Spatial-temporal data requires flexible regression models which can model the dependence of responses on space- and time-dependent covariates. In this paper, we describe a semiparametric space-time model from a Bayesian perspective. Nonlinear time dependence of covariates and the interactions among the covariates are constructed by local linear and piecewise linear models, allowing for more flexible orientation and position of the covariate plane by using time-varying basis functions. Space-varying covariate linkage coefficients are also incorporated to allow for the variation of space structures across the geographical location. The formulation accommodates uncertainty in the number and locations of the piecewise basis functions to characterize the global effects, spatially structured and unstructured random effects in relation to covariates. The proposed approach relies on variable selection-type mixture priors for uncertainty in the number and locations of basis functions and in the space-varying linkage coefficients. A simulation example is presented to evaluate the performance of the proposed approach with the competing models. A real data example is used for illustration.
AssunçãoRM (2003) Space varying coefficient models for small area data. Environmetrics, 14, 453–73.
2.
AssunçãoRMPotterJECavenaghiSM (2002) A Bayesian space varying parameter model applied to estimating fertility schedules. Statistics in Medicine, 21, 2057–2075.
3.
BanerjeeSJohnsonGA (2006) Coregionalized single and multi-resolution spatially varying growth-curve modelling. Biometrics, 62, 864–76.
4.
BesagJ (1974) Spatial interaction and statistical analysis of lattice systems. Journal of the Royal Statistical Society, Series B, 36, 192–36.
CowlesMKCarlinBP (1995) Markov Chain Monte Carlo diagnostics: A comparative review. Journal of the American Statistical Association, 91, 883–04.
7.
CuiWGeorgeEI (2008) Empirical Bayes vs. fully Bayes variable selection. Journal of Statistical Planning and Inference, 138, 888–900.
8.
DenisonDGTHolmesCCMallickBKSmithAFM (2002) Bayesian Methods for Nonlinear Classification and Regression. Chichester: John Wiley.
9.
DeyDChenMHChangeH (1997) Bayesian approach for nonlinear random effects models. Biometrics, 53, 1239–1252.
10.
DraperDKrnjajiM (2006) Bayesian model specification. Technical report, Department of Applied Mathematics and Statistics, Baskin School of Engineering, University of California, Santa Cruz.
11.
DreassiEBiggeriACatelanD (2005) Space-time models with time dependent covariates for the analysis of the temporal lag between socio-economic factors and mortality. Statistics in Medicine, 24, 1919–32.
GamermanDMoreiraARBRueH (2003) Space-varying regression models: specifications and simulation. Computational Statistics and Data Analysis, 42, 513–33.
14.
GeisserS (1984) On prior distribution for binary trials. American Statistician, 38, 244–51 (with discussion).
15.
GeisserS (1993) Predictive inference: An introduction. Chapman & Hall, London.
16.
GelfandAEDeyDChangH (1992) Model determination using predictive distributions with implementation via sampling based methods (with discussion). In BernadoJ. (eds), Bayesian Statistics 4. Oxford University Press, 147–67.
17.
GelfandABanerjeeSGamermanD (2005) Spatial process modelling for univariate and multivariate dynamic spatial data. Environmetrics, 16, 465–79.
18.
GeorgeEIMcCullochRE (1993) Variable selection via Gibbs sampling. Journal of the American Statistical Association, 88, 881–89.
19.
GilksWRBestNGTanKKC (1995) Adaptive rejection metropolis sampling within Gibbs sampling. Journal of Applied Statistics, 44, 455–72.
20.
GreenP (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82, 711–32.
HolmesCMallickB (2001) Bayesian regression with multivariate linear splines. Journal of the Royal Statistical Society, Series B, 63, 3–17.
23.
HolmesCMallickB (2003) Generalized nonlinear modelling with multivariate free-knot regression splines. Journal of the American Statistical Association, 98, 352–68.
24.
HossainMMLawsonAB (2010) Space-time Bayesian small area disease risk models: development and evaluation with a focus on cluster detection. Environmental and Ecological Statistics, 17, 73–95.
25.
KneibTFahrmeirL (2006) Structured additive regression for categorical space-time data: a mixed model approach. Biometrics, 62, 109–18.
26.
Knorr-HeldL (2000) Bayesian modelling of inseparable space-time variation in disease risk. Statistics in Medicine, 19, 2555–67.
27.
Knorr-HeldLBesagJ (1998) Modelling risk from a disease in time and space. Statistics in Medicine, 17, 2045–60.
28.
LagazioCBiggeriADreassiE (2003) Age-period-cohort models and disease mapping. Environmetrics, 14, 475–90.
29.
LagazioCDreassiEBiggeriA (2001) A hierarchical Bayesian model for space-time variation of disease risk. Statistical Modelling, 1, 17–29.
30.
LunnDJThomasABestNSpiegelhalterD (2000) WinBUGS–a Bayesian modelling framework: concepts, structure, and extensibility. Statistics and Computing, 10, 325–37.
31.
PaezMSGamermanDLandimFMPSalazarE (2008) Spatially varying dynamic coefficient models. Journal of Statistical Planning and Inference, 138, 1038–58.
PlummerM (2008) Penalized loss functions for Bayesian model comparison. Biostatistics, 9, 523–39.
34.
PlummerMBestNGCowlesKVinesK (2006) CODA: Convergence Diagnosis and Output Analysis for MCMC. R News, 6, 7–11.
35.
RichardsonSAbellánJJBestN (2006) Bayesian spatio-temporal analysis of joint patterns of male and female lung cancer risks in Yorkshire. Statistical Methods in Medical Research, 15, 385–407.
36.
RuppertD (2002) Selecting the number of knots for penalized splines. Journal of Computational and Graphical Statistics, 11, 735–57.
37.
RuppertDWandMPCarollRJ (2003) Semiparametric regression. Cambridge: Cambridge University Press.
38.
SchmidVHeldL (2004) Bayesian extrapolation of space–time trends in cancer registry data. Biometrics, 60, 1034–42.
39.
ScottJBergerJ (2006) An exploration of aspects of Bayesian multiple testing. Journal of Statistical Planning and Inference, 136, 2144–62.
40.
ScottJBergerJ (2008) Multiple testing, empirical bayes, and the variable-selection problem. Discussion Paper 2008-10, Department of Statistical Science, Duke University.
41.
SinhaDDeyDK (1997) Semiparametric Bayesian analysis of survival data. Journal of the American Statistics Association, 92, 1195–1212.
42.
SpiegelhalterDThomasABestNGilksW (1996) BUGS 0.5: Bayesian inference using Gibbs sampling manual. Cambridge, UK: MRC Biostatistics Unit, Institute of Public Health.
43.
SpiegelhalterDJBestNGCarlinBPLindeAVD (2002) Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society, Series B, 64, 1–34.
44.
UgarteMDGoicoaTIbáñezBMilitinoAF (2009) Evaluating the performance of spatio-temporal Bayesian models in disease mapping. Environmetrics, 20, 647–65.
45.
WallerLACarlinBPXiaHGelfandAE (1997) Hierarchical spatio-temporal mapping of disease rates. Journal of the American Statistical Association, 92, 607–17.
46.
ZhaoYStaudenmayerJCoullBAWandMP (2006) General design Bayesian generalized linear mixed models. Statistical Science, 21, 35–51.
