Indicator Kriging without Order Relation Violations

Abstract

Indicator kriging (IK) is a spatial interpolation technique aimed at estimating the conditional cumulative distribution function (ccdf) of a variable at an unsampled location. Obtained results form a discrete approximation to this ccdf, and its corresponding discrete probability density function (cpdf) should be a vector, where each component gives the probability of an occurrence of a class. Therefore, this vector must have positive components summing up to one, like in a composition in the simplex. This suggests a simplicial approach to IK, based on the algebraic-geometric structure of this sample space: simplicial IK actually works with log-odds. Interpolated log-odds can afterwards be easily re-expressed as the desired cpdf or ccdf. An alternative but equivalent approach may also be based on log-likelihoods. Both versions of the method avoid by construction all conventional IK standard drawbacks: estimates are always within the (0, 1) interval and present no order-relation problems (either with kriging or co-kriging). Even the modeling of indicator structural functions is clarified.

Classifying wave forecasts with model-based Geostatistics and the Aitchison distribution

“…These expressions are equivalent to simple kriging of logcontrasts from sampled locations (classified features) to unsampled locations (unclassified features). Interestingly, this coincides with the simplicial alternative to indicator kriging proposed by Tolosana-Delgado et al (2007) using the geometry of the simplex (Sect. 1) This method has the advantage of never giving impossible probabilities or order-relation problems, as conventional IK does.…”

A kriging approach based on Aitchison geometry for the characterization of particle-size curves in heterogeneous aquifers

“…On the other hand, typical geostatistical analyses [e.g., Riva et al (2006Riva et al ( , 2008Riva et al ( , 2010; Bianchi et al (2011) and references therein] treat each quantile separately (possibly introducing estimated crosscorrelations in terms of cross-variograms) and project their predictions through kriging on a computational grid. Such an approach, besides being methodologically different from the one we propose, can produce inconsistent results (Tolosana-Delgado et al 2008). For the purpose of our application, we consider the logtransformed 10th and 60th quantiles of the particle-size distribution in s; respectively indicated as D 10 ðsÞ and D 60 ðsÞ; i.e.…”

Compositional cokriging for mapping the probability risk of groundwater contamination by nitrates

“…In order to apply compositional cokriging, the definition of the indicator variable must be slightly modified by defining the generalized indicator variable (Tolosana-Delgado et al, 2008):…”

“…One way of working with compositional data is to perform a transformation, to apply the standard statistical techniques to the transformed data and to back-transform the results by using the inverse transformation. Among the different transformations, the isometric log-ratio transformation (Egozcue et al, 2003) is an orthonormal transformation that allows working with coordinates (that is real variables) and the standard geostatistical methodologies can be applied to the coordinates (Tolosana-Delgado et al, 2008). (Note that we are not talking about geographical coordinates but the expression of the experimental data as coordinates or coefficients with respect to an orthonormal basis provided by the Euclidean space structure of the simplex.…”