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 highlights how User can create a covariance function from parametric models. The library proposes many parametric covariance models. The multivariate Exponential model is one of them. .
Example: the multivariate exponential model¶
This model defines the covariance function by:
(1)¶
where the correlation function is given by:
(2)¶
and the spatial covariance matrix by:
(3)¶
with a correlation matrix, and for any .
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 variance of each marginal random variable:
It is possible to define the exponential model from the spatial covariance matrix rather than the correlation matrix :
(5)¶