Invited Session IV: High-Dimensional Spatial Data
Saturday, October 6, 10:45 a.m. – 12:45 p.m.
Session Chair: Victor De Oliveria
Efficient Time-Frequency Representations in High-Dimensional Spatial and Spatio-Temporal Models
Chris Wikle
Salient features of high-dimensional time-dependent outcomes and/or predictors may be difficult to discern through scientific or statistical examination in the time domain. Such features often become more pronounced and possibly more interpretable when considered from a time-frequency perspective. Critically, such ³spectrogram²-based representations can be considered analogous to spatial processes, which can be effectively represented by common reduced-rank methods. When combined with efficient variable selection approaches, such representations can improve prediction, classification, and interpretation of spatial and temporal responses.
Dimension Reduction and Alleviation of Confounding for Spatial Generalized Linear Mixed Models
John Hughes
Non-Gaussian spatial data are very common in many disciplines. When fitting spatial regressions for such data, one needs to account for dependence to ensure reliable inference for the regression coefficients. The spatial generalized linear mixed model (SGLMM) offers a very popular and flexible approach to modeling such data, but the SGLMM suffers from two major shortcomings: (1) variance inflation due to spatial confounding, and (2) high-dimensional spatial random effects that make fully Bayesian inference for such models computationally challenging. We propose a new model that mitigates confounding while including patterns of positive spatial dependence, i.e., attraction, in the random effects and excluding patterns of repulsion. We achieve this by exploiting what we believe to be the intrinsic geometry for these models. The utilization of this geometry permits a natural and dramatic reduction in the dimension of the random effects.
Hierarchical factor models for large spatially misaligned data: A low-rank predictive process approach.
Sudipto Banerjee and Qian Ren.
This article deals with jointly modeling a large number of geographically referenced outcomes observed over a very large number of locations. We seek to capture associations among the variables as well as the strength of spatial association for each variable. In addition, we reckon with the common setting where not all the variables have been observed over all locations, which leads to spatial misalignment. Dimension reduction is needed in two aspects: (i) the length of the vector of outcomes, and (ii) the very large number of spatial locations. Latent variable (factor) models are usually used to address the former, while low-rank spatial processes offer a rich and flexible modeling option for dealing with a large number of locations. We merge these two ideas to propose a class of hierarchical low-rank spatial factor models. Our framework pursues stochastic selection of the latent factors without resorting to complex computational strategies (such as reversible jump algorithms) by utilizing certain identifiability characterizations for the spatial factor model. A Markov chain Monte Carlo (MCMC) algorithm is developed for estimation that also deals with the spatial misalignment problem. We recover the full posterior distribution of the missing values (along with model parameters) in a Bayesian predictive framework. Various additional modeling and implementation issues are discussed as well. We illustrate our methodology with simulation experiments and an environmental data set.
Estimation and prediction in spatial models with block composite likelihoods
Jo Eidsvik
A block composite likelihood is constructed from the joint densities of pairs of adjacent spatial blocks. This allows large datasets to be split into many smaller datasets, each of which can be evaluated separately, and combined through a simple summation. Estimates for unknown parameters are obtained by maximizing the block composite likelihood function. In addition, we propose a method for optimal spatial prediction from the composite likelihood model. Asymptotic variances for both parameter estimates and predictions are computed using Godambe sandwich matrices. The approach gives considerable improvements in computational efficiency, and obviates memory problems. Synthetic and real-data examples are presented.