Abstract
This study adds to our ability to predict the unknown by empirically assessing the performance of a novel geostatistical-nonparametric hybrid technique to provide accurate predictions of the value of an attribute together with locally-relevant measures of prediction confidence, at point locations for a single realisation spatial process. The nonstationary variogram technique employed generalises a moving window kriging (MWK) model where classic variogram (CV) estimators are replaced with information-rich, geographically weighted variogram (GWV) estimators. The GWVs are constructed using kernel smoothing. The resultant and novel MWK–GWV model is compared with a standard MWK model (MWK–CV), a standard nonlinear model (Box–Cox kriging, BCK) and a standard linear model (simple kriging, SK), using four example datasets. Exploratory local analyses suggest that each dataset may benefit from a MWK application. This expectation was broadly confirmed once the models were applied. Model performance results indicate much promise in the MWK–GWV model. Situations where a MWK model is preferred to a BCK model and where a MWK–GWV model is preferred to a MWK–CV model are discussed with respect to model performance, parameterisation and complexity; and with respect to sample scale, information and heterogeneity.
Similar content being viewed by others
References
Anderes EB, Stein ML (2005) Estimating deformations of isotropic Gaussian random fields on the plane. Technical report no. 27, Centre for Integrating Statistics and Environmental Sciences, University of Chicago
Banerjee S, Gelfand AE, Finley AO, Sang H (2008) Gaussian predictive processes for large spatial datasets. J R Stat Soc B Stat Methodol 70:825–848
Brunsdon C (2001) The Comap: exploring spatial pattern via conditional distributions. Comput Environ Urban Syst 25:53–68
Cattle JA, McBratney AB, Minasny B (2002) Kriging methods evaluation for assessing the spatial distribution of urban soil lead contamination. J Environ Qual 31:1576–1588
Chilès J-P, Delfiner P (1999) Geostatistics—modelling spatial uncertainty. Wiley Interscience, New York
CLAG-Freshwaters (1995) Critical loads of acid deposition for United Kingdom freshwaters. Critical Loads Advisory Group, Sub-report on Freshwaters, ITE, Penicuik
Cressie N (1993) Statistics for spatial data. Wiley, New Jersey
Dubois G (2008) Editorial. Advances in automatic interpolation for real-time mapping. Stoch Environ Res Risk Assess 22:597–599
Fotheringham AS, Brunsdon C, Charlton M (2002) Geographically weighted regression—the analysis of spatially varying relationships. Wiley, Chichester
Fuentes M (2001) A new high frequency kriging approach for nonstationary environmental processes. Environmetrics 12:469–483
Fuentes M (2005) A formal test for nonstationarity of spatial stochastic processes. J Multivar Anal 96:30–54
Fuentes M, Guttorp P, Challenor P (2003) Statistical assessment of numerical models. Int Stat Rev 71:201–221
Goovaerts P (2001) Geostatistical modelling of uncertainty in soil science. Geoderma 103:3–26
Goovaerts P, Jacquez GM, Greiling D (2005) Exploring scale-dependent correlations between cancer mortality rates using factorial kriging and population-weighted semivariograms. Geogr Anal 37:152–182
Haas TC (1990) Lognormal and moving window methods of estimating acid deposition. J Am Stat Assoc 85:950–963
Haas TC (1996) Multivariate spatial prediction in the presence of non-linear trend and covariance non-stationarity. Environmetrics 7:145–165
Haas TC (1998) Statistical assessment of spatio-temporal pollutant trends and meteorological transport models. Atmos Environ 32:1865–1879
Haas TC (2002) New systems for modelling, estimating, and predicting a multivariate spatio-temporal process. Environmetrics 13:311–332
Harris P, Brunsdon C (2010) Exploring spatial variation and spatial relationships in a freshwater acidification critical load data set for Great Britain using geographically weighted summary statistics. Comput Geosci 36:54–70
Higdon D, Swall J, Kern J (1999) Nonstationary spatial modelling. In: Berardo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics 6. Oxford University Press, Oxford, pp 761–768
Isaaks EH, Srivastava RM (1989) An introduction to applied geostatistics. Oxford University Press, New York
Johannesson G, Cressie N (2004) Finding large-scale spatial trends in massive, global, environmental datasets. Environmetrics 15:1–44
Journel AG (1983) Nonparametric estimation of spatial distributions. Math Geol 15:445–468
Journel AG (1989) Fundamentals of geostatistics in five lessons. Short course in geology. American Geophysical Union Press, Washington, DC
Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic Press, London
Kitanidis PK, Shen K-F (1996) Geostatistical interpolation of chemical concentration. Adv Water Resour 19:369–378
Lahiri SN, Lee Y, Cressie N (2002) On asymptotic distribution and asymptotic efficiency of least squares estimators of spatial variogram parameters. J Stat Plan Inference 103:65–85
Lloyd CD (2010) Nonstationary models for exploring and mapping monthly precipitation in the United Kingdom. Int J Climatol 30:390–405
Lloyd CD, Atkinson PM (2002) Nonstationary approaches for mapping terrain and assessing prediction uncertainty. Trans GIS 6:17–30
Matheron G (1976) A simple substitute for conditional expectation: the disjunctive kriging. In: Guarascio M, David M, Huijbrechts C (eds) Advanced geostatistics for the mining industry. Reidel, Dordrecht, pp 221–236
Müller WG (1999) Least-squares fitting from the variogram cloud. Stat Probab Lett 43:93–98
Omre H (1984) The variogram and its estimation. In Verly G, David M, Journel AG, Marachal A (eds) Geostatistics for natural resources characterization. ASI. Reidel, Hingham, pp 107–125
Paciorek CJ, Schervish MJ (2006) Spatial modelling using a new class of nonstationary covariance functions. Environmetrics 17:483–506
Pardo-Igúzquiza E, Dowd P, Grimes D (2005) An automatic moving window approach for mapping meteorological data. Int J Climatol 26:665–678
Parker HM (1991) Statistical treatment of outlier data in epithermal gold deposit reserve estimation. Math Geol 23:175–199
Pebesma EJ (2004) Multivariate geostatistics in S: the gstat package. Comput Geosci 30:683–691
Ploner A, Dutter R (2000) New directions in geostatistics. J Stat Plan Inference 91:499–509
Ribeiro PJ, Diggle PJ (2001) geoR: a package for geostatistical analysis. R News 1:15–18
Richmond A (2002) Two-point declustering for weighting data pairs in experimental variogram calculations. Comput Geosci 28:231–241
Rivoirard J (2002) Weighted variograms. In Kleingold WJ, Krige DG (eds) Geostatistics 2000 Cape Town. Geostatistical Association of Southern Africa, South Africa
Sampson PD, Damian D, Guttorp P (2001) Advances in modeling and inference for environmental processes with nonstationary spatial covariance. NRCSE-TRS No 61, National Research Centre for Statistics and the Environment Technical Report Series, 2001
Stein A, Hoogerwerf M, Bouma J (1988) Use of soil map delineations to improve (co)kriging of point data on moisture deficits. Geoderma 43:163–177
Van Tooren CF, Haas TC (1993) A site investigation strategy using moving window kriging and automated semivariogram modelling. In: Contaminated soil ‘93. Kluwer Academic Press, Dordrecht, pp 609–622
Walter C, McBratney AB, Douaoui A, Minasny B (2001) Spatial prediction of topsoil salinity in the Chelif Valley, Algeria, using local ordinary kriging with local variograms versus whole-area variogram. Aust J Soil Res 39:259–272
Yamamoto JK (2000) An alternative measure of the reliability of ordinary kriging estimates. Math Geol 32:489–509
Zhang X, Eijkeren JC, Heemink AW (1995) On the weighted least-squares method for fitting a semivariogram model. Comput Geosci 21:605–608
Acknowledgements
Research presented in this paper was funded by a Strategic Research Cluster grant (07/SRC/I1168) by the Science Foundation Ireland under the National Development Plan. The authors gratefully acknowledge this support. Thanks are also due to the first author’s PhD studentship at Newcastle University and to Professor Chris Brunsdon for kindly providing the R code for the comap. We also thank the anonymous referees whose comments and insights helped improve the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Harris, P., Charlton, M. & Fotheringham, A.S. Moving window kriging with geographically weighted variograms. Stoch Environ Res Risk Assess 24, 1193–1209 (2010). https://doi.org/10.1007/s00477-010-0391-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00477-010-0391-2