1 | #ifndef _theplu_yat_statistics_roc_ |
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2 | #define _theplu_yat_statistics_roc_ |
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3 | |
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4 | // $Id: ROC.h 2720 2012-04-12 03:40:00Z peter $ |
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5 | |
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6 | /* |
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7 | Copyright (C) 2004 Peter Johansson |
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8 | Copyright (C) 2005, 2006, 2007, 2008 Jari Häkkinen, Peter Johansson |
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9 | Copyright (C) 2011, 2012 Peter Johansson |
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10 | |
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11 | This file is part of the yat library, http://dev.thep.lu.se/yat |
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12 | |
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13 | The yat library is free software; you can redistribute it and/or |
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14 | modify it under the terms of the GNU General Public License as |
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15 | published by the Free Software Foundation; either version 3 of the |
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16 | License, or (at your option) any later version. |
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17 | |
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18 | The yat library is distributed in the hope that it will be useful, |
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19 | but WITHOUT ANY WARRANTY; without even the implied warranty of |
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20 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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21 | General Public License for more details. |
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22 | |
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23 | You should have received a copy of the GNU General Public License |
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24 | along with yat. If not, see <http://www.gnu.org/licenses/>. |
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25 | */ |
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26 | |
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27 | #include "Averager.h" |
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28 | #include "yat/utility/stl_utility.h" |
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29 | #include "yat/utility/yat_assert.h" |
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30 | |
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31 | #include <gsl/gsl_randist.h> |
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32 | |
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33 | #include <map> |
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34 | #include <utility> |
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35 | |
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36 | namespace theplu { |
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37 | namespace yat { |
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38 | namespace statistics { |
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39 | |
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40 | /// |
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41 | /// @brief Reciever Operating Characteristic. |
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42 | /// |
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43 | /// As the area under an ROC curve is equivalent to Mann-Whitney U |
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44 | /// statistica, this class can be used to perform a Mann-Whitney |
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45 | /// U-test (aka Wilcoxon). |
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46 | /// |
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47 | /// \see AUC |
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48 | /// |
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49 | class ROC |
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50 | { |
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51 | |
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52 | public: |
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53 | /// |
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54 | /// @brief Default constructor |
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55 | /// |
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56 | ROC(void); |
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57 | |
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58 | /** |
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59 | \brief Add a data value. |
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60 | |
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61 | \param value data value |
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62 | \param target \c true if value belongs to class positive |
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63 | |
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64 | \param weight indicating how important the data point is. A |
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65 | zero weight implies the data point is ignored. A negative |
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66 | weight should be understood as removing a data point and thus |
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67 | typically only makes sense if there is a previously added data |
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68 | point with same \a value and \a target. |
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69 | |
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70 | */ |
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71 | void add(double value, bool target, double weight=1.0); |
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72 | |
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73 | /** |
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74 | \brief Area Under Curve, AUC |
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75 | |
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76 | \see AUC for how the area is calculated |
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77 | |
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78 | @return Area under curve. |
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79 | */ |
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80 | double area(void); |
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81 | |
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82 | /** |
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83 | \brief threshold for p_value calculation |
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84 | |
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85 | Function can used to change the minimum_size. |
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86 | |
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87 | \return reference to threshold minimum size |
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88 | */ |
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89 | unsigned int& minimum_size(void); |
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90 | |
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91 | /** |
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92 | \brief threshold for p_value calculation |
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93 | |
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94 | Threshold deciding whether p-value is computed using exact |
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95 | method or a Gaussian approximation. If both number of positive |
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96 | samples, n_pos(void), and number of negative samples, |
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97 | n_neg(void), are smaller than minimum_size the exact method is |
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98 | used. |
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99 | |
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100 | \see p_value |
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101 | |
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102 | \return const reference to minimum_size |
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103 | */ |
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104 | const unsigned int& minimum_size(void) const; |
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105 | |
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106 | /// |
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107 | /// \brief number of samples |
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108 | /// |
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109 | /// @return sum of weights |
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110 | /// |
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111 | double n(void) const; |
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112 | |
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113 | /// |
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114 | /// \brief number of negative samples |
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115 | /// |
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116 | /// @return sum of weights with negative target |
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117 | /// |
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118 | double n_neg(void) const; |
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119 | |
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120 | /// |
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121 | /// \brief number of positive samples |
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122 | /// |
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123 | /// @return sum of weights with positive target |
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124 | /// |
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125 | double n_pos(void) const; |
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126 | |
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127 | /** |
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128 | \brief One-sided P-value |
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129 | |
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130 | Calculates the one-sided p-value, i.e., probability to get this |
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131 | area (or greater) given that there is no difference |
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132 | between the two classes. |
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133 | |
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134 | \b Exact \b method: In the exact method the function goes |
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135 | through all permutations and counts what fraction for which the |
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136 | area is greater (or equal) than area in original |
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137 | permutation. In case all non-zero weights are not equal, |
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138 | iterating through all permutations is not sufficient so |
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139 | algorithm goes through all combinations instead which quickly |
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140 | becomes a large number (N!). |
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141 | |
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142 | \b Large-sample \b Approximation: When many data points are |
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143 | available, see minimum_size(), a Gaussian approximation is used |
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144 | and the p-value is calculated as |
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145 | \f[ |
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146 | P = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^z |
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147 | \exp{\left(-\frac{t^2}{2}\right)} dt |
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148 | \f] |
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149 | |
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150 | where |
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151 | |
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152 | \f[ |
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153 | z = \frac{\textrm{area} - 0.5 - 0.5/(n^+ \cdot n^-)}{s} |
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154 | \f] |
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155 | |
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156 | and |
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157 | |
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158 | \f[ |
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159 | s^2 = \frac{n+1+\sum \left(n_x \cdot (n_x^2-1)\right)} |
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160 | {12\cdot n^+\cdot n^-} |
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161 | \f] |
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162 | |
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163 | where sum runs over different data values (of ties) and \f$ n_x |
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164 | \f$ is number data points with that value. The sum is a |
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165 | correction term for ties and is zero if there are no ties. |
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166 | |
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167 | The number of samples in a group, \f$ n^+ \f$, is calculated as |
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168 | \f$ n = (\sum w)^2 / \sum w^2 \f$ |
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169 | |
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170 | \return \f$ P(a \ge \textrm{area}) \f$ |
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171 | */ |
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172 | double p_value_one_sided(void) const; |
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173 | |
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174 | /** |
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175 | \brief Two-sided p-value. |
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176 | |
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177 | Calculates the probability to get an area, \c a, equal or more |
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178 | extreme than \c area |
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179 | \f[ |
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180 | P(a \ge \textrm{max}(\textrm{area},1-\textrm{area})) + |
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181 | P(a \le \textrm{min}(\textrm{area}, 1-\textrm{area})) \f] |
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182 | |
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183 | If there are no ties, distribution of \a a is symmetric, so if |
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184 | area is greater than 0.5, this boils down to \f$ P = 2*P(a \ge |
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185 | \textrm{area}) = 2*P_\textrm{one-sided}\f$. |
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186 | |
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187 | \return two-sided p-value |
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188 | |
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189 | \see p_value_one_sided |
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190 | */ |
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191 | double p_value(void) const; |
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192 | |
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193 | /** |
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194 | \brief remove a data value |
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195 | |
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196 | A data point with identical \a value, \a target, and \a weight |
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197 | must have beed added prior calling this function; else an |
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198 | exception is thrown. |
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199 | |
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200 | \since New in yat 0.9 |
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201 | */ |
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202 | void remove(double value, bool target, double weight=1.0); |
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203 | |
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204 | /** |
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205 | @brief Set everything to zero |
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206 | */ |
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207 | void reset(void); |
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208 | |
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209 | private: |
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210 | typedef std::multimap<double, std::pair<bool, double> > Map; |
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211 | |
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212 | // struct used i count functions |
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213 | struct Weights |
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214 | { |
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215 | Weights(void); |
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216 | double small_pos; |
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217 | double small_neg; |
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218 | double tied_pos; |
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219 | double tied_neg; |
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220 | }; |
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221 | |
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222 | /// Implemented as in MatLab 13.1 |
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223 | double get_p_approx(double) const; |
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224 | |
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225 | /** |
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226 | return false if all non-zero weights are equal |
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227 | */ |
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228 | bool is_weighted(void) const; |
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229 | |
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230 | /** |
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231 | return (sum x)^2 / sum x^2 |
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232 | */ |
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233 | size_t nof_points(const Averager& a) const; |
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234 | |
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235 | /* |
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236 | Calculate probability to get an area equal (smaller) than \a |
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237 | area given the distribution of weights and ties in multimap_ |
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238 | */ |
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239 | double p_left_weighted(double area) const; |
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240 | |
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241 | /* |
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242 | Calculate probability to get an area equal (greater) than \a |
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243 | area given the distribution of weights and ties in multimap_ |
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244 | */ |
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245 | double p_right_weighted(double area) const; |
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246 | |
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247 | /* |
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248 | Count number of combinations (of N!) that gives weight sum equal |
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249 | or larger than \a threshold. |
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250 | |
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251 | Range [first, last) is used to check for ties. If, e.g., *first |
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252 | and *(first+1) are equal implies that the two largest values are |
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253 | equal. |
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254 | */ |
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255 | template <typename Iterator> |
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256 | double count(Iterator first, Iterator last, double threshold) const; |
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257 | |
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258 | /* |
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259 | Loop over all elements in \a weights and call count(7) |
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260 | */ |
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261 | template <typename Iterator> |
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262 | double count(Map& weights, Iterator iter, Iterator last, |
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263 | double threshold, double sum, const Weights& weight) const; |
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264 | |
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265 | /* |
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266 | Count number of combinations in which sum>=threshold given |
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267 | classes and weights in \a weight. Range [iter, last) is used to |
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268 | handle ties. |
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269 | */ |
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270 | template <typename Iterator> |
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271 | double count(Map& weights, Iterator iter, Iterator last, |
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272 | double threshold, double sum, Weights weight, |
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273 | const std::pair<bool, double>& entry) const; |
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274 | |
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275 | /* |
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276 | Calculates probability to get \a block number of pairs correctly |
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277 | sorted when having \a pos positive samples and \a neg negative |
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278 | samples given the distribution of ties as in [first, last). |
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279 | */ |
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280 | template<typename ForwardIterator> |
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281 | double p_exact_with_ties(ForwardIterator first, ForwardIterator last, |
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282 | double block, unsigned int pos, |
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283 | unsigned int neg) const; |
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284 | |
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285 | /** |
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286 | \return P(auc >= area) |
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287 | */ |
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288 | double p_exact_right(double area) const; |
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289 | |
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290 | /** |
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291 | \return P(auc <= area) |
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292 | */ |
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293 | double p_exact_left(double area) const; |
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294 | |
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295 | bool use_exact_method(void) const; |
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296 | |
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297 | double area_; |
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298 | bool has_ties_; |
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299 | unsigned int minimum_size_; |
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300 | Averager neg_weights_; |
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301 | Averager pos_weights_; |
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302 | Map multimap_; |
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303 | }; |
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304 | |
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305 | template<typename ForwardIterator> |
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306 | double |
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307 | ROC::p_exact_with_ties(ForwardIterator begin, ForwardIterator end, |
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308 | double block, unsigned int pos,unsigned int neg) const |
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309 | { |
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310 | if (block <= 0) |
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311 | return 1.0; |
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312 | if (block > pos*neg) |
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313 | return 0.0; |
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314 | |
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315 | ForwardIterator iter(begin); |
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316 | unsigned int n=0; |
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317 | while (iter!=end && iter->first == begin->first) { |
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318 | ++iter; |
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319 | ++n; |
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320 | } |
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321 | double result = 0; |
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322 | /* |
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323 | pos1 neg1 | n |
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324 | pos2 neg2 | |
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325 | ---- ---- ---- |
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326 | pos neg |
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327 | */ |
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328 | |
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329 | // ensure pos1 and neg2 are non-negative |
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330 | unsigned int pos1 = n - std::min(n, neg); |
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331 | // ensure pos2 and neg1 are non-negative |
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332 | unsigned int max = std::min(n, pos); |
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333 | YAT_ASSERT(pos1<=max); |
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334 | for ( ; pos1<=max; ++pos1) { |
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335 | unsigned int neg1 = n-pos1; |
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336 | YAT_ASSERT(neg1<=n); |
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337 | unsigned int pos2 = pos-pos1; |
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338 | YAT_ASSERT(pos2<=pos); |
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339 | unsigned int neg2 = neg-neg1; |
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340 | YAT_ASSERT(neg2<=neg); |
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341 | result += gsl_ran_hypergeometric_pdf(pos1, static_cast<unsigned int>(pos), |
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342 | static_cast<unsigned int>(neg), n) |
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343 | * p_exact_with_ties(iter, end, |
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344 | block - pos2*neg1 - 0.5*pos1*neg1, |
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345 | pos2, neg2); |
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346 | } |
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347 | return result; |
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348 | } |
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349 | |
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350 | |
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351 | template <typename Iterator> |
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352 | double ROC::count(Iterator first, Iterator last, double threshold) const |
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353 | { |
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354 | Map map(multimap_); |
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355 | ROC::Weights w; |
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356 | w.small_pos = pos_weights_.sum_x(); |
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357 | w.small_neg = neg_weights_.sum_x(); |
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358 | return count(map, first, last, threshold*w.small_pos*w.small_neg, 0, w); |
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359 | } |
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360 | |
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361 | |
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362 | |
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363 | template <typename Iterator> |
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364 | double ROC::count(Map& weights, Iterator iter, Iterator last, |
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365 | double threshold, double sum, const Weights& w) const |
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366 | { |
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367 | double result = 0.0; |
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368 | // loop over all elements |
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369 | for (Map::iterator i=weights.begin(); i!=weights.end(); ++i) { |
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370 | Map::value_type save = *i; |
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371 | Map::iterator hint = i; |
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372 | ++hint; |
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373 | weights.erase(i); |
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374 | result += count(weights, iter, last, threshold, sum, w, save.second); |
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375 | i = weights.insert(hint, save); |
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376 | } |
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377 | YAT_ASSERT(weights.size()); |
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378 | return result/weights.size(); |
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379 | } |
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380 | |
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381 | template <typename Iterator> |
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382 | double ROC::count(Map& weights, Iterator iter, Iterator last, |
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383 | double threshold, double sum, Weights w, |
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384 | const std::pair<bool, double>& entry) const |
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385 | { |
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386 | double tiny = 10e-10; |
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387 | |
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388 | Iterator next(iter); |
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389 | ++next; |
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390 | |
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391 | // update weights |
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392 | if (entry.first) { |
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393 | w.tied_pos += entry.second; |
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394 | w.small_pos -= entry.second; |
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395 | } |
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396 | else { |
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397 | w.tied_neg += entry.second; |
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398 | w.small_neg -= entry.second; |
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399 | } |
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400 | |
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401 | // last entry in equal range |
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402 | if (next==last || *next!=*iter) { |
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403 | sum += 0.5*w.tied_pos*w.tied_neg + w.tied_pos * w.small_neg; |
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404 | w.tied_pos=0; |
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405 | w.tied_neg=0; |
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406 | } |
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407 | |
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408 | // max sum happens if all pos values belong to current equal range |
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409 | // and none of the remaining neg values |
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410 | double max_sum = sum + 0.5*(w.tied_pos+w.small_pos)*w.tied_neg + |
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411 | (w.tied_pos+w.small_pos)*w.small_neg; |
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412 | |
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413 | if (max_sum<threshold-tiny) |
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414 | return 0.0; |
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415 | if (sum + 0.5*w.tied_pos*(w.tied_neg+w.small_neg) >= threshold-tiny) |
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416 | return 1.0; |
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417 | |
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418 | if (next!=last) |
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419 | return count(weights, next, last, threshold, sum, w); |
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420 | return 0.0; |
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421 | } |
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422 | |
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423 | }}} // of namespace statistics, yat, and theplu |
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424 | #endif |
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