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[CreaPhase.git] / octave_packages / m / statistics / base / kendall.m

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+## Copyright (C) 1995-2012 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
+## <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn  {Function File} {} kendall (@var{x})
+## @deftypefnx {Function File} {} kendall (@var{x}, @var{y})
+## @cindex Kendall's Tau
+## Compute Kendall's @var{tau}.
+##
+## For two data vectors @var{x}, @var{y} of common length @var{n},
+## Kendall's @var{tau} is the correlation of the signs of all rank
+## differences of @var{x} and @var{y}; i.e., if both @var{x} and
+## @var{y} have distinct entries, then
+##
+## @tex
+## $$ \tau = {1 \over n(n-1)} \sum_{i,j} {\rm sign}(q_i-q_j) {\rm sign}(r_i-r_j) $$
+## @end tex
+## @ifnottex
+##
+## @example
+## @group
+##          1
+## tau = -------   SUM sign (q(i) - q(j)) * sign (r(i) - r(j))
+##       n (n-1)   i,j
+## @end group
+## @end example
+##
+## @end ifnottex
+## @noindent
+## in which the
+## @tex
+## $q_i$ and $r_i$
+## @end tex
+## @ifnottex
+## @var{q}(@var{i}) and @var{r}(@var{i})
+## @end ifnottex
+## are the ranks of @var{x} and @var{y}, respectively.
+##
+## If @var{x} and @var{y} are drawn from independent distributions,
+## Kendall's @var{tau} is asymptotically normal with mean 0 and variance
+## @tex
+## ${2 (2n+5) \over 9n(n-1)}$.
+## @end tex
+## @ifnottex
+## @code{(2 * (2@var{n}+5)) / (9 * @var{n} * (@var{n}-1))}.
+## @end ifnottex
+##
+## @code{kendall (@var{x})} is equivalent to @code{kendall (@var{x},
+## @var{x})}.
+## @seealso{ranks, spearman}
+## @end deftypefn
+
+## Author: KH <Kurt.Hornik@wu-wien.ac.at>
+## Description: Kendall's rank correlation tau
+
+function tau = kendall (x, y = [])
+
+  if (nargin < 1 || nargin > 2)
+    print_usage ();
+  endif
+
+  if (   ! (isnumeric (x) || islogical (x))
+      || ! (isnumeric (y) || islogical (y)))
+    error ("kendall: X and Y must be numeric matrices or vectors");
+  endif
+
+  if (ndims (x) != 2 || ndims (y) != 2)
+    error ("kendall: X and Y must be 2-D matrices or vectors");
+  endif
+
+  if (isrow (x))
+    x = x.';
+  endif
+  [n, c] = size (x);
+
+  if (nargin == 2)
+    if (isrow (y))
+      y = y.';
+    endif
+    if (rows (y) != n)
+      error ("kendall: X and Y must have the same number of observations");
+    else
+      x = [x, y];
+    endif
+  endif
+
+  if (isa (x, 'single') || isa (y, 'single'))
+    cls = 'single';
+  else
+    cls = 'double';
+  endif
+  r   = ranks (x);
+  m   = sign (kron (r, ones (n, 1, cls)) - kron (ones (n, 1, cls), r));
+  tau = corr (m);
+
+  if (nargin == 2)
+    tau = tau(1 : c, (c + 1) : columns (x));
+  endif
+
+endfunction
+
+
+%!test
+%! x = [1:2:10];
+%! y = [100:10:149];
+%! assert (kendall (x,y), 1, 5*eps);
+%! assert (kendall (x,fliplr (y)), -1, 5*eps);
+
+%!assert (kendall (logical(1)), 1);
+%!assert (kendall (single(1)), single(1));
+
+%% Test input validation
+%!error kendall ();
+%!error kendall (1, 2, 3);
+%!error kendall (['A'; 'B']);
+%!error kendall (ones(2,1), ['A'; 'B']);
+%!error kendall (ones (2,2,2));
+%!error kendall (ones (2,2), ones (2,2,2));
+%!error kendall (ones (2,2), ones (3,2));