#include <itkGaussianDistribution.h>

Inheritance diagram for itk::Statistics::GaussianDistribution:

Collaboration diagram for itk::Statistics::GaussianDistribution:

Detailed Description

GaussianDistribution class defines the interface for a univariate Gaussian distribution (pdfs, cdfs, etc.).

GaussianDistribution provides access to the probability density function (pdf), the cumulative distribution function (cdf), and the inverse cumulative distribution function for a Gaussian distribution.

The EvaluatePDF(), EvaluateCDF, EvaluateInverseCDF() methods are all virtual, allowing algorithms to be written with an abstract interface to a distribution (with said distribution provided to the algorithm at run-time). Static methods, not requiring an instance of the distribution, are also provided. The static methods allow for optimized access to distributions when the distribution is known a priori to the algorithm.

GaussianDistributions are univariate. Multivariate versions may be provided under a separate superclass (since the parameters to the pdf and cdf would have to be vectors not scalars).

GaussianDistributions can be used for Z-score statistical tests.

Note: This work is part of the National Alliance for Medical Image Computing (NAMIC), funded by the National Institutes of Health through the NIH Roadmap for Medical Research, Grant U54 EB005149. Information on the National Centers for Biomedical Computing can be obtained from http://commonfund.nih.gov/bioinformatics.

Wiki Examples:

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Create a Gaussian distribution

Definition at line 60 of file itkGaussianDistribution.h.

Public Types
typedef SmartPointer< const Self >	ConstPointer

typedef SmartPointer< Self >	Pointer

typedef GaussianDistribution	Self

typedef ProbabilityDistribution	Superclass

Public Types inherited from itk::Statistics::ProbabilityDistribution
typedef SmartPointer< const Self >	ConstPointer

typedef Array< double >	ParametersType

typedef SmartPointer< Self >	Pointer

typedef ProbabilityDistribution	Self

typedef Object	Superclass

Public Types inherited from itk::Object
typedef SmartPointer< const Self >	ConstPointer

typedef SmartPointer< Self >	Pointer

typedef Object	Self

typedef LightObject	Superclass

Public Types inherited from itk::LightObject
typedef SmartPointer< const Self >	ConstPointer

typedef SmartPointer< Self >	Pointer

typedef LightObject	Self

Public Member Functions
virtual ::itk::LightObject::Pointer	CreateAnother (void) const

virtual double	EvaluateCDF (double x) const

virtual double	EvaluateCDF (double x, const ParametersType &) const

virtual double	EvaluateCDF (double x, double mean, double variance) const

virtual double	EvaluateInverseCDF (double p) const

virtual double	EvaluateInverseCDF (double p, const ParametersType &) const

virtual double	EvaluateInverseCDF (double p, double mean, double variance) const

virtual double	EvaluatePDF (double x) const

virtual double	EvaluatePDF (double x, const ParametersType &) const

virtual double	EvaluatePDF (double x, double mean, double variance) const

virtual double	GetMean () const

virtual const char *	GetNameOfClass () const

virtual SizeValueType	GetNumberOfParameters () const

virtual double	GetVariance () const

virtual bool	HasMean () const

virtual bool	HasVariance () const

virtual void	SetMean (double)

virtual void	SetVariance (double)

Public Member Functions inherited from itk::Statistics::ProbabilityDistribution
virtual const ParametersType &	GetParameters ()

virtual void	SetParameters (const ParametersType &params)

Public Member Functions inherited from itk::Object
unsigned long	AddObserver (const EventObject &event, Command *)

unsigned long	AddObserver (const EventObject &event, Command *) const

virtual void	DebugOff () const

virtual void	DebugOn () const

Command *	GetCommand (unsigned long tag)

bool	GetDebug () const

MetaDataDictionary &	GetMetaDataDictionary (void)

const MetaDataDictionary &	GetMetaDataDictionary (void) const

virtual ModifiedTimeType	GetMTime () const

virtual const TimeStamp &	GetTimeStamp () const

bool	HasObserver (const EventObject &event) const

void	InvokeEvent (const EventObject &)

void	InvokeEvent (const EventObject &) const

virtual void	Modified () const

virtual void	Register () const

void	RemoveAllObservers ()

void	RemoveObserver (unsigned long tag)

void	SetDebug (bool debugFlag) const

void	SetMetaDataDictionary (const MetaDataDictionary &rhs)

virtual void	SetReferenceCount (int)

virtual void	UnRegister () const

Public Member Functions inherited from itk::LightObject
virtual void	Delete ()

virtual int	GetReferenceCount () const

	itkCloneMacro (Self)

void	Print (std::ostream &os, Indent indent=0) const

Static Public Member Functions
static double	CDF (double x)

static double	CDF (double x, const ParametersType &)

static double	CDF (double x, double mean, double variance)

static double	InverseCDF (double p, const ParametersType &)

static double	InverseCDF (double p, double mean, double variance)

static Pointer	New ()

static double	PDF (double x)

static double	PDF (double x, const ParametersType &)

static double	PDF (double x, double mean, double variance)


static double	InverseCDF (double p)

Protected Member Functions
	GaussianDistribution (void)

void	PrintSelf (std::ostream &os, Indent indent) const

virtual	~GaussianDistribution (void)

Protected Member Functions inherited from itk::Statistics::ProbabilityDistribution
void	PrintSelf (std::ostream &os, Indent indent) const

	ProbabilityDistribution (void)

virtual	~ProbabilityDistribution (void)

Protected Member Functions inherited from itk::Object
	Object ()

bool	PrintObservers (std::ostream &os, Indent indent) const

virtual void	SetTimeStamp (const TimeStamp &time)

virtual	~Object ()

Protected Member Functions inherited from itk::LightObject
virtual LightObject::Pointer	InternalClone () const

	LightObject ()

virtual void	PrintHeader (std::ostream &os, Indent indent) const

virtual void	PrintTrailer (std::ostream &os, Indent indent) const

virtual	~LightObject ()

Private Member Functions
	GaussianDistribution (const Self &)

void	operator= (const Self &)

Additional Inherited Members
Protected Types inherited from itk::LightObject
typedef int	InternalReferenceCountType

Protected Attributes inherited from itk::Statistics::ProbabilityDistribution
ParametersType	m_Parameters

Member Typedef Documentation

typedef SmartPointer< const Self > itk::Statistics::GaussianDistribution::ConstPointer

Definition at line 68 of file itkGaussianDistribution.h.

typedef SmartPointer< Self > itk::Statistics::GaussianDistribution::Pointer

Definition at line 67 of file itkGaussianDistribution.h.

typedef GaussianDistribution itk::Statistics::GaussianDistribution::Self

Standard class typedefs

Definition at line 65 of file itkGaussianDistribution.h.

typedef ProbabilityDistribution itk::Statistics::GaussianDistribution::Superclass

Definition at line 66 of file itkGaussianDistribution.h.

Constructor & Destructor Documentation

itk::Statistics::GaussianDistribution::GaussianDistribution ( void )

protected

virtual itk::Statistics::GaussianDistribution::~GaussianDistribution ( void )

inlineprotectedvirtual

Definition at line 235 of file itkGaussianDistribution.h.

itk::Statistics::GaussianDistribution::GaussianDistribution ( const Self & )

private

Member Function Documentation

static double itk::Statistics::GaussianDistribution::CDF ( double x )

static

Static method to evaluate the cumulative distribution function (cdf) of a standardized (mean zero, unit variance) Gaussian. The static method provides optimized access without requiring an instance of the class. Accuracy is approximately 10^-8.

static double itk::Statistics::GaussianDistribution::CDF	(	double	x,
		const ParametersType &
	)

static

Static method to evaluate the cumulative distribution function (cdf) of a Gaussian. The parameters of the distribution are passed as a parameter vector. The ordering of the parameters is (mean, variance). The static method provides optimized access without requiring an instance of the class.

static double itk::Statistics::GaussianDistribution::CDF	(	double	x,
		double	mean,
		double	variance
	)

static

Static method to evaluate the cumulative distribution function (cdf) of a Gaussian. The parameters of the distribution are passed as separate values. The static method provides optimized access without requiring an instance of the class.

virtual::itk::LightObject::Pointer itk::Statistics::GaussianDistribution::CreateAnother ( void ) const

virtual

Create an object from an instance, potentially deferring to a factory. This method allows you to create an instance of an object that is exactly the same type as the referring object. This is useful in cases where an object has been cast back to a base class.

Reimplemented from itk::Object.

virtual double itk::Statistics::GaussianDistribution::EvaluateCDF ( double x ) const

virtual

Evaluate the cumulative distribution function (cdf). The parameters of the distribution are assigned via SetParameters().

Implements itk::Statistics::ProbabilityDistribution.

virtual double itk::Statistics::GaussianDistribution::EvaluateCDF	(	double	x,
		const ParametersType &
	)		const

virtual

Evaluate the cumulative distribution function (cdf). The parameters for the distribution are passed as a parameters vector. The ordering of the parameters is (mean, variance).

Implements itk::Statistics::ProbabilityDistribution.

virtual double itk::Statistics::GaussianDistribution::EvaluateCDF	(	double	x,
		double	mean,
		double	variance
	)		const

virtual

Evaluate the cumulative distribution function (cdf). The parameters of the distribution are passed as separate parameters.

virtual double itk::Statistics::GaussianDistribution::EvaluateInverseCDF ( double p ) const

virtual

Evaluate the inverse cumulative distribution function (inverse cdf). Parameter p must be between 0.0 and 1.0. The parameters of the distribution are assigned via SetParameters().

Implements itk::Statistics::ProbabilityDistribution.

virtual double itk::Statistics::GaussianDistribution::EvaluateInverseCDF	(	double	p,
		const ParametersType &
	)		const

virtual

Evaluate the inverse cumulative distribution function (inverse cdf). Parameter p must be between 0.0 and 1.0. The parameters for the distribution are passed as a parameters vector. The ordering of the parameters is (mean, variance).

Implements itk::Statistics::ProbabilityDistribution.

virtual double itk::Statistics::GaussianDistribution::EvaluateInverseCDF	(	double	p,
		double	mean,
		double	variance
	)		const

virtual

Evaluate the inverse cumulative distribution function (inverse cdf). Parameter p must be between 0.0 and 1.0. The parameters of the distribution are passed as separate parameters.

virtual double itk::Statistics::GaussianDistribution::EvaluatePDF ( double x ) const

virtual

Evaluate the probability density function (pdf). The parameters of the distribution are assigned via SetParameters().

Implements itk::Statistics::ProbabilityDistribution.

virtual double itk::Statistics::GaussianDistribution::EvaluatePDF	(	double	x,
		const ParametersType &
	)		const

virtual

Evaluate the probability density function (pdf). The parameters for the distribution are passed as a parameters vector. The ordering of the parameters is (mean, variance).

Implements itk::Statistics::ProbabilityDistribution.

virtual double itk::Statistics::GaussianDistribution::EvaluatePDF	(	double	x,
		double	mean,
		double	variance
	)		const

virtual

Evaluate the probability density function (pdf). The parameters of the distribution are passed as separate parameters.

virtual double itk::Statistics::GaussianDistribution::GetMean ( ) const

virtual

Get the mean of the Gaussian distribution. Defaults to 0.0. The mean is stored in position 0 of the parameters vector.

Implements itk::Statistics::ProbabilityDistribution.

virtual const char* itk::Statistics::GaussianDistribution::GetNameOfClass ( ) const

virtual

Strandard macros

Reimplemented from itk::Statistics::ProbabilityDistribution.

virtual SizeValueType itk::Statistics::GaussianDistribution::GetNumberOfParameters ( ) const

inlinevirtual

Return the number of parameters. For a univariate Gaussian, this is 2 (mean, variance).

Implements itk::Statistics::ProbabilityDistribution.

Definition at line 78 of file itkGaussianDistribution.h.

virtual double itk::Statistics::GaussianDistribution::GetVariance ( ) const

virtual

Get the variance of the Gaussian distribution. Defaults to 1.0. The variance is stored in position 1 of the parameters vector.

Implements itk::Statistics::ProbabilityDistribution.

virtual bool itk::Statistics::GaussianDistribution::HasMean ( ) const

inlinevirtual

Does this distribution have a mean?

Implements itk::Statistics::ProbabilityDistribution.

Definition at line 133 of file itkGaussianDistribution.h.

virtual bool itk::Statistics::GaussianDistribution::HasVariance ( ) const

inlinevirtual

Does this distribution have a variance?

Implements itk::Statistics::ProbabilityDistribution.

Definition at line 145 of file itkGaussianDistribution.h.

static double itk::Statistics::GaussianDistribution::InverseCDF ( double p )

static

Static method to evaluate the inverse cumulative distribution function of a standardized (mean zero, unit variance) Gaussian. The static method provides optimized access without requiring an instance of the class. Parameter p must be between 0.0 and 1.0.

THis implementation was provided by Robert W. Cox from the Biophysics Research Institute at the Medical College of Wisconsin. This function is based off of a rational polynomial approximation to the inverse Gaussian CDF which can be found in M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. John Wiley & Sons. New York. Equation 26.2.23. pg. 933. 1972.

Since the initial approximation only provides an estimate within 4.5 E-4 of the true value, 3 Newton-Raphson interations are used to refine the approximation. Accuracy is approximately 10^-8.

Let, Q(x) = (1/sqrt(2*pi)) Int_{x}^{infinity} e^{-t^2/2} dt = 0.5 * erfc(x/sqrt(2))

Given p, this function computes x such that Q(x) = p, for 0 < p < 1

Note that the Gaussian CDF is defined as P(x) = (1/sqrt(2*pi)) Int_{-infinity}{x} e^{-t^2/2} dt = 1 - Q(x)

This function has been modified to compute the inverse of P(x) instead of Q(x).

static double itk::Statistics::GaussianDistribution::InverseCDF	(	double	p,
		const ParametersType &
	)

static

Static method to evaluate the inverse cumulative distribution function of a Gaussian. The parameters of the distribution are passed as a parameter vector. The ordering of the parameters is (mean, variance). The static method provides optimized access without requiring an instance of the class. Parameter p must be between 0.0 and 1.0

static double itk::Statistics::GaussianDistribution::InverseCDF	(	double	p,
		double	mean,
		double	variance
	)

static

Static method to evaluate the inverse cumulative distribution function of a Gaussian. The parameters of the distribution are passed as separate values. The static method provides optimized access without requiring an instance of the class. Parameter p must be between 0.0 and 1.0

static Pointer itk::Statistics::GaussianDistribution::New ( )

static

Method for creation through the object factory.

void itk::Statistics::GaussianDistribution::operator= ( const Self & )

private

static double itk::Statistics::GaussianDistribution::PDF ( double x )

static

Static method to evaluate the probability density function (pdf) of a standardized (mean zero, unit variance) Gaussian. The static method provides optimized access without requiring an instance of the class.

static double itk::Statistics::GaussianDistribution::PDF	(	double	x,
		const ParametersType &
	)

static

Static method to evaluate the probability density function (pdf) of a Gaussian. The parameters of the distribution are passed as a parameter vector. The ordering of the parameters is (mean, variance). The static method provides optimized access without requiring an instance of the class.

static double itk::Statistics::GaussianDistribution::PDF	(	double	x,
		double	mean,
		double	variance
	)

static

Static method to evaluate the probability density function (pdf) of a Gaussian. The parameters of the distribution are passed as separate values. The static method provides optimized access without requiring an instance of the class.

void itk::Statistics::GaussianDistribution::PrintSelf	(	std::ostream &	os,
		Indent	indent
	)		const

protectedvirtual

Methods invoked by Print() to print information about the object including superclasses. Typically not called by the user (use Print() instead) but used in the hierarchical print process to combine the output of several classes.

Reimplemented from itk::Object.

virtual void itk::Statistics::GaussianDistribution::SetMean ( double )

virtual

Set the mean of the Gaussian distribution. Defaults to 0.0. The mean is stored in position 0 of the parameters vector.

virtual void itk::Statistics::GaussianDistribution::SetVariance ( double )

virtual

Set the variance of the Gaussian distribution. Defaults to 1.0. The variance is stored in position 1 of the parameters vector.

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

itkGaussianDistribution.h

Detailed Description

Public Types

Public Member Functions

Static Public Member Functions

Protected Member Functions

Private Member Functions

Additional Inherited Members

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation