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MbRandom.cpp
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MbRandom.cpp
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/*!
* \file
* This file contains the implementation of MbRandom, a
* class that deals with random variables.
*
* \brief Implementation of MbRandom class
*
* MrBayes version 4.0 beta
*
* (c) Copyright 2005.
* \version 4.0 beta
* \date Last modified: $Date: 2006/09/11 17:29:51 $
* \author John Huelsenbeck (1)
* \author Bret Larget (2)
* \author Paul van der Mark (3)
* \author Fredrik Ronquist (3)
* \author Donald Simon (4)
* \author (authors listed in alphabetical order)
* (1) Department of Integrative Biology, University of California, Berkeley
* (2) Departments of Botany and of Statistics, University of Wisconsin - Madison
* (3) School of Computational Science, Florida State University
* (4) Department of Mathematics/Computer Science, Duquesne University
*
* 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 (the file gpl.txt included with this
* distribution or http://www.gnu.org/licenses/gpl.txt for more
* details.
*
* $Id: MbRandom.cpp,v 1.3 2006/09/11 17:29:51 paulvdm Exp $
*/
#include "MbRandom.h"
#include <cmath>
#include <cstdlib>
#include <ctime>
#include <iostream>
#include <vector>
/*!
* Constructor for MbRandom class. This constructor does not take
* any parameters and initializes the seed using the current
* system time.
*
* \brief Constructor for MbRandom, initializing seed with system time.
* \param Takes no parameter.
* \return Returns no value.
* \throws Does not throw an error.
*/
MbRandom::MbRandom(void) {
setSeed();
initializedFacTable = false;
availableNormalRv = false;
}
/*!
* Constructor for MbRandom class. This constructor takes as a
* parameter a long integer with the user-supplied seed for the
* random number generator.
*
* \brief Constructor for MbRandom, initializing seed with a user-supplied long integer.
* \param x is a long integer with the user-supplied random number seed.
* \return Returns no value.
* \throws Does not throw an error.
*/
MbRandom::MbRandom(seedType x) {
setSeed(x, 0);
initializedFacTable = false;
availableNormalRv = false;
}
/*!
* This function generates a chi square distributed random variable.
*
* \brief Chi-square random variable.
* \param v is the degrees of freedom parameter of the chi-square distribution.
* \return Returns a chi-square distributed random variable.
* \throws Does not throw an error.
*/
double MbRandom::chiSquareRv(double v) {
/* Cast the degrees of freedom parameter as an integer. We will see
if there is a decimal remainder later. */
int n = (int)(v);
double x2;
if ( (double)(n) == v && n <= 100 )
{
/* If the degrees of freedom is an integer and less than 100, we
generate our chi-square random variable by generating v
standard normal random variables, squaring each, and taking the
sum of the squared random variables. */
x2 = 0.0;
for (int i=0; i<n; i++)
{
double x = normalRv();
x2 += x * x;
}
}
else
{
/* Otherwise, we use the relationship of the chi-square to a gamma
(it is a special case of the gamma) to generate the chi-square
random variable. */
x2 = gammaRv(v/2.0, 0.5);
}
return x2;
}
/*!
* This function calculates the probability density
* for a chi-square distributed random variable.
*
* \brief Chi-square probability density.
* \param v is the degrees of freedom parameter of the chi-square.
* \param x is the chi-square random variable.
* \return Returns the probability density.
* \throws Does not throw an error.
*/
double MbRandom::chiSquarePdf(double v, double x) {
double pdf;
if ( x < 0.0 )
{
pdf = 0.0;
}
else
{
double b = v / 2.0;
pdf = exp ( -0.5 * x ) * pow ( x, ( b - 1.0 ) ) / ( pow ( 2.0, b ) * gamma ( b ) );
}
return pdf;
}
/*!
* This function calculates the cumulative probability
* for a chi-square distributed random variable.
*
* \brief Chi-square cumulative probability.
* \param v is the degrees of freedom parameter of the chi-square.
* \param x is the chi-square random variable.
* \return Returns the cumulative probability.
* \throws Does not throw an error.
*/
double MbRandom::chiSquareCdf(double v, double x) {
return gammaCdf( v / 2.0, 0.5, x );
}
/*!
* This function calculates the natural log of the probability density
* for a chi-square distributed random variable.
*
* \brief Natural log of chi-square probability density.
* \param v is the degrees of freedom parameter of the chi-square.
* \param x is the chi-square random variable.
* \return Returns the natural log of the probability density.
* \throws Does not throw an error.
*/
double MbRandom::lnChiSquarePdf(double v, double x) {
double b = v / 2.0;
return ( -(b * log(2.0) + lnGamma(b)) - b + (b - 1.0) * log(x) );
}
/*!
* This function calculates the quantile of a chi square distribution with v
* degrees of freedom.
*
* \brief Quantile of a chi square distribution.
* \param v is the degrees of freedom of the chi square.
* \param prob is the probability up to the quantile.
* \return Returns quantile value (or -1 if in error).
* \throws Does not throw an error.
*/
double MbRandom::chiSquareQuantile(double prob, double v) {
double e = 0.5e-6, aa = 0.6931471805, p = prob,
a = 0.0, q = 0.0, p1 = 0.0, p2 = 0.0, t = 0.0,
x = 0.0, b = 0.0;
if (p < 0.000002 || p > 0.999998 || v <= 0.0)
return (-1.0);
double g = lnGamma(v/2.0);
double xx = v/2.0;
double c = xx - 1.0;
double ch = pow((p*xx*exp(g+xx*aa)), 1.0/xx);
if (v >= -1.24*log(p))
goto l1;
if (ch-e < 0)
return (ch);
goto l4;
l1:
if (v > 0.32)
goto l3;
ch = 0.4;
a = log(1.0-p);
l2:
q = ch;
p1 = 1.0+ch*(4.67+ch);
p2 = ch*(6.73+ch*(6.66+ch));
t = -0.5+(4.67+2.0*ch)/p1 - (6.73+ch*(13.32+3.0*ch))/p2;
ch -= (1.0-exp(a+g+0.5*ch+c*aa)*p2/p1)/t;
if (fabs(q/ch-1.0)-0.01 <= 0.0)
goto l4;
else
goto l2;
l3:
x = pointNormal (p);
p1 = 0.222222/v;
ch = v*pow((x*sqrt(p1)+1.0-p1), 3.0);
if (ch > 2.2*v+6.0)
ch = -2.0*(log(1.0-p)-c*log(0.5*ch)+g);
l4:
q = ch;
p1 = 0.5*ch;
if ( (t = incompleteGamma(p1, xx, g)) < 0.0 )
{
std::cerr << "Error in function \"IncompleteGamma" << std::endl;
return (-1.0);
}
p2 = p-t;
t = p2*exp(xx*aa+g+p1-c*log(ch));
b = t/ch;
a = 0.5*t-b*c;
double s1 = (210.0+a*(140.0+a*(105.0+a*(84.0+a*(70.0+60.0*a))))) / 420.0;
double s2 = (420.0+a*(735.0+a*(966.0+a*(1141.0+1278.0*a))))/2520.0;
double s3 = (210.0+a*(462.0+a*(707.0+932.0*a)))/2520.0;
double s4 = (252.0+a*(672.0+1182.0*a)+c*(294.0+a*(889.0+1740.0*a)))/5040.0;
double s5 = (84.0+264.0*a+c*(175.0+606.0*a))/2520.0;
double s6 = (120.0+c*(346.0+127.0*c))/5040.0;
ch += t*(1+0.5*t*s1-b*c*(s1-b*(s2-b*(s3-b*(s4-b*(s5-b*s6))))));
if (fabs(q/ch-1.0) > e)
goto l4;
return (ch);
}
/*!
* This function generates a gamma-distributed random variable.
*
* \brief Gamma random variable.
* \param a is the shape parameter of the gamma.
* \param b is the scale parameter of the gamma.
* \return Returns a gamma-distributed random variable.
* \throws Does not throw an error.
*/
double MbRandom::gammaRv(double a, double b) {
return (rndGamma(a) / b);
}
/*!
* This function calculates the probability density
* for a gamma-distributed random variable.
*
* \brief Gamma probability density.
* \param a is the shape parameter of the gamma.
* \param b is the scale parameter of the gamma.
* \param x is the gamma random variable.
* \return Returns the probability density.
* \throws Does not throw an error.
*/
double MbRandom::gammaPdf(double a, double b, double x) {
return (pow(b, a) / gamma(a)) * pow(x, a - 1.0) * exp(-x * b);
}
/*!
* This function calculates the cumulative probability
* for a gamma-distributed random variable.
*
* \brief Gamma cumulative probability.
* \param a is the shape parameter of the gamma.
* \param b is the scale parameter of the gamma.
* \param x is the gamma random variable.
* \return Returns the cumulative probability.
* \throws Does not throw an error.
*/
double MbRandom::gammaCdf(double a, double b, double x) {
return incompleteGamma(b*x, a, lnGamma(a));
}
/*!
* This function calculates the natural log of the probability density
* for a gamma-distributed random variable.
*
* \brief Natural log of gamma probability density.
* \param a is the shape parameter of the gamma.
* \param b is the scale parameter of the gamma.
* \param x is the gamma random variable.
* \return Returns the natural log of the probability density.
* \throws Does not throw an error.
*/
double MbRandom::lnGammaPdf(double a, double b, double x) {
return a * log(b) - lnGamma(a) + (a - 1.0) * log(x) - x * b;
}
/*!
* This function returns the quantile of a normal probability
* distribution.
*
* \brief Natural quantile.
* \param mu is the mean parameter of the normal.
* \param sigma is the variance parameter of the normal.
* \param p is the probability up to the quantile.
* \return Returns the quantile.
* \throws Does not throw an error.
*/
double MbRandom::logNormalQuantile(double mu, double sigma, double p) {
return exp( normalQuantile(mu, sigma, p) );
}
/*!
* This function calculates the cumulative probability
* for a normally-distributed random variable.
*
* \brief Normal cumulative probability.
* \param mu is the mean parameter of the normal.
* \param sigma is the variance parameter of the normal.
* \param x is the normal random variable.
* \return Returns the cumulative probability.
* \see Adams, A. G. 1969. Areas under the normal curve. Cojputer J. 12:197-198.
* \throws Does not throw an error.
*/
double MbRandom::normalCdf(double mu, double sigma, double x) {
double cdf;
double q;
double z = (x - mu) / sigma;
/* |X| <= 1.28 */
if ( fabs(z) <= 1.28 )
{
double a1 = 0.398942280444;
double a2 = 0.399903438504;
double a3 = 5.75885480458;
double a4 = 29.8213557808;
double a5 = 2.62433121679;
double a6 = 48.6959930692;
double a7 = 5.92885724438;
double y = 0.5 * z * z;
q = 0.5 - fabs(z) * ( a1 - a2 * y / ( y + a3 - a4 / ( y + a5 + a6 / ( y + a7 ) ) ) );
}
else if ( fabs(z) <= 12.7 )
{
double b0 = 0.398942280385;
double b1 = 3.8052E-08;
double b2 = 1.00000615302;
double b3 = 3.98064794E-04;
double b4 = 1.98615381364;
double b5 = 0.151679116635;
double b6 = 5.29330324926;
double b7 = 4.8385912808;
double b8 = 15.1508972451;
double b9 = 0.742380924027;
double b10 = 30.789933034;
double b11 = 3.99019417011;
double y = 0.5 * z * z;
q = exp(-y) * b0 / (fabs(z) - b1 + b2 / (fabs(z) + b3 + b4 / (fabs(z) - b5 + b6 / (fabs(z) + b7 - b8 / (fabs(z) + b9 + b10 / (fabs(z) + b11))))));
}
else
{
q = 0.0;
}
if ( z < 0.0 )
{
/* negative x */
cdf = q;
}
else
{
/* positive x */
cdf = 1.0 - q;
}
return cdf;
}
/*!
* This function returns the quantile of a normal probability
* distribution.
*
* \brief Natural quantile.
* \param mu is the mean parameter of the normal.
* \param sigma is the variance parameter of the normal.
* \param p is the probability up to the quantile.
* \return Returns the quantile.
* \throws Does not throw an error.
*/
double MbRandom::normalQuantile(double mu, double sigma, double p) {
double z = pointNormal(p);
double x = z * sigma + mu;
return x;
}
/*!
* This function generates a uniformly-distributed random variable on the interval [0,1).
* It is a version of the Marsaglia Multi-Carry.
*
* Taken from:
* Mathlib : A C Library of Special Functions
* Copyright (C) 2000, 2003 The R Development Core Team
*
* This random generator has a period of 2^60, which ensures it has the maximum
* period of 2^32 for unsigned ints (32 bit ints).
*
* \brief Uniform[0,1) random variable.
* \return Returns a uniformly-distributed random variable on the interval [0,1).
* \throws Does not throw an error.
* \see http://stat.fsu.edu/~geo/diehard.html
*/
double MbRandom::uniformRv(void) {
// Returns a pseudo-random number between 0 and 1.
I1 = 36969 * (I1 & 0177777) + (I1 >> 16);
I2 = 18000 * (I2 & 0177777) + (I2 >> 16);
return ((I1 << 16)^(I2 & 0177777)) * 2.328306437080797e-10; /*!< in [0,1) */
//return (double)rand() / (double)RAND_MAX;
}
/*!
* This function calculates the cumulative probability
* for a uniform(0,1) random variable.
*
* \brief Uniform(0,1) cumulative probability.
* \param x is the uniform random variable.
* \return Returns the cumulative probability.
* \throws Does not throw an error.
*/
double MbRandom::uniformCdf(double x) {
if ( x < 0.0 )
return 0.0;
else if ( x > 1.0 )
return 1.0;
else
return x;
}
/*!
* This function generates a Beta-distributed random variable.
*
* \brief Beta random variable.
* \param a parameter of the Beta.
* \param b parameter of the Beta.
* \return Returns the random variable.
* \throws Does not throw an error.
*/
double MbRandom::betaRv(double a, double b) {
double z0 = rndGamma( a );
double z1 = rndGamma( b );
double sum = z0 + z1;
double x = z0 / sum;
return x;
}
/*!
* This function returns the probability density for a
* Beta-distributed random variable.
*
* \brief Beta probability density.
* \param a parameter of the Beta.
* \param b parameter of the Beta.
* \return Returns the probability density.
* \throws Does not throw an error.
*/
double MbRandom::betaPdf(double a, double b, double x) {
double pdf;
if ( x < 0.0 || 1.0 < x )
pdf = 0.0;
else
pdf = pow(x, (a - 1.0)) * pow((1.0 - x), (b - 1.0)) / beta(a, b);
return pdf;
}
/*!
* This function returns the natural log of the probability density
* for a Beta-distributed random variable.
*
* \brief Beta log probability density.
* \param a parameter of the Beta.
* \param b parameter of the Beta.
* \return Returns the natural log of the probability density.
* \throws Does not throw an error.
*/
double MbRandom::lnBetaPdf(double a, double b, double x) {
return ( (lnGamma(a + b) - lnGamma(a) - lnGamma(b)) + (a - 1.0) * log(x) + (b - 1.0) * log(1.0 - x) );
}
/*!
* This function returns the cumulative probability for a
* Beta-distributed random variable.
*
* \brief Beta cumulative probability.
* \param a parameter of the Beta.
* \param b parameter of the Beta.
* \return Returns the cumulative probability.
* \throws Does not throw an error.
*/
double MbRandom::betaCdf(double a, double b, double x) {
double cdf;
if ( x <= 0.0 )
cdf = 0.0;
else if ( x <= 1.0 )
cdf = incompleteBeta(a, b, x);
else
cdf = 1.0;
return cdf;
}
/*!
* This function returns the quantile for a
* Beta-distributed random variable.
*
* \brief Beta quantile.
* \param a parameter of the Beta.
* \param b parameter of the Beta.
* \param p is the probability up to the quantile.
* \return Returns the quantile.
* \throws Does not throw an error.
*/
double MbRandom::betaQuantile(double a, double b, double p) {
# define MAXK 20
double bcoeff;
double error = 0.0001;
double errapp = 0.01;
int j;
/* estimate the solution */
double x = a / ( a + b );
double xOld = 0.0;
int loopCnt = 2;
double d[MAXK * (MAXK-1)];
while ( errapp <= fabs ( ( x - xOld ) / x ) && loopCnt != 0 )
{
xOld = x;
loopCnt--;
/* cdfX = PROB { BETA(A,B) <= X }
q = ( cdf - cdfX ) / pdfX */
double cdfX = betaCdf(a, b, x);
double pdfX = betaPdf(a, b, x);
double q = (p - cdfX) / pdfX;
/* D(N,K) = C(N,K) * Q**(N+K-1) / (N-1)! */
double t = 1.0 - x;
double s1 = q * ( b - 1.0 ) / t;
double s2 = q * ( 1.0 - a ) / x;
d[2-1+0*MAXK] = s1 + s2;
double tail = d[2-1+0*MAXK] * q / 2.0;
x = x + q + tail;
int k = 3;
while ( error < fabs ( tail / x ) && k <= MAXK )
{
/* find D(2,K-2) */
s1 = q * ((double)(k) - 2.0) * s1 / t;
s2 = q * (2.0 - (double)(k)) * s2 / x;
d[2-1+(k-2)*MAXK] = s1 + s2;
/* find D(3,K-3), D(4,K-4), D(5,K-5), ... , D(K-1,1) */
for (int i=3; i<=k-1; i++)
{
double sum2 = d[2-1+0*MAXK] * d[i-2+(k-i)*MAXK];
bcoeff = 1.0;
for ( j = 1; j <= k - i; j++ )
{
bcoeff = ( bcoeff * ( double ) ( k - i - j + 1 ) ) / ( double ) ( j );
sum2 = sum2 + bcoeff * d[2-1+j*MAXK] * d[i-2+(k-i-j)*MAXK];
}
d[i-1+(k-i)*MAXK] = sum2 + d[i-2+(k-i+1)*MAXK] / (double)(i - 1);
}
/* compute D(K,0) and use it to expand the series */
d[k-1+0*MAXK] = d[2-1+0*MAXK] * d[k-2+0*MAXK] + d[k-2+1*MAXK] / (double)(k - 1);
tail = d[k-1+0*MAXK] * q / (double)(k);
x += tail;
/* check for divergence */
if ( x <= 0.0 || 1.0 <= x )
{
std::cout << "Error in betaQuantile: The series has diverged" << std::endl;
x = -1.0;
return x;
}
k++;
}
}
return x;
# undef MAXK
}
/*!
* This function generates a Dirichlet-distributed random variable.
*
* \brief Dirichlet random variable.
* \param *a is a pointer to a vector of doubles containing the parameters of the Dirichlet.
* \param n is an integer with the number of Dirichlet prameters.
* \param *z is a pointer to a vector of doubles containing the Dirichlet random variable.
* \return Does not return a value (the random variable is initialized in the parameter z).
* \throws Does not throw an error.
*/
void MbRandom::dirichletRv(const std::vector<double> &a, std::vector<double> &z) {
int n = a.size();
double sum = 0.0;
for(int i=0; i<n; i++)
{
/* z[i] = rndGamma(a[i]) / 1.0; */
z[i] = rndGamma(a[i]);
sum += z[i];
}
for(int i=0; i<n; i++)
z[i] /= sum;
}
/*!
* This function calculates the probability density
* for a Dirichlet-distributed random variable.
*
* \brief Dirichlet probability density.
* \param *a is a pointer to a vector of doubles containing the Dirichlet parameters.
* \param *z is a pointer to a vector of doubles containing the random variables.
* \param n is the number of Dirichlet parameters/random variables.
* \return Returns the probability density.
* \throws Throws an MbException::ERROR.
*/
double MbRandom::dirichletPdf(const std::vector<double> &a, const std::vector<double> &z) {
int n = a.size();
double zSum = 0.0;
for (int i=0; i<n; i++)
zSum += z[i];
double tol = 0.0001;
if ( tol < fabs( zSum - 1.0 ) )
{
std::cout << "Fatal error in dirichletPdf" << std::endl;
exit(1);
//ui->error("Fatal error in dirichletPdf");
//throw(MbException(MbException::ERROR));
}
double aSum = 0.0;
for (int i=0; i<n; i++)
aSum += a[i];
double aProd = 1.0;
for (int i=0; i<n; i++)
aProd *= gamma(a[i]);
double pdf = gamma(aSum) / aProd;
for (int i=0; i<n; i++)
pdf = pdf * pow( z[i], a[i] - 1.0 );
return pdf;
}
/*!
* This function calculates the natural log of the probability density
* for a Dirichlet-distributed random variable.
*
* \brief Natural log of Dirichlet probability density.
* \param *a is a pointer to a vector of doubles containing the Dirichlet parameters.
* \param *z is a pointer to a vector of doubles containing the random variables.
* \param n is the number of Dirichlet parameters/random variables.
* \return Returns the natural log of the probability density.
* \throws Does not throw an error.
*/
double MbRandom::lnDirichletPdf(const std::vector<double> &a, const std::vector<double> &z) {
int n = a.size(); //!< we assume that a and z have the same size
double alpha0 = 0.0;
for (int i=0; i<n; i++)
alpha0 += a[i];
double lnP = lnGamma(alpha0);
for (int i=0; i<n; i++)
lnP -= lnGamma(a[i]);
for (int i=0; i<n; i++)
lnP += (a[i] - 1.0) * log(z[i]);
return lnP;
}
/*!
* This function generates a discrete and uniformly-distributed random
* variable on the interval (a,b).
*
* \brief Discrete uniform(a,b) random variable.
* \param a is the lower bound on the uniform.
* \param b is the upper bound on the uniform.
* \return Returns a discrete and uniformly-distributed random variable
* on the interval (a,b).
* \throws Does not throw an error.
*/
int MbRandom::discreteUniformRv(int a, int b) {
return (int)((b-a+1) * uniformRv()) + a;
}
/*!
* This function generates a Poisson-distributed random
* variable with parameter lambda.
*
* \brief Poisson(lambda) random variable.
* \param lambda the rate parameter of the Poisson.
* \return This function returns a Poisson-distributed integer.
* \throws Does not throw an error.
*/
int MbRandom::poissonRv(double lambda) {
if (lambda < 17.0)
{
if (lambda < 1.0e-6)
{
if (lambda == 0.0)
return 0;
if (lambda < 0.0)
{
/* there should be an error here */
std::cout << "Parameter negative in poisson function" << std::endl;
}
/* For extremely small lambda we calculate the probabilities of x = 1
and x = 2 (ignoring higher x). The reason for using this
method is to prevent numerical inaccuracies in other methods. */
return poissonLow(lambda);
}
else
{
/* use the inversion method */
return poissonInver(lambda);
}
}
else
{
if (lambda > 2.0e9)
{
/* there should be an error here */
std::cout << "Parameter too big in poisson function" << std::endl;
}
/* use the ratio-of-uniforms method */
return poissonRatioUniforms(lambda);
}
}
/*!
* This function calculates the cumulative probability for a
* Poisson distribution.
*
* \brief Poisson cumulative probability.
* \param lambda is the rate parameter of the Poisson.
* \param x is the value of the random variable.
* \return Returns the cumulative probability.
* \throws Does not throw an error.
*/
double MbRandom::poissonCdf(double lambda, int x) {
if ( x < 0 )
return 0.0;
double next = exp(-lambda);
double cdf = next;
for (int i=1; i<=x; i++)
{
double last = next;
next = last * lambda / (double)i;
cdf += next;
}
return cdf;
}
/*!
* This function calculates the average or median values of the
* categories for a discretized gamma distribution. The gamma is
* broken into K categories, with each category having equal
* probability. The mean or meadian value for each category is
* then calculated.
*
* \brief Calculates the mean/median values for a discretized gamma.
* \param catRate a pointer to a vector of doubles containing the nCats mean/median values.
* \param a the shape parameter of the gamma distribution.
* \param b the scale parameter of the gamma distribution.
* \param nCats the number of discrete categories.
* \param When true, the median of the category is calculated. Otherwise
the mean is used.
* \return Does not return a value.
* \throws Does not throw an error.
*/
void MbRandom::discretizeGamma(std::vector<double> &catRate, double a, double b, int nCats, bool median) {
double factor = a / b * nCats;
if (median)
{
/* the median value for each category is used to represent all of the values
in that category */
double interval = 1.0 / (2.0 * nCats);
for (int i=0; i<nCats; i++)
catRate[i] = chiSquareQuantile((i * 2.0 + 1.0) * interval, 2.0 * a) / (2.0 * b);
double t = 0.0;
for (int i=0; i<nCats; i++)
t += catRate[i];
for (int i=0; i<nCats; i++)
catRate[i] *= factor / t;
}
else
{
/* the mean value for each category is used to represent all of the values
in that category */
/* calculate the points in the gamma distribution */
for (int i=0; i<nCats-1; i++)
catRate[i] = chiSquareQuantile((i + 1.0) / nCats, 2.0 * a) / (2.0 * b);
/* calculate the cumulative values */
double lnGammaValue = lnGamma(a + 1.0);
for (int i=0; i<nCats-1; i++)
catRate[i] = incompleteGamma(catRate[i] * b, a + 1.0, lnGammaValue);
catRate[nCats-1] = 1.0;
/* calculate the relative values and rescale */
for (int i=nCats-1; i>0; i--)
{
catRate[i] -= catRate[i-1];
catRate[i] *= factor;
}
catRate[0] *= factor;
}
}
/*!
* This function calculates the beta function.
*
* \brief Beta function.
* \param a is an argument.
* \param b is an argument.
* \return Returns the value of the beta function.
* \throws Does not throw an error.
*/
double MbRandom::beta(double a, double b) {
return ( exp(lnGamma(a) + lnGamma(b) - lnGamma(a + b)) );
}
/*!
* This function calculates the gamma function for real x.
*
* \brief Gamma function.
* \param x is the argument.
* \return Returns the value of the gamma function.
* \throws Does not throw an error.
*/
double MbRandom::gamma(double x) {
double c[7] = { -1.910444077728E-03,
8.4171387781295E-04,
-5.952379913043012E-04,
7.93650793500350248E-04,
-2.777777777777681622553E-03,
8.333333333333333331554247E-02,
5.7083835261E-03 };
double fact;
int i;
int n;
double p[8] = { -1.71618513886549492533811,
2.47656508055759199108314E+01,
-3.79804256470945635097577E+02,
6.29331155312818442661052E+02,
8.66966202790413211295064E+02,
-3.14512729688483675254357E+04,
-3.61444134186911729807069E+04,
6.64561438202405440627855E+04 };
bool parity;
double q[8] = { -3.08402300119738975254353E+01,
3.15350626979604161529144E+02,
-1.01515636749021914166146E+03,
-3.10777167157231109440444E+03,
2.25381184209801510330112E+04,
4.75584627752788110767815E+03,
-1.34659959864969306392456E+05,
-1.15132259675553483497211E+05 };
double sqrtpi = 0.9189385332046727417803297;
double sum2;
double value;
double xbig = 35.040;
double xden;
double xminin = 1.18E-38;
double xnum;
double y;
double y1;
double ysq;
double z;
parity = false;
fact = 1.0;
n = 0;
y = x;
if ( y <= 0.0 )
{
/* argument negative */
y = -x;
y1 = ( double ) ( ( int ) ( y ) );
value = y - y1;
if ( value != 0.0 )
{
if ( y1 != ( double ) ( ( int ) ( y1 * 0.5 ) ) * 2.0 )
parity = true;
fact = -PI / sin(PI * value);
y = y + 1.0;
}
else
{
//value = d_huge ( );
value = HUGE_VAL;
return value;
}
}
if ( y < mbEpsilon() )
{
/* argument < EPS */
if ( xminin <= y )
{
value = 1.0 / y;
}
else
{
//value = d_huge ( );
value = HUGE_VAL;
return value;
}
}
else if ( y < 12.0 )
{
y1 = y;
/* 0.0 < argument < 1.0 */
if ( y < 1.0 )
{
z = y;
y = y + 1.0;
}
/* 1.0 < argument < 12.0, reduce argument if necessary */
else
{
n = int ( y ) - 1;
y = y - ( double ) ( n );
z = y - 1.0;
}
/* evaluate approximation for 1.0 < argument < 2.0 */
xnum = 0.0;
xden = 1.0;
for ( i = 0; i < 8; i++ )
{
xnum = ( xnum + p[i] ) * z;
xden = xden * z + q[i];
}
value = xnum / xden + 1.0;
/* adjust result for case 0.0 < argument < 1.0 */
if ( y1 < y )
{
value = value / y1;
}
/* adjust result for case 2.0 < argument < 12.0 */
else if ( y < y1 )
{
for ( i = 1; i <= n; i++ )
{
value = value * y;
y = y + 1.0;
}
}
}
else
{
/* evaluate for 12 <= argument */
if ( y <= xbig )
{
ysq = y * y;
sum2 = c[6];
for ( i = 0; i < 6; i++ )
{
sum2 = sum2 / ysq + c[i];
}
sum2 = sum2 / y - y + sqrtpi;
sum2 = sum2 + ( y - 0.5 ) * log ( y );
value = exp ( sum2 );
}
else
{
//value = d_huge ( );
value = HUGE_VAL;