- HEIGHT = 0
- CENTER
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.
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 actual parameters are in order:
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]
Copy constructor (deep copy)
Copy assignment (deep copy)
Destructor
Variable dimensionality
The analytical integrated area underneath the Gaussian. Use these functions as an alternative to the height functions.
The center ordinate of the Gaussian
The FWHM of the Gaussian is sqrt(8*variance*log(2)). The variance MUST be positive
The covariance Matrix defines the correlations between all the axes.
Functions to convert between internal Vector of parameters and the Covariance Matrix