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Groundwater Arsenic Discontinuity
Published in M. Manzurul Hassan, Arsenic in Groundwater, 2018
The term geostatistics was first introduced by Matheron (1963) on the basis of the theory of Danie Gerhardus Krige, a South African mining engineer (Krige, 1951) for estimating reserves of ore. Since then, geostatistics has been expanding as a spatial method for analyzing spatial discontinuities of geographical and environmental problems. In the 1980s, Professor John Aitchison from Hong Kong University (Aitchison, 1982) developed a compositional data analysis procedure that is nowadays known as the “log-ratio approach” (Pawlowsky-Glahn and José Egozcue, 2016). Geostatistics is currently applied in diverse disciplines for spatial continuity. An advantage of geostatistics is its use of quantitative measures of spatial correlation, commonly expressed as variograms (Uyan and Cay, 2013). A prerequisite for geostatistical interpolation is the definition of a model describing the spatial autocorrelation between the observations.
Improving geological logging of drill holes using geochemical data and data analytics for mineral exploration in the Gawler Ranges, South Australia
Published in Australian Journal of Earth Sciences, 2021
E. J. Hill, A. Fabris, Y. Uvarova, C. Tiddy
Geochemical data are compositional data, i.e. they are expressed as parts of a whole, for example, parts per million (ppm) as in the pXRF data for this study. Compositional data carry relative information, and therefore, for statistical purposes, compositional data should always be represented as ratios (Aitchison, 1986; Pawlowsky-Glahn et al., 2015). For example, ternary plots provide relative information for three elements; the same information can be conveyed by using ratios between the three elements. Log ratios are more mathematically convenient than ratios (for example, they allow direct application of linear methods) and are therefore the most widely used method for analysis of compositional data (Aitchison, 1986). Log-ratio transformation methods include the additive log-ratio and centred log-ratio (CLR) transforms developed by Aitchison (1986) and the isometric log-ratio proposed by Egozcue et al. (2003). Compositional data transformed using log ratios can be statistically analysed using the same methods as non-compositional data. In the examples presented here the CLR transform is used as it is a useful method for data exploration: it is easy to apply and to interpret and does not assume prior knowledge of geological processes. The CLR transform uses the ratio of each element to the geometric mean of all the elements (x1 … xD) in the sample (x): where g(x) is the geometric mean, given by:
Multivariate geostatistical simulation with sum and fraction constraints
Published in Applied Earth Science, 2018
Marcel Antonio Arcari Bassani, João Felipe Coimbra Leite Costa, Clayton Vernon Deutsch
One method to deal with variables that have sum constraint is based on using ratios or log-ratios of the original variables (Pawlowsky-Glahn and Olea 2004; Pawlowsky-Glahn and Egozcue 2006; Barnett and Deutsch 2012; Boisvert et al. 2013; Mery et al. 2017). This approach has been used successfully with compositional data. Compositional data represent a set of variables that sum to a constant. When the data does not sum to a constant, a filler variable may be added to the data set. The filler variable corresponds to the constant minus the sum of the remaining variables. The most common transformations for compositional data are the additive log-ratio (Aitchison 1986), centred log-ratio (Aitchison 1986) and isometric log-ratio (Egozcue et al. 2003). Manchuk et al. (2017) used the isometric log-ratio followed by PPMT to account for the sum constraint of the data set. In this paper, we used a ratio similar to the additive log-ratio. The difference is that the logarithmic was not calculated.
Probabilistic disaggregation of a spatial portfolio of exposure for natural hazard risk assessment
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2018
Rocco Custer, Kazuyoshi Nishijima
Compositional data give quantitative descriptions of parts of a whole. In a disaggregation problem, the aggregated quantity is the whole and disaggregated variables are parts of the whole and are therefore compositional variables. Seminal work on the analysis of compositional data is found in Aitchison (1986). A prominent probability distribution to model compositional data is the Dirichlet distribution (Connor and Mosimann 1969). Although it facilitates to formulate a variety of disaggregation problems, it does not provide much flexibility in the modelling of variance and correlation structure, since it implies a strictly negative correlation structure among variables. Several attempts have been made to render the Dirichlet distribution more general and flexible, see e.g. Connor and Mosimann (1969), Thomas and Jacob (2006), Ongaro, Migliorati, and Monti (2008) and Ng et al. (2009). However, the aforementioned shortcomings have yet to be completely overcome. Another popular approach to modelling compositional data is based on log-ratio transformations of compositional data, which is free from the problem of a constrained sample space and allows using standard multivariate techniques (Aitchison 1986).