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########################################################################
##
## Copyright (C) 1995-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## 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
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {@var{tau} =} kendall (@var{x})
## @deftypefnx {} {@var{tau} =} kendall (@var{x}, @var{y})
## @cindex Kendall's Tau
## Compute Kendall's
## @tex
## $\tau$.
## @end tex
## @ifnottex
## @var{tau}.
## @end ifnottex
##
## For two data vectors @var{x}, @var{y} of common length @math{N}, Kendall's
## @tex
## $\tau$
## @end tex
## @ifnottex
## @var{tau}
## @end ifnottex
## 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
## @var{tau} = ------- SUM sign (@var{q}(i) - @var{q}(j)) * sign (@var{r}(i) - @var{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}(i) and @var{r}(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
## @tex
## $\tau$
## @end tex
## @ifnottex
## @var{tau}
## @end ifnottex
## is asymptotically normal with mean 0 and variance
## @tex
## ${2 (2N+5) \over 9N(N-1)}$.
## @end tex
## @ifnottex
## @code{(2 * (2N+5)) / (9 * N * (N-1))}.
## @end ifnottex
##
## @code{kendall (@var{x})} is equivalent to @code{kendall (@var{x},
## @var{x})}.
## @seealso{ranks, spearman}
## @end deftypefn
function tau = kendall (x, y = [])
if (nargin < 1)
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 (1), NaN)
%!assert (kendall (single (1)), single (NaN))
## Test input validation
%!error <Invalid call> kendall ()
%!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))
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