GaussianND.h

Classes

GaussianND -- A Multi-dimensional Gaussian functional. (full description)

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

Interface

Public Members
GaussianND() : GaussianNDParam<T>()
explicit GaussianND(uInt ndim) : ndim<T>(ndim)
GaussianND(uInt ndim, const T &height) : GaussianNDParam<T>(ndim, height)
GaussianND(uInt ndim, const T &height, const Vector<T> &mean) : GaussianNDParam<T>(ndim, height, mean)
GaussianND(uInt ndim, const T &height, const Vector<T> &mean, const Vector<T> &variance) : Vector<T>(ndim, height, mean, variance)
GaussianND(uInt ndim, const T &height, const Vector<T> &mean, const Matrix<T> &covar) : GaussianNDParam<T>(ndim, height, mean, covar)
GaussianND(const GaussianND &other) : GaussianNDParam<T>(other)
GaussianND<T> &operator=(const GaussianND<T> &other)
virtual ~GaussianND()
virtual T eval(typename Function<T>::FunctionArg x) const
virtual Function<T> *clone() const

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. The parameters are, in order:

  1. height (1 term). No assumptions on what quantity the height represents, and it can be negative
  2. mean (ndim terms).
  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

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

Makes 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,

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

Copy constructor (deep copy)

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

Copy assignment (deep copy)

virtual ~GaussianND()

Destructor

virtual T eval(typename Function<T>::FunctionArg x) const

Evaluate the Gaussian at x.

virtual Function<T> *clone() const

Return a copy of this object from the heap. The caller is responsible for deleting this pointer.