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 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 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} )=\left[\mat{A}\mat{\Delta}( \vect{\tau} ) \right] \,\mat{R}\, \left[ \mat{\Delta}( \vect{\tau} )\mat{A}\right]

where \mat{R} \in  \mathcal{M}_{d \times d}([-1, 1]) is a correlation matrix, \mat{\Delta}( \vect{\tau} ) \in \mathcal{M}_{d \times d}(\Rset) is defined by:

(2)\mat{\Delta}( \vect{\tau} )= \mbox{Diag}(e^{-\lambda_1|\tau|/2}, \dots, e^{-\lambda_d|\tau|/2})

and \mat{A}\in \mathcal{M}_{d \times d}(\Rset) is defined by:

(3)\mat{A}= \mbox{Diag}(a_1, \dots, a_d)

with \lambda_i>0 and a_i>0 for any i.

We call \vect{a} the amplitude vector and \vect{\lambda} the scale vector. 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 \mat{\Delta}( \vect{\tau} ) that models the correlation between the marginal random variables X^i_{\vect{t}} and X^i_{\vect{t}+\vect{\tau}}:

    \Cor{X_{\vect{t}}^i,X^i_{\vect{t}+\vect{\tau}}} = e^{-\lambda_i|\tau|}

  • the matrix \mat{A} that models the variance of each marginal random variable:

    \Var{X_{\vect{t}}} = (a_1, \dots, a_d)

This model is such that:

   C_{ij}^{stat}(\vect{\tau}) & = a_ie^{-\lambda_i|\tau|/2}R_{i,j}a_je^{-\lambda_j|\tau|/2},\quad i\neq j\\
   C_{ii}^{stat}(\vect{\tau}) & = a_ie^{-\lambda_i|\tau|/2}R_{i,i}a_ie^{-\lambda_j|\tau|/2}=a_i^2e^{-\lambda_i|\tau|}

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

(6)\forall \vect{t} \in \cD,\quad \mat{C}^{spat} = \Expect{X_{\vect{t}}X^t_{\vect{t}}} = \mat{A}\,\mat{R}\, \mat{A}