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skewness.m: Use long-form of 'end' keyword in cset ab75e72c5b36. * scripts/statistics/base/skewness.m: Replace 'end' with 'endif'.
author Rik <rik@octave.org>
date Sun, 06 Oct 2013 19:48:25 -0700
parents 7004c733412f
children 8334144458a4
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
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1 ## Copyright (C) 2013 Julien Bect
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2 ## Copyright (C) 1996-2012 John W. Eaton
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3 ##
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4 ## This file is part of Octave.
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5 ##
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6 ## Octave is free software; you can redistribute it and/or modify it
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7 ## under the terms of the GNU General Public License as published by
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8 ## the Free Software Foundation; either version 3 of the License, or (at
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9 ## your option) any later version.
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10 ##
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11 ## Octave is distributed in the hope that it will be useful, but
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12 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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14 ## General Public License for more details.
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15 ##
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16 ## You should have received a copy of the GNU General Public License
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17 ## along with Octave; see the file COPYING. If not, see
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18 ## <http://www.gnu.org/licenses/>.
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19
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20 ## -*- texinfo -*-
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21 ## @deftypefn {Function File} {} skewness (@var{x})
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22 ## @deftypefnx {Function File} {} skewness (@var{x}, @var{flag})
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23 ## @deftypefnx {Function File} {} skewness (@var{x}, @var{flag}, @var{dim})
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24 ## Compute the sample skewness of the elements of @var{x}:
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25 ## @tex
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26 ## $$
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27 ## {\rm skewness} (@var{x}) = {{{1\over N}\,
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28 ## \sum_{i=1}^N (@var{x}_i - \bar{@var{x}})^3} \over \sigma^3},
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29 ## $$
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30 ## where $N$ is the length of @var{x}, $\bar{@var{x}}$ its mean and $\sigma$
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31 ## its (uncorrected) standard deviation.
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32 ## @end tex
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33 ## @ifnottex
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34 ##
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35 ## @example
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36 ## @group
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37 ## mean ((@var{x} - mean (@var{x})).^3)
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38 ## skewness (@var{X}) = ------------------------.
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39 ## std (@var{x}).^3
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40 ## @end group
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41 ## @end example
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42 ##
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43 ## @end ifnottex
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44 ##
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45 ## @noindent
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46 ## The optional argument @var{flag} controls which normalization is used.
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47 ## If @var{flag} is equal to 1 (default value, used when @var{flag} is omitted
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48 ## or empty), return the sample skewness as defined above. If @var{flag} is
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49 ## equal to 0, return the adjusted skewness coefficient instead:
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50 ## @tex
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51 ## $$
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52 ## {\rm skewness} (@var{x}) = {\sqrt{N (N - 1)} \over N - 2} \times \,
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53 ## {{{1 \over N} \sum_{i=1}^N (@var{x}_i - \bar{@var{x}})^3} \over \sigma^3}
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54 ## $$
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55 ## @end tex
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56 ## @ifnottex
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57 ##
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58 ## @example
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59 ## @group
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60 ## sqrt (N*(N-1)) mean ((@var{x} - mean (@var{x})).^3)
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61 ## skewness (@var{X}, 0) = -------------- * ------------------------.
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62 ## (N - 2) std (@var{x}).^3
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63 ## @end group
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64 ## @end example
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65 ##
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66 ## @end ifnottex
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67 ## The adjusted skewness coefficient is obtained by replacing the sample second
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68 ## and third central moments by their bias-corrected versions.
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69 ##
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70 ## If @var{x} is a matrix, or more generally a multi-dimensional array, return
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71 ## the skewness along the first non-singleton dimension. If the optional
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72 ## @var{dim} argument is given, operate along this dimension.
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73 ##
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74 ## @seealso{var, kurtosis, moment}
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75 ## @end deftypefn
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76
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77 ## Author: KH <Kurt.Hornik@wu-wien.ac.at>
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78 ## Created: 29 July 1994
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79 ## Adapted-By: jwe
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80
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81 function y = skewness (x, flag, dim)
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82
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83 if (nargin < 1) || (nargin > 3)
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84 print_usage ();
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85 endif
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86
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87 if (! (isnumeric (x) || islogical (x)))
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88 error ("skewness: X must be a numeric vector or matrix");
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89 endif
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90
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91 if (nargin < 2 || isempty (flag))
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92 flag = 1; # default: do not use the "bias corrected" version
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93 else
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94 flag = double (flag);
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95 if (flag != 0 && flag != 1)
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96 error ("skewness: FLAG must be 0 or 1");
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97 endif
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98 endif
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99
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100 nd = ndims (x);
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101 sz = size (x);
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102 if (nargin < 3)
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103 ## Find the first non-singleton dimension.
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104 (dim = find (sz > 1, 1)) || (dim = 1);
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105 else
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106 if (!(isscalar (dim) && dim == fix (dim)) || !(1 <= dim && dim <= nd))
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107 error ("skewness: DIM must be an integer and a valid dimension");
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108 endif
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109 endif
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110
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111 n = sz(dim);
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112 sz(dim) = 1;
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113
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114 x = center (x, dim); # center also promotes integer, logical to double
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115 s = std (x, 1, dim); # Normalize with 1/N
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116 y = sum (x .^ 3, dim);
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117 y ./= (n * s .^ 3);
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118 y(s == 0) = NaN;
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119
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120 ## Apply bias correction to the third central sample moment
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121 if (flag == 0)
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122 if (n > 2)
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123 y *= sqrt (n * (n - 1)) / (n - 2);
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124 else
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125 y(:) = NaN;
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126 endif
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127 endif
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128
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129 endfunction
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130
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131
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132 %!assert (skewness ([-1, 0, 1]), 0)
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133 %!assert (skewness ([-2, 0, 1]) < 0)
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134 %!assert (skewness ([-1, 0, 2]) > 0)
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135 %!assert (skewness ([-3, 0, 1]) == -1 * skewness ([-1, 0, 3]))
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136 %!assert (skewness (ones (3, 5)), NaN (1, 5))
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137
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138 %!test
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139 %! x = [0; 0; 0; 1];
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140 %! y = [x, 2*x];
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141 %! assert (skewness (y), 1.154700538379251 * [1 1], 5*eps);
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142
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143 %!assert (skewness ([1:5 10; 1:5 10], 0, 2), 1.439590274527954 * [1; 1], eps)
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144 %!assert (skewness ([1:5 10; 1:5 10], 1, 2), 1.051328089232020 * [1; 1], eps)
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145 %!assert (skewness ([1:5 10; 1:5 10], [], 2), 1.051328089232020 * [1; 1], eps)
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146
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147 ## Test behaviour on single input
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148 %!assert (skewness (single ([1:5 10])), single (1.0513283), eps ("single"))
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149 %!assert (skewness (single ([1 2]), 0), single (NaN))
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150
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151 ## Test input validation
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152 %!error skewness ()
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153 %!error skewness (1, 2, 3)
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154 %!error <X must be a numeric vector or matrix> skewness (['A'; 'B'])
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155 %!error <FLAG must be 0 or 1> skewness (1, 2)
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156 %!error <DIM must be an integer> skewness (1, [], ones (2,2))
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157 %!error <DIM must be an integer> skewness (1, [], 1.5)
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158 %!error <DIM must be .* a valid dimension> skewness (1, [], 0)
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159 %!error <DIM must be .* a valid dimension> skewness (1, [], 3)
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160