# Covariance models¶

We consider a multivariate stochastic process of dimension , where is an event, is a domain of , is a multivariate index and .

We note the random variable at index defined by and a realization of the process , for a given defined by .

If the process is a second order process, we note:

• its mean function, defined by ,
• its covariance function, defined by ,
• its correlation function, defined for all , by such that for all , .

In a general way, the covariance models write:

where:

• is the scale parameter
• id the amplitude parameter
• is the Cholesky factor of :

The correlation function may depend on additional specific parameters which are not made explicit here.

The global correlation is given by two separate correlations:

• the spatial correlation between the components of which is given by the correlation matrix and the vector of marginal variances . The spatial correlation does not depend on . For each , it links together the components of .

• the correlation between and which is given by .

• In the general case, the correlation links each component to all the components of and ;
• In some particular cases, the correlation is such that depends only on the component and that link does not depend on the component . In that case, can be defined from the scalar function by . Then, the covariance model writes: