BLU Estimators and Compositional Data

One of the principal objections to the logratio approach for the statistical analysis of compositional data has been the absence of unbiasedness and minimum variance properties of some estimators: they seem not to be BLU estimator. Using a geometric approach, we introduce the concept of metric variance and of a compositional unbiased estimator, and we show that the closed geometric mean is a c-BLU estimator (compositional best linear unbiased estimator with respect to the geometry of the simplex) of the center of the distribution of a random composition. Thus, it satisfies analogous properties to the arithmetic mean as a BLU estimator of the expected value in real space. The geometric approach used gives real meaning to the concepts of measure of central tendency and measure of dispersion and opens up a new way of understanding the statistical analysis of compositional data.     

Logratio Analysis and Compositional Distance

The concept of distance between two compositions is important in the statistical analysis of compositional data, particularly in such activities as cluster analysis and multidimensional scaling. This paper exposes the fallacies in a recent criticism of logratio-based distance measures—in particular, the misstatements that logratio methods destroy distance structures and are denominator dependent. Emphasis is on ensuring that compositional data analysis involving distance concepts satisfies certain logically necessary invariance conditions. Logratio analysis and its associated distance measures satisfy these conditions.     

Compositional Kriging: A Spatial Interpolation Method for Compositional Data

Compositional data are very common in the earth sciences. Nevertheless, little attention has been paid to the spatial interpolation of these data sets. Most interpolators do not necessarily satisfy the constant sum and nonnegativity constraints of compositional data, nor take spatial structure into account. Therefore, compositional kriging is introduced as a straightforward extension of ordinary kriging that complies with these constraints. In two case studies, the performance of compositional kriging is compared with that of the additive logratio-transform. In the first case study, compositional kriging yielded significantly more accurate predictions than the additive logratio-transform, while in the second case study the performances were comparable.     

Dealing with Zeros and Missing Values in Compositional Data Sets Using Nonparametric Imputation

The statistical analysis of compositional data based on logratios of parts is not suitable when zeros are present in a data set. Nevertheless, if there is interest in using this modeling approach, several strategies have been published in the specialized literature which can be used. In particular, substitution or imputation strategies are available for rounded zeros. In this paper, existing nonparametric imputation methods—both for the additive and the multiplicative approach—are revised and essential properties of the last method are given. For missing values a generalization of the multiplicative approach is proposed.     

Groups of Parts and Their Balances in Compositional Data Analysis

Amalgamation of parts of a composition has been extensively used as a technique of analysis to achieve reduced dimension, as was discussed during the CoDaWork'03 meeting (Girona, Spain, 2003). It was shown to be a non-linear operation in the simplex that does not preserve distances under perturbation. The discussion motivated the introduction in the present paper of concepts such as group of parts, balance between groups, and sequential binary partition, which are intended to provide tools of compositional data analysis for dimension reduction. Key concepts underlying this development are the established tools of subcomposition, coordinates in an orthogonal basis of the simplex, balancing element and, in general, the Aitchison geometry in the simplex. Main new results are: a method to analyze grouped parts of a compositional vector through the adequate coordinates in an ad hoc orthonormal basis; and the study of balances of groups of parts (inter-group analysis) as an orthogonal projection similar to that used in standard subcompositional analysis (intra-group analysis). A simulated example compares results when testing equal centers of two populations using amalgamated parts and balances; it shows that, in certain circumstances, results from both analysis can disagree.     

Outlier Detection for Compositional Data Using Robust Methods

Outlier detection based on the Mahalanobis distance (MD) requires an appropriate transformation in case of compositional data. For the family of logratio transformations (additive, centered and isometric logratio transformation) it is shown that the MDs based on classical estimates are invariant to these transformations, and that the MDs based on affine equivariant estimators of location and covariance are the same for additive and isometric logratio transformation. Moreover, for 3-dimensional compositions the data structure can be visualized by contour lines. In higher dimension the MDs of closed and opened data give an impression of the multivariate data behavior.     

New Perspectives on Water Chemistry and Compositional Data Analysis

Water chemistry is commonly investigated to determine the suitability of water for various uses. With increased knowledge of aqueous chemistry, it has become possible to interpret the evolutionary processes that determine water composition and quality. This paper presents procedures for exploring and modeling the environment using compositional data from water analysis, utilizing statistical tools in an appropriate sample space. Our procedures build on a methodology based on log-ratios initiated by John Aitchison in the early 1980's. They are not only useful for interpreting the structure of the data, but also for characterizing and modeling the influence of geochemical processes acting on the environment. The geochemistry of water samples collected from wells on Vulcano Island (one of the Aeolian Islands of the Italian province of Sicily) will be used to illustrate the techniques, although an exhaustive overview would require many different examples. Vulcano island is a quiescent volcanic area where mobilization of chemical species by weathering of volcanic rocks and input of gaseous components from fumarolic activity has produced environmental changes expressed in the composition of phreatic waters at the surface and in the shallow subsurface. Changes in the chemical composition of waters in unconfined aquifers of the northwestern part of the island around the active crater appear to be useful in understanding the natural processes at work.     

Correlation Analysis for Compositional Data

Compositional data need a special treatment prior to correlation analysis. In this paper we argue why standard transformations for compositional data are not suitable for computing correlations, and why the use of raw or log-transformed data is neither meaningful. As a solution, a procedure based on balances is outlined, leading to sensible correlation measures. The construction of the balances is demonstrated using a real data example from geochemistry. It is shown that the considered correlation measures are invariant with respect to the choice of the binary partitions forming the balances. Robust counterparts to the classical, non-robust correlation measures are introduced and applied. By using appropriate graphical representations, it is shown how the resulting correlation coefficients can be interpreted.     

Modelling Compositional Change: The Example of Chemical Weathering of Granitoid Rocks

Perturbation on the simplex is an operation which can be used to numerically describe changes in the composition of, for example, soils, sediments, or rocks. The combination of perturbation and power transformation provides a strong tool for analyzing compositional linear processes in the simplex. When the process is constrained in the sense of a well-known starting (or final) composition, noncentred principal component analysis can be used to estimate the leading perturbation vector of the process. Applying these mathematical tools to chemical major element data from a weathering profile developed on granitoid rocks allows us to model the compositional changes associated with the process of chemical weathering. The comparison of these results with the compositional linear trend defined by erosional products of several of the world's major drainage systems yields close similarities. The latter observation allows for a mathematical formulation of a global mean weathering trend within the system Al₂O₃–CaO– Na₂O– K₂O. We further demonstrate the usefulness of the approach for validating processes behind individual trends and for combining the effects of different processes which modify the composition of soils, sediments, and rocks. Alternatives to the Chemical Index of Alteration (CIA) are discussed to obtain a translation-invariant scale for the process of chemical weathering.     

The statistical study of hydraulic geometry: A new direction for compositional data analysis

Hydraulic exponents and unit hydraulic exponents are unit-sum constrained, which requires that they be analyzed by statistical methods designed for compositional data. Though uncertainties remain regarding selection of the best constraining operation and method of handling departures from the unit-sum constraint, neither category of uncertainty should be an impediment to the selection of the appropriate statistical methodology. In a small sample study, the hydraulic geometry of different types of streams were compared: (1) semi-arid: perennial vs. ephemeral; (2) tropical: Puerto Rico vs. West Malaysia; and (3) semi-arid vs. tropical (by pooling the previous data sets). All three comparisons revealed statistically significant differences in either logratio mean vectorsor logratio covariance matrices but not both. All six categories of data had logistic normal distributions. Because the derivatives at a given discharge of curvilinear hydraulic geometry relationships and hydraulic exponents on either side of the breakpoints of piecewise linear relationships are unit-sum constrained, they also can be studied by compositional methods. However, the compositional approach is limited in cases where distributions have large departures from logistic normality and for streams that have negative hydraulic exponents.     

Compositional Data Analysis: Where Are We and Where Should We Be Heading?

We take stock of the present position of compositional data analysis, of what has been achieved in the last 20 years, and then make suggestions as to what may be sensible avenues of future research. We take an uncompromisingly applied mathematical view, that the challenge of solving practical problems should motivate our theoretical research; and that any new theory should be thoroughly investigated to see if it may provide answers to previously abandoned practical considerations.     

Subcompositional Patterns in Cenozoic Volcanic Rocks of Hungary

Data selected from an extensive major element database of Cenozoic volcanic rocks (including calc-alkaline andesites, dacites, rhyolites, and alkali basalts) of Hungary are used to illustrate the detection and modeling of subcompositional patterns using a statistical analysis based on the assumption that relative differences between the observed values are more meaningful than absolute ones. In particular, two roughly linear compositional patterns (associated one to the alkaline basalts, the other to the calc-alkaline series) are revealed and evaluated, and it is shown how principal component analysis can be used to obtain the estimated subcomposition of their incidental intersection point.     

Discriminant Analysis for Compositional Data Incorporating Cell-Wise Uncertainties

Mathematical Geosciences - In the geosciences it is still uncommon to include measurement uncertainties into statistical methods such as discriminant analysis, but, especially for trace elements,...     

Compositional Data Analysis of Some Alkaline Glasses

The approach to the analysis of compositional data involving log-ratio transformation of the data has not been generally adopted by researchers wishing to analyse such data. In the context of exploratory methods of multivariate analysis, such as principal components analysis, where the hope is to identify (cluster) structure in the data, this may be because traditional methods can produce more interpretable results than the log-ratio approach. After illustrating this with an example, circumstances under which the log-ratio approach performs poorly when traditional approaches work well are identified. Log-ratio analysis can be dominated by variables having low absolute presence and high relative variation that do not contribute to, and can obscure, structure in the data. Traditional methods can detect certain kinds of structure in the data that correspond to structure on a ratio scale, after a suitable redefinition of the composition. Since traditional methods often detect such structure more directly than log-ratio analysis it can be concluded that claims that the traditional analysis is inappropriate or meaningless are exaggerated. This conclusion is based on empirical experience rather than theoretical concerns. The arguments are illustrated using compositional data for alkaline glasses, but have more general application.     

Logratios and Natural Laws in Compositional Data Analysis

The impossibility of interpreting correlations of raw compositional components and associated statistical methods has been clearly demonstrated over the last four decades and alternative statistical methodology developed. Despite this a return to the traditional use of raw components has been advocated recently and alternative methodology such as logratio analysis strongly criticized. This paper exposes the fallacies in this recent advocacy and demonstrates the constructive role that logratio analysis can play in geological compositional problems, in particular in the investigation of natural laws and in subcompositional investigations.