High-order Statistics of Spatial Random Fields: Exploring Spatial Cumulants for Modeling Complex Non-Gaussian and Non-linear Phenomena

Dimitrakopoulos, Roussos; Mustapha, Hussein; Gloaguen, Erwan

doi:10.1007/s11004-009-9258-9

Cited by 117 publications

(65 citation statements)

References 25 publications

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“…Such ineffectiveness is especially severe in geological or environmental studies, as discussed by , Strebelle (2002) and Tjelmeland and Besag (1998). In the past decade, various techniques have been developed to improve the reliability of characterizing random fields, such as bootstrapping (Kleijnen et al, 2012;Schelin and Luna, 2010;Stein, 2008, 2004;Mukul et al, 2004), copula-based methods (Kazianka, 2013;Pilz et al, 2012;Kazianka andPilz, 2010a, 2010b;Bárdossy and Li, 2008;Bárdossy, 2006), kernel-based methods (Honarkhah and Caers, 2010;Scheidt and Caers, 2010, 2009a, 2009b), Bayesian (Nieto-Barajas and Sinha, 2014Troldborg et al, 2012;Pilz et al, 2012;Kazianka andPilz, 2011, 2012), multi-point simulation methods (De Iaco, 2013;Boucher, 2009;Chugunova and Hu, 2008;Wu et al, 2008;Mirowski et al, 2008;Arpat and Caers, 2007;Zhang et al, 2006;Strebelle, 2002), multi-scale simulations using wavelets (Chatterjee and Dimitrakopoulos, 2012;Dimitrakopoulos, 2009, 2008), and spatial-cumulant-based simulation methods (Goodfellow et al, 2012;MachucaMory and Dimitrakopoulos, 2012;, Dimitrakopoulos et al, 2010.…”

Section: Introductionmentioning

confidence: 99%

A new approach to spatial data interpolation using higher-order statistics

Liu

Anh

McGree

et al. 2014

Stoch Environ Res Risk Assess

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Abstract:Interpolation techniques for spatial data have been applied frequently in various fields of geosciences. Although most conventional interpolation methods assume that it is sufficient to use first-and second-order statistics to characterize random fields, researchers have now realized that these methods cannot always provide reliable interpolation results, since geological and environmental phenomena tend to be very complex, presenting nonGaussian distribution and/or non-linear inter-variable relationship. This paper proposes a new approach to the interpolation of spatial data, which can be applied with great flexibility.Suitable cross-variable higher-order spatial statistics are developed to measure the spatial relationship between the random variable at an unsampled location and those in its neighbourhood. Given the computed cross-variable higher-order spatial statistics, the conditional probability density function (CPDF) is approximated via polynomial expansions, which is then utilized to determine the interpolated value at the unsampled location as an expectation. In addition, the uncertainty associated with the interpolation is quantified by constructing prediction intervals of interpolated values. The proposed method is applied to a mineral deposit dataset, and the results demonstrate that it outperforms kriging methods in uncertainty quantification. The introduction of the cross-variable higher-order spatial statistics noticeably improves the quality of the interpolation since it enriches the information that can be extracted from the observed data, and this benefit is substantial when working with data that are sparse or have non-trivial dependence structures.

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Section: Introductionmentioning

confidence: 99%

A new approach to spatial data interpolation using higher-order statistics

Liu

Anh

McGree

et al. 2014

Stoch Environ Res Risk Assess

View full text Add to dashboard Cite

show abstract

“…81 Dimitrakopoulos et al (2010) describe the extension of multivariate cumulants to 82 the spatial random field Z(x). Consider the third-order cumulant.…”

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confidence: 99%

“…x we may define a set of three locations {x, to that of Dimitrakopoulos et al (2010). Given such a configuration, and making the 87 ergodicity assumption that the distribution of Z(x) is independent of x, we may express 88 the third-order cumulant for the random field at these locations as a function of lag only:…”

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confidence: 99%

“…As proposed by Dimitrakopoulos et al (2010) of interest), the estimator for the third cumulant of a zero-mean random variable from a 100 set of observations at locations X is therefore…”

mentioning

confidence: 99%

“…In practice, when sampling is not on a regular array, it is necessary to allow some 189 tolerance in the definition of the lag distances (Dimitrakopoulos et al, 2010). In this study …”

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confidence: 99%

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Using third-order cumulants to investigate spatial variation: A case study on the porosity of the Bunter Sandstone

Lark

2015

Spatial Statistics

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1The multivariate cumulants characterize aspects of the spatial variability of a re-2 gionalized variable. A centred multivariate Gaussian random variable, for example, has 3 zero third-order cumulants. In this paper it is shown how the third-order cumulants can 4 be used to test the plausibility of the assumption of multivariate normality for the porosity 5 of an important formation, the Bunter Sandstone in the North Sea. The results suggest 6 that the spatial variability of this variable deviates from multivariate normality, and that 7 this assumption may lead to misleading inferences about, for example, the uncertainty 8 attached to kriging predictions.

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The Algorithmic Building Blocks

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High-order Statistics of Spatial Random Fields: Exploring Spatial Cumulants for Modeling Complex Non-Gaussian and Non-linear Phenomena

Cited by 117 publications

References 25 publications

A new approach to spatial data interpolation using higher-order statistics

A new approach to spatial data interpolation using higher-order statistics

Using third-order cumulants to investigate spatial variation: A case study on the porosity of the Bunter Sandstone

The Algorithmic Building Blocks

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