Development and evaluation of geostatistical methods for non-Euclidean-based spatial covariance matrices. (English) Zbl 1421.86017

Math. Geosci. 51, No. 6, 767-791 (2019); correction ibid. 51, No. 6, 843 (2019).
Summary: Customary and routine practice of geostatistical modeling assumes that inter-point distances are a Euclidean metric (i.e., as the crow flies) when characterizing spatial variation. There are many real-world settings, however, in which the use of a non-Euclidean distance is more appropriate, for example, in complex bodies of water. However, if such a distance is used with current semivariogram functions, the resulting spatial covariance matrices are no longer guaranteed to be positive-definite. Previous attempts to address this issue for geostatistical prediction (i.e., kriging) models transform the non-Euclidean space into a Euclidean metric, such as through multi-dimensional scaling (MDS). However, these attempts estimate spatial covariances only after distances are scaled. An alternative method is proposed to re-estimate a spatial covariance structure originally based on a non-Euclidean distance metric to ensure validity. This method is compared to the standard use of Euclidean distance, as well as a previously utilized MDS method. All methods are evaluated using cross-validation assessments on both simulated and real-world experiments. Results show a high level of bias in prediction variance for the previously developed MDS method that has not been highlighted previously. Conversely, the proposed method offers a preferred tradeoff between prediction accuracy and prediction variance and at times outperforms the existing methods for both sets of metrics. Overall results indicate that this proposed method can provide improved geostatistical predictions while ensuring valid results when the use of non-Euclidean distances is warranted.


86A32 Geostatistics
60G25 Prediction theory (aspects of stochastic processes)
62M30 Inference from spatial processes
Full Text: DOI Link


[1] Berman, JD; Breysse, PN; White, RH; Waugh, DW; Curriero, FC, Evaluating methods for spatial mapping: applications for estimating ozone concentrations across the contiguous United States, Environ Technol Innov, 3, 1-10, (2015)
[2] Bivand R, Keitt T, Rowlingson B (2016) rgdal: bindings for the geospatial data abstraction library, R package version 1.1-10 edn.
[3] Boisvert JB (2010) Geostatistics with locally varying anisotropy. University of Alberta, Edmonton
[4] Boisvert, JB; Deutsch, CV, Programs for kriging and sequential Gaussian simulation with locally varying anisotropy using non-Euclidean distances, Comput Geosci, 37, 495-510, (2011)
[5] Cheng, SH; Higham, NJ, A modified Cholesky algorithm based on a symmetric indefinite factorization, SIAM J Matrix Anal Appl, 19, 1097-1110, (1998) · Zbl 0949.65022
[6] Chesapeake Bay Program (2017) Data hub: CBP GIS datasets. Chesapeake Bay Program. https://www.chesapeakebay.net/what/data. Accessed 17 Sept 2015
[7] Congdon CD, Martin JD (2007) On using standard residuals as a metric of kriging model quality. In: Proceedings of the 48th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, Honolulu HI
[8] Core Team R (2016) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna
[9] Cressie NAC (1993) Statistics for spatial data, Revised edn. Wiley, London
[10] Curriero, FC, On the use of non-euclidean distance measures in geostatistics, Math Geol, 38, 907-926, (2006) · Zbl 1162.86320
[11] Datta, A.; Banerjee, S.; Finley, AO; Gelfand, AE, On nearest-neighbor Gaussian process models for massive spatial data, Wiley Interdiscip Rev Comput Stat, 8, 162-171, (2016)
[12] Davis, BJ; Jacobs, JM; Davis, MF; Schwab, KJ; DePaola, A.; Curriero, FC, Environmental determinants of Vibrio parahaemolyticus in the Chesapeake Bay, Appl Environ Microbiol, 83, e01117-e01147, (2017)
[13] Castillo, E.; Colosimo, BM; Tajbakhsh, SD, Geodesic gaussian processes for the parametric reconstruction of a free-form surface, Technometrics, 57, 87-99, (2015)
[14] Diggle PJ, Ribeiro PJ (2007) Model-based geostatistics. Springer series in statistics. Springer, New York
[15] ESRI (2011) ArcGIS desktop: release 10.3. Environmental Systems Research Institute, Redlands
[16] ESRI (2016) Cross Validation. esri. http://desktop.arcgis.com/en/arcmap/10.3/tools/geostatistical-analyst-toolbox/cross-validation.htm. Accessed 27 June 2016
[17] Etten JV (2015) gdistance: distances and routes on geographical grids, R package version 1.1-9 edn.
[18] Gardner, B.; Sullivan, PJ; Lembo, AJ, Predicting stream temperatures: Geostatistical model comparison using alternative distance metrics, Can J Fish Aquat Sci, 60, 344-351, (2003)
[19] Hengl, T.; Heuvelink, GB; Stein, A., A generic framework for spatial prediction of soil variables based on regression-kriging, Geoderma, 120, 75-93, (2004)
[20] Henshaw, SL; Curriero, FC; Shields, TM; Glass, GE; Strickland, PT; Breysse, PN, Geostatistics and GIS: tools for characterizing environmental contamination, J Med Syst, 28, 335-348, (2004)
[21] Higham, NJ, Computing the nearest correlation matrix: a problem from finance, IMA J Numer Anal, 22, 329-343, (2002) · Zbl 1006.65036
[22] Jeffrey, SJ; Carter, JO; Moodie, KB; Beswick, AR, Using spatial interpolation to construct a comprehensive archive of Australian climate data, Environ Model Softw, 16, 309-330, (2001)
[23] Jensen, OP; Christman, MC; Miller, TJ, Landscape-based geostatistics: a case study of the distribution of blue crab in Chesapeake Bay, Environmetrics, 17, 605-621, (2006)
[24] Kane, MJ; Emerson, J.; Weston, S., Scalable strategies for computing with massive data, J Stat Softw, 55, 1-19, (2013)
[25] Laaha, G.; Skøien, J.; Blöschl, G., Spatial prediction on river networks: comparison of top-kriging with regional regression, Hydrol Process, 28, 315-324, (2014)
[26] Little, LS; Edwards, D.; Porter, DE, Kriging in estuaries: as the crow flies, or as the fish swims?, J Exp Mar Biol Ecol, 213, 1-11, (1997)
[27] Liu, R.; Young, MT; Chen, J-C; Kaufman, JD; Chen, H., Ambient air pollution exposures and risk of Parkinson disease, Environ Health Perspect, 124, 1759, (2016)
[28] Løland, A.; Host, G., Spatial covariance modelling in a complex coastal domain by multidimensional scaling, Environmetrics, 14, 307-321, (2003)
[29] Lu B, Charlton M, Fotheringham AS (2011) Geographically Weighted Regression using a non-Euclidean distance metric with a study on London house price data. In: Procedia environmental sciences, pp 92-97. https://doi.org/10.1016/j.proenv.2011.07.017
[30] Lu, B.; Charlton, M.; Harris, P.; Fotheringham, AS, Geographically weighted regression with a non-Euclidean distance metric: a case study using hedonic house price data, Int J Geogr Inf Sci, 28, 660-681, (2014)
[31] Lucas C (2001) Computing nearest covariance and correlation matrices. M.S, Thesis, University of Manchester
[32] Maechler M (2016) sfsmisc: utilities from “Seminar fuer Statistik” ETH Zurich, R package version 1.1-0 edn.
[33] Mardia KV, Kent JT, Bibby JM (1979) Multivariate analysis. Academic Press, London
[34] Matheron, G., The theory of regionalized variables and its applications, Les Cah Morphol Math, 5, 218, (1971)
[35] Meyer D, Dimitriadou E, Hornik K, Weingessel A, Leisch F (2015) e1071: Misc Functions of the Department of Statistics, Probability Theory Group (Formerly: E1071), TU Wien, R package version 1.6-7. edn.
[36] Murphy, R.; Perlman, E.; Ball, WP; Curriero, FC, Water-distance-based Kriging in Chesapeake Bay, J Hydrol Eng, 20, 0501403, (2015)
[37] Novomestky F (2012) matrixcalc: collection of functions for matrix calculations, R package version 1.0-3 edn.
[38] Rathbun, SL, Spatial modelling in irregularly shaped regions: Kriging estuaries, Environmetrics, 9, 109-129, (1998)
[39] Ribeiro PJ, Diggle PJ (2016) geoR: analysis of geostatistical data, R package version 1.7-5.2 edn.
[40] Roweis, ST; Saul, LK, Nonlinear dimensionality reduction by locally linear embedding, Science, 290, 2323-2326, (2000)
[41] Rowlingson B, Diggle P (2015) splancs: spatial and space-time point pattern analysis, R package version 2.01-38 edn.
[42] Sampson, PD; Guttorp, P., Nonparametric estimation of nonstationary spatial covariance structure, J Am Stat Assoc, 87, 108-119, (1992)
[43] Schlather, M.; Malinowski, A.; Menck, PJ; Oesting, M.; Strokorb, K., Analysis, simulation and prediction of multivariate random fields with package RandomFields, J Stat Softw, 63, 1-25, (2015)
[44] USGS (2016) The national hydrography dataset. https://nhd.usgs.gov/index.html. Accessed 3 Dec 2016
[45] Ver Hoef, JM, Kriging models for linear networks and non-Euclidean distances: cautions and solutions, Methods Ecol Evol., (2018)
[46] Wickham H (2009) ggplot2: elegant graphics for data analysis. Springer, New York · Zbl 1170.62004
[47] Yu H, Wang X, Qing J, Nie H (2015) ArcMap raster edit suite (ARES), 0.2.1 edn. https://github.com/haoliangyu/ares
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.