Parametric stationary covariance models

Let X: \Omega \times \cD \rightarrow \Rset^d be a multivariate stationary normal process where \cD \in \Rset^n. The process is supposed to be zero mean. It is entirely defined by its covariance function C^{stat}: \cD \rightarrow \mathcal{M}_{d \times d}(\Rset), defined by C^{stat}(\vect{\tau})=\Expect{X_{\vect{s}}X_{\vect{s}+\vect{\tau}}^t} for all \vect{s}\in \Rset^n.
If the process is continuous, then \cD=\Rset^n. In the discrete case, \cD 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. C^{stat}.

The multivariate exponential model

This model defines the covariance function C^{stat} by:

(1)\forall \vect{\tau} \in \cD,\quad C^{stat}( \vect{\tau} )= \rho\left(\dfrac{\vect{\tau}}{\theta}\right)\, \mat{C^{stat}}(\vect{\tau})

where the correlation function \rho is given by:

(2)\rho(\vect{\tau} ) = e^{-\left\| \vect{\tau} \right\|_2} \quad \forall (\vect{s}, \vect{t}) \in \cD

and the spatial covariance matrix \mat{C^{stat}}(\vect{s}, \vect{t}) by:

(3)\mat{C^{stat}}(\vect{\tau})= \mbox{Diag}(\vect{\sigma}) \, \mat{R} \, \mbox{Diag}(\vect{\sigma}).

with \mat{R} \in \mathcal{M}_{d \times d}([-1, 1]) a correlation matrix , \theta_i>0 and \sigma_i>0 for any i.

The expression of C^{stat} is the combination of:

  • the matrix \mat{R} that models the spatial correlation between the components of the process X at any vertex \vect{t} (since the process is stationary):

    (4)\forall \vect{t}\in \cD,\quad \mat{R} = \Cor{X_{\vect{t}}, X_{\vect{t}}}

  • the matrix \mbox{Diag}(\vect{\sigma}) that models the variance of each marginal random variable:

    \begin{aligned} \Var{X_{\vect{t}}} = (\sigma_1, \dots, \sigma_d) \end{aligned}

It is possible to define the exponential model from the spatial covariance matrix \mat{C}^{spat} rather than the correlation matrix \mat{R} :

(5)\forall \vect{t} \in \cD,\quad \mat{C}^{spat} = \Expect{X_{\vect{t}}X^t_{\vect{t}}} = \mbox{Diag}(\vect{\sigma})\,\mat{R}\, \mbox{Diag}(\vect{\sigma})