A Hierarchical Spatiotemporal Statistical Model Motivated by Glaciology

Abstract

In this paper, we extend and analyze a Bayesian hierarchical spatio-temporal model for physical systems. A novelty is to model the discrepancy between the output of a computer simulator for a physical process and the actual process values with a multivariate random walk. For computational efficiency, linear algebra for bandwidth limited matrices is utilized, and first-order emulator inference allows for the fast emulation of a numerical partial differential equation (PDE) solver. A test scenario from a physical system motivated by glaciology is used to examine the speed and accuracy of the computational methods used, in addition to the viability of modeling assumptions. We conclude by discussing how the model and associated methodology can be applied in other physical contexts besides glaciology.

A Review of Bayesian Modelling in Glaciology

“…The model was further developed by Gopalan et al (2019), with a particular focus on computational efficiency, an issue that must be often addressed in large-scale glaciological applications. Specifically, the use of surrogate process models constructed via first-order emulators (Hooten et al, 2011), parallelisation of an approximation to the log-likelihood, and the use of sparse matrix algebra routines were shown to help alleviate computational difficulties encountered when making inference with large Bayesian hierarchical glaciology models.…”

A higher-order singular value decomposition tensor emulator for spatio-temporal simulators

“…The most basic model assumes that the error term is independent and identically Gaussian from observation to observation, though that is not a necessary requirement. For instance, Gopalan et al (2019) consider a multivariate random walk with spatio-temporal correlation.…”

“…For example, in the case where physical observations are available for the process being emulated, one can embed the deterministic point emulator in lieu of a computer simulator within a Bayesian hierarchical model as in the Bayesian calibration literature (Kennedy and O'Hagan, 2001;Brynjarsdóttir and O'Hagan, 2014;Gopalan et al, 2019). From there, uncertainty estimates for the physical parameters and predictions of the physical process can be obtained with Bayesian inference.…”

“…A Bayesian approach is then used, with MCMC for example, to make inference about θ and δ. A variation that is detailed in Gopalan et al (2019) is to replace the deterministic function δ(., ., .) with a spatio-temporal stochastic process, S i .…”

“…In situations where one has a greater knowledge of the fundamental process dynamics, it is often preferable to embed a "black box" dynamical model (simulator) within a formal statistical framework, such as is done in the computer model calibration literature (Kennedy and O'Hagan, 2001). Examples of such black box simulators include PDE models (Gopalan et al, 2019) or agent-based models that have multiple individuals interacting across time and space (e.g., Fadikar et al, 2018).…”