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

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:

**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.

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.; *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,

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.