Abstract
A fundamental decision to make during the analysis of geostatistical data is the modeling of the spatial dependence structure as stationary or non-stationary. Although second-order stationary modeling approaches have been successfully applied in geostatistical applications for decades, there is a growing interest in second-order non-stationary modeling approaches. This paper provides a review of modeling approaches allowing to take into account the second-order non-stationarity in univariate geostatistical data. One broad distinction between these modeling approaches relies on the way that the second-order non-stationarity is captured. It seems unlikely to prove that there would be the best second-order non-stationary modeling approach for all geostatistical applications. However, some of them are distinguished by their simplicity, interpretability, and flexibility.
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References
Almendral A, Abrahamsen P, Hauge R (2008). Multidimensional scaling and anisotropic covariance functions. In: Proceedings of the eight international geostatistics congress, pp 187–196
Anderes EB, Chatterjee S (2009) Consistent estimates of deformed isotropic Gaussian random fields on the plane. Ann Stat 37(5):2324–2350
Anderes EB, Stein ML (2008) Estimating deformations of isotropic Gaussian random fields on the plane. Ann Stat 36:719–741
Anderes EB, Stein ML (2011) Local likelihood estimation for nonstationary random fields. J Multivar Anal 102(3):506–520
Atkinson PM, Lloyd CD (2007) Non-stationary variogram models for geostatistical sampling optimisation: an empirical investigation using elevation data. Comput Geosci 33(10):1285–1300
Banerjee S, Gelfand AE, Knight JR, Sirmans CF (2004) Spatial modeling of house prices using normalized distance-weighted sums of stationary processes. J Bus Econ Stat 22(2):206–213
Bel L (2004) Non parametric variogram estimator: application to air pollution data. In: geoENV IV, geostatistics for environmental applications. Quantitative geology and geostatistics, vol 13. Springer, Dordrecht, pp 29–40
Boisvert JB, Deutsch CV (2011) Programs for kriging and sequential Gaussian simulation with locally varying anisotropy using non-Euclidean distances. Comput Geosci 37(4):495–510
Bolin D (2014) Spatial Matérn fields driven by non-Gaussian noise. Scand J Stat 41(3):557–579
Bolin D, Lindgren F (2011) Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping. Ann Appl Stat 5(1):523–550
Bornn L, Shaddick G, Zidek JV (2012) Modeling nonstationary processes through dimension expansion. J Am Stat Assoc 107(497):281–289
Calder CA (2008) A dynamic process convolution approach to modeling ambient particulate matter concentrations. Environmetrics 19(1):39–48
Chang Y-M, Hsu N-J, Huang H-C (2010) Semiparametric estimation and selection for nonstationary spatial covariance functions. J Comput Graph Stat 19(1):117–139
Chilès JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty. Wiley, New York
Cohen A, Jones RH (1969) Regression on a random field. J Am Stat Assoc 64:1172–1182
Cressie N, Johannesson G (2008) Fixed rank kriging for very large spatial data sets. J R Stat Soc Ser B 70:209–226
Dagbert M, David M, Crozel D, Desbarats A (1984) Computing variograms in folded strata-controlled deposits. In: Verly G, David M, Journel AG, Marechal A (eds) Geostatistics for natural resources characterization: part 1. Springer, Dordrecht, pp 71–89
Dalang RC, Khoshnevisan D (2009) A minicourse on stochastic partial differential equations, vol 1962. Springer, Dordrecht
Damian D, Sampson PD, Guttorp P (2001) Bayesian estimation of semi-parametric non-stationary spatial covariance structures. Environmetrics 12(2):161–178
D’Hondt O, López-Martínez C, Ferro-Famil L, Pottier E (2007) Spatially nonstationary anisotropic texture analysis in SAR images. IEEE Trans Geosci Remote Sens 45(12–1):3905–3918
Ecker M, De Oliveira V, Isakson H (2013) A note on a non-stationary point source spatial model. Environ Ecol Stat 20(1):59–67
Ecker MD, Oliveira VD (2008) Bayesian spatial modeling of housing prices subject to a localized externality. Commun Stat Theory Methods 37(13):2066–2078
Fouedjio F (2015) Space deformation non-stationary geostatistical approach for prediction of geological objects: case study at El Teniente Mine (Chile). Nat Resour Res 25(3):283–296
Fouedjio F, Desassis N, Rivoirard J (2016) A generalized convolution model and estimation for non-stationary random functions. Spat Stat 16:35–52
Fouedjio F, Desassis N, Romary T (2015) Estimation of space deformation model for non-stationary random functions. Spat Stat 13:45–61
Fouedjio F, Séguret S (2016) Predictive geological mapping using closed-form non-stationary covariance functions with locally varying anisotropy: case study at El Teniente Mine (Chile). Nat Resour Res. doi:10.1007/s11053-016-9293-4
Fuentes M (2001) A high frequency kriging approach for non-stationary environmental processes. Environmetrics 12(5):469–483
Fuentes M (2002a) Interpolation of nonstationary air pollution processes: a spatial spectral approach. Stat Model 2(4):281–298
Fuentes M (2002b) Spectral methods for nonstationary spatial processes. Biometrika 89(1):197–210
Fuglstad G-A, Lindgren F, Simpson D, Rue H (2015a) Exploring a new class of non-stationary spatial Gaussian random fields with varying local anisotropy. Stat Sin 25:115–133
Fuglstad G-A, Simpson D, Lindgren F, Rue H (2015b) Does non-stationary spatial data always require non-stationary random fields? Spat Stat 14C:505–531
Gosoniu L, Vounatsou P (2011) Non-stationary partition modeling of geostatistical data for malaria risk mapping. J Appl Stat 38(1):3–13
Gosoniu L, Vounatsou P, Sogoba N, Maire N, Smith T (2009) Mapping malaria risk in West Africa using a Bayesian nonparametric non-stationary model. Comput Stat Data Anal 53(9):3358–3371
Guillot G, Senoussi R, Monestiez P (2001). A positive definite estimator of the non stationary covariance of random fields. In: geoENV III, geostatistics for environmental applications. Quantitative geology and geostatistics, vol 11. Springer, Dordrecht, pp 333–344
Haas TC (1990a) Kriging and automated variogram modeling within a moving window. Atmos Environ Part A Gen Top 24(7):1759–1769
Haas TC (1990b) Lognormal and moving window methods of estimating acid deposition. J Am Stat Assoc 85(412):950–963
Harris P, Charlton M, Fotheringham AS (2010) Moving window kriging with geographically weighted variograms. Stoch Env Res Risk Assess 24(8):1193–1209
Heaton M, Katzfuss M, Berrett C (2014) Constructing valid spatial processes on the sphere using kernel convolutions. Environmetrics 25(1):2–15
Heaton MJ, Christensen WF, Terres MA (2015) Nonstationary Gaussian process models using spatial hierarchical clustering from finite differences. Technometrics. doi:10.1080/00401706.2015.1102763
Higdon D (1998) A process-convolution approach to modelling temperatures in the North Atlantic Ocean. Environ Ecol Stat 5(2):173–190
Higdon D (2002) Space and space-time modeling using process convolutions, Springer edn. Springer, London, pp 37–56
Higdon D, Swall J, Kern J (1999) Non-stationary spatial modeling. In: Bayesian statistics, vol 6. Oxford University Press, New York, pp 761–768
Hoef JMV, Peterson E, Theobald D (2006) Spatial statistical models that use flow and stream distance. Environ Ecol Stat 13(4):449–464
Holland D, Saltzman N, Cox LH, Nychka D (1999) Spatial prediction of sulfur dioxide in eastern United States. In: geoENV-II—geostatistics for environmental applications. Kluwer, Dordrecht, pp 65–76
Hughes-Oliver JM, González-Farías G (1999) Parametric covariance models for shock-induced stochastic processes. J Stat Plan Inference 77(1):51–72
Hughes-Oliver JM, González-Farías G, Lu JC, Chen D (1998a) Parametric nonstationary correlation models. Stat Probab Lett 40(3):267–278
Hughes-Oliver JM, Lu JC, Davis JC, Gyurcsik RS (1998b) Achieving uniformity in a semiconductor fabrication process using spatial modeling. J Am Stat Assoc 93(443):1252–1252
Ingebrigtsen R, Lindgren F, Steinsland I (2014) Spatial models with explanatory variables in the dependence structure. Spat Stat 8:20–38
Ingebrigtsen R, Lindgren F, Steinsland I, Martino S (2015) Estimation of a non-stationary model for annual precipitation in southern Norway using replicates of the spatial field. Spat Stat 14C:338–364
Iovleff S, Perrin O (2004) Estimating a nonstationary spatial structure using simulated annealing. J Comput Graph Stat 13(1):90–105
Kim HM, Mallick BK, Holmes CC (2005) Analyzing nonstationary spatial data using piecewise Gaussian processes. J Am Stat Assoc 100(470):653–668
Kruskal JB (1964) Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29(1):1–27
Lefebvre Polus E, De Fouquet C, Bernard-Michel C, Flipo N, Poulin M (2008) Geostatistical model for concentrations or flow rates in streams: some results. In: Geostats 2008—8th international geostatistics congress, vol 2. Santiago, Chile, pp 871–880
Legleiter CJ, Kyriakidis PC (2006) Forward and inverse transformations between cartesian and channel-fitted coordinate systems for meandering rivers. Math Geol 38(8):927–958
Liang M, Marcotte D (2015) A class of non-stationary covariance functions with compact support. Stoch Environ Res Risk Assess 30(3):1–15
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 Ser B 73(4):423–498
Løland A, Høst G (2003) Spatial covariance modelling in a complex coastal domain by multidimensional scaling. Environmetrics 14(3):307–321
Lloyd CD, Atkinson PM (2000) Interpolation elevation with locally-adaptative kriging. In: Atkinson P (ed) GIS and GeoComputation: innovations in GIS 7. Taylor & Francis, Park Drive, pp 241–253
Lloyd CD, Atkinson PM (2002) Non-stationary approaches for mapping terrain and assessing prediction uncertainty. Trans GIS 6(1):17–30
Machuca-Mory D, Deutsch C (2013) Non-stationary geostatistical modeling based on distance weighted statistics and distributions. Math Geosci 45:31–48
Magneron C, Jeannee N, Le Moine O, Bourillet JF (2010) Integrating prior knowledge and locally varying parameters with moving-geostatistics: methodology and application to bathymetric mapping. In: geoENV VII—geostatistics for environmental applications, vol. 16. Springer, Dordrecht, pp 405–415
Mardia K, Goodall C (1993) Spatial-temporal analysis of multivariate environmental monitoring data. In: Multivariate environmental statistics. Elsevier Science Publishers, Amsterdam, pp 347–386
Matérn B (1986) Spatial Variation. Lecture notes in statistics. Springer, New york
Mateu J, Fernandez-Avilas G, Montero J (2010) On a class of non-stationary, compactly supported spatial covariance functions. Stoch Environ Res Risk Assess 27(2):1–13
Matsuo T, Nychka D, Paul D (2011) Nonstationary covariance modeling for incomplete data: Monte Carlo EM approach. Comput Stat Data Anal 55(6):2059–2073
McBratney AB, Minasny B (2013) Spacebender. Spat Stat 4:57–67
Neto JHV, Schmidt AM, Guttorp P (2014) Accounting for spatially varying directional effects in spatial covariance structures. J R Stat Soc Ser C 63(1):103–122
Nott DJ, Dunsmuir WTM (2002) Estimation of nonstationary spatial covariance structure. Biometrika 89(4):819–829
Nychka D, Bandyopadhyay S, Hammerling D, Lindgren F, Sain S (2015) A multiresolution Gaussian process model for the analysis of large spatial datasets. J Comput Graph Stat 24(2):579–599
Nychka D, Saltzman N (1998) Design of air quality networks. In: Case studies in environmental statistics. Lectures notes in statistics, vol 132. Springer, New York, pp 51–76
Nychka D, Wikle C, Royle JA (2002) Multiresolution models for nonstationary spatial covariance functions. Stat Model 2(4):315–331
Oehlert GW (1993) Regional trends in sulfate wet deposition. J Am Stat Assoc 88(422):390–399
Paciorek CJ, Schervish MJ (2006) Spatial modelling using a new class of nonstationary covariance functions. Environmetrics 17(5):483–506
Perrin O, Meiring W (1999) Identifiability for non-stationary spatial structure. J Appl Probab 36(4):1244–1250
Perrin O, Meiring W (2003) Nonstationarity in Rn is second-order stationarity in R2n. J Appl Probab 40(3):815–820
Perrin O, Monestiez P (1998) Modeling of non-stationary spatial covariance structure by parametric radial basis deformations. Quantitative geology and geostatistics, vol 11. Springer, Dordrecht, pp 175–186
Perrin O, Senoussi R (2000) Reducing non-stationary random fields to stationarity and isotropy using a space deformation. Stat Probab Lett 48(1):23–32
Pintore A, Holmes C (2004) Spatially adaptive non-stationary covariance functions via spatially adaptive spectra. Technical report, University of Oxford
Porcu E, Matkowski J, Mateu J (2010) On the non-reducibility of non-stationary correlation functions to stationary ones under a class of mean-operator transformations. Stoch Env Res Risk Assess 24(5):599–610
Rasmussen C, Williams C (2006) Gaussian processes for machine learning. Adaptative computation and machine learning series. University Press Group Limited, London
Risser MD, Calder CA (2015) Regression-based covariance functions for nonstationary spatial modeling. Environmetrics 26(4):284–297
Rivest M, Marcotte D (2012) Kriging groundwater solute concentrations using flow coordinates and nonstationary covariance functions. J Hydrol 472–473:238–253
Rivest M, Marcotte D, Pasquier P (2012) Sparse data integration for the interpolation of concentration measurements using kriging in natural coordinates. J Hydrol 416–417:72–82
Sampson PD, Guttorp P (1992) Nonparametric-estimation of nonstationary spatial covariance structure. J Am Stat Assoc 87(417):108–119
Schmidt AM, Guttorp P, O’Hagan A (2011) Considering covariates in the covariance structure of spatial processes. Environmetrics 22(4):487–500
Schmidt AM, O’Hagan A (2003) Bayesian inference for non-stationary spatial covariance structure via spatial deformations. J R Stat Soc Ser B 65:743–758
Simpson D, Lindgren F, Rue H (2012) Think continuous: Markovian Gaussian models in spatial statistics. Spat Stat 1:16–29
Smith RL (1996) Estimating nonstationary spatial correlations. Technical report, University of North Carolina
Stein M (2005) Nonstationary spatial covariance functions. Technical report, University of Chicago
Stephenson J, Holmes C, Gallagher K, Pintore A (2005) A statistical technique for modelling non-stationary spatial processes. In: Geostatistics banff 2004, vols 1 and 2, vol 14. Springer, Dordrecht, pp 125–134
Vera J, Macias R, Angulo J (2008) Non-stationary spatial covariance structure estimation in oversampled domains by cluster differences scaling with spatial constraints. Stoch Env Res Risk Assess 22(1):95–106
Vera JF, Macias R, Angulo JM (2009) A latent class MDS model with spatial constraints for non-stationary spatial covariance estimation. Stoch Env Res Risk Assess 23(6):769–779
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. Soil Res 39(2):259–272
Whittle P (1954) On stationary processes in the plane. Biometrika 41(3/4):434–449
Whittle P (1963) Stochastic processes in several dimensions. Bull Int Stat Inst 40:974–994
Zhu Z, Wu Y (2010) Estimation and prediction of a class of convolution-based spatial nonstationary models for large spatial data. J Comput Graph Stat 19(1):74–95
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Fouedjio, F. Second-order non-stationary modeling approaches for univariate geostatistical data. Stoch Environ Res Risk Assess 31, 1887–1906 (2017). https://doi.org/10.1007/s00477-016-1274-y
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DOI: https://doi.org/10.1007/s00477-016-1274-y