Mercurial > octave
view scripts/statistics/kendall.m @ 31227:0dec459a4064
sparse-xpow.cc: Performance tweak for threshold selection
author | Arun Giridhar <arungiridhar@gmail.com> |
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date | Wed, 14 Sep 2022 09:59:31 -0400 |
parents | 5d3faba0342e |
children | 597f3ee61a48 |
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######################################################################## ## ## Copyright (C) 1995-2022 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 (logical (1)), 1) %!assert (kendall (single (1)), single (1)) ## 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))