Abstract
This paper presents a complete Bayesian methodology for analyzing spatial data, one which employs proper priors and features diagnostic methods in the Bayesian spatial setting. The spatial covariance structure is modeled using a rich class of covariance functions for Gaussian random fields. A general class of priors for trend, scale, and structural covariance parameters is considered. In particular, we obtain analytic results that allow easy computation of the predictive distribution for an arbitrary prior on the parameters of the covariance function using importance sampling. The computations, as well as model diagnostics and sensitivity analysis, are illustrated with a set of precipitation data.
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References
Ahmed, S. and DeMarsily, G. (1987) Comparison of geostatistical methods for estimating transmissivity using data on transmissivity and specific capacity. Water Resources Research, 23, 1717-37.
Aslanidou, H., Dey, D.K. and Sinha, D., (1998) Bayesian analysis of multivariate survival data using Monte Carlo methods. Canadian Journal of Statistics, 26, 33-48.
Beek, E.G., Stein, A., and Janssen, L.L.F. (1992) Spatial variability and interpolation of daily precipitation amount. Stochastic Hydrology and Hydraulics, 6, 304-20.
Besag, J. (1974) Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society, Series B, 36, 192-236.
Box, G.E.P. (1980) Sampling and Bayes' inference in scientific modelling and robustness. Journal of the Royal Statistical Society, Series A, 143, 383-430.
Brown, P.J., Le, N.D., and Zidek, J.V. (1994) Multivariate spatial interpolation and exposure to air pollutants. Can. J. Statist., 22, 489-509.
Chua, S.H. and Bras, R.L. (1982) Optimal estimators of mean areal precipitation in regions of orographic influence. Journal of Hydrology, 57, 23-48.
Cressie, N. and Zimmerman, D.L. (1992) On the Stability of the Geostatistical Method. Mathematical Geology, 24, 45-59.
Cressie, N. (1991) Statistics for Spatial Data, John Wiley, New Jersey.
Cui, H., Stein, A., and Myers, D.E. (1995) Extension of spatial information, Bayesian kriging and updating of prior variogram parameters. Environmetrics, 4, 373-84.
Daly, C., Neilson, R.P., and Phillips, D.L. (1994) A statistical-topographic model for mapping climatological precipitation over mountainous terrain. Journal of Applied Hydrology, 33, 140-58.
De Oliveira, V., Kedem, B., and Short, D.A. (1997) Bayesian prediction of transformed Gaussian random fields. J. Americ. Statist. Assoc., 92, 1422-33.
Dey, D.K., Gelfand, A.E., Swartz, T., and Vlachos, P. (1995) Simulation-based model checking for hierarchical models, Technical Report No. 95-29, Department of Statistics, University of Connecticut.
Dingman, S.L., Seely-Reynolds, D.M., and Reynolds, R.C. III. (1988) Application of kriging to estimating mean annual precipitation in a region of orographic influence. Water Resources Bulletin, 24, 329-39.
Geisser, S. and Eddy, W.F. (1979) A predictive approach to model selection. Journal of the American Statistical Association, 74, 153-60.
Gelfand, A.E. and Dey, D.K. (1994) Bayesian model choice: asymptotics and exact calculations. Journal of the Royal Statistical Society Series B, 56, 501-14.
Handcock, M.S. and Stein, M.L. (1993) A Bayesian analysis of kriging. Technometrics, 35, 403-10.
Hevesi, J.A., Jstok, J.D., and Flint, A.L. (1992) Precipitation estimation in mountainous terrain using multivariate geostatistics, part I and part II. Journal of Applied Meteorology, 31, 661-88.
Hjort, N.L. and Omre, H. (1994) Topics in spatial statistics. Scandinavian Journal of Statistics, 21, 289-357.
Journel, A.G. and Huijbregts, C.J. (1978) Mining Geostatistics, Academic Press, San Diego.
Kitanidis, P.K. (1986) Parameter uncertainty in estimation of spatial functions: Bayesian analysis. Water Resources Research, 22, 499-507.
Kitanidis, P.K. and Lane, R.W. (1985) Maximum likelihood parameter estimation of hydrologic spatial processes by the Gauss-Newton method. Journal of Hydrology, 79, 53-71.
Le, N.D. and Zidek, J.V. (1992) Interpolation with uncertain spatial covariances: a Bayesian alternative to kriging. Journal of Multivariate Analysis, 43, 351-74.
Le, N.D., Sun, W., and Zidek, J.V. (1997) Bayesian multivariate spatial interpolation with data missing by design. J. R. Statist. Soc. B., 59, 501-10.
Mallick, B.K. and Gelfand, A.E. (1994) Generalized linear models with unknown link functions. Biometrika, 81, 137-45.
Matérn, B. (1986) Spatial Variation, (2nd ed.)., Springer-Verlag, Berlin: New York
Meng, X.L. (1994) Posterior predictive p-values. Annals of Statistics, 22, 1142-60.
Ripley, B.D. (1981) Spatial Statistics, John Wiley, New York.
Ripley, B.D. (1988) Statistical Inference for Spatial Processes, Cambridge University Press.
Rubin, D.B. (1984) Bayesianly justifiable and relevant frequency calculations for the applied statistician. Annals of Statistics, 12, 1151-72.
Sampson, P.D. and Guttorp, P. (1992) Nonparametric estimation of nonstationary spatial covariance structure. J. Americ. Statist. Assoc., 87, 108-19.
Sánchez-Azofeifa, G.A. (1994) En el uso de estadisticas especiales para la definicion de superficies de precipitacion en cuencas tropicales, Technical report, Complex System Research Center, University of New Hampshire.
Stone, M. (1974) Cross-validatory choice and assessment of statistical predictions. Journal of the Royal Statistical Society, Series B, 36, 111-47.
Sun, W. (1998) Comparison of a cokriging method with a Bayesian alternative. Environmetrics, 9, 445-57.
Tanner, M.A. (1996) Tools for Statistical Inference, (3rd ed.). Springer-Verlag, Berlin: New York
Taylor, C.C. and Burrough, P.A. (1986) Multiscale sources of spatial variation in soil. III. Improved methods for fitting the nested model to one-dimensional semivariograms. Mathematical Geology, 18, 811-21.
Warnes, J.J. (1986) A sensitivity analysis for universal kriging. Mathematical Geology, 18, 653-76.
Zimmerman, D.L. and Harville, D.A. (1991) A random field approach to the analysis of field-plot experiments and other spatial experiments. Biometrics, 47, 223-29.
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Gaudard, M., Karson, M., Linder, E. et al. Bayesian spatial prediction. Environmental and Ecological Statistics 6, 147–171 (1999). https://doi.org/10.1023/A:1009614003692
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DOI: https://doi.org/10.1023/A:1009614003692