Generate a random variate from a noncentral chi-square $\chi^2$ distribution. More...

#include <ChiSqr.h>

Inheritance diagram for honey::ChiSqr_< Real >:

Collaboration diagram for honey::ChiSqr_< Real >:

Public Types
typedef Matrix< matrix::dynamic, matrix::dynamic, Real >	MatrixN

typedef Vec< matrix::dynamic, Real >	VecN

typedef Vec< 2, Real >	Vec2

Public Member Functions
	ChiSqr_ (optional< RandomGen & > gen, Real nu, Real lambda=0)

	ChiSqr_ (Real nu, Real lambda=0)

virtual Real	next () const
	Get next randomly distributed variate. Requires a random generator (see ctor or setGen()) More...

virtual Real	pdf (Real x) const
	Probability Density Function. More...

virtual Real	cdf (Real x) const
	Cumulative Distribution Function. More...

virtual Real	cdfInv (Real P) const
	Inverse of the CDF. More...

virtual Real	mean () const
	Calc mean. More...

virtual Real	variance () const
	Calc variance. More...

Vec2	stdDevCi (Real stdDev, Real alpha) const
	Calculate the 100*(1-alpha)% confidence interval of the standard deviation. More...

Public Member Functions inherited from honey::RandomDist< Real >
	RandomDist (optional< RandomGen & > gen=optnull)
	Construct with a random generator to use for next() More...

virtual	~RandomDist ()

virtual Real	cdfComp (Real x) const
	Complement of the CDF. More...

Real	stdDev () const
	Calc standard deviation. More...

void	setGen (RandomGen &gen)
	Set random generator to use for next() More...

RandomGen &	getGen () const
	Get random generator. More...

Static Public Member Functions
static Real	test (const VecN &observed, const VecN &expected)
	Calculate the p-value for a list of observed and expected frequencies. More...

static Real	test (const MatrixN &mat)
	Calculate the p-value for a 'contingency table', a matrix of frequencies for different categories. More...

Public Attributes
Real	nu

Real	lambda

Friends
template<class >
class	ChiSqr_

Additional Inherited Members
Protected Types inherited from honey::RandomDist< Real >
typedef Numeral< Real >::Real_	Real_

typedef Real_::DoubleType	Double_

typedef Double_::Real	Double

typedef Alge_< Real >	Alge

typedef Alge_< Double >	Alge_d

typedef Uniform_< Double >	Uniform

Protected Member Functions inherited from honey::RandomDist< Real >
Real	cdfInvFind (Real P, Real min, Real max, bool discrete=false) const
	Generic binary search algorithm to find Cdf. More...

Detailed Description

template<class Real>
class honey::ChiSqr_< Real >

Generate a random variate from a noncentral chi-square $\chi^2$ distribution.

The chi-square distribution can be used to test how likely it is that an observed distribution is due to chance. It can test the goodness of fit of observations to a model by testing a 'null hypothesis' that the observations are tied to some expected values. It can also be used to test if categorical tallies (eg. male or female for each race) within a table are related or independent.

Example

Use Pearson's chi-squared test to determine if a 6-sided die has been tampered with.
Take samples by rolling the die and observing the frequency of the 6 independent categorical variables (ie. number of times 1,2...6 appears face up)
Here the null hypothesis is that all the sides will appear face up the same number of times.

$\displaystyle \chi^2 = \sum_{i=1}^6 \frac{observedFreq_i - expectedFreq_i^2}{expectedFreq_i}$

This result, X^2, asymptotically approaches a chi-square distribution.
Let's say that the result is 12.
Next, use the Cumulative Distribution Function (CDF). The CDF is a curve that increases from 0 to 1, it tells us the probability that a sample will be <= X^2.
The number of degrees of freedom are the number of categorical variables (6) minus one.

$p = 1 - cdf(\chi^2, 6-1)$

p will be about 0.03, or 3%. This tells us that the probability of the die samples being tied to their expected values (ie. pure chance) is 3%. Less than 5% strongly suggests that the categorical variables are correlated, so the die sides are not independent, they have been tampered with.

Probability density function when $\lambda = 0$ , central chi-square distribution, same as $\mathrm{Gamma}(\nu/2, 2)$ :

$\displaystyle p(x,\nu,\lambda) = \frac{1}{2^{\nu/2} \Gamma(\nu/2)} x^{\nu/2 - 1} e^{-x/2}$ where $\Gamma$ is the gamma function.

Probability density function when $\lambda \not= 0$ , poisson weighted mix of central chi-square distributions:

$\displaystyle p(x,\nu,\lambda) = \sum_{i=0}^\infty \frac{e^{-\lambda/2} (\lambda/2)^i}{i!} \chi^2(x, \nu+2i)$

Parameters

nu	$\nu$ Number of degrees of freedom. Range > 0
lambda	$\lambda$ Non-centrality parameter. Range >= 0

Return values

x	Random variate. Range [0,inf]

Member Typedef Documentation

template<class Real >

typedef Matrix<matrix::dynamic, matrix::dynamic, Real> honey::ChiSqr_< Real >::MatrixN

template<class Real >

typedef Vec<2,Real> honey::ChiSqr_< Real >::Vec2

template<class Real >

typedef Vec<matrix::dynamic, Real> honey::ChiSqr_< Real >::VecN

Constructor & Destructor Documentation

template<class Real >

honey::ChiSqr_< Real >::ChiSqr_	(	optional< RandomGen & >	gen,
		Real	nu,
		Real	lambda = `0`
	)

inline

template<class Real >

honey::ChiSqr_< Real >::ChiSqr_	(	Real	nu,
		Real	lambda = `0`
	)

inline

Member Function Documentation

template<class Real >

Real honey::ChiSqr_< Real >::cdf ( Real x ) const

virtual

Cumulative Distribution Function.

The integral of the PDF from -inf to x.

Parameters

x	value, range depends on distribution.

Return values

P	a probability in range [0,1] that a random variate will be <= x.

Reimplemented from honey::RandomDist< Real >.

template<class Real >

Real honey::ChiSqr_< Real >::cdfInv ( Real P ) const

virtual

Inverse of the CDF.

Parameters

P	cumulative probability

Return values

x	A value that has probability P of being >= a random variate X. Also, x satisfies: cdf(x) = P.

Reimplemented from honey::RandomDist< Real >.

template<class Real >

virtual Real honey::ChiSqr_< Real >::mean ( ) const

inlinevirtual

Calc mean.

Reimplemented from honey::RandomDist< Real >.

template<class Real >

Real honey::ChiSqr_< Real >::next ( ) const

virtual

Get next randomly distributed variate. Requires a random generator (see ctor or setGen())

Reimplemented from honey::RandomDist< Real >.

template<class Real >

Real honey::ChiSqr_< Real >::pdf ( Real x ) const

virtual

Probability Density Function.

The PDF integrates to 1 over the entire range of possible values of x.

Parameters

x	value, range depends on distribution.

Return values

p	a relative likelihood in range [0,inf] that a random variate X will equal x.

Reimplemented from honey::RandomDist< Real >.

template<class Real >

Vec2 honey::ChiSqr_< Real >::stdDevCi	(	Real	stdDev,
		Real	alpha
	)		const

inline

Calculate the 100*(1-alpha)% confidence interval of the standard deviation.

template<class Real >

Real honey::ChiSqr_< Real >::test	(	const VecN &	observed,
		const VecN &	expected
	)

static

Calculate the p-value for a list of observed and expected frequencies.

template<class Real >

Real honey::ChiSqr_< Real >::test ( const MatrixN & mat )

static

Calculate the p-value for a 'contingency table', a matrix of frequencies for different categories.

A p-value below 5% is significant, meaning that deviations between the categories are probably not due to random chance. All frequencies in the matrix should be at least 5 for reliable results. The degrees of freedom in the matrix is assumed to be (rows-1)*(cols-1), that is, it is not possible to populate any part of the matrix knowing other parts.

template<class Real >

virtual Real honey::ChiSqr_< Real >::variance ( ) const

inlinevirtual

Calc variance.

Reimplemented from honey::RandomDist< Real >.

Friends And Related Function Documentation

template<class Real >

template<class >

friend class ChiSqr_

friend

Member Data Documentation

template<class Real >

Real honey::ChiSqr_< Real >::lambda

template<class Real >

Real honey::ChiSqr_< Real >::nu

The documentation for this class was generated from the following files:

src/common/Honey/Math/Random/Dist/ChiSqr.h
src/common/Honey/Math/Random/Dist/ChiSqr.cpp

Public Types

Public Member Functions

Static Public Member Functions

Public Attributes

Friends

Additional Inherited Members

Detailed Description

template<class Real> class honey::ChiSqr_< Real >

Example

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

Friends And Related Function Documentation

Member Data Documentation

template<class Real>
class honey::ChiSqr_< Real >