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// @(#)root/mathcore:$Id$
// Authors: Andras Zsenei & Lorenzo Moneta 06/2005
/**********************************************************************
* *
* Copyright (c) 2005 , LCG ROOT MathLib Team *
* *
* *
**********************************************************************/
#include "Math/Math.h"
#include "Math/SpecFuncMathCore.h"
#include <limits>
namespace ROOT {
namespace Math {
double beta_pdf(double x, double a, double b) {
if (x < 0 || x > 1.0) return 0;
if (x == 0 ) {
// need this wor Windows
if (a < 1) return std::numeric_limits<double>::infinity();
else if (a > 1) return 0;
else if ( a == 1) return b; // to avoid a nan from log(0)*0
}
if (x == 1 ) {
// need this wor Windows
if (b < 1) return std::numeric_limits<double>::infinity();
else if (b > 1) return 0;
else if ( b == 1) return a; // to avoid a nan from log(0)*0
}
return std::exp( ROOT::Math::lgamma(a + b) - ROOT::Math::lgamma(a) - ROOT::Math::lgamma(b) +
std::log(x) * (a -1.) + ROOT::Math::log1p(-x ) * (b - 1.) );
}
double binomial_pdf(unsigned int k, double p, unsigned int n) {
if (k > n) {
return 0.0;
} else {
double coeff = ROOT::Math::lgamma(n+1) - ROOT::Math::lgamma(k+1) - ROOT::Math::lgamma(n-k+1);
return std::exp(coeff + k * std::log(p) + (n - k) * ROOT::Math::log1p(-p));
}
}
double negative_binomial_pdf(unsigned int k, double p, double n) {
// impelment in term of gamma function
if (n < 0) return 0.0;
if (p < 0 || p > 1.0) return 0.0;
double coeff = ROOT::Math::lgamma(k+n) - ROOT::Math::lgamma(k+1.0) - ROOT::Math::lgamma(n);
return std::exp(coeff + n * std::log(p) + double(k) * ROOT::Math::log1p(-p));
}
double breitwigner_pdf(double x, double gamma, double x0) {
double gammahalf = gamma/2.0;
return gammahalf/(M_PI * ((x-x0)*(x-x0) + gammahalf*gammahalf));
}
double cauchy_pdf(double x, double b, double x0) {
return b/(M_PI * ((x-x0)*(x-x0) + b*b));
}
double chisquared_pdf(double x, double r, double x0) {
if ((x-x0) < 0) {
return 0.0;
}
double a = r/2 -1.;
// let return inf for case x = x0 and treat special case of r = 2 otherwise will return nan
if (x == x0 && a == 0) return 0.5;
return std::exp ((r/2 - 1) * std::log((x-x0)/2) - (x-x0)/2 - ROOT::Math::lgamma(r/2))/2;
}
double crystalball_function(double x, double alpha, double n, double sigma, double mean) {
// evaluate the crystal ball function
if (sigma < 0.) return 0.;
double z = (x - mean)/sigma;
if (alpha < 0) z = -z;
double abs_alpha = std::abs(alpha);
// double C = n/abs_alpha * 1./(n-1.) * std::exp(-alpha*alpha/2.);
// double D = std::sqrt(M_PI/2.)*(1.+ROOT::Math::erf(abs_alpha/std::sqrt(2.)));
// double N = 1./(sigma*(C+D));
if (z > - abs_alpha)
return std::exp(- 0.5 * z * z);
else {
//double A = std::pow(n/abs_alpha,n) * std::exp(-0.5*abs_alpha*abs_alpha);
double nDivAlpha = n/abs_alpha;
double AA = std::exp(-0.5*abs_alpha*abs_alpha);
double B = nDivAlpha -abs_alpha;
double arg = nDivAlpha/(B-z);
return AA * std::pow(arg,n);
}
}
double crystalball_pdf(double x, double alpha, double n, double sigma, double mean) {
// evaluation of the PDF ( is defined only for n >1)
if (sigma < 0.) return 0.;
if ( n <= 1) return std::numeric_limits<double>::quiet_NaN(); // pdf is not normalized for n <=1
double abs_alpha = std::abs(alpha);
double C = n/abs_alpha * 1./(n-1.) * std::exp(-alpha*alpha/2.);
double D = std::sqrt(M_PI/2.)*(1.+ROOT::Math::erf(abs_alpha/std::sqrt(2.)));
double N = 1./(sigma*(C+D));
return N * crystalball_function(x,alpha,n,sigma,mean);
}
double exponential_pdf(double x, double lambda, double x0) {
if ((x-x0) < 0) {
return 0.0;
} else {
return lambda * std::exp (-lambda * (x-x0));
}
}
double fdistribution_pdf(double x, double n, double m, double x0) {
// function is defined only for both n and m > 0
if (n < 0 || m < 0)
return std::numeric_limits<double>::quiet_NaN();
if ((x-x0) < 0)
return 0.0;
return std::exp((n/2) * std::log(n) + (m/2) * std::log(m) + ROOT::Math::lgamma((n+m)/2) - ROOT::Math::lgamma(n/2) - ROOT::Math::lgamma(m/2)
+ (n/2 -1) * std::log(x-x0) - ((n+m)/2) * std::log(m + n*(x-x0)) );
}
double gamma_pdf(double x, double alpha, double theta, double x0) {
if ((x-x0) < 0) {
return 0.0;
} else if ((x-x0) == 0) {
if (alpha == 1) {
return 1.0/theta;
} else {
return 0.0;
}
} else if (alpha == 1) {
return std::exp(-(x-x0)/theta)/theta;
} else {
return std::exp((alpha - 1) * std::log((x-x0)/theta) - (x-x0)/theta - ROOT::Math::lgamma(alpha))/theta;
}
}
double gaussian_pdf(double x, double sigma, double x0) {
double tmp = (x-x0)/sigma;
return (1.0/(std::sqrt(2 * M_PI) * std::fabs(sigma))) * std::exp(-tmp*tmp/2);
}
double bigaussian_pdf(double x, double y, double sigmax , double sigmay , double rho , double x0 , double y0 ) {
double u = (x-x0)/sigmax;
double v = (y-y0)/sigmay;
double c = 1. - rho*rho;
double z = u*u - 2.*rho*u*v + v*v;
return 1./(2 * M_PI * sigmax * sigmay * std::sqrt(c) ) * std::exp(- z / (2. * c) );
}
double landau_pdf(double x, double xi, double x0) {
// LANDAU pdf : algorithm from CERNLIB G110 denlan
// same algorithm is used in GSL
static double p1[5] = {0.4259894875,-0.1249762550, 0.03984243700, -0.006298287635, 0.001511162253};
static double q1[5] = {1.0 ,-0.3388260629, 0.09594393323, -0.01608042283, 0.003778942063};
static double p2[5] = {0.1788541609, 0.1173957403, 0.01488850518, -0.001394989411, 0.0001283617211};
static double q2[5] = {1.0 , 0.7428795082, 0.3153932961, 0.06694219548, 0.008790609714};
static double p3[5] = {0.1788544503, 0.09359161662,0.006325387654, 0.00006611667319,-0.000002031049101};
static double q3[5] = {1.0 , 0.6097809921, 0.2560616665, 0.04746722384, 0.006957301675};
static double p4[5] = {0.9874054407, 118.6723273, 849.2794360, -743.7792444, 427.0262186};
static double q4[5] = {1.0 , 106.8615961, 337.6496214, 2016.712389, 1597.063511};
static double p5[5] = {1.003675074, 167.5702434, 4789.711289, 21217.86767, -22324.94910};
static double q5[5] = {1.0 , 156.9424537, 3745.310488, 9834.698876, 66924.28357};
static double p6[5] = {1.000827619, 664.9143136, 62972.92665, 475554.6998, -5743609.109};
static double q6[5] = {1.0 , 651.4101098, 56974.73333, 165917.4725, -2815759.939};
static double a1[3] = {0.04166666667,-0.01996527778, 0.02709538966};
static double a2[2] = {-1.845568670,-4.284640743};
if (xi <= 0) return 0;
double v = (x - x0)/xi;
double u, ue, us, denlan;
if (v < -5.5) {
u = std::exp(v+1.0);
if (u < 1e-10) return 0.0;
ue = std::exp(-1/u);
us = std::sqrt(u);
denlan = 0.3989422803*(ue/us)*(1+(a1[0]+(a1[1]+a1[2]*u)*u)*u);
} else if(v < -1) {
u = std::exp(-v-1);
denlan = std::exp(-u)*std::sqrt(u)*
(p1[0]+(p1[1]+(p1[2]+(p1[3]+p1[4]*v)*v)*v)*v)/
(q1[0]+(q1[1]+(q1[2]+(q1[3]+q1[4]*v)*v)*v)*v);
} else if(v < 1) {
denlan = (p2[0]+(p2[1]+(p2[2]+(p2[3]+p2[4]*v)*v)*v)*v)/
(q2[0]+(q2[1]+(q2[2]+(q2[3]+q2[4]*v)*v)*v)*v);
} else if(v < 5) {
denlan = (p3[0]+(p3[1]+(p3[2]+(p3[3]+p3[4]*v)*v)*v)*v)/
(q3[0]+(q3[1]+(q3[2]+(q3[3]+q3[4]*v)*v)*v)*v);
} else if(v < 12) {
u = 1/v;
denlan = u*u*(p4[0]+(p4[1]+(p4[2]+(p4[3]+p4[4]*u)*u)*u)*u)/
(q4[0]+(q4[1]+(q4[2]+(q4[3]+q4[4]*u)*u)*u)*u);
} else if(v < 50) {
u = 1/v;
denlan = u*u*(p5[0]+(p5[1]+(p5[2]+(p5[3]+p5[4]*u)*u)*u)*u)/
(q5[0]+(q5[1]+(q5[2]+(q5[3]+q5[4]*u)*u)*u)*u);
} else if(v < 300) {
u = 1/v;
denlan = u*u*(p6[0]+(p6[1]+(p6[2]+(p6[3]+p6[4]*u)*u)*u)*u)/
(q6[0]+(q6[1]+(q6[2]+(q6[3]+q6[4]*u)*u)*u)*u);
} else {
u = 1/(v-v*std::log(v)/(v+1));
denlan = u*u*(1+(a2[0]+a2[1]*u)*u);
}
return denlan/xi;
}
double lognormal_pdf(double x, double m, double s, double x0) {
if ((x-x0) <= 0) {
return 0.0;
} else {
double tmp = (std::log((x-x0)) - m)/s;
return 1.0 / ((x-x0) * std::fabs(s) * std::sqrt(2 * M_PI)) * std::exp(-(tmp * tmp) /2);
}
}
double normal_pdf(double x, double sigma, double x0) {
double tmp = (x-x0)/sigma;
return (1.0/(std::sqrt(2 * M_PI) * std::fabs(sigma))) * std::exp(-tmp*tmp/2);
}
double poisson_pdf(unsigned int n, double mu) {
if (n > 0)
return std::exp (n*std::log(mu) - ROOT::Math::lgamma(n+1) - mu);
else {
// when n = 0 and mu = 0, 1 is returned
if (mu >= 0) return std::exp(-mu);
// return a nan for mu < 0 since it does not make sense
return std::log(mu);
}
}
double tdistribution_pdf(double x, double r, double x0) {
return (std::exp (ROOT::Math::lgamma((r + 1.0)/2.0) - ROOT::Math::lgamma(r/2.0)) / std::sqrt (M_PI * r))
* std::pow ((1.0 + (x-x0)*(x-x0)/r), -(r + 1.0)/2.0);
}
double uniform_pdf(double x, double a, double b, double x0) {
//!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! when a=b
if ((x-x0) < b && (x-x0) >= a) {
return 1.0/(b - a);
} else {
return 0.0;
}
}
} // namespace Math
} // namespace ROOT
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