Various approaches for constructing an ensemble of metamodels using local measures

Abstract

Metamodels are approximate mathematical models used as surrogates for computationally expensive simulations. Since metamodels are widely used in design space exploration and optimization, there is growing interest in developing techniques to enhance their accuracy. It has been shown that the accuracy of metamodel predictions can be increased by combining individual metamodels in the form of an ensemble. Several efforts were focused on determining the contribution (or weight factor) of a metamodel in the ensemble using global error measures. In addition, prediction variance is also used as a local error measure to determine the weight factors. This paper investigates the efficiency of using local error measures, and also presents the use of the pointwise cross validation error as a local error measure as an alternative to using prediction variance. The effectiveness of ensemble models are tested on several problems with varying dimensionality: five mathematical benchmark problems, two structural mechanics problems and an automobile crash problem. It is found that the spatial ensemble models show better performances than the global ensemble for the low-dimensional problems, while the global ensemble is a more accurate model than the spatial ensembles for the high-dimensional problems. Ensembles based on pointwise cross validation error and prediction variance provide similar accuracy. The ensemble models based on local measures reduce cross validation errors drastically, but their performances are not that impressive in reducing the error evaluated at random test points, because the pointwise cross validation error is not a good surrogate for the error at a point.