Multi - dimensional Gauss function class. More...

#include <NDGauss.h>

Public Member Functions
	CNDGauss (dword k)
	Constructor. More...

	CNDGauss (const CNDGauss &rhs)

	~CNDGauss (void)

CNDGauss &	operator= (const CNDGauss &rhs)

CNDGauss &	operator+= (const CNDGauss &rhs)

const CNDGauss	operator+ (const CNDGauss &rhs) const
	Merging operator. More...

void	clear (void)
	Clears class variables. More...

void	freeze (void)
	Freezes the state of the Gaussian function. More...

void	addPoint (Mat &point)
	Adds new n-dimensional point (sample) for Gaussian function approximation. More...

bool	empty (void) const
	Checks weather the Gaussian function is approximated. More...

void	setMu (Mat &mu)
	Sets \(\mu\). More...

Mat	getMu (void) const
	Returns \(\mu\). More...

void	setSigma (Mat &sigma)
	Sets \(\Sigma\). More...

Mat	getSigma (void) const
	Returns \(\Sigma\). More...

Mat	getSigmaInv (void) const
	Returns \(\Sigma^{-1}\). More...

long double	getAlpha (void) const
	Returns \(\alpha\). More...

double	getValue (Mat &x, Mat &aux1=Mat(), Mat &aux2=Mat(), Mat &aux3=Mat()) const
	Returns unscaled value of the Gaussian function. More...

void	setNumPoints (long nPoints)
	Sets the number of approximation points. More...

size_t	getNumPoints (void) const
	Returns the number of approximation points. More...

double	getEuclidianDistance (const Mat &x) const
	Returns the Euclidian distance between argument point x and the center of multivariate normal distribution \(\mu\). More...

double	getMahalanobisDistance (const Mat &x) const
	Returns the Mahalanobis distance between argument point x and the center of multivariate normal distribution \(\mathcal{N}(\mu,\Sigma)\). More...

double	getKullbackLeiberDivergence (CNDGauss &x) const
	Returns the Kullback-Leiber divergence from the multivariate normal distribution \(\mathcal{N}(\mu,\Sigma)\) to argument multivariate normal distribution \(\mathcal{N}_x(\mu_x,\Sigma_x)\). More...

Mat	getSample (void) const
	Returns a random vector (sample) from multivariate normal distribution. More...

Detailed Description

Multi - dimensional Gauss function class.

This class allows to approximate the distribution of random variables (samples) \( \textbf{x} \), represented as k - dinemstional points \( [x_1,x_2,\dots,x_k] \). Under the assumption, that the random variables \( \textbf{x} \) are normally distributed, the approximation is done with multivariate normal distribution: \( \mathcal{N}_k(\mu,\Sigma)= \alpha\cdot\exp\big( -\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)\big) \) whith coefficient \( \alpha = \frac{1}{(2\pi)^{k/2}|\Sigma|^{1/2}} \). The parameters \( \mu\) and \(\Sigma \) of the Gaussian function \( \mathcal{N}(\mu,\Sigma) \) may be estimated sequentially with function addPoint() or operator operator+(); also they may be set directly with functions setMu() and setSigma().

The value of the Gaussian function \( \mathcal{N}(\mu,\Sigma) \) in point \( \textbf{x} \) is achieved with functions getAlpha() and getValue():

double value = CNDGauss::getAlpha() * CNDGauss::getValue(x);

In order to generate a random sample from the distribution, given by Gaussian function \( \mathcal{N}(\mu,\Sigma) \), one uses function getSample().

Author: Sergey G. Kosov, serge.nosp@m.y.ko.nosp@m.sov@p.nosp@m.roje.nosp@m.ct-10.nosp@m..de

Definition at line 26 of file NDGauss.h.

Constructor & Destructor Documentation

DirectGraphicalModels::CNDGauss::CNDGauss ( dword k )

Constructor.

Parameters

k	dimensions

Definition at line 12 of file NDGauss.cpp.

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DirectGraphicalModels::CNDGauss::CNDGauss ( const CNDGauss & rhs )

Definition at line 29 of file NDGauss.cpp.

DirectGraphicalModels::CNDGauss::~CNDGauss ( void )

Definition at line 40 of file NDGauss.cpp.

Member Function Documentation

void DirectGraphicalModels::CNDGauss::addPoint ( Mat & point )

Adds new n-dimensional point (sample) for Gaussian function approximation.

This function is a faster approximation of the merging operator (Ref. operator+()):
\( \hat{\mu} = \frac{n\mu + p}{n + 1} \); \( \hat{\Sigma} = \frac{n\Sigma + (p-\hat{\mu})(p-\hat{\mu})^\top}{n + 1}\)

while(point) estGaussian.addPoint(point); // estimated Gauss function is updated

Parameters

point k-dimensional point: Mat(size: k x 1; type: CV_64FC1)

Definition at line 111 of file NDGauss.cpp.

void DirectGraphicalModels::CNDGauss::clear ( void )

Clears class variables.

Allows to re-use the class

Definition at line 85 of file NDGauss.cpp.

bool DirectGraphicalModels::CNDGauss::empty ( void ) const

inline

Checks weather the Gaussian function is approximated.

Return values

TRUE	if the Gaussian had at least 1 point for approximation, or
FALSE	otherwise

Definition at line 92 of file NDGauss.h.

void DirectGraphicalModels::CNDGauss::freeze ( void )

Freezes the state of the Gaussian function.

This is an optimization function, which calculates and fills internal variables need for major get- accessors of this class. If the function was not called, tese variable will be calculated in the get- accessors every tiime on call.

Definition at line 93 of file NDGauss.cpp.

long double DirectGraphicalModels::CNDGauss::getAlpha ( void ) const

Returns \(\alpha\).

Returns: The Gaussian coefficient \(\alpha\).

Definition at line 167 of file NDGauss.cpp.

double DirectGraphicalModels::CNDGauss::getEuclidianDistance ( const Mat & x ) const

Returns the Euclidian distance between argument point x and the center of multivariate normal distribution \(\mu\).

The Euclidian distance is calculated by the formula: \(D_E(\mathcal{N};x)=\sqrt{ (x-\mu)^\top(x-\mu) }\).

Parameters

x	n-dimensional point (sample): Mat(size: k x 1; type: CV_64FC1)

Returns: the Euclidian distance: \(D_E(x)\)

Definition at line 193 of file NDGauss.cpp.

double DirectGraphicalModels::CNDGauss::getKullbackLeiberDivergence ( CNDGauss & x ) const

Returns the Kullback-Leiber divergence from the multivariate normal distribution \(\mathcal{N}(\mu,\Sigma)\) to argument multivariate normal distribution \(\mathcal{N}_x(\mu_x,\Sigma_x)\).

The Kullback-Leiber divergence (or relative entropy) is calculated by the formula: \(D_{KL}(\mathcal{N};\mathcal{N}_x)=\frac{1}{2}\Big( tr(\Sigma^{-1}_{x}\Sigma) + D^{2}_{M}(\mathcal{N}_x;\mu) - k - \ln\big(\frac{\left|\Sigma\right|}{\left|\Sigma_x\right|}\big) \Big)\) and expressed in nats. Here \(D_M(\mathcal{N}_x;\mu)\) is the Mahalanobis distance beween the centers of multivariate normal distributions \(\mu_x\) and \(\mu\) with respect to \(\mathcal{N}_x\) (see getMahalanobisDistance() for more details). Please note, that it is not a symmetrical quantity, that is to say \( D_{KL}(\mathcal{N};\mathcal{N}_x) \neq D_{KL}(\mathcal{N}_x;\mathcal{N})\) and so could be hardly used as a "distance".

Parameters

x	multivariate normal distribution \(\mathcal{N}_x(\mu_x,\Sigma_x)\) of the dimension k.

Returns: the Kullback-Leiber divergence: \(D_{KL}(\mathcal{N}_x)\)

Definition at line 217 of file NDGauss.cpp.

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double DirectGraphicalModels::CNDGauss::getMahalanobisDistance ( const Mat & x ) const

Returns the Mahalanobis distance between argument point x and the center of multivariate normal distribution \(\mathcal{N}(\mu,\Sigma)\).

The Mahalanobis distance is calculated by the formula: \(D_M(\mathcal{N};x)=\sqrt{ (x-\mu)^\top\Sigma^{-1}(x-\mu) }\)

Parameters

x	n-dimensional point (sample): Mat(size: k x 1; type: CV_64FC1)

Returns: the Mahalanobis distance: \(D_M(x)\)

Definition at line 208 of file NDGauss.cpp.

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Mat DirectGraphicalModels::CNDGauss::getMu ( void ) const

inline

Returns \(\mu\).

Returns: the mean vector \(\mu\): Mat(size: k x 1; type: CV_64FC1)

Definition at line 102 of file NDGauss.h.

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size_t DirectGraphicalModels::CNDGauss::getNumPoints ( void ) const

inline

Returns the number of approximation points.

Returns: the number of sample points, used for the approximation.

Definition at line 155 of file NDGauss.h.

Mat DirectGraphicalModels::CNDGauss::getSample ( void ) const

Returns a random vector (sample) from multivariate normal distribution.

The implementation is based on the paper Generating Random Vectors from the Multivariate Normal Distribution

Returns: n-dimensional point (sample): Mat(size: k x 1; type: CV_64FC1)

Definition at line 239 of file NDGauss.cpp.

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Mat DirectGraphicalModels::CNDGauss::getSigma ( void ) const

inline

Returns \(\Sigma\).

Returns: the covariance matrix \(\Sigma\): Mat(size: k x k; type: CV_64FC1)

Definition at line 112 of file NDGauss.h.

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Mat DirectGraphicalModels::CNDGauss::getSigmaInv ( void ) const

Returns \(\Sigma^{-1}\).

Returns: The inversed covariance matrix \(\Sigma^{-1}\): Mat(size: k x k; type: CV_64FC1)

Definition at line 157 of file NDGauss.cpp.

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double DirectGraphicalModels::CNDGauss::getValue	(	Mat &	x,
		Mat &	aux1 = `Mat()`,
		Mat &	aux2 = `Mat()`,
		Mat &	aux3 = `Mat()`
	)		const

Returns unscaled value of the Gaussian function.

This function returns unscaled value of the Gaussian function, i.e. \( \exp\big( -\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)\big) \). In order to get the value of \( \mathcal{N}_k(\mu,\Sigma) \), the output of this function must be multiplied with \( \alpha \) from the getAlpha() function.

Note: Three auxilary parameters aux1, aux2 and aux3 are needed for more efficient sequential calculation and do not influent the resulting value, e.g. the code:
Mat aux1, aux2, aux3;

for (int i = 0; i < 100; i++) y[i] = getValue(x[i], aux1, aux2, aux3);

will run about two times faster, then the code:
for (int i = 0; i < 100; i++) y[i] = getValue(x[i]);

This function is PPL-safe function.

Parameters

x	n-dimensional point (sample): Mat(size: k x 1; type: CV_64FC1)
aux1	Auxilary variable
aux2	Auxilary variable
aux3	Auxilary variable

Returns: unscaled value of the Gaussian function

Definition at line 179 of file NDGauss.cpp.

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const CNDGauss DirectGraphicalModels::CNDGauss::operator+ ( const CNDGauss & rhs ) const

inline

Merging operator.

This operator merges two Gauss finctions into one:

\( \hat{\mu} = \frac{n_1\mu_1 + n_2\mu_2}{n_1 + n_2} \); \( \hat{\Sigma} = \frac{n_1(\Sigma_1 + \mu_1\mu^{\top}_{1}) + n_2(\Sigma_2 + \mu_2\mu^{\top}_{2})}{n_1 + n_2} - \hat{\mu}\hat{\mu}^\top\)

For the sequential Gauss function estimation one also may use this operator (see the code below) or addPoint() function:

CNDGauss newGaussian(point.rows);       // auxilary Gauss function
while(point) {                          // set of all sample points
    newGaussian.setMu(point);           // mu is set to be equal to the sample
    estGaussian += newGaussian;         // estimated Gauss function is updated
}

In this case \(n_2 = 1\) and \(\Sigma_2 = 0\).

Definition at line 62 of file NDGauss.h.

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CNDGauss & DirectGraphicalModels::CNDGauss::operator+= ( const CNDGauss & rhs )

Definition at line 67 of file NDGauss.cpp.

CNDGauss & DirectGraphicalModels::CNDGauss::operator= ( const CNDGauss & rhs )

Definition at line 49 of file NDGauss.cpp.

void DirectGraphicalModels::CNDGauss::setMu ( Mat & mu )

Sets \(\mu\).

Parameters

mu	the mean vector \(\mu\) : Mat(size: k x 1; type: CV_64FC1)

Definition at line 135 of file NDGauss.cpp.

void DirectGraphicalModels::CNDGauss::setNumPoints ( long nPoints )

inline

Sets the number of approximation points.

Parameters

nPoints the number of sample points, used for the approximation.

Definition at line 150 of file NDGauss.h.

void DirectGraphicalModels::CNDGauss::setSigma ( Mat & sigma )

Sets \(\Sigma\).

Parameters

sigma the covariance matrix \(\Sigma\): Diagonal Mat(size: k x k; type: CV_64FC1)

Definition at line 146 of file NDGauss.cpp.

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

D:/Projects/DGM/modules/DGM/NDGauss.h
D:/Projects/DGM/modules/DGM/NDGauss.cpp

Public Member Functions

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation