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annotate scripts/statistics/base/kurtosis.m @ 20209:d9341b422488 stable

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doc: Update more docstrings to have one sentence summary as first line. Reviewed statistics/ and @ftp/ in scripts directory. * scripts/@ftp/ascii.m, scripts/@ftp/binary.m, scripts/@ftp/cd.m, scripts/@ftp/dir.m, scripts/@ftp/ftp.m, scripts/@ftp/mget.m, scripts/@ftp/mput.m, scripts/statistics/base/center.m, scripts/statistics/base/cloglog.m, scripts/statistics/base/cov.m, scripts/statistics/base/gls.m, scripts/statistics/base/histc.m, scripts/statistics/base/iqr.m, scripts/statistics/base/kendall.m, scripts/statistics/base/kurtosis.m, scripts/statistics/base/logit.m, scripts/statistics/base/lscov.m, scripts/statistics/base/mahalanobis.m, scripts/statistics/base/mean.m, scripts/statistics/base/meansq.m, scripts/statistics/base/median.m, scripts/statistics/base/mode.m, scripts/statistics/base/moment.m, scripts/statistics/base/ols.m, scripts/statistics/base/ppplot.m, scripts/statistics/base/prctile.m, scripts/statistics/base/probit.m, scripts/statistics/base/range.m, scripts/statistics/base/ranks.m, scripts/statistics/base/run_count.m, scripts/statistics/base/runlength.m, scripts/statistics/base/skewness.m, scripts/statistics/base/spearman.m, scripts/statistics/base/statistics.m, scripts/statistics/base/std.m, scripts/statistics/base/table.m, scripts/statistics/base/var.m, scripts/statistics/base/zscore.m, scripts/statistics/distributions/betainv.m, scripts/statistics/distributions/binoinv.m, scripts/statistics/distributions/binopdf.m, scripts/statistics/distributions/cauchy_cdf.m, scripts/statistics/distributions/cauchy_inv.m, scripts/statistics/distributions/cauchy_pdf.m, scripts/statistics/distributions/chi2cdf.m, scripts/statistics/distributions/chi2inv.m, scripts/statistics/distributions/chi2pdf.m, scripts/statistics/distributions/discrete_cdf.m, scripts/statistics/distributions/discrete_inv.m, scripts/statistics/distributions/discrete_pdf.m, scripts/statistics/distributions/empirical_cdf.m, scripts/statistics/distributions/empirical_inv.m, scripts/statistics/distributions/empirical_pdf.m, scripts/statistics/distributions/expcdf.m, scripts/statistics/distributions/expinv.m, scripts/statistics/distributions/exppdf.m, scripts/statistics/distributions/finv.m, scripts/statistics/distributions/fpdf.m, scripts/statistics/distributions/gamcdf.m, scripts/statistics/distributions/gaminv.m, scripts/statistics/distributions/gampdf.m, scripts/statistics/distributions/geoinv.m, scripts/statistics/distributions/geopdf.m, scripts/statistics/distributions/hygecdf.m, scripts/statistics/distributions/hygeinv.m, scripts/statistics/distributions/hygepdf.m, scripts/statistics/distributions/kolmogorov_smirnov_cdf.m, scripts/statistics/distributions/laplace_cdf.m, scripts/statistics/distributions/laplace_inv.m, scripts/statistics/distributions/laplace_pdf.m, scripts/statistics/distributions/logistic_inv.m, scripts/statistics/distributions/logncdf.m, scripts/statistics/distributions/logninv.m, scripts/statistics/distributions/lognpdf.m, scripts/statistics/distributions/nbincdf.m, scripts/statistics/distributions/nbininv.m, scripts/statistics/distributions/nbinpdf.m, scripts/statistics/distributions/nbinrnd.m, scripts/statistics/distributions/normcdf.m, scripts/statistics/distributions/norminv.m, scripts/statistics/distributions/normpdf.m, scripts/statistics/distributions/poisscdf.m, scripts/statistics/distributions/poissinv.m, scripts/statistics/distributions/poisspdf.m, scripts/statistics/distributions/stdnormal_cdf.m, scripts/statistics/distributions/stdnormal_pdf.m, scripts/statistics/distributions/tcdf.m, scripts/statistics/distributions/tinv.m, scripts/statistics/distributions/tpdf.m, scripts/statistics/distributions/unidcdf.m, scripts/statistics/distributions/unidinv.m, scripts/statistics/distributions/unidpdf.m, scripts/statistics/distributions/unidrnd.m, scripts/statistics/distributions/unifcdf.m, scripts/statistics/distributions/unifinv.m, scripts/statistics/distributions/wblcdf.m, scripts/statistics/distributions/wblinv.m, scripts/statistics/distributions/wblpdf.m, scripts/statistics/distributions/wienrnd.m, scripts/statistics/models/private/logistic_regression_derivatives.m, scripts/statistics/models/private/logistic_regression_likelihood.m, scripts/statistics/tests/anova.m, scripts/statistics/tests/chisquare_test_homogeneity.m, scripts/statistics/tests/chisquare_test_independence.m, scripts/statistics/tests/cor_test.m, scripts/statistics/tests/f_test_regression.m, scripts/statistics/tests/hotelling_test.m, scripts/statistics/tests/kolmogorov_smirnov_test.m, scripts/statistics/tests/kolmogorov_smirnov_test_2.m, scripts/statistics/tests/kruskal_wallis_test.m, scripts/statistics/tests/manova.m, scripts/statistics/tests/mcnemar_test.m, scripts/statistics/tests/prop_test_2.m, scripts/statistics/tests/run_test.m, scripts/statistics/tests/sign_test.m, scripts/statistics/tests/t_test.m, scripts/statistics/tests/t_test_2.m, scripts/statistics/tests/t_test_regression.m, scripts/statistics/tests/u_test.m, scripts/statistics/tests/var_test.m, scripts/statistics/tests/welch_test.m, scripts/statistics/tests/wilcoxon_test.m, scripts/statistics/tests/z_test.m, scripts/statistics/tests/z_test_2.m: Update more docstrings to have one sentence summary as first line.
author Rik <rik@octave.org>
date Sun, 10 May 2015 16:07:41 -0700
parents 9fc020886ae9
children 76f67400649e
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1 ## Copyright (C) 2013-2015 Julien Bect
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2 ## Copyright (C) 1996-2015 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} {} kurtosis (@var{x})
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22 ## @deftypefnx {Function File} {} kurtosis (@var{x}, @var{flag})
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23 ## @deftypefnx {Function File} {} kurtosis (@var{x}, @var{flag}, @var{dim})
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24 ## Compute the sample kurtosis of the elements of @var{x}.
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25 ##
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26 ## The sample kurtosis is defined as
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27 ## @tex
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28 ## $$
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29 ## \kappa_1 = {{{1\over N}\,
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30 ## \sum_{i=1}^N (@var{x}_i - \bar{@var{x}})^4} \over \sigma^4},
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31 ## $$
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32 ## where $N$ is the length of @var{x}, $\bar{@var{x}}$ its mean, and $\sigma$
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33 ## its (uncorrected) standard deviation.
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34 ## @end tex
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35 ## @ifnottex
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36 ##
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37 ## @example
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38 ## @group
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39 ## mean ((@var{x} - mean (@var{x})).^4)
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40 ## k1 = ------------------------
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41 ## std (@var{x}).^4
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42 ## @end group
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43 ## @end example
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44 ##
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45 ## @end ifnottex
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46 ##
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47 ## @noindent
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48 ## The optional argument @var{flag} controls which normalization is used.
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49 ## If @var{flag} is equal to 1 (default value, used when @var{flag} is omitted
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50 ## or empty), return the sample kurtosis as defined above. If @var{flag} is
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51 ## equal to 0, return the @w{"bias-corrected"} kurtosis coefficient instead:
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52 ## @tex
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53 ## $$
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54 ## \kappa_0 = 3 + {\scriptstyle N - 1 \over \scriptstyle (N - 2)(N - 3)} \,
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55 ## \left( (N + 1)\, \kappa_1 - 3 (N - 1) \right)
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56 ## $$
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57 ## @end tex
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58 ## @ifnottex
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59 ##
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60 ## @example
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61 ## @group
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62 ## N - 1
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63 ## k0 = 3 + -------------- * ((N + 1) * k1 - 3 * (N - 1))
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64 ## (N - 2)(N - 3)
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65 ## @end group
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66 ## @end example
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67 ##
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68 ## @end ifnottex
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69 ## The bias-corrected kurtosis coefficient is obtained by replacing the sample
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70 ## second and fourth central moments by their unbiased versions. It is an
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71 ## unbiased estimate of the population kurtosis for normal populations.
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72 ##
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73 ## If @var{x} is a matrix, or more generally a multi-dimensional array, return
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74 ## the kurtosis along the first non-singleton dimension. If the optional
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75 ## @var{dim} argument is given, operate along this dimension.
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76 ##
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77 ## @seealso{var, skewness, moment}
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78 ## @end deftypefn
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79
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80 ## Author: KH <Kurt.Hornik@wu-wien.ac.at>
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81 ## Created: 29 July 1994
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82 ## Adapted-By: jwe
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83
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84 function y = kurtosis (x, flag, dim)
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85
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86 if (nargin < 1) || (nargin > 3)
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87 print_usage ();
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88 endif
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89
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90 if (! (isnumeric (x) || islogical (x)))
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91 error ("kurtosis: X must be a numeric vector or matrix");
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92 endif
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93
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94 if (nargin < 2 || isempty (flag))
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95 flag = 1; # default: do not use the "bias corrected" version
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96 else
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97 if (! isscalar (flag) || (flag != 0 && flag != 1))
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98 error ("kurtosis: FLAG must be 0 or 1");
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99 endif
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100 endif
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101
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102 nd = ndims (x);
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103 sz = size (x);
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104 if (nargin < 3)
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105 ## Find the first non-singleton dimension.
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106 (dim = find (sz > 1, 1)) || (dim = 1);
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107 else
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108 if (! (isscalar (dim) && dim == fix (dim)) || ! (1 <= dim && dim <= nd))
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109 error ("kurtosis: DIM must be an integer and a valid dimension");
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110 endif
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111 endif
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112
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113 n = sz(dim);
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114 sz(dim) = 1;
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115
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116 x = center (x, dim); # center also promotes integer, logical to double
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117 v = var (x, 1, dim); # normalize with 1/N
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118 y = sum (x .^ 4, dim);
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119 idx = (v != 0);
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120 y(idx) = y(idx) ./ (n * v(idx) .^ 2);
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121 y(! idx) = NaN;
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122
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123 ## Apply bias correction to the second and fourth central sample moment
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124 if (flag == 0)
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125 if (n > 3)
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126 C = (n - 1) / ((n - 2) * (n - 3));
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127 y = 3 + C * ((n + 1) * y - 3 * (n - 1));
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128 else
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129 y(:) = NaN;
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130 endif
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131 endif
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132
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133 endfunction
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134
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135
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136 %!test
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137 %! x = [-1; 0; 0; 0; 1];
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138 %! y = [x, 2*x];
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139 %! assert (kurtosis (y), [2.5, 2.5], sqrt (eps));
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140
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141 %!assert (kurtosis ([-3, 0, 1]) == kurtosis ([-1, 0, 3]))
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142 %!assert (kurtosis (ones (3, 5)), NaN (1, 5))
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143
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144 %!assert (kurtosis ([1:5 10; 1:5 10], 0, 2), 5.4377317925288901 * [1; 1], 8 * eps)
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145 %!assert (kurtosis ([1:5 10; 1:5 10], 1, 2), 2.9786509002956195 * [1; 1], 8 * eps)
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146 %!assert (kurtosis ([1:5 10; 1:5 10], [], 2), 2.9786509002956195 * [1; 1], 8 * eps)
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147
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148 ## Test behaviour on single input
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149 %!assert (kurtosis (single ([1:5 10])), single (2.9786513), eps ("single"))
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150 %!assert (kurtosis (single ([1 2]), 0), single (NaN))
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151
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152 ## Verify no "divide-by-zero" warnings
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153 %!test
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154 %! wstate = warning ("query", "Octave:divide-by-zero");
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155 %! warning ("on", "Octave:divide-by-zero");
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156 %! unwind_protect
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157 %! lastwarn (""); # clear last warning
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158 %! kurtosis (1);
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159 %! assert (lastwarn (), "");
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160 %! unwind_protect_cleanup
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161 %! warning (wstate, "Octave:divide-by-zero");
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162 %! end_unwind_protect
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163
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164 ## Test input validation
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165 %!error kurtosis ()
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166 %!error kurtosis (1, 2, 3)
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167 %!error <X must be a numeric vector or matrix> kurtosis (['A'; 'B'])
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168 %!error <FLAG must be 0 or 1> kurtosis (1, 2)
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169 %!error <FLAG must be 0 or 1> kurtosis (1, [1 0])
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170 %!error <DIM must be an integer> kurtosis (1, [], ones (2,2))
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171 %!error <DIM must be an integer> kurtosis (1, [], 1.5)
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172 %!error <DIM must be .* a valid dimension> kurtosis (1, [], 0)
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173 %!error <DIM must be .* a valid dimension> kurtosis (1, [], 3)