Random Mixture: affine combination of independent univariate distributions¶
A multivariate random variable may be defined as an affine transform of independent univariate random variable, as follows:
where is a deterministic vector with , a deterministic matrix and are some independent univariate distributions.
In such a case, it is possible to evaluate directly the distribution of and then to ask any request compatible with a distribution: moments, probability and cumulative density functions, quantiles (in dimension 1 only) …
Evaluation of the probability density function of the Random Mixture
As the univariate random variables are independent, the characteristic function of , denoted , is easily defined from the characteristic function of denoted as follows :
where , ,
The parameters are calibrated using the following formula:
where and , are respectively the number of standard deviations covered by the marginal distribution ( by default) and the number of marginal deviations beyond which the density is negligible ( by default).
The parameter is dynamically calibrated: we start with then we double value until the total contribution of the additional terms is negligible.
Evaluation of the moments of the Random Mixture
The relation (1) enables to evaluate all the moments of the random mixture, if mathematically defined. For example, we have:
Computation on a regular grid
The interest is to compute the density function on a regular grid. Purposes are to get an approximation quickly. The regular grid is of form:
By denoting :
for which the term is the most CPU consuming. This term rewrites:
The aim is to rewrite the previous expression as a - discrete Fourier transform, in order to apply Fast Fourier Transform (FFT) for its evaluation.
We set and and . For convenience, we introduce the functions:
We use instead of in this function to simplify expressions below.
For performance reasons, we want to use the discrete Fourier transform with the following convention in dimension 1:
which extension to dimensions 2 and 3 are respectively:
We decompose sums of on the interval into three parts:
If we already computed for dimension , then the middle term in this sum is trivial.
To compute the last sum of equation, we apply a change of variable :
To compute the first sum of equation, we apply a change of variable :
In order to compute sum from to , we multiply by and consider
In order to compute sum from to , we consider