Abstract
Suppose a two-dimensional spatial process z(x) with generalized covariance function G(x, x′) α |x − x′|2 log |x − x′| (Matheron, 1973, Adv. in Appl. Probab., 5, 439–468) is observed with error at a number of locations. This paper gives a kernel approximation to the optimal linear predictor, or kriging predictor, of z(x) under this model as the observations get increasingly dense. The approximation is in terms of a Kelvin function which itself can be easily approximated by series expansions. This generalized covariance function is of particular interest because the predictions it yields are identical to an order 2 thin plate smoothing spline. For moderate sample sizes, the kernel approximation is seen to work very well when the observations are on a square grid and fairly well when the observations come from a uniform random sample.
This manuscript was prepared using computer facilities supported in part by National Science Foundation Grants No. DMS-8601732 and DMS-8404941 to the Department of Statistics at The University of Chicago.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M. and Stegun, I. (1965). Handbook of Mathematical Functions, Dover, New York.
Akhiezer, N. I. and Glazman, I. M. (1961). Theory of Linear Operators in Hilbert Space (translated by M. Nestell), Frederick Ungar, New York.
Cogburn, R. and Davis, H. T. (1974). Periodic splines and spectral estimation, Ann. Statist., 2, 1108–1126.
Cox, D. D. (1983). Asymptotics of M-type smoothing splines, Ann. Statist., 11, 530–551.
Cox, D. D. (1984). Multivariate smoothing spline functions, SIAM J. Numer. Anal., 21, 789–813.
Delfiner, P. (1976). Linear estimation of non-stationary spatial phenomena, Advanced Geostatistics in the Mining Industry, Proceedings of NATO A.S.I. (eds. E. M. Gurascio, M. David and Ch. Huijbregts), 49–68, Reidel, Dordecht.
Dubrule, O. (1983). Two methods with different objectives: splines and kriging, Math. Geol., 15, 245–257.
Duchon, J. (1976). Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces, RAIRO Anal. Numér., 10, 5–12.
Gel'fand, I. M. and Vilenkin, N. Ya. (1964). Generalized Functions, Vol. 4, Applications of harmonic analysis (translated by A. Feinstein), Academic Press, New York.
Gröbner, W. and Hofreiter, N. (1950). Integraltafel Zweiter Teil Bestimmte Integrale, Springer, Vienna.
Journel, A. G. (1977). Kriging in terms of projections, Math. Geol., 9, 563–586.
Kimeldorf, G. and Wahba, G. (1970). A correspondence between Bayesian estimation on stochastic processes and smoothing by splines, Ann. Math. Statist., 41, 495–502.
Kimeldorf, G. and Wahba, G. (1971). Some results on Tchebycheffian Splines, J. Math. Anal. Appl., 33, 82–95.
Matheron, G. (1973). The intrinsic random functions and their applications, Adv. in Appl. Probab., 5, 439–468.
Matheron, G. (1980). Splines and kriging: their formal equivalence, Internal Report, Centre de Géostatistique, Ecole des Mines de Paris, Fontainebleau.
Meinguet, J. (1979). Multivariate interpolation at arbitrary points made simple, Z. Angew. Math. Phys., 30, 292–304.
Nosova, L. N. (1961). Tables of Thomson Functions and Their First Derivatives (translated by P. Basu), Pergamon Press, New York.
Silverman, B. W. (1984). Spline smoothing: the equivalent variable kernel method, Ann. Statist., 12, 898–916.
Silverman, B. W. (1985). Some aspects of the spline smoothing approach to nonparametric regression curve fitting, J. Roy. Statist. Soc. Ser. B, 47, 1–52.
Stein, M. L. (1988a). An application of the theory of equivalence of Gaussian measures to a prediction problem, IEEE Trans. Inform. Theory, 34, 580–582.
Stein, M. L. (1988b). Asymptotically efficient prediction of a random field with a misspecified covariance function, Ann. Statist., 16, 55–63.
Stein, M. L. and Handcock, M. S. (1989). Some asymptotic properties of kriging when the covariance function is misspecified, Math. Geol., 21, 171–190.
Wahba, G. and Wendelberger, J. (1980). Some new mathematical methods for variational objective analysis using splines and cross-validation, Monthly Weather Review, 108, 1122–1143.
Author information
Authors and Affiliations
Additional information
The author was supported by a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship.
About this article
Cite this article
Stein, M.L. A kernel approximation to the kriging predictor of a spatial process. Ann Inst Stat Math 43, 61–75 (1991). https://doi.org/10.1007/BF00116469
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00116469