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########################################################################
##
## Copyright (C) 1996-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## This file is part of Octave.
##
## Octave 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.
##
## Octave 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 Octave; see the file COPYING. If not, see
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {@var{y} =} kurtosis (@var{x})
## @deftypefnx {} {@var{y} =} kurtosis (@var{x}, @var{flag})
## @deftypefnx {} {@var{y} =} kurtosis (@var{x}, @var{flag}, @var{dim})
## Compute the sample kurtosis of the elements of @var{x}.
##
## The sample kurtosis is defined as
## @tex
## $$
## \kappa_1 = {{{1\over N}\,
## \sum_{i=1}^N (x_i - \bar{x})^4} \over \sigma^4},
## $$
## where $N$ is the length of @var{x}, $\bar{x}$ its mean, and $\sigma$
## its (uncorrected) standard deviation.
## @end tex
## @ifnottex
##
## @example
## @group
## mean ((@var{x} - mean (@var{x})).^4)
## k1 = ------------------------
## std (@var{x}).^4
## @end group
## @end example
##
## @end ifnottex
##
## @noindent
## The optional argument @var{flag} controls which normalization is used.
## If @var{flag} is equal to 1 (default value, used when @var{flag} is omitted
## or empty), return the sample kurtosis as defined above. If @var{flag} is
## equal to 0, return the @w{"bias-corrected"}@ kurtosis coefficient instead:
## @tex
## $$
## \kappa_0 = 3 + {\scriptstyle N - 1 \over \scriptstyle (N - 2)(N - 3)} \,
## \left( (N + 1)\, \kappa_1 - 3 (N - 1) \right)
## $$
## @end tex
## @ifnottex
##
## @example
## @group
## N - 1
## k0 = 3 + -------------- * ((N + 1) * k1 - 3 * (N - 1))
## (N - 2)(N - 3)
## @end group
## @end example
##
## @noindent
## where @math{N} is the length of the @var{x} vector.
##
## @end ifnottex
## The bias-corrected kurtosis coefficient is obtained by replacing the sample
## second and fourth central moments by their unbiased versions. It is an
## unbiased estimate of the population kurtosis for normal populations.
##
## If @var{x} is a matrix, or more generally a multi-dimensional array, return
## the kurtosis along the first non-singleton dimension. If the optional
## @var{dim} argument is given, operate along this dimension.
##
## @seealso{var, skewness, moment}
## @end deftypefn
function y = kurtosis (x, flag, dim)
if (nargin < 1)
print_usage ();
endif
if (! (isnumeric (x) || islogical (x)))
error ("kurtosis: X must be a numeric vector or matrix");
endif
if (nargin < 2 || isempty (flag))
flag = 1; # default: do not use the "bias corrected" version
else
if (! isscalar (flag) || (flag != 0 && flag != 1))
error ("kurtosis: FLAG must be 0 or 1");
endif
endif
nd = ndims (x);
sz = size (x);
if (nargin < 3)
## Find the first non-singleton dimension.
(dim = find (sz > 1, 1)) || (dim = 1);
else
if (! (isscalar (dim) && dim == fix (dim) && dim > 0))
error ("kurtosis: DIM must be an integer and a valid dimension");
endif
endif
n = size (x, dim);
sz(dim) = 1;
x = center (x, dim); # center also promotes integer, logical to double
v = var (x, 1, dim); # normalize with 1/N
y = sum (x .^ 4, dim);
idx = (v != 0);
y(idx) = y(idx) ./ (n * v(idx) .^ 2);
y(! idx) = NaN;
## Apply bias correction to the second and fourth central sample moment
if (flag == 0)
if (n > 3)
C = (n - 1) / ((n - 2) * (n - 3));
y = 3 + C * ((n + 1) * y - 3 * (n - 1));
else
y(:) = NaN;
endif
endif
endfunction
%!test
%! x = [-1; 0; 0; 0; 1];
%! y = [x, 2*x];
%! assert (kurtosis (y), [2.5, 2.5], sqrt (eps));
%!assert (kurtosis ([-3, 0, 1]) == kurtosis ([-1, 0, 3]))
%!assert (kurtosis (ones (3, 5)), NaN (1, 5))
%!assert (kurtosis (1, [], 3), NaN)
%!assert (kurtosis ([1:5 10; 1:5 10], 0, 2),
%! 5.4377317925288901 * [1; 1], 8 * eps)
%!assert (kurtosis ([1:5 10; 1:5 10], 1, 2),
%! 2.9786509002956195 * [1; 1], 8 * eps)
%!assert (kurtosis ([1:5 10; 1:5 10], [], 2),
%! 2.9786509002956195 * [1; 1], 8 * eps)
## Test behavior on single input
%!assert (kurtosis (single ([1:5 10])), single (2.9786513), eps ("single"))
%!assert (kurtosis (single ([1 2]), 0), single (NaN))
## Verify no warnings
%!test
%! lastwarn (""); # clear last warning
%! kurtosis (1);
%! assert (lastwarn (), "");
## Test input validation
%!error <Invalid call> kurtosis ()
%!error <X must be a numeric vector or matrix> kurtosis (['A'; 'B'])
%!error <FLAG must be 0 or 1> kurtosis (1, 2)
%!error <FLAG must be 0 or 1> kurtosis (1, [1 0])
%!error <DIM must be an integer> kurtosis (1, [], ones (2,2))
%!error <DIM must be an integer> kurtosis (1, [], 1.5)
%!error <DIM must be .* a valid dimension> kurtosis (1, [], 0)
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