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1 @c Copyright (C) 1996, 1997, 1999, 2000, 2002, 2004, 2005, 2006, |
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2 @c 2007 John W. Eaton |
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3 @c |
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4 @c This file is part of Octave. |
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5 @c |
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6 @c Octave is free software; you can redistribute it and/or modify it |
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7 @c under the terms of the GNU General Public License as published by the |
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8 @c Free Software Foundation; either version 3 of the License, or (at |
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9 @c your option) any later version. |
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10 @c |
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11 @c Octave is distributed in the hope that it will be useful, but WITHOUT |
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12 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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13 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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14 @c for more details. |
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15 @c |
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16 @c You should have received a copy of the GNU General Public License |
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17 @c along with Octave; see the file COPYING. If not, see |
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18 @c <http://www.gnu.org/licenses/>. |
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19 |
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20 @node Statistics |
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21 @chapter Statistics |
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22 |
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23 Octave has support for various statistical methods. This includes |
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24 basic descriptive statistics, statistical tests, random number generation, |
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25 and much more. |
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26 |
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27 The functions that analyze data all assume that multidimensional data |
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28 is arranged in a matrix where each row is an observation, and each |
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29 column is a variable. So, the matrix defined by |
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30 |
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31 @example |
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32 a = [ 0.9, 0.7; |
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33 0.1, 0.1; |
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34 0.5, 0.4 ]; |
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35 @end example |
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36 |
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37 @noindent |
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38 contains three observations from a two-dimensional distribution. |
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39 While this is the default data arrangement, most functions support |
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40 different arrangements. |
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41 |
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42 It should be noted that the statistics functions don't handle data |
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43 containing NaN, NA, or Inf. Such values need to be handled explicitly. |
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44 |
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45 @menu |
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46 * Descriptive Statistics:: |
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47 * Basic Statistical Functions:: |
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48 * Statistical Plots:: |
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49 * Tests:: |
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50 * Models:: |
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51 * Distributions:: |
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52 * Random Number Generation:: |
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53 @end menu |
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54 |
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55 @node Descriptive Statistics |
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56 @section Descriptive Statistics |
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57 |
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58 Octave can compute various statistics such as the moments of a data set. |
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59 |
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60 @DOCSTRING(mean) |
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61 |
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62 @DOCSTRING(median) |
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63 |
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64 @DOCSTRING(meansq) |
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65 |
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66 @DOCSTRING(std) |
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67 |
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68 @DOCSTRING(var) |
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69 |
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70 @DOCSTRING(mode) |
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71 |
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72 @DOCSTRING(cov) |
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73 |
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74 @DOCSTRING(cor) |
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75 |
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76 @DOCSTRING(corrcoef) |
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77 |
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78 @DOCSTRING(kurtosis) |
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79 |
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80 @DOCSTRING(skewness) |
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81 |
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82 @DOCSTRING(statistics) |
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83 |
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84 @DOCSTRING(moment) |
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85 |
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86 @node Basic Statistical Functions |
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87 @section Basic Statistical Functions |
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88 |
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89 Octave also supports various helpful statistical functions. |
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90 |
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91 @DOCSTRING(mahalanobis) |
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92 |
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93 @DOCSTRING(center) |
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94 |
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95 @DOCSTRING(studentize) |
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96 |
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97 @DOCSTRING(nchoosek) |
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98 |
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99 @DOCSTRING(perms) |
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100 |
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101 @DOCSTRING(values) |
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102 |
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103 @DOCSTRING(table) |
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104 |
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105 @DOCSTRING(spearman) |
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106 |
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107 @DOCSTRING(run_count) |
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108 |
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109 @DOCSTRING(ranks) |
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110 |
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111 @DOCSTRING(range) |
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112 |
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113 @DOCSTRING(probit) |
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114 |
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115 @DOCSTRING(logit) |
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116 |
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117 @DOCSTRING(cloglog) |
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118 |
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119 @DOCSTRING(kendall) |
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120 |
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121 @DOCSTRING(iqr) |
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122 |
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123 @DOCSTRING(cut) |
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124 |
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125 @node Statistical Plots |
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126 @section Statistical Plots |
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127 |
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128 @c Should hist be moved to here, or perhaps the qqplot and ppplot |
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129 @c functions should be moved to the Plotting Chapter? |
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130 |
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131 Octave can create Quantile Plots (QQ-Plots), and Probability Plots |
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132 (PP-Plots). These are simple graphical tests for determining if a |
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133 data set comes from a certain distribution. |
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134 |
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135 It is worth noticing that Octave can also show histograms of data |
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136 using the @code{hist} function as described in |
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137 @ref{Two-Dimensional Plots}. |
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138 |
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139 @DOCSTRING(qqplot) |
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140 |
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141 @DOCSTRING(ppplot) |
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142 |
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143 @node Tests |
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144 @section Tests |
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145 |
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146 Octave can perform several different statistical tests. The following |
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147 table summarizes the available tests. |
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148 |
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149 @multitable @columnfractions .4 .5 |
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150 @item @strong{Hypothesis} |
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151 @tab @strong{Test Functions} |
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152 @item Equal mean values |
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153 @tab @code{anova}, @code{hotelling_test2}, @code{t_test_2}, |
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154 @code{welch_test}, @code{wilcoxon_test}, @code{z_test_2} |
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155 @item Equal medians |
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156 @tab @code{kruskal_wallis_test}, @code{sign_test} |
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157 @item Equal variances |
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158 @tab @code{bartlett_test}, @code{manova}, @code{var_test} |
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159 @item Equal distributions |
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160 @tab @code{chisquare_test_homogeneity}, @code{kolmogorov_smirnov_test_2}, |
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161 @code{u_test} |
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162 @item Equal marginal frequencies |
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163 @tab @code{mcnemar_test} |
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164 @item Equal success probabilities |
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165 @tab @code{prop_test_2} |
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166 @item Independent observations |
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167 @tab @code{chisquare_test_independence}, @code{run_test} |
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168 @item Uncorrelated observations |
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169 @tab @code{cor_test} |
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170 @item Given mean value |
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171 @tab @code{hotelling_test}, @code{t_test}, @code{z_test} |
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172 @item Observations from given distribution |
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173 @tab @code{kolmogorov_smirnov_test} |
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174 @item Regression |
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175 @tab @code{f_test_regression}, @code{t_test_regression} |
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176 @end multitable |
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177 |
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178 The tests return a p-value that describes the outcome of the test. |
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179 Assuming that the test hypothesis is true, the p-value is the probability |
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180 of obtaining a worse result than the observed one. So large p-values |
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181 corresponds to a successful test. Usually a test hypothesis is accepted |
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182 if the p-value exceeds @math{0.05}. |
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183 |
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184 @DOCSTRING(anova) |
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185 |
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186 @DOCSTRING(bartlett_test) |
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187 |
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188 @DOCSTRING(chisquare_test_homogeneity) |
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189 |
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190 @DOCSTRING(chisquare_test_independence) |
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191 |
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192 @DOCSTRING(cor_test) |
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193 |
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194 @DOCSTRING(f_test_regression) |
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195 |
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196 @DOCSTRING(hotelling_test) |
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197 |
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198 @DOCSTRING(hotelling_test_2) |
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199 |
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200 @DOCSTRING(kolmogorov_smirnov_test) |
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201 |
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202 @DOCSTRING(kolmogorov_smirnov_test_2) |
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203 |
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204 @DOCSTRING(kruskal_wallis_test) |
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205 |
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206 @DOCSTRING(manova) |
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207 |
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208 @DOCSTRING(mcnemar_test) |
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209 |
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210 @DOCSTRING(prop_test_2) |
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211 |
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212 @DOCSTRING(run_test) |
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213 |
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214 @DOCSTRING(sign_test) |
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215 |
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216 @DOCSTRING(t_test) |
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217 |
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218 @DOCSTRING(t_test_2) |
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219 |
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220 @DOCSTRING(t_test_regression) |
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221 |
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222 @DOCSTRING(u_test) |
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223 |
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224 @DOCSTRING(var_test) |
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225 |
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226 @DOCSTRING(welch_test) |
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227 |
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228 @DOCSTRING(wilcoxon_test) |
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229 |
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230 @DOCSTRING(z_test) |
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231 |
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232 @DOCSTRING(z_test_2) |
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233 |
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234 @node Models |
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235 @section Models |
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236 |
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237 @DOCSTRING(logistic_regression) |
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238 |
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239 @node Distributions |
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240 @section Distributions |
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241 |
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242 Octave has functions for computing the Probability Density Function |
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243 (PDF), the Cumulative Distribution function (CDF), and the quantile |
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244 (the inverse of the CDF) of a large number of distributions. |
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245 |
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246 The following table summarizes the supported distributions (in |
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247 alphabetical order). |
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248 |
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249 @multitable @columnfractions .4 .2 .2 .2 |
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250 @item @strong{Distribution} |
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251 @tab @strong{PDF} |
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252 @tab @strong{CDF} |
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253 @tab @strong{Quantile} |
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254 @item Beta Distribution |
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255 @tab @code{betapdf} |
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256 @tab @code{betacdf} |
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257 @tab @code{betainv} |
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258 @item Binomial Distribution |
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259 @tab @code{binopdf} |
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260 @tab @code{binocdf} |
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261 @tab @code{binoinv} |
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262 @item Cauchy Distribution |
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263 @tab @code{cauchy_pdf} |
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264 @tab @code{cauchy_cdf} |
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265 @tab @code{cauchy_inv} |
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266 @item Chi-Square Distribution |
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267 @tab @code{chi2pdf} |
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268 @tab @code{chi2cdf} |
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269 @tab @code{chi2inv} |
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270 @item Univariate Discrete Distribution |
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271 @tab @code{discrete_pdf} |
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272 @tab @code{discrete_cdf} |
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273 @tab @code{discrete_inv} |
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274 @item Empirical Distribution |
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275 @tab @code{empirical_pdf} |
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276 @tab @code{empirical_cdf} |
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277 @tab @code{empirical_inv} |
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278 @item Exponential Distribution |
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279 @tab @code{exppdf} |
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280 @tab @code{expcdf} |
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281 @tab @code{expinv} |
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282 @item F Distribution |
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283 @tab @code{fpdf} |
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284 @tab @code{fcdf} |
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285 @tab @code{finv} |
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286 @item Gamma Distribution |
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287 @tab @code{gampdf} |
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288 @tab @code{gamcdf} |
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289 @tab @code{gaminv} |
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290 @item Geometric Distribution |
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291 @tab @code{geopdf} |
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292 @tab @code{geocdf} |
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293 @tab @code{geoinv} |
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294 @item Hypergeometric Distribution |
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295 @tab @code{hygepdf} |
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296 @tab @code{hygecdf} |
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297 @tab @code{hygeinv} |
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298 @item Kolmogorov Smirnov Distribution |
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299 @tab @emph{Not Available} |
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300 @tab @code{kolmogorov_smirnov_cdf} |
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301 @tab @emph{Not Available} |
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302 @item Laplace Distribution |
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303 @tab @code{laplace_pdf} |
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304 @tab @code{laplace_cdf} |
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305 @tab @code{laplace_inv} |
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306 @item Logistic Distribution |
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307 @tab @code{logistic_pdf} |
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308 @tab @code{logistic_cdf} |
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309 @tab @code{logistic_inv} |
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310 @item Log-Normal Distribution |
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311 @tab @code{lognpdf} |
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312 @tab @code{logncdf} |
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313 @tab @code{logninv} |
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314 @item Pascal Distribution |
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315 @tab @code{nbinpdf} |
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316 @tab @code{nbincdf} |
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317 @tab @code{nbininv} |
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318 @item Univariate Normal Distribution |
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319 @tab @code{normpdf} |
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320 @tab @code{normcdf} |
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321 @tab @code{norminv} |
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322 @item Poisson Distribution |
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323 @tab @code{poisspdf} |
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324 @tab @code{poisscdf} |
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325 @tab @code{poissinv} |
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326 @item t (Student) Distribution |
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327 @tab @code{tpdf} |
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328 @tab @code{tcdf} |
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329 @tab @code{tinv} |
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330 @item Univariate Discrete Distribution |
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331 @tab @code{unidpdf} |
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332 @tab @code{unidcdf} |
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333 @tab @code{unidinv} |
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334 @item Uniform Distribution |
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335 @tab @code{unifpdf} |
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336 @tab @code{unifcdf} |
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337 @tab @code{unifinv} |
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338 @item Weibull Distribution |
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339 @tab @code{wblpdf} |
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340 @tab @code{wblcdf} |
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341 @tab @code{wblinv} |
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342 @end multitable |
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343 |
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344 @DOCSTRING(betacdf) |
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345 |
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346 @DOCSTRING(betainv) |
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347 |
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348 @DOCSTRING(betapdf) |
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349 |
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350 @DOCSTRING(binocdf) |
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351 |
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352 @DOCSTRING(binoinv) |
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353 |
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354 @DOCSTRING(binopdf) |
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355 |
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356 @DOCSTRING(cauchy_cdf) |
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357 |
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358 @DOCSTRING(cauchy_inv) |
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359 |
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360 @DOCSTRING(cauchy_pdf) |
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361 |
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362 @DOCSTRING(chi2cdf) |
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363 |
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364 @DOCSTRING(chi2inv) |
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365 |
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366 @DOCSTRING(chi2pdf) |
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367 |
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368 @DOCSTRING(discrete_cdf) |
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369 |
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370 @DOCSTRING(discrete_inv) |
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371 |
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372 @DOCSTRING(discrete_pdf) |
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373 |
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374 @DOCSTRING(empirical_cdf) |
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375 |
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376 @DOCSTRING(empirical_inv) |
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377 |
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378 @DOCSTRING(empirical_pdf) |
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379 |
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380 @DOCSTRING(expcdf) |
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381 |
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382 @DOCSTRING(expinv) |
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383 |
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384 @DOCSTRING(exppdf) |
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385 |
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386 @DOCSTRING(fcdf) |
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387 |
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388 @DOCSTRING(finv) |
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389 |
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390 @DOCSTRING(fpdf) |
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391 |
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392 @DOCSTRING(gamcdf) |
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393 |
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394 @DOCSTRING(gaminv) |
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395 |
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396 @DOCSTRING(gampdf) |
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397 |
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398 @DOCSTRING(geocdf) |
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399 |
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400 @DOCSTRING(geoinv) |
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401 |
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402 @DOCSTRING(geopdf) |
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403 |
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404 @DOCSTRING(hygecdf) |
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405 |
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406 @DOCSTRING(hygeinv) |
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407 |
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408 @DOCSTRING(hygepdf) |
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409 |
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410 @DOCSTRING(kolmogorov_smirnov_cdf) |
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411 |
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412 @DOCSTRING(laplace_cdf) |
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413 |
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414 @DOCSTRING(laplace_inv) |
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415 |
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416 @DOCSTRING(laplace_pdf) |
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417 |
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418 @DOCSTRING(logistic_cdf) |
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419 |
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420 @DOCSTRING(logistic_inv) |
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421 |
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422 @DOCSTRING(logistic_pdf) |
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423 |
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424 @DOCSTRING(logncdf) |
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425 |
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426 @DOCSTRING(logninv) |
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427 |
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428 @DOCSTRING(lognpdf) |
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429 |
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430 @DOCSTRING(nbincdf) |
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431 |
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432 @DOCSTRING(nbininv) |
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433 |
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434 @DOCSTRING(nbinpdf) |
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435 |
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436 @DOCSTRING(normcdf) |
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437 |
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438 @DOCSTRING(norminv) |
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439 |
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440 @DOCSTRING(normpdf) |
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441 |
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442 @DOCSTRING(poisscdf) |
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443 |
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444 @DOCSTRING(poissinv) |
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445 |
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446 @DOCSTRING(poisspdf) |
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447 |
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448 @DOCSTRING(tcdf) |
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449 |
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450 @DOCSTRING(tinv) |
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451 |
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452 @DOCSTRING(tpdf) |
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453 |
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454 @DOCSTRING(unidcdf) |
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455 |
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456 @DOCSTRING(unidinv) |
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457 |
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458 @DOCSTRING(unidpdf) |
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459 |
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460 @DOCSTRING(unifcdf) |
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461 |
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462 @DOCSTRING(unifinv) |
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463 |
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464 @DOCSTRING(unifpdf) |
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465 |
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466 @DOCSTRING(wblcdf) |
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467 |
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468 @DOCSTRING(wblinv) |
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469 |
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470 @DOCSTRING(wblpdf) |
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471 |
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472 @node Random Number Generation |
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473 @section Random Number Generation |
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474 |
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475 Octave can generate random numbers from a large number of distributions. |
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476 The random number generators are based on the random number generators |
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477 described in @ref{Special Utility Matrices}. |
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478 @c Should rand, randn, rande, randp, and randg be moved to here? |
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479 |
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480 The following table summarizes the available random number generators |
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481 (in alphabetical order). |
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482 |
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483 @multitable @columnfractions .4 .3 |
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484 @item @strong{Distribution} @tab @strong{Function} |
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485 @item Beta Distribution @tab @code{betarnd} |
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486 @item Binomial Distribution @tab @code{binornd} |
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487 @item Cauchy Distribution @tab @code{cauchy_rnd} |
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488 @item Chi-Square Distribution @tab @code{chi2rnd} |
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489 @item Univariate Discrete Distribution @tab @code{discrete_rnd} |
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490 @item Empirical Distribution @tab @code{empirical_rnd} |
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491 @item Exponential Distribution @tab @code{exprnd} |
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492 @item F Distribution @tab @code{frnd} |
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493 @item Gamma Distribution @tab @code{gamrnd} |
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494 @item Geometric Distribution @tab @code{geornd} |
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495 @item Hypergeometric Distribution @tab @code{hygernd} |
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496 @item Laplace Distribution @tab @code{laplace_rnd} |
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497 @item Logistic Distribution @tab @code{logistic_rnd} |
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498 @item Log-Normal Distribution @tab @code{lognrnd} |
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499 @item Pascal Distribution @tab @code{nbinrnd} |
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500 @item Univariate Normal Distribution @tab @code{normrnd} |
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501 @item Poisson Distribution @tab @code{poissrnd} |
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502 @item t (Student) Distribution @tab @code{trnd} |
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503 @item Univariate Discrete Distribution @tab @code{unidrnd} |
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504 @item Uniform Distribution @tab @code{unifrnd} |
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505 @item Weibull Distribution @tab @code{wblrnd} |
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506 @item Wiener Process @tab @code{wienrnd} |
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507 @end multitable |
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508 |
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509 @DOCSTRING(betarnd) |
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510 |
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511 @DOCSTRING(binornd) |
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512 |
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513 @DOCSTRING(cauchy_rnd) |
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514 |
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515 @DOCSTRING(chi2rnd) |
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516 |
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517 @DOCSTRING(discrete_rnd) |
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518 |
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519 @DOCSTRING(empirical_rnd) |
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520 |
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521 @DOCSTRING(exprnd) |
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522 |
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523 @DOCSTRING(frnd) |
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524 |
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525 @DOCSTRING(gamrnd) |
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526 |
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527 @DOCSTRING(geornd) |
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528 |
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529 @DOCSTRING(hygernd) |
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530 |
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531 @DOCSTRING(laplace_rnd) |
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532 |
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533 @DOCSTRING(logistic_rnd) |
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534 |
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535 @DOCSTRING(lognrnd) |
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536 |
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537 @DOCSTRING(nbinrnd) |
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538 |
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539 @DOCSTRING(normrnd) |
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540 |
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541 @DOCSTRING(poissrnd) |
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542 |
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543 @DOCSTRING(trnd) |
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544 |
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545 @DOCSTRING(unidrnd) |
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546 |
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547 @DOCSTRING(unifrnd) |
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548 |
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549 @DOCSTRING(wblrnd) |
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550 |
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551 @DOCSTRING(wienrnd) |
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552 |
