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Bayesian inference for non-stationary spatial covariance structure via spatial deformations. (English) Zbl 1063.62034

Summary: In geostatistics it is common practice to assume that the underlying spatial process is stationary and isotropic, i.e., the spatial distribution is unchanged when the origin of the index set is translated and under rotations about the origin. However, in environmental problems, such assumptions are not realistic since local influences in the correlation structure of the spatial process may be found in the data. This paper proposes a Bayesian model to address the anisotropy problem.
Following P. D. Sampson and P. Guttorp [J. Am. Stat. Assoc. 87, 108–119 (1992)], we define the correlation function of the spatial process by reference to a latent space, denoted by \(D\), where stationarity and isotropy hold. The space where the gauged monitoring sites lie is denoted by \(G\). We adopt a Bayesian approach in which the mapping between \(G\) and \(D\) is represented by an unknown function \({\mathbf d}(\cdot)\). A Gaussian process prior distribution is defined for \({\mathbf d}(\cdot)\). Unlike the Sampson-Guttorp approach, the mapping of both gauged and ungauged sites is handled in a single framework, and predictive inferences take explicit account of uncertainty in the mapping. Markov chain Monte Carlo methods are used to obtain samples from the posterior distributions. Two examples are discussed: a simulated data set and the solar radiation data set that also was analysed by Sampson and Guttorp.

MSC:

62F15 Bayesian inference
62M30 Inference from spatial processes
62P12 Applications of statistics to environmental and related topics
65C40 Numerical analysis or methods applied to Markov chains
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References:

[1] Cressie N. A. C., Statistics for Spatial Data (1993) · Zbl 1347.62005 · doi:10.1002/9781119115151
[2] DOI: 10.1002/1099-095X(200103)12:2<161::AID-ENV452>3.0.CO;2-G · doi:10.1002/1099-095X(200103)12:2<161::AID-ENV452>3.0.CO;2-G
[3] Dawid A. P., J. R. Statist. Soc. 41 pp 1– (1979)
[4] Dryden I. L., Statistical Shape Analysis (1998) · Zbl 0901.62072
[5] Gamerman D., Markov Chain Monte Carlo-Stochastic Simulation for Bayesian Inference (1997) · Zbl 0881.62002
[6] Gilks W. R., Appl. Statist. 41 pp 337– (1992)
[7] Guttorp P., Environmetrics 5 pp 241– (1994)
[8] Guttorp P., Statistics in the Environmental & Earth Sciences pp 39– (1992)
[9] Higdon D., Bayesian Statistics 6 pp 761– (1999) · Zbl 0951.62091
[10] Iovleff G., Technical Report (1999)
[11] DOI: 10.1016/0047-259X(92)90040-M · Zbl 0762.62025 · doi:10.1016/0047-259X(92)90040-M
[12] Loader C., Statistics in the Environmental & Earth Sciences pp 52– (1992)
[13] Mardia K. V., Multivariate Environmental Statistics pp 347– (1993)
[14] Monestiez P., Technical Report (1991)
[15] O’Hagan A., Kendall’s Advanced Theory of Statistics (1994)
[16] Sampson P. D., J. Am. Statist. Ass. 87 pp 108– (1992)
[17] A. M. Schmidt (2001 ) Bayesian spatial interpolation of environmental monitoring stations .PhD Thesis. Department of Probability and Statistics, University of Sheffield, Sheffield.
[18] Smith R. L., Technical Report (1996)
[19] DOI: 10.1002/(SICI)1099-095X(199809/10)9:5<565::AID-ENV324>3.0.CO;2-S · doi:10.1002/(SICI)1099-095X(199809/10)9:5<565::AID-ENV324>3.0.CO;2-S
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