1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
|
// Pseudorandom numbers from a truncated Gaussian distribution.
//
// This implements an extension of Chopin's algorithm detailed in
// N. Chopin, "Fast simulation of truncated Gaussian distributions",
// Stat Comput (2011) 21:275-288
//
// Copyright (C) 2012 Guillaume Dollé, Vincent Mazet
// (LSIIT, CNRS/Université de Strasbourg)
// Version 2012-07-04, Contact: vincent.mazet@unistra.fr
//
// 06/07/2012:
// - first launch of rtnorm.cpp
//
// Licence: GNU General Public License Version 2
// This program is free software; you can redistribute it and/or modify it
// under the terms of the GNU General Public License as published by the
// Free Software Foundation; either version 2 of the License, or (at your
// option) any later version. This program is distributed in the hope that
// it will be useful, but WITHOUT ANY WARRANTY; without even the implied
// warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU General Public License for more details. You should have received a
// copy of the GNU General Public License along with this program; if not,
// see http://www.gnu.org/licenses/old-licenses/gpl-2.0.txt
//
// Depends: LibGSL
// OS: Unix based system
#include <cmath>
#include <iostream>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_sf_erf.h>
#include "rtnorm.hpp"
int N = 4001; // Index of the right tail
//------------------------------------------------------------
// Pseudorandom numbers from a truncated Gaussian distribution
// The Gaussian has parameters mu (default 0) and sigma (default 1)
// and is truncated on the interval [a,b].
// Returns the random variable x and its probability p(x).
std::pair<double, double> rtnorm(
gsl_rng *gen,
double a,
double b,
const double mu,
const double sigma)
{
// Design variables
double xmin = -2.00443204036; // Left bound
double xmax = 3.48672170399; // Right bound
int kmin = 5; // if kb-ka < kmin then use a rejection algorithm
double INVH = 1631.73284006; // = 1/h, h being the minimal interval range
int I0 = 3271; // = - floor(x(0)/h)
double ALPHA = 1.837877066409345; // = log(2*pi)
int xsize=sizeof(Rtnorm::x)/sizeof(double); // Length of table x
int stop = false;
double sq2 = 7.071067811865475e-1; // = 1/sqrt(2)
double sqpi = 1.772453850905516; // = sqrt(pi)
double r, z, e, ylk, simy, lbound, u, d, sim, Z, p;
int i, ka, kb, k;
// Scaling
if(mu!=0 || sigma!=1)
{
a=(a-mu)/sigma;
b=(b-mu)/sigma;
}
//-----------------------------
// Check if a < b
if(a>=b)
{
std::cerr<<"*** B must be greater than A ! ***"<<std::endl;
exit(1);
}
// Check if |a| < |b|
else if(fabs(a)>fabs(b))
r = -rtnorm(gen,-b,-a).first; // Pair (r,p)
// If a in the right tail (a > xmax), use rejection algorithm with a truncated exponential proposal
else if(a>xmax)
r = rtexp(gen,a,b);
// If a in the left tail (a < xmin), use rejection algorithm with a Gaussian proposal
else if(a<xmin)
{
while(!stop)
{
r = gsl_ran_gaussian(gen,1);
stop = (r>=a) && (r<=b);
}
}
// In other cases (xmin < a < xmax), use Chopin's algorithm
else
{
// Compute ka
i = I0 + floor(a*INVH);
ka = Rtnorm::ncell[i];
// Compute kb
(b>=xmax) ?
kb = N :
(
i = I0 + floor(b*INVH),
kb = Rtnorm::ncell[i]
);
// If |b-a| is small, use rejection algorithm with a truncated exponential proposal
if(abs(kb-ka) < kmin)
{
r = rtexp(gen,a,b);
stop = true;
}
while(!stop)
{
// Sample integer between ka and kb
k = floor(gsl_rng_uniform(gen) * (kb-ka+1)) + ka;
if(k == N)
{
// Right tail
lbound = Rtnorm::x[xsize-1];
z = -log(gsl_rng_uniform(gen));
e = -log(gsl_rng_uniform(gen));
z = z / lbound;
if ((pow(z,2) <= 2*e) && (z < b-lbound))
{
// Accept this proposition, otherwise reject
r = lbound + z;
stop = true;
}
}
else if ((k <= ka+1) || (k>=kb-1 && b<xmax))
{
// Two leftmost and rightmost regions
sim = Rtnorm::x[k] + (Rtnorm::x[k+1]-Rtnorm::x[k]) * gsl_rng_uniform(gen);
if ((sim >= a) && (sim <= b))
{
// Accept this proposition, otherwise reject
simy = Rtnorm::yu[k]*gsl_rng_uniform(gen);
if ( (simy<yl(k)) || (sim * sim + 2*log(simy) + ALPHA) < 0 )
{
r = sim;
stop = true;
}
}
}
else // All the other boxes
{
u = gsl_rng_uniform(gen);
simy = Rtnorm::yu[k] * u;
d = Rtnorm::x[k+1] - Rtnorm::x[k];
ylk = yl(k);
if(simy < ylk) // That's what happens most of the time
{
r = Rtnorm::x[k] + u*d*Rtnorm::yu[k]/ylk;
stop = true;
}
else
{
sim = Rtnorm::x[k] + d * gsl_rng_uniform(gen);
// Otherwise, check you're below the pdf curve
if((sim * sim + 2*log(simy) + ALPHA) < 0)
{
r = sim;
stop = true;
}
}
}
}
}
//-----------------------------
// Scaling
if(mu!=0 || sigma!=1)
r = r*sigma + mu;
// Compute the probability
Z = sqpi *sq2 * sigma * ( gsl_sf_erf(b*sq2) - gsl_sf_erf(a*sq2) );
p = exp(-pow((r-mu)/sigma,2)/2) / Z;
return std::pair<double,double>(r,p);
}
//------------------------------------------------------------
// Compute y_l from y_k
double yl(int k)
{
double yl0 = 0.053513975472; // y_l of the leftmost rectangle
double ylN = 0.000914116389555; // y_l of the rightmost rectangle
if (k == 0)
return yl0;
else if(k == N-1)
return ylN;
else if(k <= 1953)
return Rtnorm::yu[k-1];
else
return Rtnorm::yu[k+1];
}
//------------------------------------------------------------
// Rejection algorithm with a truncated exponential proposal
double rtexp(gsl_rng *gen, double a, double b)
{
int stop = false;
double twoasq = 2*pow(a,2);
double expab = exp(-a*(b-a)) - 1;
double z, e;
while(!stop)
{
z = log(1 + gsl_rng_uniform(gen)*expab);
e = -log(gsl_rng_uniform(gen));
stop = (twoasq*e > pow(z,2));
}
return a - z/a;
}
|
