BCMath Namespace Reference

Some useful mathematic functions. More...

Functions
double	LogGaus (double x, double mean, double sigma, bool norm)
double	LogPoisson (double x, double par)
double	ApproxBinomial (int n, int k, double p)
double	LogApproxBinomial (int n, int k, double p)
double	LogBinomFactor (int n, int k)
double	ApproxLogFact (double x)
double	LogFact (int n)
void	CacheFactorial (unsigned int n)
double	LogNoverK (int n, int k)
int	Nint (double x)
double	LogBreitWignerNonRel (double x, double mean, double Gamma, bool norm)
double	LogBreitWignerRel (double x, double mean, double Gamma)
double	LogChi2 (double x, int n)
double	LogVoigtian (double x, double sigma, double gamma)
double	LogGammaPDF (double x, double alpha, double beta)
double	LogLogNormal (double x, double mean, double sigma)
double	SplitGaussian (double *x, double *par)
double	chi2 (double *x, double *par)
void	RandomChi2 (std::vector< double > &randoms, int K)
TH1D *	ECDF (const std::vector< double > &data)
std::vector< int >	longestRuns (const std::vector< bool > &bitStream)
std::vector< double >	longestRunsChi2 (const std::vector< double > &yMeasured, const std::vector< double > &yExpected, const std::vector< double > &sigma)
double	longestRunFrequency (unsigned longestObserved, unsigned int nTrials)
double	Rvalue (const std::vector< double > &chain_means, const std::vector< double > &chain_variances, const unsigned &chain_length, const bool &strict) throw (std::invalid_argument, std::domain_error)
double	longestRunFrequency (unsigned int longestObserved, unsigned int nTrials)
p value methods
double	CorrectPValue (const double &pvalue, const unsigned &npar, const unsigned &nobservations) throw (std::domain_error)
double	FastPValue (const std::vector< unsigned > &observed, const std::vector< double > &expected, unsigned nIterations, unsigned seed) throw (std::invalid_argument)

Detailed Description

Some useful mathematic functions.

Author

Daniel Kollar

Kevin Kröninger

Jing Liu

Frederik Beaujean

Version

1.0

Date

08.2008 A namespace which encapsulates some mathematical functions necessary for BAT.

Function Documentation

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

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

Definition at line 73 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 125 of file BCMath.cxx.

void BCMath::CacheFactorial

(

unsigned int

n

)

Cache factorials for first

• n integers. The cache is filled upon first call of LogFact().

Definition at line 169 of file BCMath.cxx.

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

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 665 of file BCMath.cxx.

TH1D * BCMath::ECDF

(

const std::vector< double > &

data

)

Calculate the empirical cumulative distribution function for one dimensional data vector. For consistency, the ECDF of value smaller than the minimum observed (underflow bin) is zero, and for larger than maximum (overflow bin) it is one.

Parameters

data	the observations

Returns

histogram with normalized ECDF

Definition at line 333 of file BCMath.cxx.

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

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 690 of file BCMath.cxx.

double BCMath::LogApproxBinomial	(	int	n,
		int	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.

Definition at line 80 of file BCMath.cxx.

double BCMath::LogBinomFactor	(	int	n,
		int	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

Definition at line 101 of file BCMath.cxx.

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

Calculates the logarithm of the non-relativistic Breit-Wigner distribution.

Definition at line 225 of file BCMath.cxx.

double BCMath::LogChi2	(	double	x,
		int	n
	)

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

Definition at line 244 of file BCMath.cxx.

double BCMath::LogFact

(

int

n

)

Calculates natural logarithm of the n-factorial (n!)

Definition at line 135 of file BCMath.cxx.

double BCMath::LogGammaPDF	(	double	x,
		double	alpha,
		double	beta
	)

Returns the log of the Gamma PDF.

Definition at line 273 of file BCMath.cxx.

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

Calculate the natural logarithm of a gaussian function with mean and sigma. If norm=true (default is false) the result is multiplied by the normalization constant, i.e. divided by sqrt(2*Pi)*sigma.

Definition at line 33 of file BCMath.cxx.

double BCMath::LogLogNormal	(	double	x,
		double	mean = 0,
		double	sigma = 1
	)

Return the log of the log normal distribution

Definition at line 279 of file BCMath.cxx.

double BCMath::LogNoverK	(	int	n,
		int	k
	)

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

Definition at line 176 of file BCMath.cxx.

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

Calculate the natural logarithm of a poisson distribution.

Definition at line 57 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 262 of file BCMath.cxx.

double BCMath::longestRunFrequency	(	unsigned int	longestObserved,
		unsigned int	nTrials
	)

Find the sampling probability that, given n independent Bernoulli trials with success rate = failure rate = 1/2, the longest run of consecutive successes is greater than the longest observed run. Key idea from Burr, E.J. & Cane, G. Longest Run of Consecutive Observations Having a Specified Attribute. Biometrika 48, 461-465 (1961).

Parameters

longestObserved	actual longest run
nTrials	number of independent trials

Returns

frequency

std::vector< int > BCMath::longestRuns

(

const std::vector< bool > &

bitStream

)

Find the longest runs of zeros and ones in the bit stream

Parameters

bitStream

input sequence of boolean values

Returns

runs first entry the longest zeros run, second entry the longest ones run

Definition at line 388 of file BCMath.cxx.

std::vector< double > BCMath::longestRunsChi2	(	const std::vector< double > &	yMeasured,
		const std::vector< double > &	yExpected,
		const std::vector< double > &	sigma
	)

Find the longest success/failure run in set of norm. distributed variables. Success = observation >= expectation. Runs are weighted by the total chi^2 of all elements in the run

Parameters

yMeasured	the observations
yExpected	the expected values
sigma	the theoretical uncertainties on the expectations

Returns

runs first entry the max. weight failure run, second entry the max. success run

Definition at line 441 of file BCMath.cxx.

int BCMath::Nint

(

double

x

)

Returns the nearest integer of a double number.

Definition at line 204 of file BCMath.cxx.

void BCMath::RandomChi2	(	std::vector< double > &	randoms,
		int	K
	)

Get N random numbers distributed according to chi square function with K degrees of freedom

Definition at line 318 of file BCMath.cxx.

double BCMath::Rvalue	(	const std::vector< double > &	chain_means,
		const std::vector< double > &	chain_variances,
		const unsigned &	chain_length,
		const bool &	strict = true
	)
throw	(	std::invalid_argument,
		std::domain_error
	)

Compute the R-value according to Gelman-Rubin, GR1992 : Gelman, A. and Rubin, D.B. Inference from Iterative Simulation Using Multiple Sequences Statistical Science, Vol. 7, No. 4 (Nov. 1992), pp. 457-472

Parameters

chain_means
chain_variances
chain_length
strict	If true, use the algorithm laid forth in the paper, else use a relaxed version which generally leads to smaller R-values.

Returns

R-value

Definition at line 551 of file BCMath.cxx.

double BCMath::SplitGaussian	(	double *	x,
		double *	par
	)

Gaussian with different uncertainties to either side of the mode. Interface to use with TF1.

Parameters

x	Random variable
par	Mean, sigmadown, sigmaup

Returns

density f(x)

Definition at line 296 of file BCMath.cxx.