## Copyright (C) 1997-2015 Kurt Hornik
## Copyright (C) 2016 Dag Lyberg
## Copyright (C) 2018 John Donoghue
## Copyright (C) 2023-2024 Andreas Bertsatos <abertsatos@biol.uoa.gr>
##
## This file is part of the statistics package for GNU Octave.
##
## 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  {statistics} {@var{y} =} gppdf (@var{x}, @var{k}, @var{sigma}, @var{theta})
##
## Generalized Pareto probability density function (PDF).
##
## For each element of @var{x}, compute the probability density function (PDF)
## of the generalized Pareto distribution with shape parameter @var{k}, scale
## parameter @var{sigma}, and location parameter @var{theta}.  The size of
## @var{y} is the common size of @var{p}, @var{k}, @var{sigma}, and @var{theta}.
## A scalar input functions as a constant matrix of the same size as the other
## inputs.
##
## When @qcode{@var{k} = 0} and @qcode{@var{theta} = 0}, the Generalized Pareto
## is equivalent to the exponential distribution.  When @qcode{@var{k} > 0} and
## @code{@var{theta} = @var{k} / @var{k}} the Generalized Pareto is equivalent
## to the Pareto distribution.  The mean of the Generalized Pareto is not finite
## when @qcode{@var{k} >= 1} and the variance is not finite when
## @qcode{@var{k} >= 1/2}.  When @qcode{@var{k} >= 0}, the Generalized Pareto
## has positive density for @qcode{@var{x} > @var{theta}}, or, when
## @qcode{@var{theta} < 0}, for
## @qcode{0 <= (@var{x} - @var{theta}) / @var{sigma} <= -1 / @var{k}}.
##
## Further information about the generalized Pareto distribution can be found at
## @url{https://en.wikipedia.org/wiki/Generalized_Pareto_distribution}
##
## Input arguments must be @qcode{double} or @qcode{single}; integer, logical,
## and character arrays are rejected.  MATLAB accepts a character array and
## evaluates it at the character codes, which Octave deliberately does not,
## since a character array is an integer type and integers are refused too.
##
## With a negative shape parameter the support is the closed interval
## @math{[@var{theta}, @var{theta} - @var{sigma}/@var{k}]}, and the density at
## its upper endpoint follows the limit of the density there: @math{0} for
## @math{-1 < @var{k} < 0}, @math{1/@var{sigma}} at @math{@var{k} = -1}, and
## unbounded for @math{@var{k} < -1}.  MATLAB returns @math{0} at that endpoint
## whatever the shape, which contradicts its own @code{unifpdf}: the
## generalized Pareto with @math{@var{k} = -1} @emph{is} the uniform
## distribution on @math{[@var{theta}, @var{theta} + @var{sigma}]}, for which
## MATLAB's @code{unifpdf} returns @math{1/@var{sigma}} at the same point.  This
## implementation returns the limit, and so agrees with @code{unifpdf}.
##
## @seealso{gpcdf, gpinv, gprnd, gpfit, gplike, gpstat}
## @end deftypefn

function y = gppdf (x, k, sigma, theta)

  ## Check for valid number of input arguments
  if (nargin < 4)
    error ("gppdf: function called with too few input arguments.");
  endif

  ## Check for common size of X, K, SIGMA, and THETA
  if (! isscalar (x) || ! isscalar (k) || ! isscalar (sigma) || ! isscalar (theta))
    [err, x, k, sigma, theta] = common_size (x, k, sigma, theta);
    if (err > 0)
      error ("gppdf: X, K, SIGMA, and THETA must be of common size or scalars.");
    endif
  endif

  ## Check for X, K, SIGMA, and THETA being double or single
  if (! (isfloat (x) && isfloat (k) && isfloat (sigma) && isfloat (theta)))
    error ("gppdf: X, K, SIGMA, and THETA must be double or single.");
  endif

  ## Check for X, K, SIGMA, and THETA being reals
  if (iscomplex (x) || iscomplex (k) || iscomplex (sigma) || iscomplex (theta))
    error ("gppdf: X, K, SIGMA, and THETA must not be complex.");
  endif

  ## Check for class type
  if (isa (x, 'single') || isa (theta, 'single') || isa (sigma, 'single') ...
      || isa (k, 'single'))
    y = zeros (size (x), 'single');
  else
    y = zeros (size (x));
  endif

  ## Return NaNs for out of range values of sigma parameter
  ky = isnan (x) | ! (-Inf < theta) | ! (theta < Inf) | ...
                   ! (sigma > 0) | ! (sigma < Inf) | ...
                   ! (-Inf < k) | ! (k < Inf);
  y(ky) = NaN;

  ky = (-Inf < x) & (x < Inf) & (-Inf < theta) & (theta < Inf) & ...
        (sigma > 0) & (sigma < Inf) & (-Inf < k) & (k < Inf);
  if (isscalar (theta) && isscalar (sigma) && isscalar (k))
    z = (x - theta) / sigma;

    j = ky & (k == 0) & (z >= 0);
    if (any (j))
      y(j) = exp (-z(j));
    endif

    j = ky & (k > 0) & (z >= 0);
    if (any (j))
      y(j) = (k * z(j) + 1) .^ (-(k + 1) / k) ./ sigma;
    endif

    if (k < 0)
      j = ky & (k < 0) & (0 <= z) & (z <= -1. / k);
      if (any (j))
        y(j) = (k * z(j) + 1) .^ (-(k + 1) / k) ./ sigma;
      endif
    endif
  else
    z = (x - theta) ./ sigma;

    j = ky & (k == 0) & (z >= 0);
    if (any (j))
      y(j) = exp ( -z(j));
    endif

    j = ky & (k > 0) & (z >= 0);
    if (any (j))
      y(j) = (k(j) .* z(j) + 1) .^ (-(k(j) + 1) ./ k(j)) ...
                                      ./ sigma(j);
    endif

    if (any (k < 0))
      j = ky & (k < 0) & (0 <= z) & (z <= -1 ./ k);
      if (any (j))
        y(j) = (k(j) .* z(j) + 1) .^ (-(k(j) + 1) ./ k(j)) ...
                                        ./ sigma(j);
      endif
    endif
  endif

endfunction

%!demo
%! ## Plot various PDFs from the generalized Pareto distribution
%! x = 0:0.001:5;
%! y1 = gppdf (x, 1, 1, 0);
%! y2 = gppdf (x, 5, 1, 0);
%! y3 = gppdf (x, 20, 1, 0);
%! y4 = gppdf (x, 1, 2, 0);
%! y5 = gppdf (x, 5, 2, 0);
%! y6 = gppdf (x, 20, 2, 0);
%! plot (x, y1, '-b', x, y2, '-g', x, y3, '-r', ...
%!       x, y4, '-c', x, y5, '-m', x, y6, '-k')
%! grid on
%! xlim ([0, 5])
%! ylim ([0, 1])
%! legend ({'k = 1, σ = 1, θ = 0', 'k = 5, σ = 1, θ = 0', ...
%!          'k = 20, σ = 1, θ = 0', 'k = 1, σ = 2, θ = 0', ...
%!          'k = 5, σ = 2, θ = 0', 'k = 20, σ = 2, θ = 0'}, ...
%!         'location', 'northeast')
%! title ('Generalized Pareto PDF')
%! xlabel ('values in x')
%! ylabel ('density')

## The upper endpoint of the support, measured against R2024a 2026-08-17, which
## returns 0 for all three shapes and so disagrees with its own unifpdf.
%!test
%! ## at k = -1 the distribution is uniform, and the density is 1/sigma
%! assert_equal (gppdf (1, -1, 1, 0), 1);
%! assert_equal (gppdf (1, -1, 1, 0), unifpdf (1, 0, 1));
%! assert_equal (gppdf (2, -1, 2, 0), 0.5);
%! assert_equal (gppdf (2, -1, 2, 0), unifpdf (2, 0, 2));
%!test
%! ## between -1 and 0 the density vanishes at the endpoint
%! assert_equal (gppdf (2, -0.5, 1, 0), 0);
%!test
%! ## below -1 it diverges there
%! assert_equal (gppdf (0.5, -2, 1, 0), Inf);
%!test
%! ## and the endpoint agrees with the values approaching it
%! assert_equal (gppdf (1 - 1e-12, -1, 1, 0), 1);
%! assert_equal (gppdf (1.5, -1, 1, 0), 0);

## Test output
%!shared x, y1, y2, y3
%! x = [-Inf, -1, 0, 1/2, 1, Inf];
%! y1 = [0, 0, 1, 0.6065306597126334, 0.36787944117144233, 0];
%! y2 = [0, 0, 1, 4/9, 1/4, 0];
%! y3 = [0, 0, 1, 1, 1, 0];
%!assert_equal (gppdf (x, zeros (1,6), ones (1,6), zeros (1,6)), y1, eps)
%!assert_equal (gppdf (x, 0, 1, zeros (1,6)), y1, eps)
%!assert_equal (gppdf (x, 0, ones (1,6), 0), y1, eps)
%!assert_equal (gppdf (x, zeros (1,6), 1, 0), y1, eps)
%!assert_equal (gppdf (x, 0, 1, 0), y1, eps)
%!assert_equal (gppdf (x, 0, 1, [0, 0, 0, NaN, 0, 0]), [y1(1:3), NaN, y1(5:6)])
%!assert_equal (gppdf (x, 0, [1, 1, 1, NaN, 1, 1], 0), [y1(1:3), NaN, y1(5:6)])
%!assert_equal (gppdf (x, [0, 0, 0, NaN, 0, 0], 1, 0), [y1(1:3), NaN, y1(5:6)])
%!assert_equal (gppdf ([x(1:3), NaN, x(5:6)], 0, 1, 0), [y1(1:3), NaN, y1(5:6)])
%!assert_equal (gppdf (x, ones (1,6), ones (1,6), zeros (1,6)), y2, eps)
%!assert_equal (gppdf (x, 1, 1, zeros (1,6)), y2, eps)
%!assert_equal (gppdf (x, 1, ones (1,6), 0), y2, eps)
%!assert_equal (gppdf (x, ones (1,6), 1, 0), y2, eps)
%!assert_equal (gppdf (x, 1, 1, 0), y2, eps)
%!assert_equal (gppdf (x, 1, 1, [0, 0, 0, NaN, 0, 0]), [y2(1:3), NaN, y2(5:6)])
%!assert_equal (gppdf (x, 1, [1, 1, 1, NaN, 1, 1], 0), [y2(1:3), NaN, y2(5:6)])
%!assert_equal (gppdf (x, [1, 1, 1, NaN, 1, 1], 1, 0), [y2(1:3), NaN, y2(5:6)])
%!assert_equal (gppdf ([x(1:3), NaN, x(5:6)], 1, 1, 0), [y2(1:3), NaN, y2(5:6)])
%!assert_equal (gppdf (x, -ones (1,6), ones (1,6), zeros (1,6)), y3, eps)
%!assert_equal (gppdf (x, -1, 1, zeros (1,6)), y3, eps)
%!assert_equal (gppdf (x, -1, ones (1,6), 0), y3, eps)
%!assert_equal (gppdf (x, -ones (1,6), 1, 0), y3, eps)
%!assert_equal (gppdf (x, -1, 1, 0), y3, eps)
%!assert_equal (gppdf (x, -1, 1, [0, 0, 0, NaN, 0, 0]), [y3(1:3), NaN, y3(5:6)])
%!assert_equal (gppdf (x, -1, [1, 1, 1, NaN, 1, 1], 0), [y3(1:3), NaN, y3(5:6)])
%!assert_equal (gppdf (x, [-1, -1, -1, NaN, -1, -1], 1, 0), [y3(1:3), NaN, y3(5:6)])
%!assert_equal (gppdf ([x(1:3), NaN, x(5:6)], -1, 1, 0), [y3(1:3), NaN, y3(5:6)])

## Test class of input preserved
%!assert_equal (gppdf (single ([x, NaN]), 0, 1, 0), single ([y1, NaN]))
%!assert_equal (gppdf ([x, NaN], 0, 1, single (0)), single ([y1, NaN]))
%!assert_equal (gppdf ([x, NaN], 0, single (1), 0), single ([y1, NaN]))
%!assert_equal (gppdf ([x, NaN], single (0), 1, 0), single ([y1, NaN]))
%!assert_equal (gppdf (single ([x, NaN]), 1, 1, 0), single ([y2, NaN]))
%!assert_equal (gppdf ([x, NaN], 1, 1, single (0)), single ([y2, NaN]))
%!assert_equal (gppdf ([x, NaN], 1, single (1), 0), single ([y2, NaN]))
%!assert_equal (gppdf ([x, NaN], single (1), 1, 0), single ([y2, NaN]))
%!assert_equal (gppdf (single ([x, NaN]), -1, 1, 0), single ([y3, NaN]))
%!assert_equal (gppdf ([x, NaN], -1, 1, single (0)), single ([y3, NaN]))
%!assert_equal (gppdf ([x, NaN], -1, single (1), 0), single ([y3, NaN]))
%!assert_equal (gppdf ([x, NaN], single (-1), 1, 0), single ([y3, NaN]))

## Test input validation
%!error<gppdf: X, K, SIGMA, and THETA must be double or single.> gppdf (int32 (2), 1, 1, 0)
%!error<gppdf: X, K, SIGMA, and THETA must be double or single.> gppdf (true, 1, 1, 0)
%!error<gppdf: X, K, SIGMA, and THETA must be double or single.> gppdf ('a', 1, 1, 0)
%!error<gpcdf: function called with too few input arguments.> gpcdf ()
%!error<gpcdf: function called with too few input arguments.> gpcdf (1)
%!error<gpcdf: function called with too few input arguments.> gpcdf (1, 2)
%!error<gpcdf: function called with too few input arguments.> gpcdf (1, 2, 3)
%!error<gpcdf: X, K, SIGMA, and THETA must be of common size or scalars.> ...
%! gpcdf (ones (3), ones (2), ones (2), ones (2))
%!error<gpcdf: X, K, SIGMA, and THETA must be of common size or scalars.> ...
%! gpcdf (ones (2), ones (3), ones (2), ones (2))
%!error<gpcdf: X, K, SIGMA, and THETA must be of common size or scalars.> ...
%! gpcdf (ones (2), ones (2), ones (3), ones (2))
%!error<gpcdf: X, K, SIGMA, and THETA must be of common size or scalars.> ...
%! gpcdf (ones (2), ones (2), ones (2), ones (3))
%!error<gpcdf: X, K, SIGMA, and THETA must not be complex.> gpcdf (i, 2, 3, 4)
%!error<gpcdf: X, K, SIGMA, and THETA must not be complex.> gpcdf (1, i, 3, 4)
%!error<gpcdf: X, K, SIGMA, and THETA must not be complex.> gpcdf (1, 2, i, 4)
%!error<gpcdf: X, K, SIGMA, and THETA must not be complex.> gpcdf (1, 2, 3, i)
