Estimation of a non stationary covariance modelΒΆ

Let X: \Omega \times \cD \rightarrow \Rset^{\inputDim} be a multivariate normal process of dimension \inputDim where \cD \subset \Rset^n. X is supposed to be a second order process and we note C : \cD \times \cD \rightarrow \mathcal{M}_{\inputDim \times \inputDim}(\mathbb{R}) its covariance function. We denote (\vect{t}_0, \dots, \vect{t}_{N-1}) the vertices of the common mesh \cM and (\vect{x}_0^k, \dots, \vect{x}_{N-1}^k) the associated values of the field k. We suppose that we have K fields. We recall that the covariance function C writes:

(1)ΒΆ\forall (\vect{s}, \vect{t}) \in \cD \times \cD, \quad C(\vect{s}, \vect{t}) = \Expect{\left(X_{\vect{s}}-m(\vect{s})\right)\left(X_{\vect{t}}-m(\vect{t})\right)^t}

where the mean function m: \cD \rightarrow \Rset^{\inputDim} is defined by:

(2)ΒΆ\forall \vect{t}\in \cD , \quad m(\vect{t}) = \Expect{X_{\vect{t}}}

First, we estimate the covariance function C on the vertices of the mesh \cM. At each vertex \vect{t}_i \in \cM, we use the empirical mean estimator applied to the K fields to estimate:

  1. m(\vect{t}_i) at the vertex \vect{t}_i:

(3)ΒΆ\displaystyle \forall \vect{t}_i \in \cM, \quad m(\vect{t}_i) \simeq \frac{1}{K} \sum_{k=1}^{K} \vect{x}_i^k

  1. C(\vect{t}_i, \vect{t}_j) at the vertices (\vect{t}_i, \vect{t}_j):

(4)ΒΆ\displaystyle \forall (\vect{t}_i, \vect{t}_j) \in \cD \times \cD, \quad C(\vect{t}_i, \vect{t}_j) \simeq \frac{1}{K} \sum_{k=1}^{K} \left( \vect{x}_i^k - m(\vect{t}_i) \right) \left( \vect{x}_j^k - m(\vect{t}_j) \right)^t

Then, we build a covariance function defined on \cD \times \cD which is a piecewise constant function defined on \cD \times \cD by:

\begin{aligned} \forall (\vect{s}, \vect{t}) \in \cD \times \cD, \, C^{stat}(\vect{s}, \vect{t}) = C(\vect{t}_k, \vect{t}_l)\end{aligned}

where k is such that \vect{t}_k is the vertex of \cM the nearest to \vect{s} and \vect{t}_l the nearest to \vect{t}.