## Copyright (C) 1995-2017 Kurt Hornik ## ## 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 ## . ## -*- texinfo -*- ## @deftypefn {} {} var (@var{x}) ## @deftypefnx {} {} var (@var{x}, @var{opt}) ## @deftypefnx {} {} var (@var{x}, @var{opt}, @var{dim}) ## Compute the variance of the elements of the vector @var{x}. ## ## The variance is defined as ## @tex ## $$ ## {\rm var} (x) = \sigma^2 = {\sum_{i=1}^N (x_i - \bar{x})^2 \over N - 1} ## $$ ## where $\bar{x}$ is the mean value of $x$. ## @end tex ## @ifnottex ## ## @example ## @group ## var (x) = 1/(N-1) SUM_i (x(i) - mean(x))^2 ## @end group ## @end example ## ## @end ifnottex ## If @var{x} is a matrix, compute the variance for each column and return ## them in a row vector. ## ## The argument @var{opt} determines the type of normalization to use. ## Valid values are ## ## @table @asis ## @item 0: ## normalize with @math{N-1}, provides the best unbiased estimator of the ## variance [default] ## ## @item 1: ## normalizes with @math{N}, this provides the second moment around the mean ## @end table ## ## If @math{N==1} the value of @var{opt} is ignored and normalization by ## @math{N} is used. ## ## If the optional argument @var{dim} is given, operate along this dimension. ## @seealso{cov, std, skewness, kurtosis, moment} ## @end deftypefn ## Author: KH ## Description: Compute variance function retval = var (x, opt = 0, dim) if (nargin < 1 || nargin > 3) print_usage (); endif if (! (isnumeric (x) || islogical (x))) error ("var: X must be a numeric vector or matrix"); endif if (isempty (opt)) opt = 0; elseif (opt != 0 && opt != 1) error ("var: normalization OPT 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 ("var: DIM must be an integer and a valid dimension"); endif endif n = size (x, dim); if (isempty (x)) %% codepath for Matlab compatibility. empty x produces NaN output, but %% for ndim > 2, output depends on size of x. if ((nargin < 3) && (nd == 2) && (max (sz) < 2)) retval = NaN; else if (nargin == 3) sz(dim) = 1; else sz (find ((sz ~= 1), 1)) = 1; endif retval = NaN (sz); endif if (isa (x, "single")) retval = single (retval); endif elseif (n == 1) if (isa (x, "single")) retval = zeros (sz, "single"); else retval = zeros (sz); endif elseif (numel (x) > 0) retval = sumsq (center (x, dim), dim) / (n - 1 + opt); else error ("var: X must not be empty"); endif endfunction %!assert (var (13), 0) %!assert (var (single (13)), single (0)) %!assert (var ([1, 2, 3]), 1) %!assert (var ([1, 2, 3], 1), 2/3, eps) %!assert (var ([1, 2, 3], [], 1), [0, 0, 0]) %!assert (var ([1, 2, 3], [], 3), [0, 0, 0]) ##tests for empty input Matlab compatibility (bug #48690) %!assert (var ([]), NaN) %!assert (var (single ([])), single (NaN)) %!assert (var (ones (0, 0, 0, 0)), NaN (1, 0, 0, 0)) %!assert (var (ones (0, 0, 0, 1)), NaN (1, 0, 0, 1)) %!assert (var (ones (0, 0, 0, 2)), NaN (1, 0, 0, 2)) %!assert (var (ones (0, 0, 1, 0)), NaN (1, 0, 1, 0)) %!assert (var (ones (0, 0, 1, 1)), NaN (1, 1, 1, 1)) %!assert (var (ones (0, 0, 1, 2)), NaN (1, 0, 1, 2)) %!assert (var (ones (0, 0, 2, 0)), NaN (1, 0, 2, 0)) %!assert (var (ones (0, 0, 2, 1)), NaN (1, 0, 2, 1)) %!assert (var (ones (0, 0, 2, 2)), NaN (1, 0, 2, 2)) %!assert (var (ones (0, 1, 0, 0)), NaN (1, 1, 0, 0)) %!assert (var (ones (0, 1, 0, 1)), NaN (1, 1, 0, 1)) %!assert (var (ones (0, 1, 0, 2)), NaN (1, 1, 0, 2)) %!assert (var (ones (0, 1, 1, 0)), NaN (1, 1, 1, 0)) %!assert (var (ones (0, 1, 1, 1)), NaN (1, 1, 1, 1)) %!assert (var (ones (0, 1, 1, 2)), NaN (1, 1, 1, 2)) %!assert (var (ones (0, 1, 2, 0)), NaN (1, 1, 2, 0)) %!assert (var (ones (0, 1, 2, 1)), NaN (1, 1, 2, 1)) %!assert (var (ones (0, 1, 2, 2)), NaN (1, 1, 2, 2)) %!assert (var (ones (0, 2, 0, 0)), NaN (1, 2, 0, 0)) %!assert (var (ones (0, 2, 0, 1)), NaN (1, 2, 0, 1)) %!assert (var (ones (0, 2, 0, 2)), NaN (1, 2, 0, 2)) %!assert (var (ones (0, 2, 1, 0)), NaN (1, 2, 1, 0)) %!assert (var (ones (0, 2, 1, 1)), NaN (1, 2, 1, 1)) %!assert (var (ones (0, 2, 1, 2)), NaN (1, 2, 1, 2)) %!assert (var (ones (0, 2, 2, 0)), NaN (1, 2, 2, 0)) %!assert (var (ones (0, 2, 2, 1)), NaN (1, 2, 2, 1)) %!assert (var (ones (0, 2, 2, 2)), NaN (1, 2, 2, 2)) %!assert (var (ones (1, 0, 0, 0)), NaN (1, 1, 0, 0)) %!assert (var (ones (1, 0, 0, 1)), NaN (1, 1, 0, 1)) %!assert (var (ones (1, 0, 0, 2)), NaN (1, 1, 0, 2)) %!assert (var (ones (1, 0, 1, 0)), NaN (1, 1, 1, 0)) %!assert (var (ones (1, 0, 1, 1)), NaN (1, 1, 1, 1)) %!assert (var (ones (1, 0, 1, 2)), NaN (1, 1, 1, 2)) %!assert (var (ones (1, 0, 2, 0)), NaN (1, 1, 2, 0)) %!assert (var (ones (1, 0, 2, 1)), NaN (1, 1, 2, 1)) %!assert (var (ones (1, 0, 2, 2)), NaN (1, 1, 2, 2)) %!assert (var (ones (1, 1, 0, 0)), NaN (1, 1, 1, 0)) %!assert (var (ones (1, 1, 0, 1)), NaN (1, 1, 1, 1)) %!assert (var (ones (1, 1, 0, 2)), NaN (1, 1, 1, 2)) %!assert (var (ones (1, 1, 1, 0)), NaN (1, 1, 1, 1)) %!assert (var (ones (1, 1, 2, 0)), NaN (1, 1, 1, 0)) %!assert (var (ones (1, 2, 0, 0)), NaN (1, 1, 0, 0)) %!assert (var (ones (1, 2, 0, 1)), NaN (1, 1, 0, 1)) %!assert (var (ones (1, 2, 0, 2)), NaN (1, 1, 0, 2)) %!assert (var (ones (1, 2, 1, 0)), NaN (1, 1, 1, 0)) %!assert (var (ones (1, 2, 2, 0)), NaN (1, 1, 2, 0)) %!assert (var (ones (2, 0, 0, 0)), NaN (1, 0, 0, 0)) %!assert (var (ones (2, 0, 0, 1)), NaN (1, 0, 0, 1)) %!assert (var (ones (2, 0, 0, 2)), NaN (1, 0, 0, 2)) %!assert (var (ones (2, 0, 1, 0)), NaN (1, 0, 1, 0)) %!assert (var (ones (2, 0, 1, 1)), NaN (1, 0, 1, 1)) %!assert (var (ones (2, 0, 1, 2)), NaN (1, 0, 1, 2)) %!assert (var (ones (2, 0, 2, 0)), NaN (1, 0, 2, 0)) %!assert (var (ones (2, 0, 2, 1)), NaN (1, 0, 2, 1)) %!assert (var (ones (2, 0, 2, 2)), NaN (1, 0, 2, 2)) %!assert (var (ones (2, 1, 0, 0)), NaN (1, 1, 0, 0)) %!assert (var (ones (2, 1, 0, 1)), NaN (1, 1, 0, 1)) %!assert (var (ones (2, 1, 0, 2)), NaN (1, 1, 0, 2)) %!assert (var (ones (2, 1, 1, 0)), NaN (1, 1, 1, 0)) %!assert (var (ones (2, 1, 2, 0)), NaN (1, 1, 2, 0)) %!assert (var (ones (2, 2, 0, 0)), NaN (1, 2, 0, 0)) %!assert (var (ones (2, 2, 0, 1)), NaN (1, 2, 0, 1)) %!assert (var (ones (2, 2, 0, 2)), NaN (1, 2, 0, 2)) %!assert (var (ones (2, 2, 1, 0)), NaN (1, 2, 1, 0)) %!assert (var (ones (2, 2, 2, 0)), NaN (1, 2, 2, 0)) %!assert (var (ones (1, 1, 0, 0, 0)), NaN (1, 1, 1, 0, 0)) %!assert (var (ones (1, 1, 1, 1, 0)), NaN (1, 1, 1, 1, 1)) %!assert (var (ones (2, 1, 1, 1, 0)), NaN (1, 1, 1, 1, 0)) %!assert (var (ones (1, 2, 1, 1, 0)), NaN (1, 1, 1, 1, 0)) %!assert (var (ones (1, 3, 0, 2)), NaN (1, 1, 0, 2)) %!assert (var (single (ones (1, 3, 0, 2))), single (NaN (1, 1, 0, 2))) %!assert (var ([], 0, 1), NaN (1, 0)) %!assert (var ([], 0, 2), NaN (0, 1)) %!assert (var ([], 0, 3), []) %!assert (var (ones (1, 0), 0, 1), NaN (1, 0)) %!assert (var (ones (1, 0), 0, 2), NaN) %!assert (var (ones (1, 0), 0, 3), NaN (1, 0)) %!assert (var (ones (0, 1), 0, 1), NaN) %!assert (var (ones (0, 1), 0, 2), NaN (0, 1)) %!assert (var (ones (0, 1), 0, 3), NaN (0, 1)) %!assert (var ([], 1, 1), NaN (1, 0)) %!assert (var ([], 1, 2), NaN (0, 1)) %!assert (var ([], 1, 3), []) %!assert (var (ones (1, 0), 1, 1), NaN (1, 0)) %!assert (var (ones (1, 0), 1, 2), NaN) %!assert (var (ones (1, 0), 1, 3), NaN (1, 0)) %!assert (var (ones (0, 1), 1, 1), NaN) %!assert (var (ones (0, 1), 1, 2), NaN (0, 1)) %!assert (var (ones (0, 1), 1, 3), NaN (0, 1)) ## Test input validation %!error var () %!error var (1, 2, 3, 4) %!error var (['A'; 'B']) %!error var (1, -1) %!error skewness (1, 2) %!error skewness (1, [1 0]) %!error var (1, [], ones (2, 2)) %!error var (1, [], 1.5) %!error var (1, [], 0)