A Multi-dimensional Gaussian parameter handling. More...

#include <GaussianNDParam.h>

Inheritance diagram for casa::GaussianNDParam< T >:

Public Types
enum	{ HEIGHT, CENTER }
Public Member Functions
	GaussianNDParam ()
	Constructs a Gaussian using the indicated height, mean, variance & covariance.
	GaussianNDParam (uInt ndim)
	GaussianNDParam (uInt ndim, const T &height)
	GaussianNDParam (uInt ndim, const T &height, const Vector< T > &mean)
	GaussianNDParam (uInt ndim, const T &height, const Vector< T > &mean, const Vector< T > &variance)
	GaussianNDParam (uInt ndim, const T &height, const Vector< T > &mean, const Matrix< T > &covar)
	GaussianNDParam (const GaussianNDParam &other)
	Copy constructor (deep copy)
template<class W >
	GaussianNDParam (const GaussianNDParam< W > &other)
GaussianNDParam< T > &	operator= (const GaussianNDParam< T > &other)
	Copy assignment (deep copy)
virtual	~GaussianNDParam ()
	Destructor.
virtual const String &	name () const
	Give name of function.
virtual uInt	ndim () const
	Variable dimensionality.
T	height () const
	Get or set the peak height of the Gaussian.
void	setHeight (const T &height)
T	flux () const
	The analytical integrated area underneath the Gaussian.
void	setFlux (const T &flux)
Vector< T >	mean () const
	The center ordinate of the Gaussian.
void	setMean (const Vector< T > &mean)
Vector< T >	variance () const
	The FWHM of the Gaussian is `sqrt(8variancelog(2))`.
void	setVariance (const Vector< T > &variance)
Matrix< T >	covariance () const
	The covariance Matrix defines the correlations between all the axes.
void	setCovariance (const Matrix< T > &covar)
Protected Member Functions
void	repack (Matrix< T > &covar) const
	Functions to convert between internal Vector of parameters and the Covariance Matrix.
void	unpack (const Matrix< T > &covar)
Protected Attributes
uInt	itsDim

T	itsFlux2Hgt
	factor to convert from flux to height

Detailed Description

template<class T>
class casa::GaussianNDParam< T >

A Multi-dimensional Gaussian parameter handling.

Intended use:

Internal

 <h3>Review Status</h3><dl><dt>Reviewed By:<dd>UNKNOWN<dt>Date Reviewed:<dd>before2004/08/25<dt>Test programs:<dd>tGaussianND<dt>Demo programs:<dd>dGaussianND</dl>

Prerequisite

FunctionParam class
Function class

Synopsis

A GaussianND is used to calculate Gaussian functions of any dimension. A Gaussian1D class exists which is more appropriate for one dimensional Gaussian functions, and a Gaussian2D class exists for two dimensional functions.

A statistical description of the multi-dimensional Gaussian is used (see Kendall & Stuart "The Advanced Theory of Statistics"). A Gaussian is defined in terms of its height, mean (which is the location of the peak value), variance, (a measure of the width of the Gaussian), and covariance which skews the distribution with respect to the Axes.

In the general description the variance and covariance are specified using a covariance matrix. This is defined as (for a 4 dimensional Gaussian):

     V = |     s1*s1 r12*s1*s2 r13*s1*s3 r14*s1*s4 | 
         | r12*s1*s2     s2*s2 r23*s2*s3 r24*s2*s4 |
         | r13*s1*s3 r23*s2*s3     s3*s3 r34*s3*s4 |
         | r14*s1*s4 r24*s2*s4 r34*s3*s4     s4*s4 |

where s1 (sigma1) is the standard deviation of the Gaussian with respect to the first axis, and r12 (rho12) is the correlation between the the first and second axis. The correlation MUST be between -1 and 1, and this class checks this as well as ensuring that the diagonal is positive.

Warning: It is possible to have symmetric matrices that are of the above described form (ie; symmetric with -1 <= rho(ij) <=1) that do not generate a Gaussian function; This is because the Matrix is NOT positive definite (The limits on rho(ij) are upper limits); This class does check that the covariance Matrix is positive definite and will throw an exception (AipsError) if it is not;

The covariance Matrix can be specified by only its upper or lower triangular regions (ie. with zeros in the other triangle), otherwise it MUST be symmetric.

The Gaussian that is constructed from this covariance Matrix (V), along with mean (u) and height (h) is:

     f(x) = h*exp( -1/2 * (x-u) * V^(-1) * (x-u))

where x, and u are vectors whose length is the dimensionality of the Gaussian and V^(-1) is the inverse of the covariance Matrix defined above. For a two dimensional Gaussian with zero mean this expression reduces to:

    f(x) = h*exp(-1/(2*(1-r12^2))*(x1^2/s1^2 - 2*r12*x1*x2/(s1*s2) + x2^2/s2^2))

The amplitude of the Gaussian can be defined in two ways, either using the peak height (as is done in the constructors, and the setHeight function) or using the setFlux function. The flux in this context is the analytic integral of the Gaussian over all dimensions. Using the setFlux function does not modify the shape of the Gaussian just its height.

All the parameters of the Gaussian except its dimensionality can be modified using the set/get functions.

The parameter interface (see FunctionParam class), is used to provide an interface to the Fitting classes. There are always 4 parameter sets.
Warning: Note that the actual variance/covariance parameters are the inverse matrix of the variance/covariance matrix given by the user
; The actual parameters are in order:

height (1 term). No assumptions on what quantity the height represents, and it can be negative (enumerated by HEIGHT)
mean (ndim terms) (enumerated by CENTER).
variance (ndim terms). The variance is always positive, and an exception (AipsError) will be thrown if you try to set a negative value.
covariance (ndim*(ndim-1)/2 terms) The order is (assuming ndim=5) v12,v13,v14,v15,v23,v24,v25,v34,v35,v45. The restrictions described above for the covariance (ie. -1 < r12 < +1) are enforced.

Example

Construct a two dimensional Gaussian with mean=(0,1), variance=(.1,7) and height = 1;

    uInt ndim = 2;
    Float height = 1;
    Vector<Float> mean(ndim); mean(0) = 0, mean(1) = 1;
    Vector<Float> variance(ndim); variance(0) = .1, variance(1) = 7;
    GaussianND<Float> g(ndim, height, mean, variance); 
    Vector<Float> x(ndim); x = 0;
    cout << "g("<< x <<") = " << g(x) <<endl; // g([0,0])=1*exp(-1/2*1/7);
    x(1)++;
    cout << "g("<< x <<") = " <<g(x) <<endl;  // g([0,1])= 1
    cout << "Height: " << g.height() <<endl;    // Height: 1
    cout << "Flux: " << g.flux() << endl;       // Flux: 2*Pi*Sqrt(.1*7)
    cout << "Mean: " << g.mean() << endl;  // Mean: [0, -1]
    cout << "Variance: " << g.variance() <<endl;  // Variance: [.1, 7]
    cout << "Covariance: "<< g.covariance()<<endl;// Covariance: [.1, 0]
                                                          //             [0,  7]
    g.setFlux(1);
    cout << "g("<< x <<") = " <<g(x) <<endl;  //g([0,1])=1/(2*Pi*Sqrt(.7))
    cout << "Height: " << g.height() <<endl;    // Height: 1/(2*Pi*Sqrt(.7))
    cout << "Flux: " << g.flux() << endl;       // Flux: 1
    cout << "Mean: " << g.mean() << endl;  // Mean: [0, -1]
    cout << "Variance: " << g.variance() <<endl;  // Variance: [.1, 7]
    cout << "Covariance: "<< g.covariance()<<endl;// Covariance: [.1, 0]
                                                  //             [0,  7]

Motivation

A Gaussian Functional was needed for modeling the sky with a series of components. It was later realised that it was too general and Gaussian2D was written.

Template Type Argument Requirements (T)

T should have standard numerical operators and exp() function. Current implementation only tested for real types.

To Do

Nothing I know off, apart from possible optimization

Definition at line 182 of file GaussianNDParam.h.

Member Enumeration Documentation

template<class T>

anonymous enum

Enumerator:

HEIGHT
CENTER

Definition at line 186 of file GaussianNDParam.h.

Constructor & Destructor Documentation

template<class T>

casa::GaussianNDParam< T >::GaussianNDParam ( )

Constructs a Gaussian using the indicated height, mean, variance & covariance.

ndim defaults to 2, mean defaults to 0, height to Pi^(-ndim/2) (the flux is unity) variance defaults to 1.0, covariance defaults to 0.0,

template<class T>

casa::GaussianNDParam< T >::GaussianNDParam ( uInt ndim ) [explicit]

template<class T>

casa::GaussianNDParam< T >::GaussianNDParam	(	uInt	ndim,
		const T &	height
	)

template<class T>

casa::GaussianNDParam< T >::GaussianNDParam	(	uInt	ndim,
		const T &	height,
		const Vector< T > &	mean
	)

template<class T>

casa::GaussianNDParam< T >::GaussianNDParam	(	uInt	ndim,
		const T &	height,
		const Vector< T > &	mean,
		const Vector< T > &	variance
	)

template<class T>

casa::GaussianNDParam< T >::GaussianNDParam	(	uInt	ndim,
		const T &	height,
		const Vector< T > &	mean,
		const Matrix< T > &	covar
	)

template<class T>

casa::GaussianNDParam< T >::GaussianNDParam ( const GaussianNDParam< T > & other )

Copy constructor (deep copy)

template<class T>

template<class W >

casa::GaussianNDParam< T >::GaussianNDParam ( const GaussianNDParam< W > & other ) [inline]

Definition at line 211 of file GaussianNDParam.h.

template<class T>

virtual casa::GaussianNDParam< T >::~GaussianNDParam ( ) [virtual]

Destructor.

Member Function Documentation

template<class T>

Matrix<T> casa::GaussianNDParam< T >::covariance ( ) const

The covariance Matrix defines the correlations between all the axes.

template<class T>

T casa::GaussianNDParam< T >::flux ( ) const

The analytical integrated area underneath the Gaussian.

Use these functions as an alternative to the height functions.

template<class T>

T casa::GaussianNDParam< T >::height ( ) const [inline]

Get or set the peak height of the Gaussian.

Definition at line 234 of file GaussianNDParam.h.

References casa::GaussianNDParam< T >::HEIGHT, and casa::Function< T >::param_p.

Referenced by casa::GaussianNDParam< T >::setHeight().

template<class T>

Vector<T> casa::GaussianNDParam< T >::mean ( ) const

The center ordinate of the Gaussian.

template<class T>

virtual const String& casa::GaussianNDParam< T >::name ( ) const [inline, virtual]

Give name of function.

Reimplemented from casa::Function< T >.

Definition at line 226 of file GaussianNDParam.h.

template<class T>

virtual uInt casa::GaussianNDParam< T >::ndim ( ) const [inline, virtual]

Variable dimensionality.

Implements casa::Function< T >.

Definition at line 230 of file GaussianNDParam.h.

References casa::GaussianNDParam< T >::itsDim.

template<class T>

GaussianNDParam<T>& casa::GaussianNDParam< T >::operator= ( const GaussianNDParam< T > & other )

Copy assignment (deep copy)

template<class T>

void casa::GaussianNDParam< T >::repack ( Matrix< T > & covar ) const [protected]

Functions to convert between internal Vector of parameters and the Covariance Matrix.

template<class T>

void casa::GaussianNDParam< T >::setCovariance ( const Matrix< T > & covar )

template<class T>

void casa::GaussianNDParam< T >::setFlux ( const T & flux )

template<class T>

void casa::GaussianNDParam< T >::setHeight ( const T & height ) [inline]

Definition at line 235 of file GaussianNDParam.h.

References casa::GaussianNDParam< T >::HEIGHT, casa::GaussianNDParam< T >::height(), and casa::Function< T >::param_p.

template<class T>

void casa::GaussianNDParam< T >::setMean ( const Vector< T > & mean )

template<class T>

void casa::GaussianNDParam< T >::setVariance ( const Vector< T > & variance )

template<class T>

void casa::GaussianNDParam< T >::unpack ( const Matrix< T > & covar ) [protected]

template<class T>

Vector<T> casa::GaussianNDParam< T >::variance ( ) const

The FWHM of the Gaussian is sqrt(8*variance*log(2)).

The variance MUST be positive

Member Data Documentation

template<class T>

uInt casa::GaussianNDParam< T >::itsDim [protected]

dimensionality

Definition at line 267 of file GaussianNDParam.h.

Referenced by casa::GaussianNDParam< T >::ndim().

template<class T>

T casa::GaussianNDParam< T >::itsFlux2Hgt [protected]

factor to convert from flux to height

Definition at line 269 of file GaussianNDParam.h.

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

casacore/scimath/Functionals/GaussianNDParam.h

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

template<class T> class casa::GaussianNDParam< T >

Intended use:

Prerequisite

Synopsis

Example

Motivation

Template Type Argument Requirements (T)

To Do

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

template<class T>
class casa::GaussianNDParam< T >