Properties and comparison of some Kriging sub-model aggregation methods

TL;DR

This paper analyzes various Kriging aggregation methods, highlighting the advantages of nested Kriging in terms of consistency, interpretability, and computational efficiency, especially for large datasets and extended models.

Contribution

It provides a theoretical comparison of aggregation methods, demonstrating the superior properties of nested Kriging over simpler approaches.

Findings

Aggregation methods ignoring covariance can be inconsistent.

Nested Kriging is consistent and interpretable as an exact conditional distribution.

Efficient computation of conditional covariances is possible with nested Kriging.

Abstract

Kriging is a widely employed technique, in particular for computer experiments, in machine learning or in geostatistics. An important challenge for Kriging is the computational burden when the data set is large. This article focuses on a class of methods aiming at decreasing this computational cost, consisting in aggregating Kriging predictors based on smaller data subsets. It proves that aggregation methods that ignore the covariancebetween sub-models can yield an inconsistent final Kriging prediction. In contrast, a theoretical study of the nested Kriging method shows additional attractive properties for it: First, this predictor is consistent, second it can be interpreted as an exact conditional distribution for a modified process and third, the conditional covariances given the observations can be computed efficiently. This article also includes a theoretical and numerical analysis of how the assignment of the observation points to the sub-models can affect the prediction ability of the aggregated model. Finally, the nested Kriging method is extended to measurement errors and to universal Kriging.