Unmixing compositional data with Bayesian techniques

A general problem in compositional data analysis is the unmixing of a composition into a series of pure endmembers. In its most complex version, one does neither know the composition of these endmembers, nor their relative contribution to each observed composition. The problem is particularly cumbersome if the number of endmembers is larger than the number of observed components. This contribution proposes a possible solution of this under-determined problem. The proposed method starts assuming that the endmember composition is known. Then, a geometric characterization of the problem allows to nd the set of possible endmember proportions compatible with the observed composition. Within this set any solution may be valid, but some are more likely than other. To use this idea and choose the "most likely" solution in each case, the problem can be tackled with Bayesian Markov-Chain Monte-Carlo techniques. Finally, once we are familiar with MCMC, it is quite traightforward to allow the endmember compositions to randomly vary, and use the same MCMC to estimate the endmember composition most compatible with the studied data.Peer ReviewedPostprint (published version

Guía para el análisis espacial de datos composicionales

El tratamiento de bases de datos composicionales (proporciones, porcentajes, concentraciones, etc.) con dependencia espacial debe hacerse atendiendo a las características matemáticas de éstos: una composición válida debe tener todas las componentes positivas y su suma debe ser igual o menor a un total (1, 100%, etc.). En general, esto se consigue de forma razonablemente fácil transformando la composición mediante series de log-cocientes de componentes. Para estudiar la variabilidad espacial de una composición se recomienda estimar y modelar el variaciograma: el conjunto de variogramas de todos los log-cocientes de un par de componentes. El variaciograma contiene toda la información necesaria para caracterizar una composición con estacionaridad intrínseca, y se puede modelar con herramientas habituales de la geoestadística, como el modelo de coregionalización lineal. Además, se pueden estudiar las propiedades del modelo e inferir relaciones entre componentes y posibles procesos vinculados a alguna escala espacial concreta. Finalmente, interpolar la composición y generar mapas es tarea sencilla con las herramientas existentes de krigeado y simulación: estas técnicas y conceptos deben aplicarse a un conjunto de log-cocientes cualquiera, tal que exista una transformación invertible entre éllos y las componentes de la composición original. Dealing with spatially-dependent compositional databases (including proportions, data in percentages, concentrations etc) should pay heed to the mathematical properties of these kinds of data: a valid composition must have positive components whose sum is at most a constant (1, 100% etc.). Generally speaking this is easily done by working on a set of log-ratios of components rather than using the raw data. To study the spatial variability of these databases it is best to estimate and model the lr-variograms, i.e. the set of variograms of all possible pairwise log-ratios of components in the composition. Such lr-variograms contain all the information necessary to deal with intrinsic stationary compositions and may be modelled with standard geostatistical tools such as the linear model of coregionalization. Moreover, the properties of the model can be studied and relationships inferred between components and possible processes linked to a given spatial scale. Finally, component-by-component interpolation and mapping is straightforward with existing kriging and simulation techniques: these tools and concepts should be applied to any set of invertible component log-ratios, i.e. log-ratio transformations, in such a way that the original composition can be recovered from the transformed data and vice versa.Postprint (published version

Grain-size control on petrographic composition of sediments: compositional regression and rounded zeroes

It is well-known that sediment composition strongly depends on grain size. A number of studies have tried to quantify this relationship focusing on the sand fraction, but only very limited data exists covering wider grain size ranges. Geologists have a clear conceptual model of the relation between grain size and sediment petrograpic composition, typically displayed in evolution diagrams. We chose a classical model covering grain sizes from fine gravel to clay, and distinguishing five types of grains (rock fragments, poly- and mono crystalline quartz, feldspar and mica/clay). A compositional linear process is fitted here to a digitized version of this model, by (i) applying classical regression to the set of all pairwise log-ratios of the 5-part composition against grain size, and (ii) looking for the compositions that best approximate the set of estimated parameters, one acting as slope and one as intercept. The method is useful even in the presence of several missing values. The linear fit suggests that the relative influence of the processes controlling the relationship between grain size and sediment composition is constant along most of the grain size spectrum.Postprint (published version

Geochemistry versus grain-size relations of sediments in the light of comminution, chemical alteration, and contrasting source rocks

Around 170 sediment samples from glacial and proximal glacio-fluvial deposits have been analysed for their geochemical composition. Samples derive from two strongly contrasting source areas (granitoids vs. amphibolite) and cover a broad grain-size range from coarse sand to clay. Following descriptive data evaluation, the relation of sediment geochemical composition versus grain size is modelled using linear regression techniques in the Aitchison geometry of the simplex in order to (i) describe the effects of comminution on the composition of individual grain size fractions, (ii) describe the influence of inherited mineral-specific grain-size distributions for contrasting source rocks, and (iii) to test for any potential influence of chemical weathering. Results indicate strong overall grain-size control on sediment composition that is largely reflecting the greater grain-size control on mineralogy. Comminution leads to overall strong enrichment of sheet silicates in the fine-grained fraction at the expense of quartz and, less pronounced, feldspars. Specific elements such as Ca, P, and Ti related to certain minerals do not follow this general trend and clearly indicate source-rock dependent enrichment of certain minerals (e.g. apatite, Ti-minerals) in medium grain-size fractions. Estimates of mineral compositions obtained from a geometric endmember approach support these conclusions. Chemical weathering is shown to be negligible

Geochemistry versus grain-size relations of sediments in the light of comminution, chemical alteration, and contrasting source rocks

Around 170 sediment samples from glacial and proximal glacio-fluvial deposits have been analysed for their geochemical composition. Samples derive from two strongly contrasting source areas (granitoids vs. amphibolite) and cover a broad grain-size range from coarse sand to clay. Following descriptive data evaluation, the relation of sediment geochemical composition versus grain size is modelled using linear regression techniques in the Aitchison geometry of the simplex in order to (i) describe the effects of comminution on the composition of individual grain size fractions, (ii) describe the influence of inherited mineral-specific grain-size distributions for contrasting source rocks, and (iii) to test for any potential influence of chemical weathering. Results indicate strong overall grain-size control on sediment composition that is largely reflecting the greater grain-size control on mineralogy. Comminution leads to overall strong enrichment of sheet silicates in the fine-grained fraction at the expense of quartz and, less pronounced, feldspars. Specific elements such as Ca, P, and Ti related to certain minerals do not follow this general trend and clearly indicate source-rock dependent enrichment of certain minerals (e.g. apatite, Ti-minerals) in medium grain-size fractions. Estimates of mineral compositions obtained from a geometric endmember approach support these conclusions. Chemical weathering is shown to be negligible.Peer ReviewedPostprint (published version

Pluviometric regionalization of Catalunya: a compositional data methodology

The aim of this paper is to introduce a methodology for de¯ning groups from regionalized com- positional data, through a hierarchical clustering algorithm aware of both the spatial dependence and the compositional character of the data set. This method is used to de¯ne a regionalization of Catalunya (NE Spain) with respect to its precipitation patterns in the Winter season. This region is characterized by a highly contrasted topography, which plays a dominant role in the spatial distribution of precipitation. Each rain gauge station is characterized by the relative frequencies of occurrence of six intervals of daily precipitation amount (classes ranging from \no rain" for precipitation below 3 mm, to \heavy storm" above 50 mm). Recognizing that frequencies are com-positional data, the spatial dependence of this data set has been characterized by variograms of the set of all pair-wise log-ratios, in the fashion of the variation matrix. Then, a Mahalanobis distance between stations has been de¯ned using these variograms to ensure that gauges with high spatial correlation get smaller distances. This spatially-dependent distance criterion has been used in a Ward hierarhical cluster method to de¯ne the regions. Results reveal 5 quite homogeneous groups of stations, which can be mostly ascribed a physical meaning. Finally, possible links to regional circulation patterns are discussed.Postprint (published version

The compositional meaning of a detection limit

Conclusions The chemical interpretation of the detection limit and its stochastic model counterpart has thus different consequences for the statistical analysis than we would expect from the word by word interpretation of “below detection limit” as a concentration below some limit. The state of the art model on BDL compositional analysis is biased. Some ideas are not directly applicable to the true effects of measurement errors near the detection limit. Even the basic principles like subcompositional coherence and the requirement of the independence of the analysis from the total are not fully valid near the detection limit.Peer ReviewedPostprint (published version