Hostname: page-component-7dd5485656-pnlb5 Total loading time: 0.001 Render date: 2025-10-30T02:29:38.933Z Has data issue: false hasContentIssue false
  • English
  • Français

Interpolation of spatial data – A stochastic or a deterministic problem?

Published online by Cambridge University Press:  07 February 2013

M. SCHEUERER ,
R. SCHABACK  and
M. SCHLATHER
M. SCHEUERER
Affiliation:
Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 294, D-69120 Heidelberg, Germany email: michael.scheuerer@uni-heidelberg.de
R. SCHABACK
Affiliation:
Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, Lotzestr, 16-18, D-37083 Göttingen, Germany email: schaback@math.uni-goettingen.de
M. SCHLATHER
Affiliation:
Institut für Mathematik, Universität Mannheim, A5, 6, D-68131 Mannheim, Germany email: schlather@math.uni-mannheim.de
Get access Rights & Permissions [Opens in a new window]

Abstract

Interpolation of spatial data is a very general mathematical problem with various applications. In geostatistics, it is assumed that the underlying structure of the data is a stochastic process which leads to an interpolation procedure known as kriging. This method is mathematically equivalent to kernel interpolation, a method used in numerical analysis for the same problem, but derived under completely different modelling assumptions. In this paper we present the two approaches and discuss their modelling assumptions, notions of optimality and different concepts to quantify the interpolation accuracy. Their relation is much closer than has been appreciated so far, and even results on convergence rates of kernel interpolants can be translated to the geostatistical framework. We sketch different answers obtained in the two fields concerning the issue of kernel misspecification, present some methods for kernel selection and discuss the scope of these methods with a data example from the computer experiments literature.

Keywords

spatial interpolationgeostatisticskernel interpolation

Information

Type
A Survey in Mathematics for Industry
Information
European Journal of Applied Mathematics , Volume 24 , Issue 4 , August 2013 , pp. 601 - 629
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Anderes, E. (2010) On the consistent separation of scale and variance for Gaussian random fields. Ann. Statist. 38 (2), 870–893.Google Scholar
[2]Anderson, J. L. (1996) A method for producing and evaluating probabilistic forecasts from ensemble model integrations. J. Clim. 9, 1518–1530.2.0.CO;2>CrossRefGoogle Scholar
[3]Banerjee, S., Gelfand, A. E., Finley, A. O. & Sang, H. (2008) Gaussian predictive process models for large spatial datasets. J. R. Statist. Soc B 70 (4), 825–848.Google Scholar
[4]Berlinet, A. & Thomas-Agnan, C. (2004) Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer, Berlin, Germany.Google Scholar
[5]Caragea, P. & Smith, R. L. (2007) Asymptotic properties of computationally efficient alternative estimators for a class of multivariate normal models. J. Multivariate Anal. 98 (7), 1417–1440.Google Scholar
[6]Carlson, R. E. & Foley, T. A. (1991) The parameter R 2 in multiquadric interpolation. Comp. Math. Appl. 21, 29–42.Google Scholar
[7]Chauvet, P., Pailleux, J. & Chilès, J.-P. (1976) Analyse objective des champs météorologiques par cokrigeage. La Météorologie, 6ième Série 4, 37–54.Google Scholar
[8]Chilès, J.-P. (1976) How to adapt kriging to non-classical problems: three case studies. In: Guarascio, M., David, M. & Huijbregts, C. (editors), Advanced Geostatistics in the Mining Industry, D. Reidel, Dordrecht, Holland, pp. 69–89.Google Scholar
[9]Chilès, J.-P. & Delfiner, P. (1999) Geostatistics. Modeling Spatial Uncertainty, John Wiley, New York.Google Scholar
[10]Cressie, N. (1989) Geostatistics. Am. Stat. 43 (4), 197–202.Google Scholar
[11]Cressie, N. (1993) Statistics for Spatial Data (rev. ed. edition), Wiley, New York.Google Scholar
[12]Cressie, N. & Johannesson, G. (2008) Fixed rank kriging for very large spatial data sets. J. R. Statist. Soc B 70 (1), 209–226.Google Scholar
[13]Dahlhaus, R. & Künsch, H. R. (1987) Edge effects and efficient parameter estimation for stationary random fields. Biometrika 74 (4), 877–882.Google Scholar
[14]de Marchi, S. (2003) On optimal center locations for radial basis function interpolation: Computational aspects. Rend. Sem. Mat. Torino 61 (3), 343–358.Google Scholar
[15]de Marchi, S. & Schaback, R. (2010) Stability of kernel-based interpolation. Adv. Comput. Math. 32, 155–161.Google Scholar
[16]de Marchi, S., Schaback, R. & Wendland, H. (2005) Near-optimal data-independent point locations for radial basis functions. Adv. Comput. Math. 23 (3), 317–330.Google Scholar
[17]Diggle, P. J. & Ribeiro, P. J. (2007) Model-Based Geostatistics, Springer, Berlin, Germany.Google Scholar
[18]Duchon, J. (1976) Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. Adv. RAIRO Anal. Num. 10, 5–12.Google Scholar
[19]Duchon, J. (1977) Splines minimizing rotation invariant seminorms in Sobolev spaces. In: Schempp, W. & Zeller, K. (editors), Constructive Theory of Functions of Several Variables, Springer-Verlag, Berlin, Germay, pp. 85–100.Google Scholar
[20]Duchon, J. (1978) Sur l'erreur d'interpolation des fonctions de plusieurs variables par les Dm-splines. Adv. RAIRO Anal. Num. 12, 325–334.Google Scholar
[21]Eidsvik, J., Finley, S., Banerjee, S. & Rue, H. (2010) Approximate Bayesian Inference for Large Spatial Datasets using Predictive Process Models. Technical Report 9, Department of Mathematical Sciences, Norwegian University of Science and Technology, Norway.Google Scholar
[22]Evans, L. C. (2002) Partial Differential Equations, American Mathematical Society, Providence, RI.Google Scholar
[23]Fasshauer, G. E. (2007) Meshfree Approximation Methods with Matlab, World Scientific, Singapore.Google Scholar
[24]Fasshauer, G. E. & Zhang, J. G. (2007) On choosing “optimal” shape parameters for RBF approximation. Numer. Algorithms 45, 345–368.Google Scholar
[25]Foley, T. A. (1987) Interpolation and approximation of 3-D and 4-D scattered data. Comput. Math. Appl. 13, 711–740.Google Scholar
[26]Fornberg, B., Driscoll, T. A., Wright, G. & Charles, R. (2002) Observations on the behaviour of radial basis function approximations near boundaries. Comput. Math. Appl. 43, 473–490.Google Scholar
[27]Fornberg, B. & Wright, G. (2004) Stable computation of multiquadric interpolants for all values of the shape parameter. Comput. Math. Appl. 47, 497–523.Google Scholar
[28]Franke, R. (1982) Scattered data interpolation: Tests of some methods. Math. Comput. 38, 181–200.Google Scholar
[29]Fuentes, M. (2008) Approximate likelihood for large irregular spaced spatial data. J. Am. Stat. Assoc. 102 (477), 321–331.Google Scholar
[30]Furrer, R., Genton, M. G. & Nychka, D. (2006) Covariance tapering for interpolation of large spatial datasets. J. Comput. Graph. Stat. 15 (3), 502–523.Google Scholar
[31]Gneiting, T. & Schlather, M. (2004) Stochastic models that separate fractal dimension and the Hurst effect. SIAM Rev. 46 (2), 269–282.Google Scholar
[32]Guyon, X. (1982) Parameter estimation for a stationary process on a d-dimensional lattice. Biometrika 69 (1), 95–105.Google Scholar
[33]Handcock, M. S. & Stein, M. L. (1993) A Bayesian analysis of kriging. Technometrics 35 (4), 403–410.Google Scholar
[34]Handcock, M. S. & Wallis, J. R. (1994) An approach to statistical spatial-temporal modeling of meteorological fields (with discussion). J. Am. Stat. Assoc. 89 (7), 368–390.Google Scholar
[35]Hardy, R. L. (1971) Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res. 76, 1905–1915.Google Scholar
[36]Harville, D. A. (1974) Bayesian inference for variance components using only error contrasts. Biometrika 61, 383–385.Google Scholar
[37]Heyde, C. C. (1997) Quasi-Likelihood and Its Application, Springer, New York.Google Scholar
[38]Ibragimov, I. A. & Rozanov, Y. A. (1978) Gaussian Random Processes, Aries, A. B. (trans.), Springer, New York.Google Scholar
[39]Iske, A. (2000) Optimal Distributions of Centers for Radial Basis Function Methods. Technical Report M0004, Technische Universität München, Munich, Germany.Google Scholar
[40]Joseph, V. R., Hung, Y. & Sudjianto, A. (2008) Blind kriging: A new method for developing metamodels. J. Mech. Des. 130 (3), 1–8.Google Scholar
[41]Journel, A. G. (1982) The indicator approach to estimation of spatial distributions. In Proceedings of the 17th APCOM International Symposium, New York, pp. 793–806.Google Scholar
[42]Kimeldorf, G. S. & Wahba, G. (1970) A correspondence between Bayesian estimation on stochastic processes and smoothing by splines. Ann. Math. Stat. 41 (2), 495–502.Google Scholar
[43]Kitanidis, P. K. (1983) Statistical estimation of polynomial generalized covariance functions and hydrologic applications. Water Resour. Res. 19 (4), 909–921.Google Scholar
[44]Kitanidis, P. K. (1997) Introduction to Geostatistics: Applications in Hydrology, Cambridge University Press, New York.Google Scholar
[45]Lajaunie, C. & Béjaoui, R. (1991) Sur le Krigeage des Fonctions Complexes. Technical Report N-23/91/G, Centre de Géostatistique, Ecole des Mines de Paris, Fontainebleau.Google Scholar
[46]Lim, S. C. & Teo, L. P. (2010) Analytic and asymptotic properties of multivariate generalized Linniks probability densities. J. Fourier Anal. Appl. 16, 715–747.Google Scholar
[47]Lindgren, F., Rue, H. & Lindström, J. (2011) An explicit link between Gaussian fields and Gaussian Markov random fields: The stochastic partial differential equation approach. J. R. Stat. Soc B 73 (4), 423–498.Google Scholar
[48]Madych, W. R. & Nelson, S. A. (1988) Multivariate interpolation and conditionally positive definite functions. Approx. Theory Appl. 4, 77–89.Google Scholar
[49]Mardia, K. V. & Marshall, R. J. (1984) Maximum likelihood estimation of models for residual covariance in spatial statistics. Biometrika 71, 135–146.Google Scholar
[50]Matérn, B. (1986) Spatial Variation, 2nd ed., Lecture Notes in Statistics, Vol. 36, Springer-Verlag, Berlin, Germany.Google Scholar
[51]Matheron, G. (1971) The Theory of Regionalized Variables and its Applications. Technical Report, Cahiers du Centre de Morphologie Mathḿatique de Fontainebleau, Ecole des Mines de Paris.Google Scholar
[52]Matheron, G. (1973) The intrinsic random functions and their applications. Adv. Appl. Prob. 5, 439–468.Google Scholar
[53]Matheron, G. (1973) Le Krigeage Disjonctive. Technical Report N-360, Centre de Géostatistique, Ecole des Mines de Paris.Google Scholar
[54]Matheron, G. (1976) A simple substitute for conditional expectation: The disjunctive kriging. In: Guarascio, M., David, M. & Huijbregts, C. (editors), Advanced Geostatistics in the Mining Industry, Reidel, Dordrecht, Netherland, pp. 221–236.Google Scholar
[55]Micchelli, C. A. (1986) Interpolation of scattered data: Distance matrices and conditionally positive definite functions. Constr. Approx. 2, 11–22.Google Scholar
[56]Morris, M. D., Mitchell, T. J. & Ylvisaker, D. (1993) Bayesian design and analysis of computer experiments: Use of derivatives in surface prediction. Technometrics 35 (3), 243–255.Google Scholar
[57]Myers, D. E. (1992) Kriging, cokriging, radial basis functions and the role of positive definiteness. Comput. Math. Appl. 24 (12), 139–148.Google Scholar
[58]Narcowich, F. J., Ward, J. D. & Wendland, H. (2006) Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions. Constr. Approx. 24, 175–186.Google Scholar
[59]Omre, H. & Halvorsen, K. B. (1989) The Bayesian bridge between simple and universal kriging. Math. Geol. 21 (7), 767–786.Google Scholar
[60]Putter, H. & Young, G. A. (2001) On the effect of covariance function estimation on the accuracy of kriging predictors. Bernoulli 7 (3), 421–438.Google Scholar
[61]R Development Core Team. (2011) R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria.Google Scholar
[62]Rasmussen, C. E. & Williams, C. K. I. (2006) Gaussian Processes for Machine Learning, MIT Press, Boston, MA.Google Scholar
[63]Rippa, S. (1999) An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv. Comput. Math. 11, 193–210.Google Scholar
[64]Ritter, K. (2000) Average-Case Analysis of Numerical Problems, Lecture Notes in Mathematics, No. 1733, Springer, New York.Google Scholar
[65]Schaback, R. (1993) Comparison of radial basis function interpolants. In: Jetter, K. & Utreras, F. (editors), Multivariate Approximation: From CAGD to Wavelets, World Scientific, London, pp. 293–305.Google Scholar
[66]Schaback, R. (1995) Error estimates and condition numbers for radial basis function interpolation. Adv. Comput. Math. 3, 251–264.Google Scholar
[67]Schaback, R. (1997) Native Hilbert spaces for radial basis functions I. In: New Developments in Approximation Theory, International Series of Numerical Mathematics, No. 132, Birkhauser Verlag, Berlin, Germany, pp. 255–282.Google Scholar
[68]Schaback, R. (2011) The missing Wendland functions. Adv. Comput. Math. 34, 67–81.Google Scholar
[69]Schaback, R. (2011) Kernel-Based Meshless Methods. Technical report, Georg-August-Universität Göttingen, Göttingen, Germany.Google Scholar
[70]Schaback, R. & Wendland, H. (2006) Kernel techniques: From machine learning to meshless methods. Acta Numer. 15, 543–639.Google Scholar
[71]Scheuerer, M. (2010) Regularity of the sample paths of a general second-order random field. Stoch. Proc. Appl. 120, 1879–1897.Google Scholar
[72]Scheuerer, M. (2011) An alternative procedure for selecting a good value for the parameter c in RBF-interpolation. Adv. Comput. Math. 34 (1), 105–126.Google Scholar
[73]Schlather, M. (2001) RandomFields: Contributed extension package to R for the simulation of Gaussian and max-stable random fields. URL: cran.r-project.org.Google Scholar
[74]Seeger, M. (2004) Gaussian processes for machine learning. Int. J. Neural Syst. 14, 1–38.Google Scholar
[75]Stein, M. L. (1988) Asymptotically efficient prediction of a random field with a misspecified covariance function. Ann. Stat. 16, 55–63.Google Scholar
[76]Stein, M. L. (1990) A comparison of generalized cross validation and modified maximum likelihood for estimating the parameters of a stochastic process. Ann. Stat. 18 (3), 1139–1157.Google Scholar
[77]Stein, M. L. (1999) Interpolation of Spatial Data, Springer, New York.Google Scholar
[78]Stein, M. L. (2004) Equivalence of Gaussian measures for some nonstationary random fields. J. Stat. Plann. Inference 123, 1–11.Google Scholar
[79]Stein, M. L., Chi, Z. & Welty, L. J. (2004) Approximating likelihoods for large spatial data sets. J. R. Stat. Soc B 66 (2), 275–296.Google Scholar
[80]van der Vaart, A. W. & van Zanten, J. H. (2011) Information rates of nonparametric Gaussian process methods. J. Mach. Learn. Res. 12, 2095–2119.Google Scholar
[81]Vecchia, A. V. (1988) Estimation and model identification for continuous spatial processes. J. R. Stat. Soc B 50, 297–312.Google Scholar
[82]Wahba, G. (1985) A comparison of gcv and gml for choosing the smoothing parameter in the generalized spline smoothing problem. Ann. Stat. 13 (1), 1378–1402.Google Scholar
[83]Wahba, G. (1990) Spline Models for Observational Data, SIAM, Philadelphia, PA.Google Scholar
[84]Wendland, H. (1995) Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 389–396.Google Scholar
[85]Wendland, H. (2005) Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK.Google Scholar
[86]Wendland, H. & Rieger, C. (2005) Approximate interpolation with applications to selecting smoothing parameters. Numer. Math. 101, 729–748.Google Scholar
[87]Wu, Z. (1992) Hermite-Birkhoff interpolation of scattered data by radial basis functions. Approx. Theory Appl. 8 (2), 1–10.Google Scholar
[88]Zhang, H. (2004) Inconsistent estimation and asymptotically equivalent interpolations in model-based geostatistics. J. Am. Stat. Assoc. 99, 250–261.Google Scholar