# Estimation of a stationary covariance model¶

Let be a multivariate stationary normal process of dimension . We only treat here the case where the domain is of dimension 1: (). If the process is continuous, then . In the discrete case, is a lattice. is supposed a second order process with zero mean. It is entirely defined by its covariance function , defined by for all . In addition, we suppose that its spectral density function is defined, where is the set of -dimensional positive definite hermitian matrices. The objective is to estimate from a field or a sample of fields from the process , using first the estimation of the spectral density function and then mapping into using the inversion relation (9), when it is possible. As the mesh is a time grid (), the fields can be interpreted as time series. The estimation algorithm is outlined hereafter. Let be the time grid on which the process is observed and let also be independent realizations of or segments of one realization of the process. Using (9), the covariance function writes:

(1)

where is the element of the matrix and the one of . The integral (1) is approximated by its evaluation on the finite domain :

(2)

Let us consider the partition of the domain as follows:

• is subdivided into segments = with

• be the frequency step,

• be the frequencies on which the spectral density is computed, with

The equation (2) can be rewritten as:

We focus on the integral on each subdomain . Using numerical approximation, we have:

must match with frequency values with respect to the Shannon criteria. Thus the temporal domain of estimation is the following:

• is the time step, such as

• = is subdivided into segments = with

• be the time values on which the covariance is estimated,

The estimate of the covariance value at time value depends on the quantities of form:

(3)

We develop the expression of and and we get:

and:

We denote :

Finally, we get the following expression for integral in (3):

It follows that:

(4)

Examples: