Spatial Statistics


Estimated map of a bauxite ore from sample location first time used for my MSc module on geostatistics in 1979. Image K.V. Mardia.



Consider information gathered at different geographical locations, e.g. pollution levels at different monitoring stations, or rates of illness in different counties. A common property of spatial data is “spatial continuity”, which means that measurements at nearby locations will tend to be more similar than measurements at distant locations. A key problem of interest in this area involves the prediction of the process at new sites, given its values at a collection of data sites. This type of work can be extended to spatial-temporal statistics, where the information is gathered repeatedly at different times. Spatial-temporal modelling allows prediction not only at new locations but at different times, for example, making forecasts.


Some of the key contributions: maximum likelihood estimation in kriging, kriging with derivative information and intrinsic random field, and the link between kriging and thin plate spline.


The first paper on the MLE for kriging is Mardia (1980).


It is well known that the maximum likelihood (ML) method is a powerful statistical tool in estimation for parametric models. The method of maximum likelihood was introduced by R.A. Fisher as early as 1931 but its application in geostatistics for inference of variogram parameters has been slow and it was first introduced by Mardia (1980) - incidentally, this paper was presented to the Geological Congress in Paris in 1980! Mardia (1984) and Mardia (1990) have considered the problem in detail.


There has been a very closed scrutiny of the behaviour of the MLE for the spatial linear model over the years but still it remains to be the most powerful method after taking some precautionary measures; that is, it cannot be used blindly as a black box!


Edited Volume:

2001 Functional and Spatial Data Analysis, co-editors R. G. Aykroyd. LASR Proceedings: Leeds University Press, 2001.

Papers in Journals:

1980 Mardia, K.V. (1980). Some statistical inference problems in Kriging II: Theory. In: Proceedings 26th International Geology Congress Sciences de la Terre: Advances in Automatic Processing and Mathematical Models in Geology, Series “Informatique Geologie”, 15, pp.113-131.

1984 Maximum likelihood estimation of models for residual covariance in spatial regression (with R. J. Marshall). Biometrika 71, 135-146.

1987 Some minimum norm quadratic estimators of the components of spatial covariance (with R. J. Marshall). Mathematical Geology 17, 517-525.

1988 Multi-dimensional multivariate Gaussian Markov random fields. J. Multivar. Anal. 24, 265-284.

1990 Maximum likelihood estimation for spatial models. In Proceedings of Spatial Statistics: Past, Present and Future. Inst. of Mathematical Geography, ed. D. A. Griffith. Michigan Document Services, pp 203-225.

1996 Spectral and circulant approximations to the likelihood for stationary Gaussian random fields (with J. T. Kent). J. Statist. Plan. Inf. 50, 379-394.

1996 Conditional cyclic Markov random fields (with J. T. Kent and A. N. Walder). Adv. Appl. Probab. 28, 1-12.

1996 Kriging and splines with derivative information (with J. T. Kent, C. R. Goodall and J. A. Little). Biometrika 83, 207-221.

1999 On bias in maximum likelihood estimators (with H. R. Southworth and C. C. Taylor). J. Statist. Plan. Inf. 76, 31-39.

2006 Intrinsic random fields and image deformations (with Bookstein, F.L.; Kent, J.T.; Meyer, C.R.). Journal of Mathematical Imaging and Vision, 26, pp.59-71.

Papers in edited volumes:

1994 Link between kriging and thin plate splines (with J. T. Kent). In Probability, Statistics and Optimization (F. P. Kelly ed.). Wiley, New York, pp 325-339.

2005 A comparison of spatio-temporal Bayesian models for reconstruction of rainfall fields in a cloud seeding experiment (with S. K. Sahu, G. Jona Lasini, and A. Orasi). J. Math. Statist. 1 (4), 273-281.