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Interpolation of Spatial Data: Some Theory for Kriging (Springer Series in Statistics)

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Michael Stein got his Ph.D. in Statistics from Stanford University under the direction of Paul Switzer. I also studied at Stanford years earlier and also learned about kriging from Switzer. Kriging is a very popular technique for interpolation of spatial data between measurement points. It is an optimal linear technique when the spatial covariance structure is known. It has many practical applications to pollution data, geological data etc. Stein develops the theory as far as he can for the case when the covariance structure is unknown and must be estimated based on the measurement data.
The theoretical development requires some advanced mathematical knowledge on the part of the reader including advanced probability, Fourier analysis and Hilbert spaces. The second order properties of random fields and results on Gaussian measures needed for the development of key results are covered in Chapter 2. Those interested in the practical aspects of kriging can omit the proofs and just concentrate on the results. Chapter 6 provides important practical information.

Although difficult to digest, a careful reading of the book will provide insight into what is good and what is bad about the way kriging is commonly implemented. The bootstrap approach to assessing the accuracy of kriging predictions is briefly discussed in section 6.8 page 202.

This text concentrates on Stein's development of fixed domain asymptotics. It does not provide a broad overview of kriging. That can be found in Noel Cressie's book. It also does not deal with other aspects of interpolation such as nonlinear interpolation, estimation for non-Gaussian processes or the connections with splines.

Nevertheless this is a landmark text that should be on the shelf of any statistician interested in spatial data.

It is not an easy book, but the book has very interesting ideas about kriging. In particular, it studies deeply the situation where the number of data points increases on a fixed interval.

Several chapters are not easy to read because of the material, also is not comprehensive about kriging; but despite that,
it has a lot of interesting results and it is worth of reading.