d), 2)) .^ (1 ./ theta));
# Check bounds
k = find (! (theta >= 1) | ! (theta < inf));
case {"frank"}
# The Frank family
p = -log (1 + (prod (expm1 (repmat (-theta, 1, d) .* x), 2)) ./ (expm1 (-theta) .^ (d - 1))) ./ theta;
# Product copula at columns where theta == 0
k = find (theta == 0);
if (any (k))
p(k) = prod (x(k, :), 2);
endif
# Check bounds
if (d > 2)
k = find (! (theta > 0) | ! (theta < inf));
else
k = find (! (theta > -inf) | ! (theta < inf));
endif
case {"amh"}
# The Ali-Mikhail-Haq family
p = (theta - 1) ./ (theta - prod ((1 + repmat (theta, 1, d) .* (x - 1)) ./ x, 2));
# Check bounds
if (d > 2)
k = find (! (theta >= 0) | ! (theta < 1));
else
k = find (! (theta >= -1) | ! (theta < 1));
endif
case {"fgm"}
# The Farlie-Gumbel-Morgenstern family
# All binary combinations
bcomb = logical (floor (mod (((0:(2 .^ d - 1))' * 2 .^ ((1 - d):0)), 2)));
ecomb = ones (size (bcomb));
ecomb(bcomb) = -1;
# Summation over all combinations of order >= 2
bcomb = bcomb(sum (bcomb, 2) >= 2, end:-1:1);
# Linear constraints matrix
ac = zeros (size (ecomb, 1), size (bcomb, 1));
# Matrix to compute p
ap = zeros (size (x, 1), size (bcomb, 1));
for i = 1:size (bcomb, 1)
ac(:, i) = -prod (ecomb(:, bcomb(i, :)), 2);
ap(:, i) = prod (1 - x(:, bcomb(i, :)), 2);
endfor
p = prod (x, 2) .* (1 + sum (ap .* theta, 2));
# Check linear constraints
k = false (n, 1);
for i = 1:n
k(i) = any (ac * theta(i, :)' > 1);
endfor
endswitch
# Out of bounds parameters
if (any (k))
p(k) = NaN;
endif
endif
endfunction
%!test
%! x = [0.2:0.2:0.6; 0.2:0.2:0.6];
%! theta = [1; 2];
%! p = copulacdf ("Clayton", x, theta);
%! expected_p = [0.1395; 0.1767];
%! assert (p, expected_p, 0.001);
%!test
%! x = [0.2:0.2:0.6; 0.2:0.2:0.6];
%! p = copulacdf ("Gumbel", x, 2);
%! expected_p = [0.1464; 0.1464];
%! assert (p, expected_p, 0.001);
%!test
%! x = [0.2:0.2:0.6; 0.2:0.2:0.6];
%! theta = [1; 2];
%! p = copulacdf ("Frank", x, theta);
%! expected_p = [0.0699; 0.0930];
%! assert (p, expected_p, 0.001);
%!test
%! x = [0.2:0.2:0.6; 0.2:0.2:0.6];
%! theta = [0.3; 0.7];
%! p = copulacdf ("AMH", x, theta);
%! expected_p = [0.0629; 0.0959];
%! assert (p, expected_p, 0.001);
%!test
%! x = [0.2:0.2:0.6; 0.2:0.1:0.4];
%! theta = [0.2, 0.1, 0.1, 0.05];
%! p = copulacdf ("FGM", x, theta);
%! expected_p = [0.0558; 0.0293];
%! assert (p, expected_p, 0.001);
|