/// Student's distribution probability function S(t,n), for n degrees of freedom.
/// Student's distribution probability is used in the test of whether two observed
/// distributions have the same mean. S(t,n) is the probability, for n degrees of
/// freedom, that a statistic t (measuring the observed difference of means)
/// would be smaller than the observed value if the means were in fact the same.
/// Two means are significantly different if, e.g. S(t,n) > 0.99;
/// in other words 1-S(t,n) is the significance level at which the hypothesis
/// that the means are equal is disproved.
/// @param double t input statistic value
/// @param int n degrees of freedom, n > 0
/// @return Student's distribution probability P(t,n)
double StudentsDistProbability(const double& t, const int& n) throw(Exception)
{
if(n <= 0) {
Exception e("Non-positive degrees of freedom in StudentsDistribution()");
GPSTK_THROW(e);
}
return (1.0 - incompleteBeta(double(n)/(t*t+double(n)),double(n)/2,0.5));
}
/// F-distribution cumulative distribution function FDistCDF(F,n1,n2) F>=0 n1,n2>0.
/// This function occurs in the statistical test of whether two observed samples
/// have the same variance. If F is the ratio of the observed dispersion (variance)
/// of the first sample to that of the second, where the first sample has n1
/// degrees of freedom and the second has n2 degrees of freedom, then this function
/// returns the probability that F would be as large as it is if the first
/// sample's distribution has smaller variance than the second's. In other words,
/// FDistCDF(f,n1,n2) is the significance level at which the hypothesis
/// "sample 1 has smaller variance than sample 2" can be rejected.
/// A small numerical value implies a significant rejection, in turn implying
/// high confidence in the hypothesis "sample 1 has variance greater than or equal
/// to that of sample 2".
/// Ref http://www.itl.nist.gov/div898/handbook/ 1.3.6.6.5
/// @param F input statistic value, the ratio variance1/variance2, F >= 0
/// @param n1 degrees of freedom of first sample, n1 > 0
/// @param n2 degrees of freedom of second sample, n2 > 0
/// @return probability that the sample is less than F.
double FDistCDF(const double& F, const int& n1, const int& n2)
throw(Exception)
{
if(F < 0) GPSTK_THROW(Exception("Negative statistic"));
if(n1 <= 0 || n2 <= 0) GPSTK_THROW(Exception("Non-positive degree of freedom"));
try {
return (1.0 - incompleteBeta(double(n2)/(double(n2)+double(n1)*F),
double(n2)/2.0,double(n1)/2.0));
}
catch(Exception& e) { GPSTK_RETHROW(e); }
}
/// Cumulative Distribution Function CDF() for Student-t-distribution CDF.
/// If X is a random variable following a normal distribution with mean zero and
/// variance unity, and chisq is a random variable following an independent
/// chi-square distribution with n degrees of freedom, then the distribution of
/// the ratio X/sqrt(chisq/n) is called Student's t-distribution with n degrees
/// of freedom. The probability that |X/sqrt(chisq/n)| will be less than a fixed
/// constant t is StudentCDF(t,n);
/// Ref http://www.itl.nist.gov/div898/handbook/ 1.3.6.6.4
/// Abramowitz and Stegun 26.7.1
/// @param t input statistic value
/// @param n degrees of freedom of first sample, n > 0
/// @return probability that the sample is less than X.
double StudentsCDF(const double& t, const int& n)
throw(Exception)
{
if(n <= 0) GPSTK_THROW(Exception("Non-positive degree of freedom"));
try {
// NB StudentsCDF(-t,n) = 1.0-StudentsCDF(t,n);
double x = 0.5*incompleteBeta(double(n)/(t*t+double(n)),double(n)/2,0.5);
if(t >= 0.0) return (1.0 - x);
return (x);
}
catch(Exception& e) { GPSTK_RETHROW(e); }
}
/// F distribution probability function F(f,n1,n2), f>=0, n1,n2>0
/// This function occurs in the statistical test of whether two observed samples
/// have the same variance. If f is the ratio of the observed dispersion of the
/// first sample to that of the second one, where the first sample has n1 degrees
/// of freedom and the second has n2 degrees of freedom, then this function
/// returns the probability that f would be as large as it is if the first
/// sample's distribution has smaller variance than the second's. In other words,
/// FDistribution(f,n1,n2) is the significance level at which the hypothesis
/// "sample 1 has smaller variance than sample 2" can be rejected.
/// A small numerical value implies a significant rejection, in turn implying
/// high confidence in the hypothesis "sample 1 has variance greater than or equal
/// to that of sample 2".
/// @param double f input statistic value, the ratio variance1/variance2, f >= 0
/// @param int n1 degrees of freedom of first sample, n1 > 0
/// @param int n2 degrees of freedom of second sample, n2 > 0
/// @return F distribution F(f,n1,n2)
double FDistProbability(const double& f, const int& n1, const int& n2)
throw(Exception)
{
if(f < 0) {
Exception e("Negative statistic in FDistribution()");
GPSTK_THROW(e);
}
if(n1 <= 0 || n2 <= 0) {
Exception e("Non-positive degrees of freedom in FDistribution()");
GPSTK_THROW(e);
}
return incompleteBeta(double(n2)/(double(n2)+double(n1)*f),
double(n2)/2.0,double(n1)/2.0);
}
/******************************
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;
}