GaussianNDParam.h

Classes

GaussianNDParam -- A Multi-dimensional Gaussian parameter handling. (full description)

template<class T> class GaussianNDParam : public Function<T>

Types

enum

HEIGHT = 0
CENTER

Interface

Public Members
GaussianNDParam()
explicit 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 Vector<T> &covar)
GaussianNDParam(const GaussianNDParam &other)
GaussianNDParam<T> &operator=(const GaussianNDParam<T> &other)
virtual ~GaussianNDParam()
virtual uInt ndim() const
T height() const
void setHeight(const T &height)
T flux() const
void setFlux(const T &flux)
Vector<T> mean() const
void setMean(const Vector<T> &mean)
Vector<T> variance() const
void setVariance(const Vector<T> &variance)
Matrix<T> covariance() const
void setCovariance(const Matrix<T> &covar)
Protected Members
void repack(Matrix<T> &covar) const
void unpack(const Matrix<T> &covar)

Description

Review Status

Programs:
Demos:
Tests:

Prerequisite

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:
  1. height (1 term). No assumptions on what quantity the height represents, and it can be negative (enumerated by HEIGHT)
  2. mean (ndim terms) (enumerated by CENTER).
  3. variance (ndim terms). The variance is always positive, and an exception (AipsError) will be thrown if you try to set a negative value.
  4. 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)

To Do

Member Description

enum

GaussianNDParam()
explicit 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 Vector<T> &covar)

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,

GaussianNDParam(const GaussianNDParam &other)

Copy constructor (deep copy)

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

Copy assignment (deep copy)

virtual ~GaussianNDParam()

Destructor

virtual uInt ndim() const

Variable dimensionality

T height() const
void setHeight(const T &height)

Get or set the peak height of the Gaussian

T flux() const
void setFlux(const T &flux)

The analytical integrated area underneath the Gaussian. Use these functions as an alternative to the height functions.

Vector<T> mean() const
void setMean(const Vector<T> &mean)

The center ordinate of the Gaussian

Vector<T> variance() const
void setVariance(const Vector<T> &variance)

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

Matrix<T> covariance() const
void setCovariance(const Matrix<T> &covar)

The covariance Matrix defines the correlations between all the axes.

void repack(Matrix<T> &covar) const
void unpack(const Matrix<T> &covar)

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