# Parametric stationary covariance models¶

Let be a multivariate stationary normal process where . The process is supposed to be zero mean. It is entirely defined by its covariance function , defined by for all .
If the process is continuous, then . In the discrete case, is a lattice.
This use case illustrates how the User can create a covariance function from parametric models. The library proposes the multivariate Exponential model as one of the possible parametric models for the covariance function .

The multivariate exponential model

This model defines the covariance function by:

(1)

where is a correlation matrix, is defined by:

(2)

and is defined by:

(3)

with and for any .

We call the amplitude vector and the scale vector. The expression of is the combination of:

• the matrix that models the spatial correlation between the components of the process at any vertex (since the process is stationary):

(4)

• the matrix that models the correlation between the marginal random variables and :

• the matrix that models the variance of each marginal random variable:

This model is such that:

(5)

It is possible to define the exponential model from the spatial covariance matrix rather than the correlation matrix :

(6)