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1 /* The MIT License
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2
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3 Copyright (C) 2010, 2013 Genome Research Ltd.
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4 Copyright (C) 2011 Attractive Chaos <attractor@live.co.uk>
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5
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6 Permission is hereby granted, free of charge, to any person obtaining
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7 a copy of this software and associated documentation files (the
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8 "Software"), to deal in the Software without restriction, including
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9 without limitation the rights to use, copy, modify, merge, publish,
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10 distribute, sublicense, and/or sell copies of the Software, and to
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11 permit persons to whom the Software is furnished to do so, subject to
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12 the following conditions:
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13
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14 The above copyright notice and this permission notice shall be
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15 included in all copies or substantial portions of the Software.
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16
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17 THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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18 EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
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19 MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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20 NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS
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21 BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN
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22 ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
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23 CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
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24 SOFTWARE.
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25 */
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26
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27 #include <math.h>
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28 #include <stdlib.h>
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29 #include "htslib/kfunc.h"
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30
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31 /* Log gamma function
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32 * \log{\Gamma(z)}
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33 * AS245, 2nd algorithm, http://lib.stat.cmu.edu/apstat/245
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34 */
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35 double kf_lgamma(double z)
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36 {
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37 double x = 0;
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38 x += 0.1659470187408462e-06 / (z+7);
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39 x += 0.9934937113930748e-05 / (z+6);
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40 x -= 0.1385710331296526 / (z+5);
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41 x += 12.50734324009056 / (z+4);
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42 x -= 176.6150291498386 / (z+3);
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43 x += 771.3234287757674 / (z+2);
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44 x -= 1259.139216722289 / (z+1);
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45 x += 676.5203681218835 / z;
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46 x += 0.9999999999995183;
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47 return log(x) - 5.58106146679532777 - z + (z-0.5) * log(z+6.5);
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48 }
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49
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50 /* complementary error function
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51 * \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2} dt
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52 * AS66, 2nd algorithm, http://lib.stat.cmu.edu/apstat/66
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53 */
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54 double kf_erfc(double x)
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55 {
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56 const double p0 = 220.2068679123761;
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57 const double p1 = 221.2135961699311;
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58 const double p2 = 112.0792914978709;
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59 const double p3 = 33.912866078383;
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60 const double p4 = 6.37396220353165;
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61 const double p5 = .7003830644436881;
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62 const double p6 = .03526249659989109;
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63 const double q0 = 440.4137358247522;
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64 const double q1 = 793.8265125199484;
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65 const double q2 = 637.3336333788311;
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66 const double q3 = 296.5642487796737;
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67 const double q4 = 86.78073220294608;
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68 const double q5 = 16.06417757920695;
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69 const double q6 = 1.755667163182642;
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70 const double q7 = .08838834764831844;
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71 double expntl, z, p;
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72 z = fabs(x) * M_SQRT2;
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73 if (z > 37.) return x > 0.? 0. : 2.;
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74 expntl = exp(z * z * - .5);
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75 if (z < 10. / M_SQRT2) // for small z
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76 p = expntl * ((((((p6 * z + p5) * z + p4) * z + p3) * z + p2) * z + p1) * z + p0)
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77 / (((((((q7 * z + q6) * z + q5) * z + q4) * z + q3) * z + q2) * z + q1) * z + q0);
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78 else p = expntl / 2.506628274631001 / (z + 1. / (z + 2. / (z + 3. / (z + 4. / (z + .65)))));
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79 return x > 0.? 2. * p : 2. * (1. - p);
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80 }
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81
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82 /* The following computes regularized incomplete gamma functions.
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83 * Formulas are taken from Wiki, with additional input from Numerical
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84 * Recipes in C (for modified Lentz's algorithm) and AS245
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85 * (http://lib.stat.cmu.edu/apstat/245).
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86 *
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87 * A good online calculator is available at:
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88 *
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89 * http://www.danielsoper.com/statcalc/calc23.aspx
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90 *
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91 * It calculates upper incomplete gamma function, which equals
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92 * kf_gammaq(s,z)*tgamma(s).
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93 */
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94
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95 #define KF_GAMMA_EPS 1e-14
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96 #define KF_TINY 1e-290
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97
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98 // regularized lower incomplete gamma function, by series expansion
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99 static double _kf_gammap(double s, double z)
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100 {
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101 double sum, x;
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102 int k;
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103 for (k = 1, sum = x = 1.; k < 100; ++k) {
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104 sum += (x *= z / (s + k));
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105 if (x / sum < KF_GAMMA_EPS) break;
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106 }
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107 return exp(s * log(z) - z - kf_lgamma(s + 1.) + log(sum));
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108 }
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109 // regularized upper incomplete gamma function, by continued fraction
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110 static double _kf_gammaq(double s, double z)
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111 {
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112 int j;
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113 double C, D, f;
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114 f = 1. + z - s; C = f; D = 0.;
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115 // Modified Lentz's algorithm for computing continued fraction
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116 // See Numerical Recipes in C, 2nd edition, section 5.2
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117 for (j = 1; j < 100; ++j) {
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118 double a = j * (s - j), b = (j<<1) + 1 + z - s, d;
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119 D = b + a * D;
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120 if (D < KF_TINY) D = KF_TINY;
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121 C = b + a / C;
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122 if (C < KF_TINY) C = KF_TINY;
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123 D = 1. / D;
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124 d = C * D;
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125 f *= d;
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126 if (fabs(d - 1.) < KF_GAMMA_EPS) break;
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127 }
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128 return exp(s * log(z) - z - kf_lgamma(s) - log(f));
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129 }
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130
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131 double kf_gammap(double s, double z)
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132 {
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133 return z <= 1. || z < s? _kf_gammap(s, z) : 1. - _kf_gammaq(s, z);
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134 }
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135
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136 double kf_gammaq(double s, double z)
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137 {
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138 return z <= 1. || z < s? 1. - _kf_gammap(s, z) : _kf_gammaq(s, z);
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139 }
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140
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141 /* Regularized incomplete beta function. The method is taken from
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142 * Numerical Recipe in C, 2nd edition, section 6.4. The following web
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143 * page calculates the incomplete beta function, which equals
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144 * kf_betai(a,b,x) * gamma(a) * gamma(b) / gamma(a+b):
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145 *
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146 * http://www.danielsoper.com/statcalc/calc36.aspx
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147 */
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148 static double kf_betai_aux(double a, double b, double x)
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149 {
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150 double C, D, f;
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151 int j;
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152 if (x == 0.) return 0.;
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153 if (x == 1.) return 1.;
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154 f = 1.; C = f; D = 0.;
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155 // Modified Lentz's algorithm for computing continued fraction
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156 for (j = 1; j < 200; ++j) {
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157 double aa, d;
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158 int m = j>>1;
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159 aa = (j&1)? -(a + m) * (a + b + m) * x / ((a + 2*m) * (a + 2*m + 1))
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160 : m * (b - m) * x / ((a + 2*m - 1) * (a + 2*m));
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161 D = 1. + aa * D;
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162 if (D < KF_TINY) D = KF_TINY;
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163 C = 1. + aa / C;
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164 if (C < KF_TINY) C = KF_TINY;
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165 D = 1. / D;
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166 d = C * D;
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167 f *= d;
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168 if (fabs(d - 1.) < KF_GAMMA_EPS) break;
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169 }
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170 return exp(kf_lgamma(a+b) - kf_lgamma(a) - kf_lgamma(b) + a * log(x) + b * log(1.-x)) / a / f;
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171 }
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172 double kf_betai(double a, double b, double x)
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173 {
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174 return x < (a + 1.) / (a + b + 2.)? kf_betai_aux(a, b, x) : 1. - kf_betai_aux(b, a, 1. - x);
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175 }
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176
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177 #ifdef KF_MAIN
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178 #include <stdio.h>
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179 int main(int argc, char *argv[])
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180 {
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181 double x = 5.5, y = 3;
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182 double a, b;
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183 printf("erfc(%lg): %lg, %lg\n", x, erfc(x), kf_erfc(x));
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184 printf("upper-gamma(%lg,%lg): %lg\n", x, y, kf_gammaq(y, x)*tgamma(y));
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185 a = 2; b = 2; x = 0.5;
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186 printf("incomplete-beta(%lg,%lg,%lg): %lg\n", a, b, x, kf_betai(a, b, x) / exp(kf_lgamma(a+b) - kf_lgamma(a) - kf_lgamma(b)));
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187 return 0;
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188 }
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189 #endif
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190
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191
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192 // log\binom{n}{k}
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193 static double lbinom(int n, int k)
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194 {
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195 if (k == 0 || n == k) return 0;
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196 return lgamma(n+1) - lgamma(k+1) - lgamma(n-k+1);
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197 }
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198
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199 // n11 n12 | n1_
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200 // n21 n22 | n2_
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201 //-----------+----
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202 // n_1 n_2 | n
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203
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204 // hypergeometric distribution
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205 static double hypergeo(int n11, int n1_, int n_1, int n)
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206 {
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207 return exp(lbinom(n1_, n11) + lbinom(n-n1_, n_1-n11) - lbinom(n, n_1));
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208 }
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209
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210 typedef struct {
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211 int n11, n1_, n_1, n;
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212 double p;
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213 } hgacc_t;
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214
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215 // incremental version of hypergenometric distribution
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216 static double hypergeo_acc(int n11, int n1_, int n_1, int n, hgacc_t *aux)
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217 {
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218 if (n1_ || n_1 || n) {
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219 aux->n11 = n11; aux->n1_ = n1_; aux->n_1 = n_1; aux->n = n;
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220 } else { // then only n11 changed; the rest fixed
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221 if (n11%11 && n11 + aux->n - aux->n1_ - aux->n_1) {
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222 if (n11 == aux->n11 + 1) { // incremental
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223 aux->p *= (double)(aux->n1_ - aux->n11) / n11
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224 * (aux->n_1 - aux->n11) / (n11 + aux->n - aux->n1_ - aux->n_1);
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225 aux->n11 = n11;
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226 return aux->p;
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227 }
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228 if (n11 == aux->n11 - 1) { // incremental
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229 aux->p *= (double)aux->n11 / (aux->n1_ - n11)
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230 * (aux->n11 + aux->n - aux->n1_ - aux->n_1) / (aux->n_1 - n11);
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231 aux->n11 = n11;
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232 return aux->p;
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233 }
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234 }
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235 aux->n11 = n11;
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236 }
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237 aux->p = hypergeo(aux->n11, aux->n1_, aux->n_1, aux->n);
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238 return aux->p;
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239 }
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240
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241 double kt_fisher_exact(int n11, int n12, int n21, int n22, double *_left, double *_right, double *two)
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242 {
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243 int i, j, max, min;
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244 double p, q, left, right;
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245 hgacc_t aux;
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246 int n1_, n_1, n;
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247
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248 n1_ = n11 + n12; n_1 = n11 + n21; n = n11 + n12 + n21 + n22; // calculate n1_, n_1 and n
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249 max = (n_1 < n1_) ? n_1 : n1_; // max n11, for right tail
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250 min = n1_ + n_1 - n; // not sure why n11-n22 is used instead of min(n_1,n1_)
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251 if (min < 0) min = 0; // min n11, for left tail
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252 *two = *_left = *_right = 1.;
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253 if (min == max) return 1.; // no need to do test
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254 q = hypergeo_acc(n11, n1_, n_1, n, &aux); // the probability of the current table
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255 // left tail
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256 p = hypergeo_acc(min, 0, 0, 0, &aux);
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257 for (left = 0., i = min + 1; p < 0.99999999 * q && i<=max; ++i) // loop until underflow
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258 left += p, p = hypergeo_acc(i, 0, 0, 0, &aux);
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259 --i;
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260 if (p < 1.00000001 * q) left += p;
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261 else --i;
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262 // right tail
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263 p = hypergeo_acc(max, 0, 0, 0, &aux);
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264 for (right = 0., j = max - 1; p < 0.99999999 * q && j>=0; --j) // loop until underflow
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265 right += p, p = hypergeo_acc(j, 0, 0, 0, &aux);
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266 ++j;
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267 if (p < 1.00000001 * q) right += p;
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268 else ++j;
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269 // two-tail
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270 *two = left + right;
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271 if (*two > 1.) *two = 1.;
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272 // adjust left and right
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273 if (abs(i - n11) < abs(j - n11)) right = 1. - left + q;
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274 else left = 1.0 - right + q;
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275 *_left = left; *_right = right;
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276 return q;
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277 }
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278
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279
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280
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