Some useful mathematic functions. More...

Functions
double	ApproxBinomial (unsigned n, unsigned k, double p)
	Calculates Binomial probability using approximations for factorial calculations if calculation for number greater than 20 required using the BCMath::ApproxLogFact function. More...

double	ApproxLogFact (double x)
	Calculates natural logarithm of the n-factorial (n!) using Srinivasa Ramanujan approximation log(n!) = nlog(n) - n + log(n(1. More...

unsigned	CacheFactorials (unsigned n)
	Cache factorials for first. More...

double	LogBinomFactor (unsigned n, unsigned k)
	Calculates natural logarithm of the Binomial factor "n over k" using approximations for factorial calculations if calculation for number greater than 20 required using the BCMath::ApproxLogFact function. More...

double	LogBinomFactorExact (unsigned n, unsigned k)
	Calculates natural logarithm of the Binomial factor "n over k". More...

double	LogFact (unsigned n)
	Calculates natural logarithm of the n-factorial (n!)

int	Nint (double x)
	Returns the nearest integer of a double number. More...

Functions for log likelihoods
double	LogGaus (double x, double mean=0, double sigma=1, bool norm=false)
	Calculate the natural logarithm of a normal distribution function. More...

double	LogSplitGaus (double x, double mode, double sigma_below, double sigma_above, bool norm=false)
	Calculate the natural logarithm of a normal distribution function with different variances below and above the mode. More...

double	LogPoisson (double x, double lambda)
	Calculate the natural logarithm of a poisson distribution. More...

double	LogApproxBinomial (unsigned n, unsigned k, double p)
	Calculates natural logarithm of the Binomial probability using approximations for factorial calculations if calculation for number greater than 20 required using the BCMath::ApproxLogFact function. More...

double	LogBreitWignerNonRel (double x, double mean, double Gamma, bool norm=false)
	Calculates the logarithm of the nonrelativistic Breit-Wigner distribution. More...

double	LogBreitWignerRel (double x, double mean, double Gamma)
	Calculates the logarithm of the relativistic Breit-Wigner distribution. More...

double	LogChi2 (double x, int n)
	Calculates the logarithm of chi square function: chi2(double x; size_t n)

double	LogVoigtian (double x, double sigma, double gamma)
	Calculates the logarithm of normalized voigtian function: voigtian(double x, double sigma, double gamma) More...

double	LogGammaPDF (double x, double alpha, double beta)
	Returns the log of the Gamma PDF.

double	LogLogNormal (double x, double mean=0, double sigma=1)
	Return the log of the log normal distribution.

p value methods
double	CorrectPValue (const double &pvalue, const unsigned &npar, const unsigned &nobservations)
	Correct a p value by transforming to a chi^2 with dof=nobservations, then back to a pvalue with dof reduced by number of fit parameters. More...

double	FastPValue (const std::vector< unsigned > &observed, const std::vector< double > &expected, unsigned nIterations=1e5, unsigned seed=0)
	Calculate the p value using fast MCMC for a histogram and the likelihood as test statistic. More...

Detailed Description

Some useful mathematic functions.

Author: Frederik Beaujean; Daniel Greenwald; Daniel Kollar; Kevin Kröninger; Jing Liu

Version: 1.0

Date: 08.2008

A namespace which encapsulates some mathematical functions necessary for BAT.

Function Documentation

double BCMath::ApproxBinomial	(	unsigned	n,
		unsigned	k,
		double	p
	)

Calculates Binomial probability using approximations for factorial calculations if calculation for number greater than 20 required using the BCMath::ApproxLogFact function.

Parameters

n	number of trials.
k	number of successes.
p	probability of success in a trial.

Definition at line 97 of file BCMath.cxx.

double BCMath::ApproxLogFact ( double x )

Calculates natural logarithm of the n-factorial (n!) using Srinivasa Ramanujan approximation log(n!) = n*log(n) - n + log(n*(1.

+4.*n*(1.+2.*n)))/6. + log(PI)/2. if n > 20. If n <= 20 it uses BCMath::LogFact to calculate it exactly.

Definition at line 130 of file BCMath.cxx.

unsigned BCMath::CacheFactorials ( unsigned n )

Cache factorials for first.

n integers.

Definition at line 156 of file BCMath.cxx.

double BCMath::CorrectPValue	(	const double &	pvalue,
		const unsigned &	npar,
		const unsigned &	nobservations
	)

Correct a p value by transforming to a chi^2 with dof=nobservations, then back to a pvalue with dof reduced by number of fit parameters.

Parameters

pvalue	The p value to correct.
npar	The number of fit parameters.
nobservations	The number of data points.

Returns: corrected p value

Definition at line 285 of file BCMath.cxx.

double BCMath::FastPValue	(	const std::vector< unsigned > &	observed,
		const std::vector< double > &	expected,
		unsigned	nIterations = `1e5`,
		unsigned	seed = `0`
	)

Calculate the p value using fast MCMC for a histogram and the likelihood as test statistic.

The method is explained in the appendix of http://arxiv.org/abs/1011.1674

Parameters

observed	The counts of observed events
expected	The expected number of events (Poisson means)
nIterations	Controls number of pseudo data sets
seed	Set to nonzero value for reproducible results.

Returns: The p value

Definition at line 310 of file BCMath.cxx.

double BCMath::LogApproxBinomial	(	unsigned	n,
		unsigned	k,
		double	p
	)

Calculates natural logarithm of the Binomial probability using approximations for factorial calculations if calculation for number greater than 20 required using the BCMath::ApproxLogFact function.

Parameters

n	number of trials.
k	number of successes.
p	probability of success in a trial.

Definition at line 79 of file BCMath.cxx.

double BCMath::LogBinomFactor	(	unsigned	n,
		unsigned	k
	)

Calculates natural logarithm of the Binomial factor "n over k" using approximations for factorial calculations if calculation for number greater than 20 required using the BCMath::ApproxLogFact function.

Even for large numbers the calculation is performed precisely, if n-k < 5

Parameters

n	upper value in binomial coefficient
k	lower value in binomial coefficient

Definition at line 111 of file BCMath.cxx.

double BCMath::LogBinomFactorExact	(	unsigned	n,
		unsigned	k
	)

Calculates natural logarithm of the Binomial factor "n over k".

Definition at line 191 of file BCMath.cxx.

double BCMath::LogBreitWignerNonRel	(	double	x,
		double	mean,
		double	Gamma,
		bool	norm = `false`
	)

Calculates the logarithm of the nonrelativistic Breit-Wigner distribution.

Definition at line 214 of file BCMath.cxx.

double BCMath::LogBreitWignerRel	(	double	x,
		double	mean,
		double	Gamma
	)

Calculates the logarithm of the relativistic Breit-Wigner distribution.

Definition at line 226 of file BCMath.cxx.

double BCMath::LogGaus	(	double	x,
		double	mean = `0`,
		double	sigma = `1`,
		bool	norm = `false`
	)

Calculate the natural logarithm of a normal distribution function.

Parameters

x	point to be evaluated.
mean	Mean.
sigma	Standard deviation.
norm	flag for including normalization constant 1/sqrt(2*pi)/sigma

Definition at line 24 of file BCMath.cxx.

double BCMath::LogPoisson	(	double	x,
		double	lambda
	)

Calculate the natural logarithm of a poisson distribution.

Parameters

x	number of occurances.
lambda	expected number of occurances.

Definition at line 53 of file BCMath.cxx.

double BCMath::LogSplitGaus	(	double	x,
		double	mode,
		double	sigma_below,
		double	sigma_above,
		bool	norm = `false`
	)

Calculate the natural logarithm of a normal distribution function with different variances below and above the mode.

Parameters

x	point to be evaluated.
mode	Mode of the function.
sigma_below	Standard deviation below mode.
sigma_above	Standard deviation above mode.
norm	flag for including normalization constant sqrt(2/pi)/(sigma_below+sigma_above)

Definition at line 44 of file BCMath.cxx.

double BCMath::LogVoigtian	(	double	x,
		double	sigma,
		double	gamma
	)

Calculates the logarithm of normalized voigtian function: voigtian(double x, double sigma, double gamma)

voigtian is a convolution of the following two functions: gaussian(x) = 1/(sqrt(2*pi)*sigma) * exp(x*x/(2*sigma*sigma) and lorentz(x) = (1/pi)*(gamma/2) / (x*x + (gamma/2)*(gamma/2))

it is singly peaked at x=0. The width of the peak is decided by sigma and gamma, so they should be positive.

Definition at line 251 of file BCMath.cxx.

int BCMath::Nint ( double x )

Returns the nearest integer of a double number.

Definition at line 185 of file BCMath.cxx.

Functions

Detailed Description

Function Documentation