Generalized Gaussian Probability Density Function
The Generalized Gaussian density has the following form:
where (rho) is the "shape parameter". The density is plotted in the following figure:
Matlab code used to generate this figure is available here: ggplot.m.
Adding an arbitrary location parameter, , and inverse scale parameter, , the density has the form,
Matlab code used to generate this figure is available here: ggplot2.m.
Generating Random Samples
Samples from the Generalized Gaussian can be generated by a transformation of Gamma random samples, using the fact that if is a distributed random variable, and is an independent random variable taking the value -1 or +1 with equal probability, then,
is distributed . That is,
where the density of is written in a non-standard but suggestive form.
Matlab code to generate random variates from the Generalized Gaussian density with parameters as described here is here:
As an example, we generate random samples from the example Generalized Gaussian densities shown above.
Matlab code used to generate this figure is available here: ggplot3.m.
A more general family of densities can be constructed from mixtures of Generalized Gaussians. A mixture density, , is made up of constituent densities together with probabilities associated with each constituent density.
The densities have different forms, or parameter values. A random variable with a mixture density can be thought of as being generated by a two-part process: first a decision is made as to which constituent density to draw from, where the density is chosen with probability , then the value of the random variable is drawn from the chosen density. Independent repetitions of this process result in a sample having the mixture density .
As an example consider the density,
Matlab code used to generate these figures is available here: ggplot4.m.