--- /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} =} 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);
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