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view scripts/statistics/skewness.m @ 29358:0a5b15007766 stable
update Octave Project Developers copyright for the new year
In files that have the "Octave Project Developers" copyright notice,
update for 2021.
author | John W. Eaton <jwe@octave.org> |
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date | Wed, 10 Feb 2021 09:52:15 -0500 |
parents | 9f9ac219896d |
children | 7854d5752dd2 |
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######################################################################## ## ## Copyright (C) 1996-2021 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 {} {} skewness (@var{x}) ## @deftypefnx {} {} skewness (@var{x}, @var{flag}) ## @deftypefnx {} {} 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) || (nargin > 3) 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 skewness () %!error skewness (1, 2, 3) %!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)