Localized/Shrinkage Kriging Predictors
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Localized/Shrinkage Kriging Predictors Zeytu Gashaw Asfaw1 · Henning Omre2
Received: 31 January 2015 / Accepted: 9 November 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com
Abstract The objective of the study is to improve the robustness and flexibility of spatial kriging predictors with respect to deviations from spatial stationarity assumptions. A predictor based on a non-stationary Gaussian random field is defined. The model parameters are inferred in an empirical Bayesian setting, using observations in a local neighborhood and a prior model assessed from the global set of observations. The localized predictor exhibits a shrinkage effect and is termed a localized/shrinkage kriging predictor. The predictor is compared to traditional localized kriging predictors in a case study on observations of annual accumulated precipitation. A cross-validation criterion is used in the comparison. The shrinkage predictor appears as clearly preferable to the traditional kriging predictors. A simulation study on prediction in non-stationary Gaussian random fields is conducted. The results from this study confirm that the shrinkage predictor is preferable to the traditional one. Moreover, the cross-validation criterion is found to be suitable for selection of the local neighborhood in the predictor. Lastly, the computational demands of localized predictors are very modest, hence these localized/shrinkage predictors are suitable for large scale spatial prediction problems. Keywords Spatial statistics · Gaussian random fields · Bayesian inference · Conjugate models
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Zeytu Gashaw Asfaw [email protected] Henning Omre [email protected]
1
School of Mathematical and Statistical Sciences, Hawassa University, Hawassa, Ethiopia
2
Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway
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Math Geosci
1 Introduction Spatial prediction is required in many applications, and examples can be found in natural resource mapping, meteorology and image analysis. Consider a regionalized variable {r (x); x ∈ D ⊂ m } where r (x) ∈ 1 is the variable of interest and x is a reference variable in the domain D. The challenge is to predict the regionalized variable from a set of observations ro = [r (x1 ), . . . , r (xn )]; x1 , . . . , xn ∈ D. In the current study, the predictors are defined in a probabilistic setting and associated predictor uncertainties can also be obtained. The classical probabilistic approach to spatial prediction is kriging (Journel and Huijbregts 1978; Chiles and Delfiner 2009). The traditional ordinary kriging predictor is based on a stationary model for the regionalized variable, with spatially constant expectation and variance, and a translation invariant spatial correlation function. The localized predictors, local neighborhood kriging (Chiles and Delfiner 2009) can be defined to robustify the predictor with respect to deviations from the stationarity assumptions. The major challenge in using localized predictors is to defin
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