--- /dev/null
+## 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} =} copulapdf (@var{family}, @var{x}, @var{theta})
+## Compute the probability density function of a copula family.
+##
+## @subheading Arguments
+##
+## @itemize @bullet
+## @item
+## @var{family} is the copula family name. Currently, @var{family} can
+## be @code{'Clayton'} for the Clayton family, @code{'Gumbel'} for the
+## Gumbel-Hougaard family, @code{'Frank'} for the Frank family, or
+## @code{'AMH'} for the Ali-Mikhail-Haq family.
+##
+## @item
+## @var{x} is the support where each row corresponds to an observation.
+##
+## @item
+## @var{theta} is the parameter of the copula. The elements of
+## @var{theta} must be greater than or equal to @code{-1} for the
+## Clayton family, greater than or equal to @code{1} for the
+## Gumbel-Hougaard family, arbitrary for the Frank family, and greater
+## than or equal to @code{-1} and lower than @code{1} for the
+## Ali-Mikhail-Haq family. Moreover, @var{theta} must be non-negative
+## for dimensions greater than @code{2}. @var{theta} must be a column
+## vector with the same number of rows as @var{x} or be scalar.
+## @end itemize
+##
+## @subheading Return values
+##
+## @itemize @bullet
+## @item
+## @var{p} is the probability density 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 = copulapdf ("Clayton", x, theta)
+## @end group
+##
+## @group
+## p = copulapdf ("Gumbel", x, 2)
+## @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: PDF of a copula family
+
+function p = copulapdf (family, x, theta)
+
+ # Check arguments
+ if (nargin != 3)
+ print_usage ();
+ endif
+
+ if (! ischar (family))
+ error ("copulapdf: family must be one of 'Clayton', 'Gumbel', 'Frank', and 'AMH'");
+ endif
+
+ if (! isempty (x) && ! ismatrix (x))
+ error ("copulapdf: x must be a numeric matrix");
+ endif
+
+ [n, d] = size (x);
+
+ if (! isvector (theta) || (! isscalar (theta) && size (theta, 1) != n))
+ error ("copulapdf: theta must be a column vector with the same number of rows as x or be scalar");
+ endif
+
+ if (n == 0)
+ # Input is empty
+ p = zeros (0, 1);
+ else
+ if (n > 1 && isscalar (theta))
+ theta = repmat (theta, n, 1);
+ endif
+
+ # Truncate input to unit hypercube
+ x(x < 0) = 0;
+ x(x > 1) = 1;
+
+ # Compute the cumulative distribution function according to family
+ lowerarg = lower (family);
+
+ if (strcmp (lowerarg, "clayton"))
+ # The Clayton family
+ log_cdf = -log (max (sum (x .^ (repmat (-theta, 1, d)), 2) - d + 1, 0)) ./ theta;
+ p = prod (repmat (theta, 1, d) .* repmat (0:(d - 1), n, 1) + 1, 2) .* exp ((1 + theta .* d) .* log_cdf - (theta + 1) .* sum (log (x), 2));
+ # Product copula at columns where theta == 0
+ k = find (theta == 0);
+ if (any (k))
+ p(k) = 1;
+ endif
+ # Check theta
+ if (d > 2)
+ k = find (! (theta >= 0) | ! (theta < inf));
+ else
+ k = find (! (theta >= -1) | ! (theta < inf));
+ endif
+ elseif (strcmp (lowerarg, "gumbel"))
+ # The Gumbel-Hougaard family
+ g = sum ((-log (x)) .^ repmat (theta, 1, d), 2);
+ c = exp (-g .^ (1 ./ theta));
+ p = ((prod (-log (x), 2)) .^ (theta - 1)) ./ prod (x, 2) .* c .* (g .^ (2 ./ theta - 2) + (theta - 1) .* g .^ (1 ./ theta - 2));
+ # Check theta
+ k = find (! (theta >= 1) | ! (theta < inf));
+ elseif (strcmp (lowerarg, "frank"))
+ # The Frank family
+ if (d != 2)
+ error ("copulapdf: Frank copula PDF implemented as bivariate only");
+ endif
+ p = (theta .* exp (theta .* (1 + sum (x, 2))) .* (exp (theta) - 1))./ (exp (theta) - exp (theta + theta .* x(:, 1)) + exp (theta .* sum (x, 2)) - exp (theta + theta .* x(:, 2))) .^ 2;
+ # Product copula at columns where theta == 0
+ k = find (theta == 0);
+ if (any (k))
+ p(k) = 1;
+ endif
+ # Check theta
+ k = find (! (theta > -inf) | ! (theta < inf));
+ elseif (strcmp (lowerarg, "amh"))
+ # The Ali-Mikhail-Haq family
+ if (d != 2)
+ error ("copulapdf: Ali-Mikhail-Haq copula PDF implemented as bivariate only");
+ endif
+ z = theta .* prod (x - 1, 2) - 1;
+ p = (theta .* (1 - sum (x, 2) - prod (x, 2) - z) - 1) ./ (z .^ 3);
+ # Check theta
+ k = find (! (theta >= -1) | ! (theta < 1));
+ else
+ error ("copulapdf: unknown copula family '%s'", family);
+ endif
+
+ 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 = copulapdf ("Clayton", x, theta);
+%! expected_p = [0.9872; 0.7295];
+%! assert (p, expected_p, 0.001);
+
+%!test
+%! x = [0.2:0.2:0.6; 0.2:0.2:0.6];
+%! p = copulapdf ("Gumbel", x, 2);
+%! expected_p = [0.9468; 0.9468];
+%! assert (p, expected_p, 0.001);
+
+%!test
+%! x = [0.2, 0.6; 0.2, 0.6];
+%! theta = [1; 2];
+%! p = copulapdf ("Frank", x, theta);
+%! expected_p = [0.9378; 0.8678];
+%! assert (p, expected_p, 0.001);
+
+%!test
+%! x = [0.2, 0.6; 0.2, 0.6];
+%! theta = [0.3; 0.7];
+%! p = copulapdf ("AMH", x, theta);
+%! expected_p = [0.9540; 0.8577];
+%! assert (p, expected_p, 0.001);
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