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Penalized structured additive regression for space-time data: a Bayesian perspective. (English) Zbl 1073.62025

Summary: We propose extensions of penalized spline generalized additive models for analyzing space-time regression data and study them from a Bayesian perspective. Nonlinear effects of continuous covariates and time trends are modelled through Bayesian versions of penalized splines, while correlated spatial effects follow a Markov random field prior. This allows to treat all functions and effects within a unified general framework by assigning appropriate priors with different forms and degrees of smoothness. Inference can be performed either with full (FB) or empirical Bayes (EB) posterior analysis. FB inference using MCMC techniques is a slight extension of previous work. For EB inference, a computationally efficient solution is developed on the basis of a generalized linear mixed model representation.
The second approach can be viewed as posterior mode estimation and is closely reated to penalized likelihood estimation in a frequentist setting. Variance components, corresponding to inverse smoothing parameters, are then estimated by marginal likelihood. We carefully compare both inferential procedures in simulation studies and illustrate them through data applications. The methodology is available in the open domain statistical package Bayes \(X\) and as an S-plus/R function.

MSC:

62F15 Bayesian inference
62M40 Random fields; image analysis
62G08 Nonparametric regression and quantile regression
65C40 Numerical analysis or methods applied to Markov chains
62C12 Empirical decision procedures; empirical Bayes procedures
62J10 Analysis of variance and covariance (ANOVA)
62J12 Generalized linear models (logistic models)

Software:

BayesX; S-PLUS; SemiPar; R
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