Gaussian Predictive Process Models for Large Spatial Data Sets

Abstract

SummaryWith scientific data available at geocoded locations, investigators are increasingly turning to spatial process models for carrying out statistical inference. Over the last decade, hierarchical models implemented through Markov chain Monte Carlo methods have become especially popular for spatial modelling, given their flexibility and power to fit models that would be infeasible with classical methods as well as their avoidance of possibly inappropriate asymptotics. However, fitting hierarchical spatial models often involves expensive matrix decompositions whose computational complexity increases in cubic order with the number of spatial locations, rendering such models infeasible for large spatial data sets. This computational burden is exacerbated in multivariate settings with several spatially dependent response variables. It is also aggravated when data are collected at frequent time points and spatiotemporal process models are used. With regard to this challenge, our contribution is to work with what we call predictive process models for spatial and spatiotemporal data. Every spatial (or spatiotemporal) process induces a predictive process model (in fact, arbitrarily many of them). The latter models project process realizations of the former to a lower dimensional subspace, thereby reducing the computational burden. Hence, we achieve the flexibility to accommodate non-stationary, non-Gaussian, possibly multivariate, possibly spatiotemporal processes in the context of large data sets. We discuss attractive theoretical properties of these predictive processes. We also provide a computational template encompassing these diverse settings. Finally, we illustrate the approach with simulated and real data sets.

Predictive comparison of joint longitudinal-survival modeling: a case study illustrating competing approaches

“…as suggested by Tsiatis and Davidian (2004) as an avenue of unexplored research. See also Kneib (2006) and Banerjee et al (2008) for related ideas. We consider this approach in Sect.…”

Review on statistical methods for large spatial Gaussian data

“…The methods in the second category seek to approximate the spatial process {Z(s)} by a lower dimensional space process {Z(s)} with the use of smoothing techniques, such as kernel convolutions, moving averages, low rank splines, or basis functions, see e.g., Wikle and Cressie (1999), Lin et al (2000), Kammann and Wand (2003), Paciorek (2007), and Banerjee et al (2008). For example, Banerjee et al (2008) considered a set of knots s * = {s * 1 , .…”

Comparing and selecting spatial predictors using local criteria

“…The spatial mixed effects model incorporates spatial dependence using a random linear combination of spatial basis functions. The number of terms in this linear combination is specified to be much smaller than the number of data points n, which is often called a "reduced-rank" approach to spatial prediction (Cressie and Johannesson, 2008;Banerjee et al, 2008;Finley et al, 2009;Wikle, 2010). An advantage of a reduced-rank approach is that the resulting FRK predictor can be computed very quickly.…”

“…To compute the FRK predictor, we used Matlab code provided by The Ohio State University's Spatial Statistics and Environmental Statistics (SSES) website (Katzfuss and Cressie, 2011b). Banerjee et al (2008) and Finley et al (2009) also use a reduced-rank approach to define a spatialpredictor called the modified predictive process (MPP). Their approach is to first predict random effects, which they call the predictive process.…”