Meta-Kriging: Scalable Bayesian Modeling and Inference for Massive Spatial Datasets

Abstract

Spatial process models for analyzing geostatistical data entail computations that become prohibitive as the number of spatial locations becomes large. There is a burgeoning literature on approaches for analyzing large spatial datasets. In this article, we propose a divide-and-conquer strategy within the Bayesian paradigm. We partition the data into subsets, analyze each subset using a Bayesian spatial process model and then obtain approximate posterior inference for the entire dataset by combining the individual posterior distributions from each subset. Importantly, as often desired in spatial analysis, we offer full posterior predictive inference at arbitrary locations for the outcome as well as the residual spatial surface after accounting for spatially oriented predictors. We call this approach “Spatial Meta-Kriging” (SMK). We do not need to store the entire data in one processor, and this leads to superior scalability. We demonstrate SMK with various spatial regression models including Gaussian processes and tapered Gaussian processes. The approach is intuitive, easy to implement, and is supported by theoretical results presented in the supplementary material available online. Empirical illustrations are provided using different simulation experiments and a geostatistical analysis of Pacific Ocean sea surface temperature data.

Prediction for Big Data Through Kriging: Small Sequential and One-Shot Designs

“…These criteria are also discussed in Kleijnen (2015, p. 120). Coverage is used in Guhaniyogi and Banerjee (2018, Fig. 2).…”

“…We also refer to Bradley et al (2016), Chilès and Desassis (2018), Damianou (2015), Guinness (2018), Meng and Ng (2016), Nickson et al (2015), Sung et al (2016), Tzeng and Huang (2018), and Van Stein et al (2017). We, however, focus on the method that is detailed in Gramacy (2016) and Gramacy and Apley (2015), and compare our methods with Gramacy's method (Gramacy's method is also compared with a new method in Guhaniyogi and Banerjee (2018)). We observe that two NN variations are used in Gramacy (2016, Section 3.1) and Gramacy and Apley (2015, Fig.…”

“…For a further discussion of frequentist and Bayesian approaches we also refer to Efron (2015) There are many sophisticated methods to analyze Kriging models for big data. Guhaniyogi & Banerjee (2018) states: "There is a burgeoning literature on approaches for analyzing large spatial datasets"; that article brie ‡y discusses several approaches, and proposes a new Bayesian approach. We also refer to Bradley et al (2016), Chilès and Desassis (2018), Damianou (2015), Guinness (2018), Meng and Ng (2016), Nickson et al (2015), Sung et al (2016), Tzeng and Huang (2018), and Van Stein et al (2017).…”

“…Furthermore, we assume that the simulation is run on a traditional personal computer; i.e., we do not consider parallel computing architectures; such architectures are considered in Gramacy (2016), , Guhaniyogi and Banerjee (2018), Guinness (2018), Gutiérrez de Ravé et al (2014), Morgan et al (2017), Van Stein et al (2017), and Xu et al (2016).…”

“…We notice that Gramacy (2016) also allows output that has noise with a small constant "intrinsic" variance; in future research we may investigate intrinsic noise with big variances that vary with the input combinations; also see Section 6. We use a frequentist approach (instead of a Bayesian approach, which is used in Angelino et al (2016), Damianou (2015), Guhaniyogi and Banerjee (2018), Gramacy (2016), Nickson et al (2015), and Pronzato and Rendas (2017)). An advantage of such a frequentist approach is that we do not need to specify a prior distribution for the Kriging (hyper)parameters; we …nd it hard to specify such a prior distribution.…”