#include <itkGaussianDistribution.h>

Inheritance diagram for itk::Statistics::GaussianDistribution:

Collaboration diagram for itk::Statistics::GaussianDistribution:

Public Types
using	ConstPointer = SmartPointer< const Self >

using	Pointer = SmartPointer< Self >

using	Self = GaussianDistribution

using	Superclass = ProbabilityDistribution

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

using	ParametersType = Array< double >

using	Pointer = SmartPointer< Self >

using	Self = ProbabilityDistribution

using	Superclass = Object

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

using	Pointer = SmartPointer< Self >

using	Self = Object

using	Superclass = LightObject

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

using	Pointer = SmartPointer< Self >

using	Self = LightObject

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

double	EvaluateCDF (double x) const override

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

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

double	EvaluateInverseCDF (double p) const override

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

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

double	EvaluatePDF (double x) const override

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

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

double	GetMean () const override

virtual const char *	GetNameOfClass () const

SizeValueType	GetNumberOfParameters () const override

double	GetVariance () const override

bool	HasMean () const override

bool	HasVariance () const override

virtual void	SetMean (double)

virtual void	SetVariance (double)

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

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

unsigned long	AddObserver (const EventObject &event, std::function< void(const EventObject &)> function) const

virtual void	DebugOff () const

virtual void	DebugOn () const

Command *	GetCommand (unsigned long tag)

bool	GetDebug () const

MetaDataDictionary &	GetMetaDataDictionary ()

const MetaDataDictionary &	GetMetaDataDictionary () 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

void	Register () const override

void	RemoveAllObservers ()

void	RemoveObserver (unsigned long tag)

void	SetDebug (bool debugFlag) const

void	SetReferenceCount (int) override

void	UnRegister () const noexcept override

void	SetMetaDataDictionary (const MetaDataDictionary &rhs)

void	SetMetaDataDictionary (MetaDataDictionary &&rrhs)

virtual void	SetObjectName (std::string _arg)

virtual const std::string &	GetObjectName () const

Public Member Functions inherited from itk::LightObject
Pointer	Clone () const

virtual void	Delete ()

virtual int	GetReferenceCount () const

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 Pointer	New ()

static double	PDF (double x)

static double	PDF (double x, const ParametersType &)

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

Static Public Member Functions inherited from itk::Object
static bool	GetGlobalWarningDisplay ()

static void	GlobalWarningDisplayOff ()

static void	GlobalWarningDisplayOn ()

static Pointer	New ()

static void	SetGlobalWarningDisplay (bool val)

Static Public Member Functions inherited from itk::LightObject
static void	BreakOnError ()

static Pointer	New ()

static double	InverseCDF (double p)

static double	InverseCDF (double p, const ParametersType &)

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

	GaussianDistribution ()

	~GaussianDistribution () override=default

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

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

	ProbabilityDistribution ()

	~ProbabilityDistribution () override

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

	~Object () override

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

virtual void	SetTimeStamp (const TimeStamp &timeStamp)

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 ()

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

Protected Attributes inherited from itk::LightObject
std::atomic< int >	m_ReferenceCount

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.

ITK Sphinx Examples:

All ITK Sphinx Examples

Create Gaussian Distribution

Definition at line 61 of file itkGaussianDistribution.h.

Member Typedef Documentation

◆ ConstPointer

using itk::Statistics::GaussianDistribution::ConstPointer = SmartPointer<const Self>

Definition at line 70 of file itkGaussianDistribution.h.

◆ Pointer

using itk::Statistics::GaussianDistribution::Pointer = SmartPointer<Self>

Definition at line 69 of file itkGaussianDistribution.h.

◆ Self

using itk::Statistics::GaussianDistribution::Self = GaussianDistribution

Standard class type aliases

Definition at line 67 of file itkGaussianDistribution.h.

◆ Superclass

using itk::Statistics::GaussianDistribution::Superclass = ProbabilityDistribution

Definition at line 68 of file itkGaussianDistribution.h.

Constructor & Destructor Documentation

◆ GaussianDistribution()

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

protected

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 iterations 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).

◆ ~GaussianDistribution()

itk::Statistics::GaussianDistribution::~GaussianDistribution ( )

overrideprotecteddefault

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 iterations 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).

Member Function Documentation

◆ CDF() [1/3]

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.

◆ CDF() [2/3]

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.

◆ CDF() [3/3]

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.

◆ CreateAnother()

virtual::itk::LightObject::Pointer itk::Statistics::GaussianDistribution::CreateAnother ( ) 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.

◆ EvaluateCDF() [1/3]

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

overridevirtual

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

Implements itk::Statistics::ProbabilityDistribution.

◆ EvaluateCDF() [2/3]

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

overridevirtual

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.

◆ EvaluateCDF() [3/3]

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.

◆ EvaluateInverseCDF() [1/3]

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

overridevirtual

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.

◆ EvaluateInverseCDF() [2/3]

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

overridevirtual

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.

◆ EvaluateInverseCDF() [3/3]

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.

◆ EvaluatePDF() [1/3]

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

overridevirtual

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

Implements itk::Statistics::ProbabilityDistribution.

◆ EvaluatePDF() [2/3]

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

overridevirtual

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.

◆ EvaluatePDF() [3/3]

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.

◆ GetMean()

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

overridevirtual

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.

◆ GetNameOfClass()

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

virtual

Strandard macros

Reimplemented from itk::Statistics::ProbabilityDistribution.

◆ GetNumberOfParameters()

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

inlineoverridevirtual

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

Implements itk::Statistics::ProbabilityDistribution.

Definition at line 81 of file itkGaussianDistribution.h.

◆ GetVariance()

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

overridevirtual

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.

◆ HasMean()

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

inlineoverridevirtual

Does this distribution have a mean?

Implements itk::Statistics::ProbabilityDistribution.

Definition at line 149 of file itkGaussianDistribution.h.

◆ HasVariance()

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

inlineoverridevirtual

Does this distribution have a variance?

Implements itk::Statistics::ProbabilityDistribution.

Definition at line 167 of file itkGaussianDistribution.h.

◆ InverseCDF() [1/3]

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 iterations 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).

◆ InverseCDF() [2/3]

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

◆ InverseCDF() [3/3]

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

◆ New()

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

static

Method for creation through the object factory.

Examples: SphinxExamples/src/Numerics/Statistics/CreateGaussianDistribution/Code.cxx.

◆ PDF() [1/3]

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.

◆ PDF() [2/3]

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.

◆ PDF() [3/3]

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.

◆ PrintSelf()

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

overrideprotectedvirtual

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 iterations 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).

Reimplemented from itk::Object.

◆ SetMean()

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.

◆ SetVariance()

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

Public Types

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Detailed Description

Member Typedef Documentation

◆ ConstPointer

◆ Pointer

◆ Self

◆ Superclass

Constructor & Destructor Documentation

◆ GaussianDistribution()

◆ ~GaussianDistribution()

Member Function Documentation

◆ CDF() [1/3]

◆ CDF() [2/3]

◆ CDF() [3/3]

◆ CreateAnother()

◆ EvaluateCDF() [1/3]

◆ EvaluateCDF() [2/3]

◆ EvaluateCDF() [3/3]

◆ EvaluateInverseCDF() [1/3]

◆ EvaluateInverseCDF() [2/3]

◆ EvaluateInverseCDF() [3/3]

◆ EvaluatePDF() [1/3]

◆ EvaluatePDF() [2/3]

◆ EvaluatePDF() [3/3]

◆ GetMean()

◆ GetNameOfClass()

◆ GetNumberOfParameters()

◆ GetVariance()

◆ HasMean()

◆ HasVariance()

◆ InverseCDF() [1/3]

◆ InverseCDF() [2/3]

◆ InverseCDF() [3/3]

◆ New()

◆ PDF() [1/3]

◆ PDF() [2/3]

◆ PDF() [3/3]

◆ PrintSelf()

◆ SetMean()

◆ SetVariance()