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1 /* This code is in the public domain */ |
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2 |
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3 /* Needs the following defines: |
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4 * NAN: value to return for Not-A-Number |
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5 * RUNI: uniform generator on (0,1) |
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6 * RNOR: normal generator |
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7 * LGAMMA: log gamma function |
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8 * INFINITE: function to test whether a value is infinite |
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9 */ |
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10 |
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11 #if defined (HAVE_CONFIG_H) |
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12 #include <config.h> |
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13 #endif |
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14 |
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15 #include <math.h> |
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16 #include <stdio.h> |
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17 |
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18 #include "f77-fcn.h" |
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19 #include "lo-ieee.h" |
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20 #include "lo-error.h" |
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21 #include "randmtzig.h" |
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22 #include "randpoisson.h" |
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23 |
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24 #undef NAN |
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25 #define NAN octave_NaN |
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26 #define INFINITE lo_ieee_isinf |
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27 #define RUNI oct_randu() |
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28 #define RNOR oct_randn() |
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29 #define LGAMMA xlgamma |
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30 |
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31 F77_RET_T |
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32 F77_FUNC (dlgams, DLGAMS) (const double *, double *, double *); |
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33 |
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34 static double |
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35 xlgamma (double x) |
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36 { |
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37 double result; |
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38 double sgngam; |
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39 |
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40 if (lo_ieee_isnan (x)) |
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41 result = x; |
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42 else if (x <= 0 || lo_ieee_isinf (x)) |
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43 result = octave_Inf; |
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44 else |
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45 F77_XFCN (dlgams, DLGAMS, (&x, &result, &sgngam)); |
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46 |
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47 return result; |
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48 } |
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49 |
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50 /* ---- pprsc.c from Stadloeber's winrand --- */ |
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51 |
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52 #include <math.h> |
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53 |
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54 /* flogfak(k) = ln(k!) */ |
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55 static double |
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56 flogfak (double k) |
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57 { |
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58 #define C0 9.18938533204672742e-01 |
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59 #define C1 8.33333333333333333e-02 |
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60 #define C3 -2.77777777777777778e-03 |
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61 #define C5 7.93650793650793651e-04 |
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62 #define C7 -5.95238095238095238e-04 |
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63 |
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64 static double logfak[30L] = { |
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65 0.00000000000000000, 0.00000000000000000, 0.69314718055994531, |
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66 1.79175946922805500, 3.17805383034794562, 4.78749174278204599, |
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67 6.57925121201010100, 8.52516136106541430, 10.60460290274525023, |
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68 12.80182748008146961, 15.10441257307551530, 17.50230784587388584, |
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69 19.98721449566188615, 22.55216385312342289, 25.19122118273868150, |
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70 27.89927138384089157, 30.67186010608067280, 33.50507345013688888, |
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71 36.39544520803305358, 39.33988418719949404, 42.33561646075348503, |
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72 45.38013889847690803, 48.47118135183522388, 51.60667556776437357, |
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73 54.78472939811231919, 58.00360522298051994, 61.26170176100200198, |
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74 64.55753862700633106, 67.88974313718153498, 71.25703896716800901 |
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75 }; |
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76 |
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77 double r, rr; |
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78 |
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79 if (k >= 30.0) |
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80 { |
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81 r = 1.0 / k; |
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82 rr = r * r; |
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83 return ((k + 0.5)*log(k) - k + C0 + r*(C1 + rr*(C3 + rr*(C5 + rr*C7)))); |
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84 } |
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85 else |
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86 return (logfak[(int)k]); |
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87 } |
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88 |
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89 |
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90 /****************************************************************** |
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91 * * |
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92 * Poisson Distribution - Patchwork Rejection/Inversion * |
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93 * * |
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94 ****************************************************************** |
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95 * * |
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96 * For parameter my < 10 Tabulated Inversion is applied. * |
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97 * For my >= 10 Patchwork Rejection is employed: * |
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98 * The area below the histogram function f(x) is rearranged in * |
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99 * its body by certain point reflections. Within a large center * |
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100 * interval variates are sampled efficiently by rejection from * |
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101 * uniform hats. Rectangular immediate acceptance regions speed * |
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102 * up the generation. The remaining tails are covered by * |
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103 * exponential functions. * |
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104 * * |
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105 ****************************************************************** |
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106 * * |
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107 * FUNCTION : - pprsc samples a random number from the Poisson * |
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108 * distribution with parameter my > 0. * |
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109 * REFERENCE : - H. Zechner (1994): Efficient sampling from * |
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110 * continuous and discrete unimodal distributions, * |
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111 * Doctoral Dissertation, 156 pp., Technical * |
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112 * University Graz, Austria. * |
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113 * SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with * |
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114 * unsigned long integer *seed. * |
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115 * * |
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116 * Implemented by H. Zechner, January 1994 * |
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117 * Revised by F. Niederl, July 1994 * |
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118 * * |
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119 ******************************************************************/ |
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120 |
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121 static double |
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122 f (double k, double l_nu, double c_pm) |
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123 { |
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124 return exp(k * l_nu - flogfak(k) - c_pm); |
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125 } |
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126 |
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127 static double |
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128 pprsc (double my) |
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129 { |
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130 static double my_last = -1.0; |
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131 static double m, k2, k4, k1, k5; |
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132 static double dl, dr, r1, r2, r4, r5, ll, lr, l_my, c_pm, |
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133 f1, f2, f4, f5, p1, p2, p3, p4, p5, p6; |
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134 double Dk, X, Y; |
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135 double Ds, U, V, W; |
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136 |
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137 if (my != my_last) |
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138 { /* set-up */ |
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139 my_last = my; |
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140 /* approximate deviation of reflection points k2, k4 from my - 1/2 */ |
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141 Ds = sqrt(my + 0.25); |
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142 |
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143 /* mode m, reflection points k2 and k4, and points k1 and k5, */ |
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144 /* which delimit the centre region of h(x) */ |
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145 m = floor(my); |
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146 k2 = ceil(my - 0.5 - Ds); |
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147 k4 = floor(my - 0.5 + Ds); |
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148 k1 = k2 + k2 - m + 1L; |
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149 k5 = k4 + k4 - m; |
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150 |
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151 /* range width of the critical left and right centre region */ |
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152 dl = (k2 - k1); |
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153 dr = (k5 - k4); |
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154 |
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155 /* recurrence constants r(k)=p(k)/p(k-1) at k = k1, k2, k4+1, k5+1 */ |
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156 r1 = my / k1; |
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157 r2 = my / k2; |
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158 r4 = my / (k4 + 1.0); |
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159 r5 = my / (k5 + 1.0); |
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160 |
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161 /* reciprocal values of the scale parameters of exp. tail envelope */ |
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162 ll = log(r1); /* expon. tail left */ |
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163 lr = -log(r5); /* expon. tail right*/ |
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164 |
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165 /* Poisson constants, necessary for computing function values f(k) */ |
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166 l_my = log(my); |
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167 c_pm = m * l_my - flogfak(m); |
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168 |
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169 /* function values f(k) = p(k)/p(m) at k = k2, k4, k1, k5 */ |
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170 f2 = f(k2, l_my, c_pm); |
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171 f4 = f(k4, l_my, c_pm); |
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172 f1 = f(k1, l_my, c_pm); |
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173 f5 = f(k5, l_my, c_pm); |
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174 |
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175 /* area of the two centre and the two exponential tail regions */ |
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176 /* area of the two immediate acceptance regions between k2, k4 */ |
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177 p1 = f2 * (dl + 1.0); /* immed. left */ |
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178 p2 = f2 * dl + p1; /* centre left */ |
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179 p3 = f4 * (dr + 1.0) + p2; /* immed. right */ |
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180 p4 = f4 * dr + p3; /* centre right */ |
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181 p5 = f1 / ll + p4; /* exp. tail left */ |
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182 p6 = f5 / lr + p5; /* exp. tail right*/ |
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183 } |
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184 |
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185 for (;;) |
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186 { |
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187 /* generate uniform number U -- U(0, p6) */ |
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188 /* case distinction corresponding to U */ |
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189 if ((U = RUNI * p6) < p2) |
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190 { /* centre left */ |
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191 |
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192 /* immediate acceptance region |
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193 R2 = [k2, m) *[0, f2), X = k2, ... m -1 */ |
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194 if ((V = U - p1) < 0.0) return(k2 + floor(U/f2)); |
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195 /* immediate acceptance region |
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196 R1 = [k1, k2)*[0, f1), X = k1, ... k2-1 */ |
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197 if ((W = V / dl) < f1 ) return(k1 + floor(V/f1)); |
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198 |
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199 /* computation of candidate X < k2, and its counterpart Y > k2 */ |
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200 /* either squeeze-acceptance of X or acceptance-rejection of Y */ |
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201 Dk = floor(dl * RUNI) + 1.0; |
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202 if (W <= f2 - Dk * (f2 - f2/r2)) |
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203 { /* quick accept of */ |
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204 return(k2 - Dk); /* X = k2 - Dk */ |
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205 } |
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206 if ((V = f2 + f2 - W) < 1.0) |
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207 { /* quick reject of Y*/ |
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208 Y = k2 + Dk; |
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209 if (V <= f2 + Dk * (1.0 - f2)/(dl + 1.0)) |
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210 { /* quick accept of */ |
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211 return(Y); /* Y = k2 + Dk */ |
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212 } |
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213 if (V <= f(Y, l_my, c_pm)) return(Y); /* final accept of Y*/ |
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214 } |
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215 X = k2 - Dk; |
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216 } |
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217 else if (U < p4) |
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218 { /* centre right */ |
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219 /* immediate acceptance region |
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220 R3 = [m, k4+1)*[0, f4), X = m, ... k4 */ |
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221 if ((V = U - p3) < 0.0) return(k4 - floor((U - p2)/f4)); |
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222 /* immediate acceptance region |
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223 R4 = [k4+1, k5+1)*[0, f5) */ |
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224 if ((W = V / dr) < f5 ) return(k5 - floor(V/f5)); |
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225 |
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226 /* computation of candidate X > k4, and its counterpart Y < k4 */ |
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227 /* either squeeze-acceptance of X or acceptance-rejection of Y */ |
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228 Dk = floor(dr * RUNI) + 1.0; |
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229 if (W <= f4 - Dk * (f4 - f4*r4)) |
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230 { /* quick accept of */ |
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231 return(k4 + Dk); /* X = k4 + Dk */ |
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232 } |
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233 if ((V = f4 + f4 - W) < 1.0) |
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234 { /* quick reject of Y*/ |
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235 Y = k4 - Dk; |
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236 if (V <= f4 + Dk * (1.0 - f4)/ dr) |
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237 { /* quick accept of */ |
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238 return(Y); /* Y = k4 - Dk */ |
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239 } |
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240 if (V <= f(Y, l_my, c_pm)) return(Y); /* final accept of Y*/ |
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241 } |
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242 X = k4 + Dk; |
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243 } |
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244 else |
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245 { |
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246 W = RUNI; |
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247 if (U < p5) |
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248 { /* expon. tail left */ |
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249 Dk = floor(1.0 - log(W)/ll); |
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250 if ((X = k1 - Dk) < 0L) continue; /* 0 <= X <= k1 - 1 */ |
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251 W *= (U - p4) * ll; /* W -- U(0, h(x)) */ |
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252 if (W <= f1 - Dk * (f1 - f1/r1)) |
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253 return(X); /* quick accept of X*/ |
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254 } |
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255 else |
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256 { /* expon. tail right*/ |
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257 Dk = floor(1.0 - log(W)/lr); |
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258 X = k5 + Dk; /* X >= k5 + 1 */ |
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259 W *= (U - p5) * lr; /* W -- U(0, h(x)) */ |
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260 if (W <= f5 - Dk * (f5 - f5*r5)) |
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261 return(X); /* quick accept of X*/ |
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262 } |
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263 } |
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264 |
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265 /* acceptance-rejection test of candidate X from the original area */ |
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266 /* test, whether W <= f(k), with W = U*h(x) and U -- U(0, 1)*/ |
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267 /* log f(X) = (X - m)*log(my) - log X! + log m! */ |
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268 if (log(W) <= X * l_my - flogfak(X) - c_pm) return(X); |
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269 } |
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270 } |
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271 /* ---- pprsc.c end ------ */ |
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272 |
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273 |
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274 /* The remainder of the file is by Paul Kienzle */ |
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275 |
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276 /* Given uniform u, find x such that CDF(L,x)==u. Return x. */ |
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277 static void |
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278 poisson_cdf_lookup(double lambda, double *p, size_t n) |
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279 { |
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280 /* Table size is predicated on the maximum value of lambda |
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281 * we want to store in the table, and the maximum value of |
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282 * returned by the uniform random number generator on [0,1). |
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283 * With lambda==10 and u_max = 1 - 1/(2^32+1), we |
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284 * have poisson_pdf(lambda,36) < 1-u_max. If instead our |
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285 * generator uses more bits of mantissa or returns a value |
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286 * in the range [0,1], then for lambda==10 we need a table |
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287 * size of 46 instead. For long doubles, the table size |
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288 * will need to be longer still. */ |
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289 #define TABLESIZE 46 |
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290 double t[TABLESIZE]; |
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291 |
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292 /* Precompute the table for the u up to and including 0.458. |
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293 * We will almost certainly need it. */ |
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294 int intlambda = (int)floor(lambda); |
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295 double P; |
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296 int tableidx; |
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297 size_t i = n; |
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298 |
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299 t[0] = P = exp(-lambda); |
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300 for (tableidx = 1; tableidx <= intlambda; tableidx++) { |
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301 P = P*lambda/(double)tableidx; |
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302 t[tableidx] = t[tableidx-1] + P; |
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303 } |
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304 |
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305 while (i-- > 0) { |
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306 double u = RUNI; |
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307 |
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308 /* If u > 0.458 we know we can jump to floor(lambda) before |
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309 * comparing (this observation is based on Stadlober's winrand |
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310 * code). For lambda >= 1, this will be a win. Lambda < 1 |
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311 * is already fast, so adding an extra comparison is not a |
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312 * problem. */ |
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313 int k = (u > 0.458 ? intlambda : 0); |
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314 |
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315 /* We aren't using a for loop here because when we find the |
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316 * right k we want to jump to the next iteration of the |
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317 * outer loop, and the continue statement will only work for |
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318 * the inner loop. */ |
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319 nextk: |
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320 if ( u <= t[k] ) { |
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321 p[i] = (double) k; |
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322 continue; |
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323 } |
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324 if (++k < tableidx) goto nextk; |
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325 |
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326 /* We only need high values of the table very rarely so we |
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327 * don't automatically compute the entire table. */ |
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328 while (tableidx < TABLESIZE) { |
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329 P = P*lambda/(double)tableidx; |
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330 t[tableidx] = t[tableidx-1] + P; |
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331 /* Make sure we converge to 1.0 just in case u is uniform |
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332 * on [0,1] rather than [0,1). */ |
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333 if (t[tableidx] == t[tableidx-1]) t[tableidx] = 1.0; |
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334 tableidx++; |
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335 if (u <= t[tableidx-1]) break; |
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336 } |
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337 |
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338 /* We are assuming that the table size is big enough here. |
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339 * This should be true even if RUNI is returning values in |
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340 * the range [0,1] rather than [0,1). |
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341 */ |
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342 p[i] = (double)(tableidx-1); |
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343 } |
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344 } |
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345 |
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346 /* From Press, et al., Numerical Recipes */ |
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347 static void |
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348 poisson_rejection (double lambda, double *p, size_t n) |
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349 { |
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350 double sq = sqrt(2.0*lambda); |
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351 double alxm = log(lambda); |
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352 double g = lambda*alxm - LGAMMA(lambda+1.0); |
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353 size_t i; |
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354 |
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355 for (i = 0; i < n; i++) |
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356 { |
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357 double y, em, t; |
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358 do { |
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359 do { |
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360 y = tan(M_PI*RUNI); |
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361 em = sq * y + lambda; |
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362 } while (em < 0.0); |
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363 em = floor(em); |
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364 t = 0.9*(1.0+y*y)*exp(em*alxm-flogfak(em)-g); |
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365 } while (RUNI > t); |
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366 p[i] = em; |
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367 } |
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368 } |
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369 |
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370 /* The cutoff of L <= 1e8 in the following two functions before using |
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371 * the normal approximation is based on: |
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372 * > L=1e8; x=floor(linspace(0,2*L,1000)); |
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373 * > max(abs(normal_pdf(x,L,L)-poisson_pdf(x,L))) |
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374 * ans = 1.1376e-28 |
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375 * For L=1e7, the max is around 1e-9, which is within the step size of RUNI. |
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376 * For L>1e10 the pprsc function breaks down, as I saw from the histogram |
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377 * of a large sample, so 1e8 is both small enough and large enough. */ |
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378 |
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379 /* Generate a set of poisson numbers with the same distribution */ |
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380 void |
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381 oct_fill_randp (double L, octave_idx_type n, double *p) |
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382 { |
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383 octave_idx_type i; |
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384 if (L < 0.0 || INFINITE(L)) |
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385 { |
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386 for (i=0; i<n; i++) |
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387 p[i] = NAN; |
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388 } |
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389 else if (L <= 10.0) |
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390 { |
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391 poisson_cdf_lookup(L, p, n); |
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392 } |
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393 else if (L <= 1e8) |
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394 { |
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395 for (i=0; i<n; i++) |
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396 p[i] = pprsc(L); |
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397 } |
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398 else |
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399 { |
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400 /* normal approximation: from Phys. Rev. D (1994) v50 p1284 */ |
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401 const double sqrtL = sqrt(L); |
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402 for (i = 0; i < L; i++) |
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403 { |
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404 p[i] = floor(RNOR*sqrtL + L + 0.5); |
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405 if (p[i] < 0.0) |
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406 p[i] = 0.0; /* will probably never happen */ |
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407 } |
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408 } |
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409 } |
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410 |
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411 /* Generate one poisson variate */ |
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412 double |
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413 oct_randp (double L) |
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414 { |
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415 double ret; |
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416 if (L < 0.0) ret = NAN; |
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417 else if (L <= 12.0) { |
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418 /* From Press, et al. Numerical recipes */ |
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419 double g = exp(-L); |
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420 int em = -1; |
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421 double t = 1.0; |
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422 do { |
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423 ++em; |
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424 t *= RUNI; |
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425 } while (t > g); |
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426 ret = em; |
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427 } else if (L <= 1e8) { |
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428 /* numerical recipes */ |
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429 poisson_rejection(L, &ret, 1); |
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430 } else if (INFINITE(L)) { |
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5775
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431 /* FIXME R uses NaN, but the normal approx. suggests that as |
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5742
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432 * limit should be inf. Which is correct? */ |
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433 ret = NAN; |
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434 } else { |
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435 /* normal approximation: from Phys. Rev. D (1994) v50 p1284 */ |
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436 ret = floor(RNOR*sqrt(L) + L + 0.5); |
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437 if (ret < 0.0) ret = 0.0; /* will probably never happen */ |
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438 } |
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439 return ret; |
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440 } |
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441 |
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442 /* |
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443 ;;; Local Variables: *** |
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444 ;;; mode: C *** |
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445 ;;; End: *** |
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446 */ |
