Stochastic Gradient MCMC for Massive Geostatistical Data: The Vecchia approximation

Table of Links

Abstract and 1 Introduction

1.1 Methods to handle large spatial datasets

1.2 Review of stochastic gradient methods

2 Matern Gaussian Process Model and its Approximations

2.1 The Vecchia approximation

3 The SG-MCMC Algorithm and 3.1 SG Langevin Dynamics

3.2 Derivation of gradients and Fisher information for SGRLD

4 Simulation Study and 4.1 Data generation

4.2 Competing methods and metrics

4.3 Results

5 Analysis of Global Ocean Temperature Data

6 Discussion, Acknowledgements, and References

Appendix A.1: Computational Details

Appendix A.2: Additional Results

2.1 The Vecchia approximation

For any set of spatial locations, the joint distribution of Y can be written as a product of univariate conditional distributions, which can then be approximated by a Vecchia approximation (Vecchia, 1988; Stein et al., 2004; Datta et al., 2016; Katzfuss and Guinness, 2021):

Let p(β, θ) be the prior distribution on the regression and covariance parameters. Using (5) we can write the posterior as (ignoring a constant that does not depend on the parameters)

Hence the log-likelihood and log-posterior of the parameters {β, θ} can be written as a sum of conditional normal log-densities, where the conditioning set is at most of size m. The cost of computing the log-posterior in (6) is linear in n and cubic in m.

Using (8), we can construct an unbiased estimate of the gradient of the Vecchia log-posterior based on a minibatch of the data: