+++ /dev/null
-/* cdf/betadistinv.c
- *
- * Copyright (C) 2004 Free Software Foundation, Inc.
- *
- *
- * 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, write to the Free Software
- * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
- */
-
-/*
- * Invert the Beta distribution.
- *
- * References:
- *
- * Roger W. Abernathy and Robert P. Smith. "Applying Series Expansion
- * to the Inverse Beta Distribution to Find Percentiles of the F-Distribution,"
- * ACM Transactions on Mathematical Software, volume 19, number 4, December 1993,
- * pages 474-480.
- *
- * G.W. Hill and A.W. Davis. "Generalized asymptotic expansions of a
- * Cornish-Fisher type," Annals of Mathematical Statistics, volume 39, number 8,
- * August 1968, pages 1264-1273.
- */
-
-#include <math.h>
-#include <gsl/gsl_math.h>
-#include <gsl/gsl_errno.h>
-#include <gsl/gsl_sf_gamma.h>
-#include <gsl/gsl_cdf.h>
-#include <gsl/gsl_randist.h>
-#include "gsl-extras.h"
-
-#define BETAINV_INIT_ERR .001
-#define BETADISTINV_N_TERMS 3
-#define BETADISTINV_MAXITER 20
-
-static double
-s_bisect (double x, double y)
-{
- double result = GSL_MIN(x,y) + fabs(x - y) / 2.0;
- return result;
-}
-static double
-new_guess_P ( double old_guess, double x, double y,
- double prob, double a, double b)
-{
- double result;
- double p_hat;
- double end_point;
-
- p_hat = gsl_cdf_beta_P(old_guess, a, b);
- if (p_hat < prob)
- {
- end_point = GSL_MAX(x,y);
- }
- else if ( p_hat > prob )
- {
- end_point = GSL_MIN(x,y);
- }
- else
- {
- end_point = old_guess;
- }
- result = s_bisect(old_guess, end_point);
-
- return result;
-}
-
-static double
-new_guess_Q ( double old_guess, double x, double y,
- double prob, double a, double b)
-{
- double result;
- double q_hat;
- double end_point;
-
- q_hat = gsl_cdf_beta_Q(old_guess, a, b);
- if (q_hat >= prob)
- {
- end_point = GSL_MAX(x,y);
- }
- else if ( q_hat < prob )
- {
- end_point = GSL_MIN(x,y);
- }
- else
- {
- end_point = old_guess;
- }
- result = s_bisect(old_guess, end_point);
-
- return result;
-}
-
-/*
- * The get_corn_fish_* functions below return the first
- * three terms of the Cornish-Fisher expansion
- * without recursion. The recursive functions
- * make the code more legible when higher order coefficients
- * are used, but terms beyond the cubic do not
- * improve accuracy.
- */
- /*
- * Linear coefficient for the
- * Cornish-Fisher expansion.
- */
-static double
-get_corn_fish_lin (const double x, const double a, const double b)
-{
- double result;
-
- result = gsl_ran_beta_pdf (x, a, b);
- if(result > 0)
- {
- result = 1.0 / result;
- }
- else
- {
- result = GSL_DBL_MAX;
- }
-
- return result;
-}
- /*
- * Quadratic coefficient for the
- * Cornish-Fisher expansion.
- */
-static double
-get_corn_fish_quad (const double x, const double a, const double b)
-{
- double result;
- double gam_ab;
- double gam_a;
- double gam_b;
- double num;
- double den;
-
- gam_ab = gsl_sf_lngamma(a + b);
- gam_a = gsl_sf_lngamma (a);
- gam_b = gsl_sf_lngamma (b);
- num = exp(2 * (gam_a + gam_b - gam_ab)) * (1 - a + x * (b + a - 2));
- den = 2.0 * pow ( x, 2*a - 1 ) * pow ( 1 - x, 2 * b - 1 );
- if ( fabs(den) > 0.0)
- {
- result = num / den;
- }
- else
- {
- result = 0.0;
- }
-
- return result;
-}
-/*
- * The cubic term for the Cornish-Fisher expansion.
- * Theoretically, use of this term should give a better approximation,
- * but in practice inclusion of the cubic term worsens the
- * iterative procedure in gsl_cdf_beta_Pinv and gsl_cdf_beta_Qinv
- * for extreme values of p, a or b.
- */
-#if 0
-static double
-get_corn_fish_cube (const double x, const double a, const double b)
-{
- double result;
- double am1 = a - 1.0;
- double am2 = a - 2.0;
- double apbm2 = a+b-2.0;
- double apbm3 = a+b-3.0;
- double apbm4 = a+b-4.0;
- double ab1ab2 = am1 * am2;
- double tmp;
- double num;
- double den;
-
- num = (am1 - x * apbm2) * (am1 - x * apbm2);
- tmp = ab1ab2 - x * (apbm2 * ( ab1ab2 * apbm4 + 1) + x * apbm2 * apbm3);
- num += tmp;
- tmp = gsl_ran_beta_pdf(x,a,b);
- den = 2.0 * x * x * (1 - x) * (1 - x) * pow(tmp,3.0);
- if ( fabs(den) > 0.0)
- {
- result = num / den;
- }
- else
- {
- result = 0.0;
- }
-
- return result;
-}
-#endif
-/*
- * The Cornish-Fisher coefficients can be defined recursivley,
- * starting with the nth derivative of s_psi = -f'(x)/f(x),
- * where f is the beta density.
- *
- * The section below was commented out since
- * the recursive generation of the coeficients did
- * not improve the accuracy of the directly coded
- * the first three coefficients.
- */
-#if 0
-static double
-s_d_psi (double x, double a, double b, int n)
-{
- double result;
- double np1 = (double) n + 1;
- double asgn;
- double bsgn;
- double bm1 = b - 1.0;
- double am1 = a - 1.0;
- double mx = 1.0 - x;
-
- asgn = (n % 2) ? 1.0:-1.0;
- bsgn = (n % 2) ? -1.0:1.0;
- result = gsl_sf_gamma(np1) * ((bsgn * bm1 / (pow(mx, np1)))
- + (asgn * am1 / (pow(x,np1))));
- return result;
-}
-/*
- * nth derivative of a coefficient with respect
- * to x.
- */
-static double
-get_d_coeff ( double x, double a,
- double b, double n, double k)
-{
- double result;
- double d_psi;
- double k_fac;
- double i_fac;
- double kmi_fac;
- double i;
-
- if (n == 2)
- {
- result = s_d_psi(x, a, b, k);
- }
- else
- {
- result = 0.0;
- for (i = 0.0; i < (k+1); i++)
- {
- k_fac = gsl_sf_lngamma(k+1.0);
- i_fac = gsl_sf_lngamma(i+1.0);
- kmi_fac = gsl_sf_lngamma(k-i+1.0);
-
- result += exp(k_fac - i_fac - kmi_fac)
- * get_d_coeff( x, a, b, 2.0, i)
- * get_d_coeff( x, a, b, (n - 1.0), (k - i));
- }
- result += get_d_coeff ( x, a, b, (n-1.0), (k+1.0));
- }
-
- return result;
-}
-/*
- * Cornish-Fisher coefficient.
- */
-static double
-get_corn_fish (double c, double x,
- double a, double b, double n)
-{
- double result;
- double dc;
- double c_prev;
-
- if(n == 1.0)
- {
- result = 1;
- }
- else if (n==2.0)
- {
- result = s_d_psi(x, a, b, 0);
- }
- else
- {
- dc = get_d_coeff(x, a, b, (n-1.0), 1.0);
- c_prev = get_corn_fish(c, x, a, b, (n-1.0));
- result = (n-1) * s_d_psi(x,a,b,0) * c_prev + dc;
- }
- return result;
-}
-#endif
-
-double
-gslextras_cdf_beta_Pinv ( const double p, const double a, const double b)
-{
- double result;
- double state;
- double beta_result;
- double lower = 0.0;
- double upper = 1.0;
- double c1;
- double c2;
-#if 0
- double c3;
-#endif
- double frac1;
- double frac2;
- double frac3;
- double frac4;
- double p0;
- double p1;
- double p2;
- double tmp;
- double err;
- double abserr;
- double relerr;
- double min_err;
- int n_iter = 0;
-
- if ( p < 0.0 )
- {
- GSLEXTRAS_CDF_ERROR("p < 0", GSL_EDOM);
- }
- if ( p > 1.0 )
- {
- GSLEXTRAS_CDF_ERROR("p > 1",GSL_EDOM);
- }
- if ( a < 0.0 )
- {
- GSLEXTRAS_CDF_ERROR ("a < 0", GSL_EDOM );
- }
- if ( b < 0.0 )
- {
- GSLEXTRAS_CDF_ERROR ( "b < 0", GSL_EDOM );
- }
- if ( p == 0.0 )
- {
- return 0.0;
- }
- if ( p == 1.0 )
- {
- return 1.0;
- }
-
- if (p > (1.0 - GSL_DBL_EPSILON))
- {
- /*
- * When p is close to 1.0, the bisection
- * works better with gsl_cdf_Q.
- */
- state = gslextras_cdf_beta_Qinv ( p, a, b);
- result = 1.0 - state;
- return result;
- }
- if (p < GSL_DBL_EPSILON )
- {
- /*
- * Start at a small value and rise until
- * we are above the correct result. This
- * avoids overflow. When p is very close to
- * 0, an initial state value of a/(a+b) will
- * cause the interpolating polynomial
- * to overflow.
- */
- upper = GSL_DBL_MIN;
- beta_result = gsl_cdf_beta_P (upper, a, b);
- while (beta_result < p)
- {
- lower = upper;
- upper *= 4.0;
- beta_result = gsl_cdf_beta_P (upper, a, b);
- }
- state = (lower + upper) / 2.0;
- }
- else
- {
- /*
- * First guess is the expected value.
- */
- lower = 0.0;
- upper = 1.0;
- state = a/(a+b);
- beta_result = gsl_cdf_beta_P (state, a, b);
- }
- err = beta_result - p;
- abserr = fabs(err);
- relerr = abserr / p;
- while ( relerr > BETAINV_INIT_ERR)
- {
- tmp = new_guess_P ( state, lower, upper,
- p, a, b);
- lower = ( tmp < state ) ? lower:state;
- upper = ( tmp < state ) ? state:upper;
- state = tmp;
- beta_result = gsl_cdf_beta_P (state, a, b);
- err = p - beta_result;
- abserr = fabs(err);
- relerr = abserr / p;
- }
-
- result = state;
- min_err = relerr;
- /*
- * Use a second order Lagrange interpolating
- * polynomial to get closer before switching to
- * the iterative method.
- */
- p0 = gsl_cdf_beta_P (lower, a, b);
- p1 = gsl_cdf_beta_P (state, a, b);
- p2 = gsl_cdf_beta_P (upper, a, b);
- if( p0 < p1 && p1 < p2)
- {
- frac1 = (p - p2) / (p0 - p1);
- frac2 = (p - p1) / (p0 - p2);
- frac3 = (p - p0) / (p1 - p2);
- frac4 = (p - p0) * (p - p1) / ((p2 - p0) * (p2 - p1));
- state = frac1 * (frac2 * lower - frac3 * state)
- + frac4 * upper;
-
- beta_result = gsl_cdf_beta_P( state, a, b);
- err = p - beta_result;
- abserr = fabs(err);
- relerr = abserr / p;
- if (relerr < min_err)
- {
- result = state;
- min_err = relerr;
- }
- else
- {
- /*
- * Lagrange polynomial failed to reduce the error.
- * This will happen with a very skewed beta density.
- * Undo previous steps.
- */
- state = result;
- beta_result = gsl_cdf_beta_P(state,a,b);
- err = p - beta_result;
- abserr = fabs(err);
- relerr = abserr / p;
- }
- }
-
- n_iter = 0;
-
- /*
- * Newton-type iteration using the terms from the
- * Cornish-Fisher expansion. If only the first term
- * of the expansion is used, this is Newton's method.
- */
- while ( relerr > GSL_DBL_EPSILON && n_iter < BETADISTINV_MAXITER)
- {
- n_iter++;
- c1 = get_corn_fish_lin (state, a, b);
- c2 = get_corn_fish_quad (state, a, b);
- /*
- * The cubic term does not help, and can can
- * harm the approximation for extreme values of
- * p, a, or b.
- */
-#if 0
- c3 = get_corn_fish_cube (state, a, b);
- state += err * (c1 + (err / 2.0 ) * (c2 + c3 * err / 3.0));
-#endif
- state += err * (c1 + (c2 * err / 2.0 ));
- /*
- * The section below which is commented out uses
- * a recursive function to get the coefficients.
- * The recursion makes coding higher-order terms
- * easier, but did not improve the result beyond
- * the use of three terms. Since explicitly coding
- * those three terms in the get_corn_fish_* functions
- * was not difficult, the recursion was abandoned.
- */
-#if 0
- coeff = 1.0;
- for(i = 1.0; i < BETADISTINV_N_TERMS; i += 1.0)
- {
- i_fac *= i;
- coeff = get_corn_fish (coeff, prior_state, a, b, i);
- state += coeff * pow(err, i) /
- (i_fac * pow (gsl_ran_beta_pdf(prior_state,a,b), i));
- }
-#endif
- beta_result = gsl_cdf_beta_P ( state, a, b );
- err = p - beta_result;
- abserr = fabs(err);
- relerr = abserr / p;
- if (relerr < min_err)
- {
- result = state;
- min_err = relerr;
- }
- }
-
- return result;
-}
-
-double
-gslextras_cdf_beta_Qinv (double q, double a, double b)
-{
- double result;
- double state;
- double beta_result;
- double lower = 0.0;
- double upper = 1.0;
- double c1;
- double c2;
-#if 0
- double c3;
-#endif
- double p0;
- double p1;
- double p2;
- double frac1;
- double frac2;
- double frac3;
- double frac4;
- double tmp;
- double err;
- double abserr;
- double relerr;
- double min_err;
- int n_iter = 0;
-
- if ( q < 0.0 )
- {
- GSLEXTRAS_CDF_ERROR("q < 0", GSL_EDOM);
- }
- if ( q > 1.0 )
- {
- GSLEXTRAS_CDF_ERROR("q > 1",GSL_EDOM);
- }
- if ( a < 0.0 )
- {
- GSLEXTRAS_CDF_ERROR ("a < 0", GSL_EDOM );
- }
- if ( b < 0.0 )
- {
- GSLEXTRAS_CDF_ERROR ( "b < 0", GSL_EDOM );
- }
- if ( q == 0.0 )
- {
- return 1.0;
- }
- if ( q == 1.0 )
- {
- return 0.0;
- }
-
- if ( q < GSL_DBL_EPSILON )
- {
- /*
- * When q is close to 0, the bisection
- * and interpolation done in the rest of
- * this routine will not give the correct
- * value within double precision, so
- * gsl_cdf_beta_Qinv is called instead.
- */
- state = gslextras_cdf_beta_Pinv ( q, a, b);
- result = 1.0 - state;
- return result;
- }
- if ( q > 1.0 - GSL_DBL_EPSILON )
- {
- /*
- * Make the initial guess close to 0.0.
- */
- upper = GSL_DBL_MIN;
- beta_result = gsl_cdf_beta_Q ( upper, a, b);
- while (beta_result > q )
- {
- lower = upper;
- upper *= 4.0;
- beta_result = gsl_cdf_beta_Q ( upper, a, b);
- }
- state = (upper + lower) / 2.0;
- }
- else
- {
- /* Bisection to get an initial approximation.
- * First guess is the expected value.
- */
- state = a/(a+b);
- lower = 0.0;
- upper = 1.0;
- }
- beta_result = gsl_cdf_beta_Q (state, a, b);
- err = beta_result - q;
- abserr = fabs(err);
- relerr = abserr / q;
- while ( relerr > BETAINV_INIT_ERR)
- {
- n_iter++;
- tmp = new_guess_Q ( state, lower, upper,
- q, a, b);
- lower = ( tmp < state ) ? lower:state;
- upper = ( tmp < state ) ? state:upper;
- state = tmp;
- beta_result = gsl_cdf_beta_Q (state, a, b);
- err = q - beta_result;
- abserr = fabs(err);
- relerr = abserr / q;
- }
- result = state;
- min_err = relerr;
-
- /*
- * Use a second order Lagrange interpolating
- * polynomial to get closer before switching to
- * the iterative method.
- */
- p0 = gsl_cdf_beta_Q (lower, a, b);
- p1 = gsl_cdf_beta_Q (state, a, b);
- p2 = gsl_cdf_beta_Q (upper, a, b);
- if(p0 > p1 && p1 > p2)
- {
- frac1 = (q - p2) / (p0 - p1);
- frac2 = (q - p1) / (p0 - p2);
- frac3 = (q - p0) / (p1 - p2);
- frac4 = (q - p0) * (q - p1) / ((p2 - p0) * (p2 - p1));
- state = frac1 * (frac2 * lower - frac3 * state)
- + frac4 * upper;
- beta_result = gsl_cdf_beta_Q( state, a, b);
- err = beta_result - q;
- abserr = fabs(err);
- relerr = abserr / q;
- if (relerr < min_err)
- {
- result = state;
- min_err = relerr;
- }
- else
- {
- /*
- * Lagrange polynomial failed to reduce the error.
- * This will happen with a very skewed beta density.
- * Undo previous steps.
- */
- state = result;
- beta_result = gsl_cdf_beta_P(state,a,b);
- err = q - beta_result;
- abserr = fabs(err);
- relerr = abserr / q;
- }
- }
-
- /*
- * Iteration using the terms from the
- * Cornish-Fisher expansion. If only the first term
- * of the expansion is used, this is Newton's method.
- */
-
- n_iter = 0;
- while ( relerr > GSL_DBL_EPSILON && n_iter < BETADISTINV_MAXITER)
- {
- n_iter++;
- c1 = get_corn_fish_lin (state, a, b);
- c2 = get_corn_fish_quad (state, a, b);
- /*
- * The cubic term does not help, and can harm
- * the approximation for extreme values of p, a and b.
- */
-#if 0
- c3 = get_corn_fish_cube (state, a, b);
- state += err * (c1 + (err / 2.0 ) * (c2 + c3 * err / 3.0));
-#endif
- state += err * (c1 + (c2 * err / 2.0 ));
- beta_result = gsl_cdf_beta_Q ( state, a, b );
- err = beta_result - q;
- abserr = fabs(err);
- relerr = abserr / q;
- if (relerr < min_err)
- {
- result = state;
- min_err = relerr;
- }
- }
-
- return result;
-}
projects / pspp-builds.git / blobdiff