scripts/statistics/base/kurtosis.m

## Copyright (C) 1996, 1997, 1998, 1999, 2000, 2002, 2004, 2005, 2006,
##               2007, 2008, 2009 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
## <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn {Function File} {} kurtosis (@var{x}, @var{dim})
## If @var{x} is a vector of length @math{N}, return the kurtosis
## @tex
## $$
##  {\rm kurtosis} (x) = {1\over N \sigma(x)^4} \sum_{i=1}^N (x_i-\bar{x})^4 - 3
## $$
## where $\bar{x}$ is the mean value of $x$.
## @end tex
## @ifnottex
##
## @example
## kurtosis (x) = N^(-1) std(x)^(-4) sum ((x - mean(x)).^4) - 3
## @end example
## @end ifnottex
##
## @noindent
## of @var{x}.  If @var{x} is a matrix, return the kurtosis over the
## first non-singleton dimension.  The optional argument @var{dim}
## can be given to force the kurtosis to be given over that 
## dimension.
## @end deftypefn

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

function retval = kurtosis (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 ("kurtosis: dim must be an integer and valid dimension");
    endif
  endif
  
  if (! ismatrix (x))
    error ("kurtosis: x has to be a matrix or a vector");
  endif

  c = sz(dim);
  sz(dim) = 1;
  idx = ones (1, nd);
  idx(dim) = c;
  x = x - repmat (mean (x, dim), idx);
  retval = zeros (sz);
  s = std (x, [], dim);
  x = sum(x.^4, dim);
  ind = find (s > 0);
  retval(ind) = x(ind) ./ (c * s(ind) .^ 4) - 3;

endfunction

%!test
%! x = [-1; 0; 0; 0; 1];
%! y = [x, 2*x];
%! assert(all (abs (kurtosis (y) - [-1.4, -1.4]) < sqrt (eps)));

%!error kurtosis ();

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