1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
|
## Copyright (C) 1995-2015 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))
|
