Permutation and Grouping Methods for Sharpening Gaussian Process Approximations
Joseph Guinness · Technometrics · 2018
Joseph Guinnessa* http://orcid.org/0000-0003-0564-6639a Department of Statistics, North Carolina State University, Raleigh, NCCONTACT Joseph Guinness [email protected] versions of one or more of the figures in the article can be found online at www.tandfonline.com/r/TECH.Supplementary materials for this article are available online. Please go to http://www.tandfonline.com/r/TECH.ABSTRACTVecchia’s approximate likelihood for Gaussian process parameters depends on how the observations are ordered, which has been cited as a deficiency. This article takes the alternative standpoint that the ordering can be tuned to sharpen the approximations. Indeed, the first part of the article includes a systematic study of how ordering affects the accuracy of Vecchia’s approximation. We demonstrate the surprising result that random orderings can give dramatically sharper approximations than default coordinate-based orderings. Additional ordering schemes are described and analyzed numerically, including orderings capable of improving on random orderings. The second contribution of this article is a new automatic method for grouping calculations of components of the approximation. The grouping methods simultaneously improve approximation accuracy and reduce computational burden. In common settings, reordering combined with grouping reduces Kullback–Leibler divergence from the target model by more than a factor of 60 compared to ungrouped approximations with default ordering. The claims are supported by theory and numerical results with comparisons to other approximations, including tapered covariances and stochastic partial differential equations. Computational details are provided, including the use of the approximations for prediction and conditional simulation. An application to space-time satellite data is presented.