Testing invariance for random field modelling

The application of the collocation theory to the prediction of some random field functional heavily depends on the knowledge of the covariance function. Whether we include the estimation of the covariance into a unique theoretical set up with the prediction of the signal, or we do it separately in a more traditional way, this step can be performed only under the assumption of some stochastic invariance of the field; under such a hypothesis, in fact, we can use a single sample to produce the empirical covariance estimate. However one easily realizes that the empirical estimator provides numerical answers even if applied to signals whose stochastic invariance is out of question. This calls for some criterion to decide, at least a posteriori, whether the hypothesis, on which the covariance estimation is based, is likely to be true or not. Three different testing procedures were considered and applied to simulated data. The results obtained are discussed in the paper; they lead to a procedure, at least in a simplified context, which verifies whether the sample has or not a deterministic linear trend. A more general result is finally presented, that is how to retrieve an estimate of the empirical covariance function of the signal, correcting the one estimated on the residues of the linear regression.