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##
## Copyright (C) 1995-2021 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or .
##
## This file is part of Octave.
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## Octave is free software: you can redistribute it and/or modify it
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########################################################################
## -*- texinfo -*-
## @deftypefn {} {} var (@var{x})
## @deftypefnx {} {} var (@var{x}, @var{w})
## @deftypefnx {} {} var (@var{x}, @var{w}, @var{dim})
## @deftypefnx {} {} var (@var{x}, @var{w}, @qcode{"ALL"})
## 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 @var{x} and $N$ is the number of
## elements of @var{x}.
##
## @end tex
## @ifnottex
##
## @example
## @group
## var (@var{x}) = 1/(N-1) SUM_i (@var{x}(i) - mean(@var{x}))^2
## @end group
## @end example
##
## @noindent
## where @math{N} is the length of the @var{x} vector.
##
## @end ifnottex
## If @var{x} is a matrix, compute the variance for each column and return
## them in a row vector.
##
## The argument @var{w} determines the weighting scheme to use.
## Valid values are
##
## @table @asis
## @item 0:
## normalize with @math{N-1}, provides the square root of the best unbiased
## estimator of the variance [default]
##
## @item 1:
## normalize with @math{N}, this provides the square root of the second
## moment around the mean
##
## @item a vector:
## compute the weighted variance with nonnegative scalar weights,
## the length of @var{w} must be equal to the size of @var{x} along dimension
## @var{dim}
## @end table
##
## If @math{N} is equal to 1 the value of @var{opt} is ignored and
## normalization by @math{N} is used.
##
## The optional variable @var{dim} can be used to force @code{var} to operate
## over the specified dimension. @var{dim} can either be a scalar dimension or
## a vector of non-repeating dimensions over which to operate. Dimensions are
## positive integers. When @var{dim} is a vector, @var{w} must be either 0 or
## 1, and the variance is calculated over the array slice defined by @var{dim}.
##
## Specifying dimension @qcode{"ALL"} will force @code{var} to operate
## on all elements of @var{x}, when @var{w} is either 0 or 1, and is equivalent
## to @code{var (@var{x}(:))}.
## @seealso{cov, std, skewness, kurtosis, moment}
## @end deftypefn
function retval = var (x, w = 0, dim)
if (nargin < 1)
print_usage ();
elseif (nargin < 3)
dim = [];
endif
if (! (isnumeric (x) || islogical (x)))
error ("var: X must be a numeric vector or matrix");
endif
if (isempty (w))
w = 0;
elseif (! isvector (w) ||
! isnumeric (w) ||
(isvector (w) && any (w < 0)) ||
(isscalar (w) && (w != 0 && w != 1)))
error ("var: W must be 0 or 1 or a vector of positive integers");
endif
nd = ndims (x);
sz = size (x);
emptydimflag = false;
if (isempty (dim))
emptydimflag = true; ## Compatibliity hack for empty x, ndims==2
## Find the first non-singleton dimension.
(dim = find (sz != 1, 1)) || (dim = 1);
else
if (! (isscalar (dim) && dim == fix (dim) && dim > 0))
if (isvector (dim) &&
isnumeric (dim) &&
all (dim > 0) &&
all (rem (dim, 1) == 0))
if (dim != unique (dim, "stable"))
error (["var: vector DIM must contain non-repeating positive"...
"integers"]);
endif
## Check W
if (! isscalar (w))
error ("var: W must be either 0 or 1 when DIM is a vector");
endif
## Reshape X to compute the variance over an array slice
if (iscolumn (dim))
dim = transpose (dim);
endif
collapsed_dims = dim;
dim = dim(end);
## Permute X to cluster the dimensions to collapse
highest_dim = max ([nd, collapsed_dims]);
perm_start = perm_end = [1:highest_dim];
perm_start(dim:end) = [];
perm_start(ismember (perm_start, collapsed_dims)) = [];
perm_end(1:dim) = [];
perm_end(ismember (perm_end, collapsed_dims)) = [];
perm = [perm_start, collapsed_dims, perm_end];
x = permute (x, perm);
## Collapse the given dimensions
newshape = ones (1, highest_dim);
newshape(1:nd) = sz;
newshape(collapsed_dims(1:(end - 1))) = 1;
newshape(dim) = prod (sz(collapsed_dims));
## New X with collapsed dimensions
x = reshape (x, newshape);
elseif (ischar (dim) &&
strcmp (tolower (dim), "all"))
## Check W
if (! isscalar (w))
error ("var: W must be either 0 or 1 when using 'ALL' as dimension");
endif
## "ALL" equals to collapsing all elements to a single vector
x = x(:);
dim = 1;
sz = size (x);
else
error ("var: DIM must be a positive integer scalar, vector, or 'all'");
endif
endif
endif
if (isempty (x))
if (emptydimflag && isequal (sz, [0 0]))
retval = NaN;
else
output_size = sz;
output_size(dim) = 1;
retval = NaN(output_size);
endif
else
n = size (x, dim);
if (n == 1)
if (isa (x, "single"))
retval = zeros (sz, "single");
else
retval = zeros (sz);
endif
else
if (isscalar (w))
retval = sumsq (center (x, dim), dim) / (n - 1 + w);
else
## Weighted variance
if (length (w) != n)
error (["var: the length of W must be equal to the size of X "...
"along which the variance is calculated"]);
else
if ((dim == 1 && rows (w) == 1) ||
(dim == 2 && columns (w) == 1))
w = transpose (w);
elseif (dim > 2)
newdims = [(ones (1, (dim - 1))), (length (w))];
w = reshape (w, newdims);
endif
den = sum (w);
mu = sum (w .* x, dim) ./ sum (w);
retval = sum (w .* ((x .- mu) .^ 2), dim) / den;
endif
endif
endif
endif
endfunction
##%!assert (var ([]), NaN)
%!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])
%!assert (var ([1:7], [1:7]), 3)
%!assert (var ([(eye (3))], [1:3]), [.1389 .2222 .25], 1e-4)
%!assert (var (ones (2,2,2), [1:2], 3), [(zeros (2,2))])
%!assert (var ([1 2; 3 4], 0, 'all'), var ([1:4]))
%!assert (var (reshape ([1:8], 2, 2, 2), 0, [1 3]), [5.6667 5.6667], 1e-4)
##Test empty inputs
%!assert (var ([]), NaN)
%!assert (var ([],[],1), NaN(1,0))
%!assert (var ([],[],2), NaN(0,1))
%!assert (var ([],[],3), [])
%!assert (var (ones (0,1)), NaN)
%!assert (var (ones (1,0)), NaN)
%!assert (var (ones (1,0), [], 1), NaN(1,0))
%!assert (var (ones (1,0), [], 2), NaN)
%!assert (var (ones (1,0), [], 3), NaN(1,0))
%!assert (var (ones (0,1)), NaN)
%!assert (var (ones (0,1), [], 1), NaN)
%!assert (var (ones (0,1), [], 2), NaN(0,1))
%!assert (var (ones (0,1), [], 3), NaN(0,1))
%!assert (var (ones (1,3,0,2)), NaN(1,1,0,2))
%!assert (var (ones (1,3,0,2), [], 1), NaN(1,3,0,2))
%!assert (var (ones (1,3,0,2), [], 2), NaN(1,1,0,2))
%!assert (var (ones (1,3,0,2), [], 3), NaN(1,3,1,2))
%!assert (var (ones (1,3,0,2), [], 4), NaN(1,3,0))
## Test input validation
%!error var ()
%!error var (['A'; 'B'])
%!error var (1, 2)
%!error var ([1 2], [-1 0])
%!error var ([1 2], eye (2))
%!error var (ones (2, 2), [1 2], [1 2])
%!error var ([1 2], [1 2], 'all')
%!error var ([1 2], [1 2 3])
%!error var (1, [], ones (2,2))
%!error var (1, [], 1.5)
%!error var (1, [], 0)