view scripts/statistics/skewness.m @ 30997:5330efaf9476

Add optional second output to var and std (bug #62395) * scripts/statistics/var.m: Add optional second output containing the mean used to calculate the variance. Move weight isempty check ahead of vector dimension isscalar check to avoid triggering incompatability error. Add BISTs testing second output with different calling options. Add BIST testing empty value passed as variance weight treated as zero. Add new output behavior to docstring, and update function definitions to show the primary variable. * scripts/statistics/std.m: Add passthrough for second output from var when std called with two outputs. Add BISTs testing second output with different calling options. Update docstring noting new output behavior. * etc/NEWS.8.md: Note output changes to var and std under Matlab Compatability.
author Nicholas R. Jankowski <jankowski.nicholas@gmail.com>
date Thu, 12 May 2022 13:10:52 -0400
parents 5d3faba0342e
children 597f3ee61a48
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########################################################################
##
## Copyright (C) 1996-2022 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)