## Copyright (C) 1996-2017 John W. Eaton
##
## 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 {} {} std (@var{x})
## @deftypefnx {} {} std (@var{x}, @var{opt})
## @deftypefnx {} {} std (@var{x}, @var{opt}, @var{dim})
## Compute the standard deviation of the elements of the vector @var{x}.
##
## The standard deviation is defined as
## @tex
## $$
## {\rm std} (x) = \sigma = \sqrt{{\sum_{i=1}^N (x_i - \bar{x})^2 \over N - 1}}
## $$
## where $\bar{x}$ is the mean value of $x$ and $N$ is the number of elements.
## @end tex
## @ifnottex
##
## @example
## @group
## std (x) = sqrt ( 1/(N-1) SUM_i (x(i) - mean(x))^2 )
## @end group
## @end example
##
## @noindent
## where @math{N} is the number of elements.
## @end ifnottex
##
## If @var{x} is a matrix, compute the standard deviation 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 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
## @end table
##
## If the optional argument @var{dim} is given, operate along this dimension.
## @seealso{var, range, iqr, mean, median}
## @end deftypefn
## Author: jwe
function retval = std (x, opt = 0, dim)
if (nargin < 1 || nargin > 3)
print_usage ();
endif
if (! (isnumeric (x) || islogical (x)))
error ("std: X must be a numeric vector or matrix");
endif
if (isempty (opt))
opt = 0;
elseif (! isscalar (opt) || (opt != 0 && opt != 1))
error ("std: 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 ("std: 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
else
retval = sqrt (sumsq (center (x, dim), dim) / (n - 1 + opt));
endif
endfunction
%!test
%! x = ones (10, 2);
%! y = [1, 3];
%! assert (std (x), [0, 0]);
%! assert (std (y), sqrt (2), sqrt (eps));
%! assert (std (x, 0, 2), zeros (10, 1));
%!assert (std (ones (3, 1, 2), 0, 2), zeros (3, 1, 2))
%!assert (std ([1 2], 0), sqrt (2)/2, 5*eps)
%!assert (std ([1 2], 1), 0.5, 5*eps)
%!assert (std (1), 0)
%!assert (std (single (1)), single (0))
%!assert (std ([1 2 3], [], 3), [0 0 0])
##tests for empty input Matlab compatibility (bug #48690)
%!assert (std ([]), NaN)
%!assert (std (single ([])), single (NaN))
%!assert (std (ones (0, 0, 0, 0)), NaN (1, 0, 0, 0))
%!assert (std (ones (0, 0, 0, 1)), NaN (1, 0, 0, 1))
%!assert (std (ones (0, 0, 0, 2)), NaN (1, 0, 0, 2))
%!assert (std (ones (0, 0, 1, 0)), NaN (1, 0, 1, 0))
%!assert (std (ones (0, 0, 1, 1)), NaN (1, 1, 1, 1))
%!assert (std (ones (0, 0, 1, 2)), NaN (1, 0, 1, 2))
%!assert (std (ones (0, 0, 2, 0)), NaN (1, 0, 2, 0))
%!assert (std (ones (0, 0, 2, 1)), NaN (1, 0, 2, 1))
%!assert (std (ones (0, 0, 2, 2)), NaN (1, 0, 2, 2))
%!assert (std (ones (0, 1, 0, 0)), NaN (1, 1, 0, 0))
%!assert (std (ones (0, 1, 0, 1)), NaN (1, 1, 0, 1))
%!assert (std (ones (0, 1, 0, 2)), NaN (1, 1, 0, 2))
%!assert (std (ones (0, 1, 1, 0)), NaN (1, 1, 1, 0))
%!assert (std (ones (0, 1, 1, 1)), NaN (1, 1, 1, 1))
%!assert (std (ones (0, 1, 1, 2)), NaN (1, 1, 1, 2))
%!assert (std (ones (0, 1, 2, 0)), NaN (1, 1, 2, 0))
%!assert (std (ones (0, 1, 2, 1)), NaN (1, 1, 2, 1))
%!assert (std (ones (0, 1, 2, 2)), NaN (1, 1, 2, 2))
%!assert (std (ones (0, 2, 0, 0)), NaN (1, 2, 0, 0))
%!assert (std (ones (0, 2, 0, 1)), NaN (1, 2, 0, 1))
%!assert (std (ones (0, 2, 0, 2)), NaN (1, 2, 0, 2))
%!assert (std (ones (0, 2, 1, 0)), NaN (1, 2, 1, 0))
%!assert (std (ones (0, 2, 1, 1)), NaN (1, 2, 1, 1))
%!assert (std (ones (0, 2, 1, 2)), NaN (1, 2, 1, 2))
%!assert (std (ones (0, 2, 2, 0)), NaN (1, 2, 2, 0))
%!assert (std (ones (0, 2, 2, 1)), NaN (1, 2, 2, 1))
%!assert (std (ones (0, 2, 2, 2)), NaN (1, 2, 2, 2))
%!assert (std (ones (1, 0, 0, 0)), NaN (1, 1, 0, 0))
%!assert (std (ones (1, 0, 0, 1)), NaN (1, 1, 0, 1))
%!assert (std (ones (1, 0, 0, 2)), NaN (1, 1, 0, 2))
%!assert (std (ones (1, 0, 1, 0)), NaN (1, 1, 1, 0))
%!assert (std (ones (1, 0, 1, 1)), NaN (1, 1, 1, 1))
%!assert (std (ones (1, 0, 1, 2)), NaN (1, 1, 1, 2))
%!assert (std (ones (1, 0, 2, 0)), NaN (1, 1, 2, 0))
%!assert (std (ones (1, 0, 2, 1)), NaN (1, 1, 2, 1))
%!assert (std (ones (1, 0, 2, 2)), NaN (1, 1, 2, 2))
%!assert (std (ones (1, 1, 0, 0)), NaN (1, 1, 1, 0))
%!assert (std (ones (1, 1, 0, 1)), NaN (1, 1, 1, 1))
%!assert (std (ones (1, 1, 0, 2)), NaN (1, 1, 1, 2))
%!assert (std (ones (1, 1, 1, 0)), NaN (1, 1, 1, 1))
%!assert (std (ones (1, 1, 2, 0)), NaN (1, 1, 1, 0))
%!assert (std (ones (1, 2, 0, 0)), NaN (1, 1, 0, 0))
%!assert (std (ones (1, 2, 0, 1)), NaN (1, 1, 0, 1))
%!assert (std (ones (1, 2, 0, 2)), NaN (1, 1, 0, 2))
%!assert (std (ones (1, 2, 1, 0)), NaN (1, 1, 1, 0))
%!assert (std (ones (1, 2, 2, 0)), NaN (1, 1, 2, 0))
%!assert (std (ones (2, 0, 0, 0)), NaN (1, 0, 0, 0))
%!assert (std (ones (2, 0, 0, 1)), NaN (1, 0, 0, 1))
%!assert (std (ones (2, 0, 0, 2)), NaN (1, 0, 0, 2))
%!assert (std (ones (2, 0, 1, 0)), NaN (1, 0, 1, 0))
%!assert (std (ones (2, 0, 1, 1)), NaN (1, 0, 1, 1))
%!assert (std (ones (2, 0, 1, 2)), NaN (1, 0, 1, 2))
%!assert (std (ones (2, 0, 2, 0)), NaN (1, 0, 2, 0))
%!assert (std (ones (2, 0, 2, 1)), NaN (1, 0, 2, 1))
%!assert (std (ones (2, 0, 2, 2)), NaN (1, 0, 2, 2))
%!assert (std (ones (2, 1, 0, 0)), NaN (1, 1, 0, 0))
%!assert (std (ones (2, 1, 0, 1)), NaN (1, 1, 0, 1))
%!assert (std (ones (2, 1, 0, 2)), NaN (1, 1, 0, 2))
%!assert (std (ones (2, 1, 1, 0)), NaN (1, 1, 1, 0))
%!assert (std (ones (2, 1, 2, 0)), NaN (1, 1, 2, 0))
%!assert (std (ones (2, 2, 0, 0)), NaN (1, 2, 0, 0))
%!assert (std (ones (2, 2, 0, 1)), NaN (1, 2, 0, 1))
%!assert (std (ones (2, 2, 0, 2)), NaN (1, 2, 0, 2))
%!assert (std (ones (2, 2, 1, 0)), NaN (1, 2, 1, 0))
%!assert (std (ones (2, 2, 2, 0)), NaN (1, 2, 2, 0))
%!assert (std (ones (1, 1, 0, 0, 0)), NaN (1, 1, 1, 0, 0))
%!assert (std (ones (1, 1, 1, 1, 0)), NaN (1, 1, 1, 1, 1))
%!assert (std (ones (2, 1, 1, 1, 0)), NaN (1, 1, 1, 1, 0))
%!assert (std (ones (1, 2, 1, 1, 0)), NaN (1, 1, 1, 1, 0))
%!assert (std (ones (1, 3, 0, 2)), NaN (1, 1, 0, 2))
%!assert (std (single (ones (1, 3, 0, 2))), single (NaN (1, 1, 0, 2)))
%!assert (std ([], 0, 1), NaN (1, 0))
%!assert (std ([], 0, 2), NaN (0, 1))
%!assert (std ([], 0, 3), [])
%!assert (std (ones (1, 0), 0, 1), NaN (1, 0))
%!assert (std (ones (1, 0), 0, 2), NaN)
%!assert (std (ones (1, 0), 0, 3), NaN (1, 0))
%!assert (std (ones (0, 1), 0, 1), NaN)
%!assert (std (ones (0, 1), 0, 2), NaN (0, 1))
%!assert (std (ones (0, 1), 0, 3), NaN (0, 1))
%!assert (std ([], 1, 1), NaN (1, 0))
%!assert (std ([], 1, 2), NaN (0, 1))
%!assert (std ([], 1, 3), [])
%!assert (std (ones (1, 0), 1, 1), NaN (1, 0))
%!assert (std (ones (1, 0), 1, 2), NaN)
%!assert (std (ones (1, 0), 1, 3), NaN (1, 0))
%!assert (std (ones (0, 1), 1, 1), NaN)
%!assert (std (ones (0, 1), 1, 2), NaN (0, 1))
%!assert (std (ones (0, 1), 1, 3), NaN (0, 1))
## Test input validation
%!error std ()
%!error std (1, 2, 3, 4)
%!error std (['A'; 'B'])
%!error std (1, 2)
%!error std (1, [], ones (2, 2))
%!error std (1, [], 1.5)
%!error std (1, [], 0)