Penalized additive regression for space-time data: a Bayesian perspective

We propose extensions of penalized spline generalized additive
models for analysing space-time regression data and study them from
a Bayesian perspective. Non-linear 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 own 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 related to penalized
likelihood estimation in a frequentist setting. Variance
components, corresponding to smoothing parameters, are then
estimated by using marginal likelihood. We carefully compare both
inferential procedures in simulation studies and illustrate them
through real data applications. The methodology is available in the
open domain statistical package BayesX and as an S-plus/R function.