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// $Id: betaUtilities.cpp 962 2006-11-07 15:13:34Z privmane $
#include "definitions.h"
#include "betaUtilities.h"
#include "gammaUtilities.h"
#include "logFile.h"
#include "errorMsg.h"
#include <cmath>
/******************************
Computes the inverse of the beta CDF: given a prob. value, calculates the x for which
the integral over 0 to x of beta CDF = prob.
Adapted from:
1. Majumder and Bhattacharjee (1973) App. Stat. 22(3) 411-414
and the corrections:
2. Cran et al. (1977) App. Stat. 26(1) 111-114
3. Berry et al. (1990) App. Stat. 39(2) 309-310
and another adaptation made in the code of Yang (tools.c)
****************************/
MDOUBLE inverseCDFBeta(MDOUBLE a, MDOUBLE b, MDOUBLE prob){
if(a<0 || b<0 || prob<0 || prob>1) {
errorMsg::reportError("error in inverseCDFBeta,illegal parameter");
}
if (prob == 0 || prob == 1)
return prob;
int maxIter=100;
MDOUBLE epsilonLow=1e-300;
MDOUBLE fpu=3e-308;
/****** changing the tail direction (prob=1-prob)*/
bool tail=false;
MDOUBLE probA=prob;
if (prob > 0.5) {
prob = 1.0 - prob;
tail = true;
MDOUBLE tmp=a;
a=b;
b=tmp;
}
MDOUBLE lnBetaVal=betaln(a,b);
MDOUBLE x;
/****** calculating chi square evaluator */
MDOUBLE r = sqrt(-log(prob * prob));
MDOUBLE y = r - (2.30753+0.27061*r)/(1.+ (0.99229+0.04481*r) * r);
MDOUBLE chiSquare = 1.0/(9.0 * b);
chiSquare = b*2 * pow(1.0 - chiSquare + y * sqrt(chiSquare), 3.0);
// MDOUBLE chiSquare2=gammq(b,prob/2.0); //chi square valued of prob with 2q df
MDOUBLE T=(4.0*a+2.0*b-2)/chiSquare;
/****** initializing x0 */
if (a > 1.0 && b > 1.0) {
r = (y * y - 3.) / 6.;
MDOUBLE s = 1. / (a*2. - 1.);
MDOUBLE t = 1. / (b*2. - 1.);
MDOUBLE h = 2. / (s + t);
MDOUBLE w = y * sqrt(h + r) / h - (t - s) * (r + 5./6. - 2./(3.*h));
x = a / (a + b * exp(w + w));
}
else {
if (chiSquare<0){
x=exp((log(b*(1-prob))+lnBetaVal)/b);
}
else if (T<1){
x=exp((log(prob*a)+lnBetaVal)/a);
}
else {
x=(T-1.0)/(T+1.0);
}
}
if(x<=fpu || x>=1-2.22e-16) x=(prob+0.5)/2; // 0<x<1 but to avoid underflow a little smaller
/****** iterating with a modified version of newton-raphson */
MDOUBLE adj, newX=x, prev=0;
MDOUBLE yprev = 0.;
adj = 1.;
MDOUBLE eps = pow(10., -13. - 2.5/(probA * probA) - 0.5/(probA *probA));
eps = (eps>epsilonLow?eps:epsilonLow);
for (int i=0; i<maxIter; i++) {
y = incompleteBeta(a,b,x);
y = (y - prob) *
exp(lnBetaVal + (1.0-a) * log(x) + (1.0-b) * log(1.0 - x)); //the classical newton-raphson formula
if (y * yprev <= 0)
prev = (fabs(adj)>fpu?fabs(adj):fpu);
MDOUBLE g = 1;
for (int j=0; j<maxIter; j++) {
adj = g * y;
if (fabs(adj) < prev) {
newX = x - adj; // new x
if (newX >= 0. && newX <= 1.) {
if (prev <= eps || fabs(y) <= eps) return(tail?1.0-x:x);;
if (newX != 0. && newX != 1.0) break;
}
}
g /= 3.;
}
if (fabs(newX-x)<fpu)
return (tail?1.0-x:x);;
x = newX;
yprev = y;
}
return (tail?1.0-x:x);
}
/******************************
Computes the average r value in percentile k whose boundaries are leftBound and rightBound
****************************/
MDOUBLE computeAverage_r(MDOUBLE leftBound, MDOUBLE rightBound, MDOUBLE alpha, MDOUBLE beta, int k){
MDOUBLE tmp;
tmp= incompleteBeta(alpha+1,beta,rightBound) - incompleteBeta(alpha+1,beta,leftBound);
tmp= (tmp*alpha/(alpha+beta))*k;
return tmp;
}
/******************************
Computes the integral from 0 to x over the beta CDF:
(1/Beta(alpha,beta))x^(alpha-1)*(1-x)^(beta-1) where
Beta(a,b)=Gamma(a)*Gamma(b)/Gamma(a+b)
****************************/
MDOUBLE incompleteBeta(MDOUBLE alpha, MDOUBLE beta, MDOUBLE x){
MDOUBLE tmp;
if (x<0 || x>1) {
LOG(5,<<"Error in function incompleteBeta : invalid x = "<<x<<" alpha = "<<alpha<<" beta= "<<beta<<endl);
errorMsg::reportError("Error in function incompleteBeta : invalid x");
}
if (x==0 || x==1) tmp=0.0;
else tmp=exp(alpha*log(x)+beta*log(1-x)-betaln(alpha,beta));
if (x<((alpha+1)/(alpha+beta+2))) return tmp*betacf(alpha,beta,x)/alpha;
return 1-tmp*betacf(beta,alpha,1-x)/beta;
}
MDOUBLE betacf(MDOUBLE a, MDOUBLE b, MDOUBLE x){
int m, m2;
MDOUBLE aa,c,d,del,h,qab,qam,qap;
qab = a+b;
qap = a+1;
qam = a-1;
c=1;
d=1-qab*x/qap;
if (fabs(d)<FPMIN) d=FPMIN;
d=1.0/d;
h=d;
for(m=1;m<=ITMAX;m++){
m2=2*m;
aa=m*(b-m)*x/((qam+m2)*(a+m2));
d = 1.0+aa*d;
if (fabs(d)<FPMIN) d = FPMIN;
c=1.0 + aa/c;
if (fabs(c)<FPMIN) c = FPMIN;
d = 1.0/d;
h *= d*c;
aa = -(a+m)*(qab+m)*x/((a+m2)*(qap+m2));
d = 1.0+aa*d;
if (fabs(d)<FPMIN) d = FPMIN;
c = 1.0 + aa/c;
if (fabs(c)<FPMIN) c = FPMIN;
d = 1.0/d;
del = d*c;
h*=del;
if (fabs(del-1.0) <= EPS) break;
}
if (m > ITMAX) LOG(5,<<"Error in function betacf : alpha || beta big ||MAXIT small"<<endl);
return h;
}
MDOUBLE betaln(MDOUBLE alpha, MDOUBLE beta){
return gammln(alpha)+gammln(beta)-gammln(alpha+beta);
}
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