scripts/statistics/base/skewness.m

## Copyright (C) 1996, 1997, 1998, 1999, 2000, 2002, 2004, 2005, 2006,
##               2007, 2008, 2009 John W. Eaton
##
## This file is part of Octave.
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## -*- texinfo -*-
## @deftypefn {Function File} {} skewness (@var{x}, @var{dim})
## If @var{x} is a vector of length @math{n}, return the skewness
## @tex
## $$
## {\rm skewness} (x) = {1\over N \sigma(x)^3} \sum_{i=1}^N (x_i-\bar{x})^3
## $$
## where $\bar{x}$ is the mean value of $x$.
## @end tex
## @ifnottex
##
## @example
## skewness (x) = N^(-1) std(x)^(-3) sum ((x - mean(x)).^3)
## @end example
## @end ifnottex
##
## @noindent
## of @var{x}.  If @var{x} is a matrix, return the skewness along the
## first non-singleton dimension of the matrix.  If the optional
## @var{dim} argument is given, operate along this dimension.
## @end deftypefn

## Author: KH <Kurt.Hornik@wu-wien.ac.at>
## Created: 29 July 1994
## Adapted-By: jwe

function retval = skewness (x, dim)

  if (nargin != 1 && nargin != 2)
    print_usage ();
  endif

  nd = ndims (x);
  sz = size (x);
  if (nargin != 2)
    ## Find the first non-singleton dimension.
    dim  = 1;
    while (dim < nd + 1 && sz(dim) == 1)
      dim = dim + 1;
    endwhile
    if (dim > nd)
      dim = 1;
    endif
  else
    if (! (isscalar (dim) && dim == round (dim))
	&& dim > 0
	&& dim < (nd + 1))
      error ("skewness: dim must be an integer and valid dimension");
    endif
  endif

  if (! ismatrix (x))
    error ("skewness: x has to be a matrix or a vector");
  endif

  c = sz(dim);
  idx = ones (1, nd);
  idx (dim) = c;
  x = x - repmat (mean (x, dim), idx);
  sz(dim) = 1;
  retval = zeros (sz);
  s = std (x, [], dim);
  ind = find (s > 0);
  x = sum (x .^ 3, dim);
  retval(ind) = x(ind) ./ (c * s(ind) .^ 3);
  
endfunction

%!error skewness ();

%!error skewness (1, 2, 3);