A Geostatistical Framework for Estimating Compositional Data Avoiding Bias in Back-transformation

Abstract

Estimation of some mineral deposits involves chemical species or a granulometric mass balance that constitute a closed constant sum (e.g., 100%). Data that add up to a constant are known as compositional data (CODA). Classical geostatistical estimation methods (e.g., kriging) are not satisfactory when CODA are used, since bias is expected when estimated mean block values are back-transformed to the original space. CODA methods use nonlinear transformations, and when the transformed data are interpolated, they cannot be returned directly to the space of the original data. If these averages are back-transformed using the inverse function, bias is generated. To avoid this bias, this article proposes geostatistical simulation of the isometric logratio ratio (ilr) transformations back-transforming point simulated values (instead of block estimations), with the averaging being postponed to the end of the process. The results show that, in addition to maintaining the mass balance and the correlations among the variables, the means (E-types) of the simulations satisfactorily reproduce the statistical characteristics of the grades without any sort of bias. A complete case study of a major bauxite deposit illustrates the methodology.

Co-simulated Size Number: An Elegant Novel Algorithm for Identification of Multivariate Geochemical Anomalies

“…This reason is advocated, because the original variables are just a subgroup of a whole family known as compositional dataset, and the original de-clustered correlation still can be considered as an approximation of inherent dependency among the compositional data and may be examined as criteria for comparison between alternative geostatistical simulation methodologies. Sixth, there are several case studies based on geostatistical simulation of log-ratio transformation of original variables and contrasting to other methods that the validation parts are mainly based on conventional statistical parameters (e.g., Rubio et al 2016;Van den Boogaart et al 2017;Hosseini and Asghari 2019 and references therein).…”

A simultaneous spatial autoregressive model for compositional data

“…For example, land cover data contain information about different land use shares and the statistical unit is a subdivision of a territory; among the many papers that treat this type of data (see Leininger et al, 2013;Overmars et al, 2003;Yoshida and Tsutsumi, 2018;Pirzamanbein et al, 2018). Another instance is in geochemistry where data consist of composition of mineral deposits into chemical elements at different locations in geographical space, see for example Rubio et al (2016) who study sediments in an artic lake or Filzmoser et al (2010) who examine the Kola moss layer composition data from the R package StatDA (Filzmoser, 2020). This is also the case in political economy for electoral data containing the vote shares by party in a multiparty election for a list of administrative subdivisions of a territory as in Katz and King (1999) or for data about turnout rates as in Borghesi and Bouchaud (2010).…”

“…The challenge for modelling such data is to accomodate at the same time their compositional and spatial nature. For spatial data with a continuous domain, Pawlowsky and Burger (1992); Pawlowsky-Glahn and Egozcue (2016); Rubio et al (2016); Martins et al (2016) adopt a geostatistical approach. On the other hand, conditional autoregressive models have been developed for the multivariate regression framework (MCAR), e.g.…”