Covariance modelsΒΆ

We consider X: \Omega \times\cD \mapsto \Rset^d a multivariate stochastic process of dimension d, where \omega \in \Omega is an event, \cD is a domain of \Rset^n, \vect{t}\in \cD is a multivariate index and X(\omega, \vect{t}) \in \Rset^d.

We note X_{\vect{t}}: \Omega \rightarrow \Rset^d the random variable at index \vect{t} \in \cD defined by X_{\vect{t}}(\omega)=X(\omega, \vect{t}) and X(\omega): \cD  \mapsto \Rset^d a realization of the process X, for a given \omega \in \Omega defined by X(\omega)(\vect{t})=X(\omega, \vect{t}).

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

  • m : \cD \mapsto  \Rset^d its mean function, defined by m(\vect{t})=\Expect{X_{\vect{t}}},
  • C : \cD \times \cD \mapsto  \cS_d^+(\Rset) its covariance function, defined by C(\vect{s}, \vect{t})=\Expect{(X_{\vect{s}}-m(\vect{s}))\Tr{(X_{\vect{t}}-m(\vect{t}))}},
  • R : \cD \times \cD \mapsto  \cS_d^+(\Rset) its correlation function, defined for all (\vect{s}, \vect{t}), by R(\vect{s}, \vect{t}) such that for all (i,j), R_{ij}(\vect{s}, \vect{t})=C_{ij}(\vect{s}, \vect{t})/\sqrt{C_{ii}(\vect{s}, \vect{t})C_{jj}(\vect{s}, \vect{t})}.

In a general way, the covariance models write:

C(\vect{s}, \vect{t}) = \mat{L}_{\rho}\left(\dfrac{\vect{s}}{\theta},
                        \dfrac{\vect{t}}{\theta}\right)\,
                        \mbox{Diag}(\vect{\sigma}) \, \mat{R} \,
                        \mbox{Diag}(\vect{\sigma}) \,
                        \Tr{\mat{L}}_{\rho}\left(\dfrac{\vect{s}}{\theta},
                        \dfrac{\vect{t}}{\theta}\right), \quad
                        \forall (\vect{s}, \vect{t}) \in \cD

where:

  • \vect{\theta} \in \Rset^n is the scale parameter
  • \vect{\sigma} \in \Rset^d id the amplitude parameter
  • \mat{L}_{\rho}(\vect{s}, \vect{t}) is the Cholesky factor of \mat{\rho}(\vect{s}, \vect{t}):

\mat{L}_{\rho}(\vect{s}, \vect{t})\,\Tr{\mat{L}_{\rho}(\vect{s}, \vect{t})}
= \mat{\rho}(\vect{s}, \vect{t})

The correlation function \mat{\rho} 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 X_{\vect{t}} which is given by the correlation matrix \mat{R} \in \cS_d^+(\Rset) and the vector of marginal variances \vect{\sigma} \in \Rset^d. The spatial correlation does not depend on \vect{t} \in \cD. For each \vect{t}, it links together the components of X_{\vect{t}}.

  • the correlation between X_{\vect{s}} and X_{\vect{t}} which is given by \mat{\rho}(\vect{s}, \vect{t}).

    • In the general case, the correlation links each component X^i_{\vect{t}} to all the components of X_{\vect{s}} and \mat{\rho}(\vect{s}, \vect{t}) \in \cS_d^+(\Rset);
    • In some particular cases, the correlation is such that X^i_{\vect{t}} depends only on the component X^i_{\vect{s}} and that link does not depend on the component i. In that case, \mat{\rho}(\vect{s}, \vect{t}) can be defined from the scalar function \rho(\vect{s}, \vect{t}) by \mat{\rho}(\vect{s}, \vect{t}) = \rho(\vect{s}, \vect{t})\, \mat{I}_d. Then, the covariance model writes:

C(\vect{s}, \vect{t}) = \rho\left(\dfrac{\vect{s}}{\theta},
                                  \dfrac{\vect{t}}{\theta}\right)\,
                        \mbox{Diag}(\vect{\sigma}) \, \mat{R} \,
                        \mbox{Diag}(\vect{\sigma}), \quad
                        \forall (\vect{s}, \vect{t}) \in \cD