Helmut Waldl

Efficient prediction designs for random fields

For estimation and predictions of random fields, it is increasingly acknowledged that the kriging variance may be a poor representative of true uncertainty. Experimental designs based on more elaborate criteria that are appropriate for empirical kriging (EK) are then often non-space-filling and very costly to determine. In this paper, we investigate the possibility of using a compound criterion inspired by an equivalence theorem type relation to build designs quasi-optimal for the EK variance when space-filling designs become unsuitable. Two algorithms are proposed, one relying on stochastic optimization to explicitly identify the Pareto front, whereas the second uses the surrogate criteria as local heuristic to choose the points at which the (costly) true EK variance is effectively computed. We illustrate the performance of the algorithms presented on both a simple simulated example and a real oceanographic dataset.

Relations Between Designs for Prediction and Estimation in Random Fields: An Illustrative Case

Two approaches are considered to design experiments for a correlated random field when the objective is to obtain precise predictions over the whole experimental domain. Both take the uncertainty of the estimated parameters of the correlation structure of the random field into account. The first one corresponds to a compound D-optimality criterion for both the trend and covariance parameters. The second one relies on an approximation of the mean squared prediction error already proposed in the literature. It is conjectured, and shown on a paradigmatic example, that for some particular settings both approaches yield similar optimal designs, thereby revealing a sort of accordance between the two criteria for random fields. However, our example also shows that a strict equivalence theorem as in the uncorrelated case is not achievable. As a side issue we cast doubts on the ubiquity of equidistant space-filling designs.

On the ML Estimation of the Correlation Parameter in Generalized Exponential Models

Maximum likelihood estimation (ML) in Generalized Exponential models (GE) is an awkward task in many applications. Especially for small n and big r we observe a large bias of the ML of r. From the theoretical point of view this is not surprising since σ2 and r are not simultaneously identifiable [1]. We develop a bias reduction method and illustrate its effectiveness.