1 /* cdf/tdistinv.c
2 *
3 * Copyright (C) 2007 Brian Gough
4 * Copyright (C) 2002 Jason H. Stover.
5 *
6 * This program is free software; you can redistribute it and/or modify
7 * it under the terms of the GNU General Public License as published by
8 * the Free Software Foundation; either version 3 of the License, or (at
9 * your option) any later version.
10 *
11 * This program is distributed in the hope that it will be useful, but
12 * WITHOUT ANY WARRANTY; without even the implied warranty of
13 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
14 * General Public License for more details.
15 *
16 * You should have received a copy of the GNU General Public License
17 * along with this program; if not, write to the Free Software
18 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA.
19 */
20
21 #include "gsl__config.h"
22 #include <math.h>
23 #include "gsl_cdf.h"
24 #include "gsl_math.h"
25 #include "gsl_randist.h"
26 #include "gsl_sf_gamma.h"
27
28 #include <stdio.h>
29
30 static double
inv_cornish_fisher(double z,double nu)31 inv_cornish_fisher (double z, double nu)
32 {
33 double a = 1 / (nu - 0.5);
34 double b = 48.0 / (a * a);
35
36 double cf1 = z * (3 + z * z);
37 double cf2 = z * (945 + z * z * (360 + z * z * (63 + z * z * 4)));
38
39 double y = z - cf1 / b + cf2 / (10 * b * b);
40
41 double t = GSL_SIGN (z) * sqrt (nu * expm1 (a * y * y));
42
43 return t;
44 }
45
46
47 double
gsl_cdf_tdist_Pinv(const double P,const double nu)48 gsl_cdf_tdist_Pinv (const double P, const double nu)
49 {
50 double x, ptail;
51
52 if (P == 1.0)
53 {
54 return GSL_POSINF;
55 }
56 else if (P == 0.0)
57 {
58 return GSL_NEGINF;
59 }
60
61 if (nu == 1.0)
62 {
63 x = tan (M_PI * (P - 0.5));
64 }
65 else if (nu == 2.0)
66 {
67 double a = 2 * P - 1;
68 x = a / sqrt (2 * (1 - a * a));
69 }
70
71 ptail = (P < 0.5) ? P : 1 - P;
72
73 if (sqrt (M_PI * nu / 2) * ptail > pow (0.05, nu / 2))
74 {
75 double xg = gsl_cdf_ugaussian_Pinv (P);
76 x = inv_cornish_fisher (xg, nu);
77 }
78 else
79 {
80 /* Use an asymptotic expansion of the tail of integral */
81
82 double beta = gsl_sf_beta (0.5, nu / 2);
83
84 if (P < 0.5)
85 {
86 x = -sqrt (nu) * pow (beta * nu * P, -1.0 / nu);
87 }
88 else
89 {
90 x = sqrt (nu) * pow (beta * nu * (1 - P), -1.0 / nu);
91 }
92
93 /* Correct nu -> nu/(1+nu/x^2) in the leading term to account
94 for higher order terms. This avoids overestimating x, which
95 makes the iteration unstable due to the rapidly decreasing
96 tails of the distribution. */
97
98 x /= sqrt (1 + nu / (x * x));
99 }
100
101 {
102 double dP, phi;
103 unsigned int n = 0;
104
105 start:
106 dP = P - gsl_cdf_tdist_P (x, nu);
107 phi = gsl_ran_tdist_pdf (x, nu);
108
109 if (dP == 0.0 || n++ > 32)
110 goto end;
111
112 {
113 double lambda = dP / phi;
114 double step0 = lambda;
115 double step1 = ((nu + 1) * x / (x * x + nu)) * (lambda * lambda / 4.0);
116
117 double step = step0;
118
119 if (fabs (step1) < fabs (step0))
120 {
121 step += step1;
122 }
123
124 if (P > 0.5 && x + step < 0)
125 x /= 2;
126 else if (P < 0.5 && x + step > 0)
127 x /= 2;
128 else
129 x += step;
130
131 if (fabs (step) > 1e-10 * fabs (x))
132 goto start;
133 }
134
135 end:
136 if (fabs(dP) > GSL_SQRT_DBL_EPSILON * P)
137 {
138 GSL_ERROR_VAL("inverse failed to converge", GSL_EFAILED, GSL_NAN);
139 }
140
141 return x;
142 }
143 }
144
145 double
gsl_cdf_tdist_Qinv(const double Q,const double nu)146 gsl_cdf_tdist_Qinv (const double Q, const double nu)
147 {
148 double x, qtail;
149
150 if (Q == 0.0)
151 {
152 return GSL_POSINF;
153 }
154 else if (Q == 1.0)
155 {
156 return GSL_NEGINF;
157 }
158
159 if (nu == 1.0)
160 {
161 x = tan (M_PI * (0.5 - Q));
162 }
163 else if (nu == 2.0)
164 {
165 double a = 2 * (1 - Q) - 1;
166 x = a / sqrt (2 * (1 - a * a));
167 }
168
169 qtail = (Q < 0.5) ? Q : 1 - Q;
170
171 if (sqrt (M_PI * nu / 2) * qtail > pow (0.05, nu / 2))
172 {
173 double xg = gsl_cdf_ugaussian_Qinv (Q);
174 x = inv_cornish_fisher (xg, nu);
175 }
176 else
177 {
178 /* Use an asymptotic expansion of the tail of integral */
179
180 double beta = gsl_sf_beta (0.5, nu / 2);
181
182 if (Q < 0.5)
183 {
184 x = sqrt (nu) * pow (beta * nu * Q, -1.0 / nu);
185 }
186 else
187 {
188 x = -sqrt (nu) * pow (beta * nu * (1 - Q), -1.0 / nu);
189 }
190
191 /* Correct nu -> nu/(1+nu/x^2) in the leading term to account
192 for higher order terms. This avoids overestimating x, which
193 makes the iteration unstable due to the rapidly decreasing
194 tails of the distribution. */
195
196 x /= sqrt (1 + nu / (x * x));
197 }
198
199 {
200 double dQ, phi;
201 unsigned int n = 0;
202
203 start:
204 dQ = Q - gsl_cdf_tdist_Q (x, nu);
205 phi = gsl_ran_tdist_pdf (x, nu);
206
207 if (dQ == 0.0 || n++ > 32)
208 goto end;
209
210 {
211 double lambda = - dQ / phi;
212 double step0 = lambda;
213 double step1 = ((nu + 1) * x / (x * x + nu)) * (lambda * lambda / 4.0);
214
215 double step = step0;
216
217 if (fabs (step1) < fabs (step0))
218 {
219 step += step1;
220 }
221
222 if (Q < 0.5 && x + step < 0)
223 x /= 2;
224 else if (Q > 0.5 && x + step > 0)
225 x /= 2;
226 else
227 x += step;
228
229 if (fabs (step) > 1e-10 * fabs (x))
230 goto start;
231 }
232 }
233
234 end:
235
236 return x;
237 }
238