Efficient emulators of computer experiments using compactly supported correlation functions, with an application to cosmology

Abstract

Statistical emulators of computer simulators have proven to be useful in a variety of applications. The widely adopted model for emulator building, using a Gaussian process model with strictly positive correlation function, is computationally intractable when the number of simulator evaluations is large. We propose a new model that uses a combination of low-order regression terms and compactly supported correlation functions to recreate the desired predictive behavior of the emulator at a fraction of the computational cost. Following the usual approach of taking the correlation to be a product of correlations in each input dimension, we show how to impose restrictions on the ranges of the correlations, giving sparsity, while also allowing the ranges to trade off against one another, thereby giving good predictive performance. We illustrate the method using data from a computer simulator of photometric redshift with 20,000 simulator evaluations and 80,000 predictions.Simulation of complex systems has become commonplace in scientific studies. Frequently, simulators (or computer models) are computationally demanding, and relatively few evaluations can be performed. In other cases, the computer models are fast to evaluate but are not readily available to all

laGP: Large-Scale Spatial Modeling via Local Approximate Gaussian Processes inR

“…An orthogonal approach is to perform approximate GP regression, and a common theme in that literature is sparsity, leading to fast matrix decompositions (e.g., Kaufman et al 2012;Sang and Huang 2012). Again, the expansion of capability is one-to-two orders of magnitude, albeit without tapping supercomputer resources which is more practical for most applications.…”

“…Again, the expansion of capability is one-to-two orders of magnitude, albeit without tapping supercomputer resources which is more practical for most applications. For example, Kaufman et al (2012) reported on an experiment with N = 20000. Some approaches in a similar vein include fixed rank kriging (Cressie and Johannesson 2008) and using '''pseudo-inputs" (Snelson and Ghahramani 2006).…”

“…The borehole experiment (Worley 1987;Morris, Mitchell, and Ylvisaker 1993) involves an 8-dimensional input space, and our use of it here follows the setup of Kaufman et al (2012); more details can be found therein. The response y is given by…”

“…Gramacy and Apley (2015)'s similar experiments included variations on the method of compactly supported covariances (CSC) (Kaufman et al 2012) yielding estimators with similar accuracies, but requiring at least an order magnitude more compute time. In fact, they commented that N = 10000 was the limit that CSC could accommodate on their machine due to memory swapping issues.…”

Gaussian processes for computer experiments

“…This becomes problematic when n goes beyond 10 3 −10 4 , and there is currently a large body of literature proposing alternative computation procedures for larger values of n [17,30,31,55].…”