Similarity Based Kriging: A dimension-free approach for prediction and adaptive experimental design

Abstract:

Kriging has recently been revived as a powerful tool for predicting the outcome of costly numerical experiments and for guiding simulator evaluations, be it for optimization, inversion, or uncertainty quantification purposes. One key requirement for a successful application of Kriging is to formulate a suitable covariance (or variogram) model reflecting the (dis)similarity between locations or “inputs”, in the sense that two inputs with a high covariance (or a low variogram) value are expected to correspond to close outcomes. Here we present an approach where the inputs can be very high-dimensional, like curves and maps, or even not identifiable as vectors, like shapes or geological scenarios. We extend the Kriging methodology to a very general set-up where a collection of inputs is given together with some measure of similarity between them, the response of interest is known only for a subset of the inputs, and predictions of the response are needed for the remaining inputs. The proposed Similarity Based Kriging (SBK) method relies on the combination of Multi-Dimensional Scaling (MDS) together with some well-chosen class of positive definite kernels. Possible degrees of freedom that govern the similarity measure and the MDS algorithms employed as well as the selected kernel make it a very versatile approach. Provided a relevant (dis)similarity measure between inputs is available, SBK appears as a simple and powerful method for predicting the outcome of complex experiments in a dimension-free set-up. After a global introduction to SBK, we will present results that concern the optimal tuning of MDS and kernel parameters. The potential of SBK for prediction and adaptive experimental design in geosciences will finally be illustrated using data sets from aquifer modelling and beyond.