Multivariate Student's t Distribution More...

#include <tDist.h>

Inheritance diagram for Multivariate::tDistribution:

Public Member Functions
	tDistribution (unsigned int Dimension, unsigned int DegFreedom, const Eigen::VectorXd &locVect, const Eigen::MatrixXd &ScaleMatr)
	Construct a multivariate student's t with the given parameters. More...

	tDistribution (unsigned int Dimension=1U, unsigned int DegFreedom=1U)
	Construct a multivariate standardized student's t. More...

Eigen::MatrixXd	ExtractSamples (unsigned int NumSamples) const
	Generates multiple simulations from the distribution. More...

Eigen::MatrixXd	ExtractSamplesCDF (unsigned int NumSamples) const
	Extracts samples from the distribution and returns their marginal CDF. More...

double	GetCumulativeDesity (const Eigen::VectorXd &Coordinates) const
	Computes the cumulative density function of the distribution in correspondence of the supplied coordinates. More...

unsigned int	GetDegreesOfFreedom () const
	Get the degrees of freedom of the distribution. More...

double	GetDensity (const Eigen::VectorXd &Coordinates) const
	Computes the probability density function of the distribution in correspondence of the supplied coordinates. More...

const Eigen::VectorXd &	GetLocationVector () const
	Get the location vector of the distribution. More...

Eigen::VectorXd	GetQuantile (double Prob) const
	Computes the inverse cumulative density function of the distribution in correspondence of the supplied probability. More...

const Eigen::MatrixXd &	GetScaleMatrix () const
	Get the scale matrix of the distribution. More...

Eigen::MatrixXd	GetVarMatrix () const
	Computes the variance matrix of the distribution. More...

bool	SetDegreesOfFreedom (unsigned int a)
	Set the degrees of freedom of the distribution. More...

bool	SetDimension (unsigned int Dimension)
	Set the dimensionality of the distribution. More...

bool	SetLocationVector (const Eigen::VectorXd &mVect)
	Set the expected values vector. More...

bool	SetLocationVector (const std::vector< double > &mVect)
	Set the location vector. More...

bool	SetScaleMatrix (const Eigen::MatrixXd &SclMatr)
	Set the scale matrix of the distribution. More...

bool	SetScaleMatrix (const std::vector< double > &mVect, bool RowWise=true)
	Set the scale matrix of the distribution. More...

Public Member Functions inherited from Multivariate::AbstarctDistribution
virtual Eigen::RowVectorXd	ExtractSample () const
	Generates a single simulation from the distribution. More...

virtual Eigen::RowVectorXd	ExtractSampleCDF () const
	Generates a single simulation from the distribution and returns its marginal CDF. More...

virtual std::vector< double >	ExtractSampleCDFVect () const
	Generates a single simulation from the distribution and returns its marginal CDF. More...

virtual std::map< unsigned int, std::vector< double > >	ExtractSamplesCDFMap (unsigned int NumSamples) const
	Extracts samples from the distribution and returns their marginal CDF. More...

virtual std::map< unsigned int, std::vector< double > >	ExtractSamplesMap (unsigned int NumSamples) const
	Generates multiple simulations from the distribution. More...

virtual std::vector< double >	ExtractSampleVector () const
	Generates a single simulation from the distribution. More...

virtual double	GetCumulativeDesity (const std::vector< double > &Coordinates) const
	Computes the cumulative density function of the distribution in correspondence of the supplied coordinates. More...

unsigned int	GetCurrentSeed () const
	Get the random number generator seed. More...

virtual double	GetDensity (const std::vector< double > &Coordinates) const
	Computes the probability density function of the distribution in correspondence of the supplied coordinates. More...

unsigned int	GetDimension () const
	Get the dimensionality of the distribution. More...

virtual std::vector< double >	GetQuantileVector (double Prob) const
	Computes the inverse cumulative density function of the distribution in correspondence of the supplied probability. More...

bool	IsValid () const
	Check if the distribution is valid. More...

void	SetRandomSeed (unsigned int NewSeed)
	Set the random number generator seed. More...

virtual double *	GetQuantileArray (double Prob)
	Computes the inverse cumulative density function of the distribution in correspondence of the supplied probability. More...

virtual double	GetCumulativeDesity (double *Coordinates) const
	Computes the cumulative density function of the distribution in correspondence of the supplied coordinates. More...

virtual double	GetDensity (double *Coordinates) const
	Computes the probability density function of the distribution in correspondence of the supplied coordinates. More...

virtual double *	ExtractSampleArray () const
	Generates a single simulation from the distribution. More...

virtual double **	ExtractSamplesMatix (unsigned int NumSamples) const
	Generates a single simulation from the distribution. More...

virtual double *	ExtractSampleCDFArray () const
	Generates a single simulation from the distribution and returns its marginal CDF. More...

virtual double **	ExtractSamplesCDFMatix (unsigned int NumSamples) const
	Generates a single simulation from the distribution and returns its marginal CDF. More...

Friends
template<class F , class T >
void	boost::math::tools::detail::handle_zero_derivative (F f, T &last_f0, const T &f0, T &delta, T &result, T &guess, const T &min, const T &max)

template<class F , class T >
T	boost::math::tools::newton_raphson_iterate (F f, T guess, T min, T max, int digits)

template<class F , class T >
T	boost::math::tools::newton_raphson_iterate (F f, T guess, T min, T max, int digits, boost::uintmax_t &max_iter)

Detailed Description

Multivariate Student's t Distribution

This class provides the functionality of calculating the probability density value, cumulative probability density value and generate random samples from a multivariate Student's t distribution.

Defining:

\( \Gamma() \) as the Gamma function
\( k \) as the dimensionality of the distribution
\( v \) as the degrees of freedom of the distribution
\( \boldsymbol{\mu}=[\mu_1 \cdots \mu_k] \) as the location vector
\( \boldsymbol{\Sigma}=\begin{bmatrix} \sigma^2_1 & \cdots & \sigma_{1,k}\\ \vdots & \ddots & \vdots \\ \sigma_{k,1} & \cdots & \sigma^2_k \end{bmatrix} \) as the scale matrix

The multivariate student's t distribution funtion is defined as: \( f(\textbf{x})= \frac{\Gamma(\frac{v+k}{2})}{\Gamma(\frac{v}{2}) v^{k/2} \pi^{k/2} |\boldsymbol{\Sigma}|^{1/2} (1+ \frac{1}{v} (\textbf{x}-\boldsymbol{\mu})' \boldsymbol{\Sigma}^{-1} (\textbf{x}-\boldsymbol{\mu}))^{\frac{v+k}{2}}} \)

The algorithm for cdf calculation is based on A. Genz & F. Bretz (2002)

If you construct multiple instances of this class, to avoid the generated samples to be the same, you should supply a different seed. To do so, for example, you can call MyDistribution.SetRandomSeed(MyDistribution.GetCurrentSeed()+1U);

Please refer to the Examples page for usage examples.

Remarks: This class is re-entrant

Date: November 2013

License: This program is free software: you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details.

Here, you can find a copy of the GNU Lesser General Public License. Alternatively, see gnu.org.

Constructor & Destructor Documentation

tDistribution::tDistribution	(	unsigned int	Dimension,
		unsigned int	DegFreedom,
		const Eigen::VectorXd &	locVect,
		const Eigen::MatrixXd &	ScaleMatr
	)

Construct a multivariate student's t with the given parameters.

Parameters

Dimension	The dimensionality of the distribution (supports also univariate student's t distributions in case this is 1)
DegFreedom	The degrees of freedom of the distribution
locVect	The location vector
ScaleMatr	The scale matrix

Construct a multivariate student's t distribution.

In case:

The Dimension is 0
The Degrees of freedom are 0
The location vector has a number of elements different from the Dimension
The ScaleMatr is not square
The ScaleMatr is not symmetric
The ScaleMatr is not semi-positive definite
The ScaleMatr has a number of rows different from the Dimension

The class will be considered invalid (it can be checked using IsValid()) and won't produce any result until the problem is fixed.

tDistribution::tDistribution	(	unsigned int	Dimension = `1U`,
		unsigned int	DegFreedom = `1U`
	)

Construct a multivariate standardized student's t.

Parameters

Dimension	The dimensionality of the multivariate normal (supports also univariate Gaussian distributions in case this is 1)
DegFreedom	The degrees of freedom of the distribution

Construct a multivariate student's t distribution with all location parameters set to 0 and scale matrix set to the identity.

In case the Dimension or the DegFreedom are 0 the class will be considered invalid (it can be checked using IsValid()) and won't produce any result until the problem is fixed.

If no argument is specified, a univariate standardized student's t with 1 degree of freedom, is created

Member Function Documentation

Eigen::MatrixXd tDistribution::ExtractSamples ( unsigned int NumSamples ) const

virtual

Generates multiple simulations from the distribution.

Parameters

NumSamples The number of simulation to run

Returns: A matrix with columns equal to the dimensionality of the distribution and rows equal to the number of simulations

This function generates NumSamples simulation from the current distribution and returns them in matrix form.

If NumSamples is 0 or the distribution is invalid, a null matrix is returned

Implements Multivariate::AbstarctDistribution.

Eigen::MatrixXd tDistribution::ExtractSamplesCDF ( unsigned int NumSamples ) const

virtual

Extracts samples from the distribution and returns their marginal CDF.

Parameters

NumSamples The number of simulation to run

Returns: A matrix with columns equal to the dimensionality of the distribution and rows equal to the number of simulations

This function generates NumSamples simulation from the current distribution, computes the marginal cumulative density function for each of them and returns them in matrix form.

This function simulates extractions from a t copula

If NumSamples is 0 or the distribution is invalid, a null matrix is returned

Implements Multivariate::AbstarctDistribution.

double tDistribution::GetCumulativeDesity ( const Eigen::VectorXd & Coordinates ) const

virtual

Computes the cumulative density function of the distribution in correspondence of the supplied coordinates.

Parameters

Coordinates A vector containing the coordinates of the point for which the cdf should be computed

Returns: The value of the cumulative density function

This function computes the cumulative density function of the current distribution associated with the given coordinates.

If the number of elements in Coordinates is different from the dimensionality of the distribution or the distribution is invalid, -1 is returned.

Todo:: Add the Genz algorithm.
The Gentz algorithm described in Genz & Bretz (2002) should be implemented here.
Prof Genz suggested to use equation 13 and 14 to calculate T then use the algorithm on page 9 and then switch to quasi monte carlo points for the algorithm at page 14.

Implements Multivariate::AbstarctDistribution.

unsigned int Multivariate::tDistribution::GetDegreesOfFreedom ( ) const

inline

Get the degrees of freedom of the distribution.

Returns: The current degrees of freedom of the distribution

See Also: SetDegreesOfFreedom()

double tDistribution::GetDensity ( const Eigen::VectorXd & Coordinates ) const

virtual

Computes the probability density function of the distribution in correspondence of the supplied coordinates.

Parameters

Coordinates A vector containing the coordinates of the point for which the pdf should be computed

Returns: The value of the probability density function

This function computes the probability density function of the current distribution associated with the given coordinates.

If the number of elements in Coordinates is different from the dimensionality of the distribution or the distribution is invalid, -1 is returned

Implements Multivariate::AbstarctDistribution.

const Eigen::VectorXd& Multivariate::tDistribution::GetLocationVector ( ) const

inline

Get the location vector of the distribution.

Returns: The current location vector of the distribution

See Also: SetLocationVector(const Eigen::VectorXd&)

Eigen::VectorXd tDistribution::GetQuantile ( double Prob ) const

virtual

Computes the inverse cumulative density function of the distribution in correspondence of the supplied probability.

Parameters

Prob	The probability for which the corresponding quantile must be found

Returns: A vector containing the coordinates of the quantile

This function computes the inverse cumulative density function of the current distribution associated with the given probability.

The solution is unique only in the univariate case.
Generally the system of equations \( F^{-1}(Coordinates_1 \cdots Coordinates_k)=Prob \) has k-1 degrees of freedom, where k is the dimensionality of the distribution.
The additional restriction imposed to get to an unique solution is that each coordinate has equal distance from it's mean.

If the probability supplied is greater than 1, less than 0 or the distribution is invalid, an empty vector is returned.

Implements Multivariate::AbstarctDistribution.

const Eigen::MatrixXd& Multivariate::tDistribution::GetScaleMatrix ( ) const

inline

Get the scale matrix of the distribution.

Returns: The current scale matrix of the distribution

See Also: SetScaleMatrix(const Eigen::MatrixXd&)

Eigen::MatrixXd tDistribution::GetVarMatrix ( ) const

Computes the variance matrix of the distribution.

Returns: The variance matrix

This function computes the variance matrix of the current distribution.

Defining:

\( v \) as the number of degrees of freedom of the distribution
\( \boldsymbol{\Sigma}=\begin{bmatrix} \sigma^2_1 & \cdots & \sigma_{1,k}\\ \vdots & \ddots & \vdots \\ \sigma_{k,1} & \cdots & \sigma^2_k \end{bmatrix} \) as the scale matrix

The variance matrix is defined as \( \frac{v}{v-2} \boldsymbol{\Sigma} \forall v>2 \)

If the degrees of freedom are less than 3 or the distribution is invalid, a null matrix is returned

bool tDistribution::SetDegreesOfFreedom ( unsigned int a )

Set the degrees of freedom of the distribution.

Parameters

a	The number of degrees of freedom of the distribution

Returns: A boolean determining if the degrees of freedom of the distribution were changed successfully

This function tries to set the degrees of freedom of the distribution to the new ones.

In case a is less than 1 the degrees of freedom of the distribution will not be changed and this function will return false

See Also: GetDegreesOfFreedom()

bool tDistribution::SetDimension ( unsigned int Dimension )

virtual

Set the dimensionality of the distribution.

Parameters

Dimension the new dimensionality of the distribution

Returns: A boolean determining if the dimensionality was changed successfully

This function will try to change the dimensionality of the distribution (e.g. 2 for bivariate, 3 for trivariate, etc.)

All the components of the location vector will be reset to 0 and the scale matrix will default to an identity matrix, degrees of freedom wil not be changed

If the argument passed is 0 the dimensionality will not be changed and the function will return false

See Also: GetDimension()

Implements Multivariate::AbstarctDistribution.

bool Multivariate::tDistribution::SetLocationVector ( const Eigen::VectorXd & mVect )

inline

Set the expected values vector.

Parameters

mVect the vector of new values for the location vector

Returns: A boolean determining if the location vector was changed successfully

This function attempts to set the location vector to the new values supplied.

If the degrees of freedom are more than 1, this vector is the mean vector

If the dimension of the vector is different from the dimension of the distribution the location vector is not changed and false is returned

See Also: GetLocationVector()

bool tDistribution::SetLocationVector ( const std::vector< double > & mVect )

Set the location vector.

Parameters

mVect the vector of new values for the location vector

This is an overloaded version of SetLocationVector(const Eigen::VectorXd&)

bool tDistribution::SetScaleMatrix ( const Eigen::MatrixXd & SclMatr )

Set the scale matrix of the distribution.

Parameters

SclMatr the new scale matrix of the distribution

Returns: A boolean determining if the scale matrix of the distribution was changed successfully

This function tries to set the scale matrix of the distribution to the new one.

In case:

The scale matrix is not square
The scale matrix is not symmetric
The scale matrix is not semi-positive definite
The scale matrix has a number of rows different from the Dimension

The scale matrix of the distribution will not be changed and this function will return false

See Also: GetScaleMatrix()

bool tDistribution::SetScaleMatrix	(	const std::vector< double > &	mVect,
		bool	RowWise = `true`
	)

Set the scale matrix of the distribution.

Parameters

mVect	a vector containing the elements of the new scale matrix of the distribution
RowWise	if it's set to true (the default) the matrix will be filled by row. If it's false it will be filled by columns

Returns: A boolean determining if the scale matrix of the distribution was changed successfully

This function tries to set the scale matrix of the distribution to the new one.

Constructs square a matrix with number of rows equal to the dimensionality of the distribution, it is then filled with the values supplied in order according to the RowWise parameter

In case:

The vector size is different from the square of the distribution dimensionality
The scale matrix is not symmetric
The scale matrix is not semi-positive definite
The scale matrix has a number of rows different from the Dimension

The scale matrix of the distribution will not be changed and this function will return false

See Also: GetScaleMatrix()

Public Member Functions

Friends

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation