Gaussian Process Modeling of Heterogeneity and Discontinuities Using Voronoi Tessellations

Abstract

Many methods for modeling functions over high-dimensional spaces assume global smoothness properties; such assumptions are often violated in practice. We introduce a method for modeling functions that display heterogeneity or contain discontinuities. The heterogeneity is dealt with by using a combination of Voronoi tessellation, to partition the input space, and separate Gaussian processes to model the function over different regions of the partitioned space. The proposed method is highly flexible since it allows the Voronoi cells to combine to form regions, which enables nonconvex and disconnected regions to be considered. In such problems, identifying the borders between regions is often of great importance and we propose an adaptive sampling method to gain extra information along such borders. The method is illustrated by simulated examples and an application to real data, in which we see improvements in prediction error over the commonly used stationary Gaussian process and other nonstationary variations. In our application, a computationally expensive computer model that simulates the formation of clouds is investigated, the proposed method more accurately predicts the underlying process at unobserved locations than existing emulation methods. Supplementary materials for this article are available online.

KEYWORDS:

• Emulation
• Gaussian process
• Locating discontinuities
• Voronoi tessellation

Acknowledgments

The authors are grateful for the constructive comments of the editors and referees that have led to improvements in this article.

Supplementary Materials

Application details Further details of the cloud modeling example from Section 6 and details of a second example analyzing ammonia levels in the USA. (pdf document)

John E Crowther - Martin Clarke Research Foundation;

Additional information

Funding

The first author gratefully acknowledges the financial support of the John E Crowther–Martin Clarke Research Foundation.