annotate scripts/statistics/kendall.m @ 33639:65b1cf6aa60a bytecode-interpreter tip

maint: Merge default to bytecode-interpreter
author Arun Giridhar <arungiridhar@gmail.com>
date Mon, 03 Jun 2024 14:38:47 -0400
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1 ########################################################################
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2 ##
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3 ## Copyright (C) 1995-2024 The Octave Project Developers
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4 ##
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5 ## See the file COPYRIGHT.md in the top-level directory of this
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6 ## distribution or <https://octave.org/copyright/>.
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7 ##
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8 ## This file is part of Octave.
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9 ##
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10 ## Octave is free software: you can redistribute it and/or modify it
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11 ## under the terms of the GNU General Public License as published by
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12 ## the Free Software Foundation, either version 3 of the License, or
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13 ## (at your option) any later version.
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14 ##
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15 ## Octave is distributed in the hope that it will be useful, but
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16 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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17 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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18 ## GNU General Public License for more details.
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19 ##
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20 ## You should have received a copy of the GNU General Public License
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21 ## along with Octave; see the file COPYING. If not, see
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22 ## <https://www.gnu.org/licenses/>.
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23 ##
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24 ########################################################################
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25
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26 ## -*- texinfo -*-
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27 ## @deftypefn {} {@var{tau} =} kendall (@var{x})
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28 ## @deftypefnx {} {@var{tau} =} kendall (@var{x}, @var{y})
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29 ## @cindex Kendall's Tau
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30 ## Compute Kendall's
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31 ## @tex
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32 ## $\tau$.
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33 ## @end tex
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34 ## @ifnottex
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35 ## @var{tau}.
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36 ## @end ifnottex
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37 ##
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38 ## For two data vectors @var{x}, @var{y} of common length @math{N}, Kendall's
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39 ## @tex
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40 ## $\tau$
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41 ## @end tex
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42 ## @ifnottex
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43 ## @var{tau}
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44 ## @end ifnottex
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45 ## is the correlation of the signs of all rank differences of
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46 ## @var{x} and @var{y}; i.e., if both @var{x} and @var{y} have distinct
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47 ## entries, then
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48 ##
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49 ## @tex
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50 ## $$ \tau = {1 \over N(N-1)} \sum_{i,j} {\rm sign}(q_i-q_j) \, {\rm sign}(r_i-r_j) $$
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51 ## @end tex
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52 ## @ifnottex
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53 ##
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54 ## @example
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55 ## @group
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56 ## 1
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57 ## @var{tau} = ------- SUM sign (@var{q}(i) - @var{q}(j)) * sign (@var{r}(i) - @var{r}(j))
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58 ## N (N-1) i,j
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59 ## @end group
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60 ## @end example
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61 ##
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62 ## @end ifnottex
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63 ## @noindent
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64 ## in which the
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65 ## @tex
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66 ## $q_i$ and $r_i$
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67 ## @end tex
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68 ## @ifnottex
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69 ## @var{q}(i) and @var{r}(i)
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70 ## @end ifnottex
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71 ## are the ranks of @var{x} and @var{y}, respectively.
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72 ##
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73 ## If @var{x} and @var{y} are drawn from independent distributions,
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74 ## Kendall's
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75 ## @tex
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76 ## $\tau$
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77 ## @end tex
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78 ## @ifnottex
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79 ## @var{tau}
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80 ## @end ifnottex
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81 ## is asymptotically normal with mean 0 and variance
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82 ## @tex
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83 ## ${2 (2N+5) \over 9N(N-1)}$.
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84 ## @end tex
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85 ## @ifnottex
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86 ## @code{(2 * (2N+5)) / (9 * N * (N-1))}.
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87 ## @end ifnottex
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88 ##
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89 ## @code{kendall (@var{x})} is equivalent to @code{kendall (@var{x},
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90 ## @var{x})}.
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91 ## @seealso{ranks, spearman}
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92 ## @end deftypefn
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93
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94 function tau = kendall (x, y = [])
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95
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96 if (nargin < 1)
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97 print_usage ();
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98 endif
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99
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100 if ( ! (isnumeric (x) || islogical (x))
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101 || ! (isnumeric (y) || islogical (y)))
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102 error ("kendall: X and Y must be numeric matrices or vectors");
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103 endif
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104
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105 if (ndims (x) != 2 || ndims (y) != 2)
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106 error ("kendall: X and Y must be 2-D matrices or vectors");
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107 endif
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108
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109 if (isrow (x))
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110 x = x.';
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111 endif
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112 [n, c] = size (x);
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113
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114 if (nargin == 2)
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115 if (isrow (y))
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116 y = y.';
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117 endif
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118 if (rows (y) != n)
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119 error ("kendall: X and Y must have the same number of observations");
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120 else
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121 x = [x, y];
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122 endif
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123 endif
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124
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125 if (isa (x, "single") || isa (y, "single"))
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126 cls = "single";
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127 else
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128 cls = "double";
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129 endif
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130 r = ranks (x);
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131 m = sign (kron (r, ones (n, 1, cls)) - kron (ones (n, 1, cls), r));
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132 tau = corr (m);
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133
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134 if (nargin == 2)
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135 tau = tau(1 : c, (c + 1) : columns (x));
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136 endif
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137
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138 endfunction
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139
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140
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141 %!test
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142 %! x = [1:2:10];
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143 %! y = [100:10:149];
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144 %! assert (kendall (x,y), 1, 5*eps);
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145 %! assert (kendall (x,fliplr (y)), -1, 5*eps);
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146
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147 %!assert (kendall (1), NaN)
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148 %!assert (kendall (single (1)), single (NaN))
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149
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150 ## Test input validation
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151 %!error <Invalid call> kendall ()
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152 %!error kendall (['A'; 'B'])
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153 %!error kendall (ones (2,1), ['A'; 'B'])
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154 %!error kendall (ones (2,2,2))
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155 %!error kendall (ones (2,2), ones (2,2,2))
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156 %!error kendall (ones (2,2), ones (3,2))