Covariance models¶

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

We note $X_{\vect{t}}: \Omega \rightarrow \Rset^{\inputDim}$ the random variable at index $\vect{t} \in \cD$ defined by $X_{\vect{t}}(\omega)=X(\omega, \vect{t})$ and $X(\omega): \cD \mapsto \Rset^{\inputDim}$ 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:

$m : \cD \mapsto \Rset^{\inputDim}$ its mean function, defined by $m(\vect{t})=\Expect{X_{\vect{t}}}$ ,
$C : \cD \times \cD \mapsto \cS_{\inputDim}^+(\Rset)$ its covariance function, defined by $C(\vect{s}, \vect{t})=\Expect{(X_{\vect{s}}-m(\vect{s}))\Tr{(X_{\vect{t}}-m(\vect{t}))}}$ ,
$R : \cD \times \cD \mapsto \cS_{\inputDim}^+(\Rset)$ 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:

$\vect{\theta} \in \Rset^{\sampleSize}$ is the scale parameter
$\vect{\sigma} \in \Rset^{\inputDim}$ 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 $\mat{R} \in \cS_{\inputDim}^+(\Rset)$ and the vector of marginal variances $\vect{\sigma} \in \Rset^{\inputDim}$ . 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 $\mat{\rho}(\vect{s}, \vect{t}) \in \cS_{\inputDim}^+(\Rset)$ ;

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 $\mat{\rho}(\vect{s}, \vect{t}) = \rho(\vect{s}, \vect{t})\, \mat{I}_{\inputDim}$ . Then, the covariance model writes:

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

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Covariance models¶