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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} =} skewness (@var{x})
## @deftypefnx {} {@var{y} =} skewness (@var{x}, @var{flag})
## @deftypefnx {} {@var{y} =} skewness (@var{x}, @var{flag}, @var{dim})
## Compute the sample skewness of the elements of @var{x}.
##
## The sample skewness is defined as
## @tex
## $$
## {\rm skewness} (@var{x}) = {{{1\over N}\,
## \sum_{i=1}^N (x_i - \bar{x})^3} \over \sigma^3},
## $$
## 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})).^3)
## skewness (@var{X}) = ------------------------.
## std (@var{x}).^3
## @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 skewness as defined above. If @var{flag} is
## equal to 0, return the adjusted skewness coefficient instead:
## @tex
## $$
## {\rm skewness} (@var{x}) = {\sqrt{N (N - 1)} \over N - 2} \times \,
## {{{1 \over N} \sum_{i=1}^N (x_i - \bar{x})^3} \over \sigma^3}
## $$
## @end tex
## @ifnottex
##
## @example
## @group
## sqrt (N*(N-1)) mean ((@var{x} - mean (@var{x})).^3)
## skewness (@var{X}, 0) = -------------- * ------------------------.
## (N - 2) std (@var{x}).^3
## @end group
## @end example
##
## @noindent
## where @math{N} is the length of the @var{x} vector.
##
## @end ifnottex
## The adjusted skewness coefficient is obtained by replacing the sample second
## and third central moments by their bias-corrected versions.
##
## If @var{x} is a matrix, or more generally a multi-dimensional array, return
## the skewness along the first non-singleton dimension. If the optional
## @var{dim} argument is given, operate along this dimension.
##
## @seealso{var, kurtosis, moment}
## @end deftypefn
function y = skewness (x, flag, dim)
if (nargin < 1)
print_usage ();
endif
if (! (isnumeric (x) || islogical (x)))
error ("skewness: X must be a numeric vector or matrix");
endif
if (nargin < 2 || isempty (flag))
flag = 1; # default: do not use the "bias corrected" version
elseif (! isscalar (flag) || (flag != 0 && flag != 1))
error ("skewness: FLAG must be 0 or 1");
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 ("skewness: 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
s = std (x, 1, dim); # Normalize with 1/N
y = sum (x .^ 3, dim);
idx = (s != 0);
y(idx) ./= (n * s(idx) .^ 3);
y(! idx) = NaN;
## Apply bias correction to the second and third central sample moment
if (flag == 0)
if (n > 2)
y *= sqrt (n * (n - 1)) / (n - 2);
else
y(:) = NaN;
endif
endif
endfunction
%!assert (skewness ([-1, 0, 1]), 0)
%!assert (skewness ([-2, 0, 1]) < 0)
%!assert (skewness ([-1, 0, 2]) > 0)
%!assert (skewness ([-3, 0, 1]) == -1 * skewness ([-1, 0, 3]))
%!assert (skewness (ones (3, 5)), NaN (1, 5))
%!assert (skewness (1, [], 3), NaN)
%!test
%! x = [0; 0; 0; 1];
%! y = [x, 2*x];
%! assert (skewness (y), 1.154700538379251 * [1 1], 5*eps);
%!assert (skewness ([1:5 10; 1:5 10], 0, 2), 1.439590274527954 * [1; 1], eps)
%!assert (skewness ([1:5 10; 1:5 10], 1, 2), 1.051328089232020 * [1; 1], 2*eps)
%!assert (skewness ([1:5 10; 1:5 10], [], 2), 1.051328089232020 * [1; 1], 2*eps)
## Test behavior on single input
%!assert (skewness (single ([1:5 10])), single (1.0513283), eps ("single"))
%!assert (skewness (single ([1 2]), 0), single (NaN))
## Verify no warnings
%!test
%! lastwarn (""); # clear last warning
%! skewness (1);
%! assert (lastwarn (), "");
## Test input validation
%!error <Invalid call> skewness ()
%!error <X must be a numeric vector or matrix> skewness (['A'; 'B'])
%!error <FLAG must be 0 or 1> skewness (1, 2)
%!error <FLAG must be 0 or 1> skewness (1, [1 0])
%!error <DIM must be an integer> skewness (1, [], ones (2,2))
%!error <DIM must be an integer> skewness (1, [], 1.5)
%!error <DIM must be .* a valid dimension> skewness (1, [], 0)
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