Abstract
The term geostatistics identifies the part of spatial statistics which is concerned with continuous spatial variation, in the following sense. The scientific focus is to study a spatial phenomenon, s(x)say, which exists throughout a continuous spatial region A ⊂ ℝ2 and can be treated as if it were a realisation of a stochastic process S(·) = {S(x): x ∈ A}. In general, S(·) is not directly observable. Instead, the available data consist of measurements Y 1,..., Y n taken at locations x 1,..., x n sampled within A, and Y i is a noisy version of S(x i ). We shall assume either that the sampling design for x 1,..., x n is deterministic or that it is stochastic but independent of the process S(·), and all analyses are carried out conditionally on x 1,...,x n .
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References
Adler, R. J. (1981). The Geometry of Random Fields, Wiley, New York.
Berger, J. O., De Oliveira, V. and Sansb, B. (2001). Objective Bayesian analysis of spatially correlated data, Journal of the American Statistical Association 96: 1361–1374.
Besag, J., York, J. and Mollié, A. (1991). Bayesian image restoration, with two applications in spatial statistics (with discussion), Annals of the Institute of Statistical Mathematics 43: 1–59.
Breslow, N. E. and Clayton, D. G. (1993). Approximate inference in generalized linear mixed models, Journal of the American Statistical Association 88: 9–25.
Brix, A. and Diggle, P. J. (2001). Spatio-temporal prediction for log-Gaussian Cox processes, Journal of the Royal Statistical Society, Series B 63: 823–841.
Brown, P. E., Diggle, P. J., Lord, M. E. and Young, P. C. (2001). Space-time calibration of radar rainfall data, Applied Statistics 50: 221–241.
Brown, P. E., Khrensen, K. F., Roberts, G. O. and Tonellato, S. (2000). Blur-generated non-separable space-time models, Journal of the Royal Statistical Society, Series B 62: 847–860.
Brown, P. J., Le, N. D. and Zidek, J. V. (1994). Multivariate spatial interpolation and exposure to air polutants, Canadian Journal of Statistics 22: 489–509.
Chilés, J. P. and Delfiner, P. (1999). Geostatistics; Modeling Spatial Uncertainty, Wiley, New York.
Christensen, O. F., Diggle, P. J. and Ribeiro Jr, P. J. (2001). Analysing positive-valued spatial data: the transformed Gaussian model, in P. Monestiez, D. Allard and R. Froidevaux (eds), GeoENV III–Geostatistics for Environmental Applications, Vol. 11 of Quantitative Geology and Geostatistics, Kluwer, Dordrecht, pp. 287–298.
Christensen, O. F., Moller, J. and Waagepetersen, R. (2001). Geometric ergodicity of Metropolis Hastings algorithms for conditional simulation in generalised linear mixed models, Methodology and Computing in Applied Probability 3: 309–327.
Christensen, O. F. and Waagepetersen, R. P. (2002). Bayesian prediction of spatial count data using generalized linear mixed models, Biometrics 58: 280–286.
Cramer, H. and Leadbetter, M. R. (1967). Stationary and Related Processes, Wiley, New York.
Cressie, N. (1993). Statistics for Spatial Data — revised edition, Wiley, New York.
De Oliveira, V. (2000). Bayesian prediction of clipped Gaussian random fields, Computational Statistics and Data Analysis 34: 299–314.
De Oliveira, V. and Ecker, M. D. (2002). Bayesian hot spot detection in the presence of a spatial trend: application to total nitrogen concentration in the Chesapeake Bay, Environmetrics 13: 85–101.
De Oliveira, V., Kedem, B. and Short, D. A. (1997). Bayesian prediction of transformed Gaussian random fields, Journal of the American Statistical Association 92: 1422–1433.
Diggle, P. J., Harper, L. and Simon, S. (1997). Geostatistical analysis of residual contamination from nuclear weapons testing, in V. Barnett and F. Turkman (eds), Statistics for the Environment 3: Pollution Assesment and Control, Wiley, Chichester, pp. 89–107.
Diggle, P. J. and Ribeiro Jr, P. J. (2002). Bayesian inference in Gaussian model based geostatistics, Geographical and Environmental Modelling 6. To appear.
Diggle, P. J., Tawn, J. A. and Moyeed, R. A. (1998). Model based geostatistics (with discussion), Applied Statistics 47: 299–350.
Geyer, C. J. (1992). Practical Markov chain Monte Carlo (with discussion), Statistical Science 7: 473–511.
Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation, Oxford University Press, New York.
Gotway, C. A. and Stroup, W. W. (1997). A generalized linear model approach to spatial data analysis and prediction, Journal of Agricultural, Biological and Environmental Statistics 2: 157–178.
Handcock, M. S. and Stein, M. L. (1993). A Bayesian analysis of kriging, Technometrics 35: 403–410.
Handcock, M. S. and Wallis, J. R. (1994). An approach to statistical spatial-temporal modeling of meteorological fields, Journal of the American Statistical Association 89: 368–390.
Heagerty, P. J. and Lele, S. R. (1998). A composite likelihood approach to binary spatial data, Journal of the American Statistical Association 93: 1099–1111.
Ihaka, R. and Gentleman, R. (1996). R: A language for data analysis and graphics, Journal of Computatioanl and Graphical Statistics 5: 299–314.
Journel, A. G. and Huijbregts, C. J. (1978). Mining Geostatistics, Academic Press, London.
Kent, J. T. (1989). Continuity properties of random fields, Annals of Probability 17: 1432–1440.
Kitanidis, P. K. (1983). Statistical estimation of polynomial generalized covariance functions and hydrological applications., Water Resources Research 22: 499–507.
Kitanidis, P. K. (1986). Parameter uncertainty in estimation of spatial functions: Bayesian analysis, Water Resources Research 22: 499–507.
Krige, D. G. (1951). A statistical approach to some basic mine valuation problems on the Witwatersrand, Journal of the Chemical, Metallurgical and Mining Society of South Africa 52: 119–139.
Le, N. D., Sun, W. and Zidek, J. V. (1997). Bayesian multivariate spatial interpolation with data missing by design, Journal of the Royal Statistical Society, Series B 59: 501–510.
Le, N. D. and Zidek, J. V. (1992). Interpolation with uncertain covariances: a Bayesian alternative to kriging, Journal of Multivariate Analysis 43: 351–374.
Mardia, K. V. and Watkins, A. J. (1989). On multimodality of the likelihood in the spatial linear model, Biometrika 76: 289–296.
Matérn, B. (1960). Spatial Variation. Meddelanden fran Statens Skogforskningsinstitut, Band 49, No. 5.
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, second edn, Chapman and Hall, London.
Natarajan, R. and Kass, R. E. (2000). Bayesian methods for generalized linear mixed models, Journal of the American Statistical Association 95: 227–237.
O’Hagan, A. (1994). Bayesian Inference,Vol. 2b of Kendall’s Advanced Theory of Statistics,Edward Arnold.
Omre, H. and Halvorsen, K. B. (1989). The Bayesian bridge between simple and universal kriging, Mathematical Geology 21: 767–786.
Patterson, H. D. and Thompson, R. (1971). Recovery of inter-block infor- mation when block sizes are unequal, Biometrika 58: 545–554.
Ribeiro Jr, P. J. and Diggle, P. J. (1999). Bayesian inference in Gaussian model-based geostatistics, Tech. Report ST-99–09,Lancaster University.
Ribeiro Jr, P. J. and Diggle, P. J. (2001). geoR: a package for geostatistical analysis, R News 1/2: 15–18. Available from: http://www.rproject.org/doc/Rnews.
Ripley, B. D. (1981). Spatial Statistics, Chapman and Hall, New York.
Sampson, P. D. and Guttorp, P. (1992). Nonparametric estimation of non-stationary spatial covariance structure, Journal of the American Statistical Association 87: 108–119.
Schlather, M. (2001). On the second-order characteristics of marked point processes, Bernoulli 7: 99–117.
Stein, M. L. (1999). Interpolation of Spatial Data: Some Theory for Kriging, Springer Verlag, New York.
Wälder, O. and Stoyan, D. (1996). On variograms in point process statistics, Biometrical Journal 38: 895–905.
Wälder, O. and Stoyan, D. (1998). On variograms in point process statistics: Erratum, Biometrical Journal 40: 109.
Warnes, J. J. and Ripley, B. D. (1987). Problems with likelihood estimation of covariance functions of spatial Gaussian processes, Biometrika 74: 640–642.
Whittle, P. (1954). On stationary processes in the plane, Biometrika 41: 434–449.
Whittle, P. (1962). Topographic correlation, power-law covariance functions, and diffusion, Biometrika 49: 305–314.
Whittle, P. (1963). Stochastic processes in several dimensions, Bulletin of the International Statistical Institute 40: 974–994.
Wikle, C. K. and Cressie, N. (1999). A dimension-reduced approach to space-time Kalman filtering, Biometrika 86: 815–829.
Zhang, H. (2002). On estimation and prediction for spatial generalised linear mixed models, Biometrics 58: 129–136.
Zidek, J. V., Sun, W. and Le, N. D. (2000). Designing and integrating composite networks for monitoring multivariate Gaussian pollution field, Applied Statistics 49: 63–79.
Zimmerman, D. L. (1989). Computationally efficient restricted maximum likelihood estimation of generalized covariance functions, Mathematical Geology 21: 655–672.
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Diggle, P.J., Ribeiro, P.J., Christensen, O.F. (2003). An Introduction to Model-Based Geostatistics. In: Møller, J. (eds) Spatial Statistics and Computational Methods. Lecture Notes in Statistics, vol 173. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21811-3_2
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DOI: https://doi.org/10.1007/978-0-387-21811-3_2
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