alr approach for replacing values below the detection limit

All of the imputation techniques usually applied for replacing values below the detection limit in compositional data sets have adverse effects on the variability. In this work we propose a modification of the EM algorithm that is applied using the additive log-ratio transformation. This new strategy is applied to a compositional data set and the results are compared with the usual imputation techniques

A Factor analysis of hidrochemical composition of Llobregat river basin

Hydrogeological research usually includes some statistical studies devised to elucidate mean background state, characterise relationships among different hydrochemical parameters, and show the influence of human activities. These goals are achieved either by means of a statistical approach or by mixing models between end-members. Compositional data analysis has proved to be effective with the first approach, but there is no commonly accepted solution to the end-member problem in a compositional framework. We present here a possible solution based on factor analysis of compositions illustrated with a case study. We find two factors on the compositional bi-plot fitting two non-centered orthogonal axes to the most representative variables. Each one of these axes defines a subcomposition, grouping those variables that lay nearest to it. With each subcomposition a log-contrast is computed and rewritten as an equilibrium equation. These two factors can be interpreted as the isometric log-ratio coordinates (ilr) of three hidden components, that can be plotted in a ternary diagram. These hidden components might be interpreted as end-members. We have analysed 14 molarities in 31 sampling stations all along the Llobregat River and its tributaries, with a monthly measure during two years. We have obtained a bi-plot with a 57% of explained total variance, from which we have extracted two factors: factor G, reflecting geological background enhanced by potash mining; and factor A, essentially controlled by urban and/or farming wastewater. Graphical representation of these two factors allows us to identify three extreme samples, corresponding to pristine waters, potash mining influence and urban sewage influence. To confirm this, we have available analysis of diffused and widespread point sources identified in the area: springs, potash mining lixiviates, sewage, and fertilisers. Each one of these sources shows a clear link with one of the extreme samples, except fertilisers due to the heterogeneity of their composition. This approach is a useful tool to distinguish end-members, and characterise them, an issue generally difficult to solve. It is worth note that the end-member composition cannot be fully estimated but only characterised through log-ratio relationships among components. Moreover, the influence of each endmember in a given sample must be evaluated in relative terms of the other samples. These limitations are intrinsic to the relative nature of compositional data

Thermodynamics and log–contrast analysis in fluid geochemistry

There are two principal chemical concepts that are important for studying the natural environment. The first one is thermodynamics, which describes whether a system is at equilibrium or can spontaneously change by chemical reactions. The second main concept is how fast chemical reactions (kinetics or rate of chemical change) take place whenever they start. In this work we examine a natural system in which both thermodynamics and kinetic factors are important in determining the abundance of NH+4 , NO−2 and NO−3 in superficial waters. Samples were collected in the Arno Basin (Tuscany, Italy), a system in which natural and antrophic effects both contribute to highly modify the chemical composition of water. Thermodynamical modelling based on the reduction-oxidation reactions involving the passage NH+4 -> NO−2 -> NO−3 in equilibrium conditions has allowed to determine the Eh redox potential values able to characterise the state of each sample and, consequently, of the fluid environment from which it was drawn. Just as pH expresses the concentration of H+ in solution, redox potential is used to express the tendency of an environment to receive or supply electrons. In this context, oxic environments, as those of river systems, are said to have a high redox potential because O2 is available as an electron acceptor. Principles of thermodynamics and chemical kinetics allow to obtain a model that often does not completely describe the reality of natural systems. Chemical reactions may indeed fail to achieve equilibrium because the products escape from the site of the rection or because reactions involving the trasformation are very slow, so that non-equilibrium conditions exist for long periods. Moreover, reaction rates can be sensitive to poorly understood catalytic effects or to surface effects, while variables as concentration (a large number of chemical species can coexist and interact concurrently), temperature and pressure can have large gradients in natural systems. By taking into account this, data of 91 water samples have been modelled by using statistical methodologies for compositional data. The application of log–contrast analysis has allowed to obtain statistical parameters to be correlated with the calculated Eh values. In this way, natural conditions in which chemical equilibrium is hypothesised, as well as underlying fast reactions, are compared with those described by a stochastic approach

On the use of principal components in contemporany population genetics: a case study

In human Population Genetics, routine applications of principal component techniques are often required. Population biologists make widespread use of certain discrete classifications of human samples into haplotypes, the monophyletic units of phylogenetic trees constructed from several single nucleotide bimorphisms hierarchically ordered. Compositional frequencies of the haplotypes are recorded within the different samples. Principal component techniques are then required as a dimension-reducing strategy to bring the dimension of the problem to a manageable level, say two, to allow for graphical analysis. Population biologists at large are not aware of the special features of compositional data and normally make use of the crude covariance of compositional relative frequencies to construct principal components. In this short note we present our experience with using traditional linear principal components or compositional principal components based on logratios, with reference to a specific dataset

Searching for consensus in ahp-group decision making. A bayesian perspective

This paper sets out to identify the initial positions of the different decision makers who intervene in a group decision making process with a reduced number of actors, and to establish possible consensus paths between these actors. As a methodological support, it employs one of the most widely-known multicriteria decision techniques, namely, the Analytic Hierarchy Process (AHP). Assuming that the judgements elicited by the decision makers follow the so-called multiplicative model (Crawford and Williams, 1985; Altuzarra et al., 1997; Laininen and Hämäläinen, 2003) with log-normal errors and unknown variance, a Bayesian approach is used in the estimation of the relative priorities of the alternatives being compared. These priorities, estimated by way of the median of the posterior distribution and normalised in a distributive manner (priorities add up to one), are a clear example of compositional data that will be used in the search for consensus between the actors involved in the resolution of the problem through the use of Multidimensional Scaling tools

Comparison studies of tin deposits (Priamurye, Russia) using the Aitchison’s methodology

A study of tin deposits from Priamurye (Russia) is performed to analyze the differences between them based on their origin and also on commercial criteria. A particular analysis based on their vertical zonality is also given for samples from Solnechnoe deposit. All the statistical analysis are made on the subcomposition formed by seven trace elements in cassiterite (In, Sc, Be, W, Nb, Ti and V) using the Aitchison’ methodology of analysis of compositional data

The chemical variability at the surface of Mars: implications for exogenic processes

The chemical composition of sediments and rocks, as well as their distribution at the Martian surface, represent a long term archive of processes, which have formed the planetary surface. A survey of chemical compositions by means of Compositional Data Analysis represents a valuable tool to extract direct evidence for weathering processes and allows to quantify weathering and sedimentation rates. clr-biplot techniques are applied for visualization of chemical relationships across the surface (“chemical maps”). The variability among individual suites of data is further analyzed by means of clr-PCA, in order to extract chemical alteration vectors between fresh rocks and their crusts and for an assessment of different source reservoirs accessible to soil formation. Both techniques are applied to elucidate the influence of remote weathering by combined analysis of several soil forming branches. Vector analysis in the Simplex provides the opportunity to study atmosphere surface interactions, including the role and composition of volcanic gases

When zero doesn't mean it and other geomathematical mischief

There is almost not a case in exploration geology, where the studied data doesn’t includes below detection limits and/or zero values, and since most of the geological data responds to lognormal distributions, these “zero data” represent a mathematical challenge for the interpretation. We need to start by recognizing that there are zero values in geology. For example the amount of quartz in a foyaite (nepheline syenite) is zero, since quartz cannot co-exists with nepheline. Another common essential zero is a North azimuth, however we can always change that zero for the value of 360°. These are known as “Essential zeros”, but what can we do with “Rounded zeros” that are the result of below the detection limit of the equipment? Amalgamation, e.g. adding Na2O and K2O, as total alkalis is a solution, but sometimes we need to differentiate between a sodic and a potassic alteration. Pre-classification into groups requires a good knowledge of the distribution of the data and the geochemical characteristics of the groups which is not always available. Considering the zero values equal to the limit of detection of the used equipment will generate spurious distributions, especially in ternary diagrams. Same situation will occur if we replace the zero values by a small amount using non-parametric or parametric techniques (imputation). The method that we are proposing takes into consideration the well known relationships between some elements. For example, in copper porphyry deposits, there is always a good direct correlation between the copper values and the molybdenum ones, but while copper will always be above the limit of detection, many of the molybdenum values will be “rounded zeros”. So, we will take the lower quartile of the real molybdenum values and establish a regression equation with copper, and then we will estimate the “rounded” zero values of molybdenum by their corresponding copper values. The method could be applied to any type of data, provided we establish first their correlation dependency. One of the main advantages of this method is that we do not obtain a fixed value for the “rounded zeros”, but one that depends on the value of the other variable. Key words: compositional data analysis, treatment of zeros, essential zeros, rounded zeros, correlation dependency

Discrete and continuous compositions

This paper examines a dataset which is modeled well by the Poisson-Log Normal process and by this process mixed with Log Normal data, which are both turned into compositions. This generates compositional data that has zeros without any need for conditional models or assuming that there is missing or censored data that needs adjustment. It also enables us to model dependence on covariates and within the composition

News from compositions, the R package

The R-package “compositions”is a tool for advanced compositional analysis. Its basic functionality has seen some conceptual improvement, containing now some facilities to work with and represent ilr bases built from balances, and an elaborated subsys- tem for dealing with several kinds of irregular data: (rounded or structural) zeroes, incomplete observations and outliers. The general approach to these irregularities is based on subcompositions: for an irregular datum, one can distinguish a “regular” sub- composition (where all parts are actually observed and the datum behaves typically) and a “problematic” subcomposition (with those unobserved, zero or rounded parts, or else where the datum shows an erratic or atypical behaviour). Systematic classification schemes are proposed for both outliers and missing values (including zeros) focusing on the nature of irregularities in the datum subcomposition(s). To compute statistics with values missing at random and structural zeros, a projection approach is implemented: a given datum contributes to the estimation of the desired parameters only on the subcompositon where it was observed. For data sets with values below the detection limit, two different approaches are provided: the well-known imputation technique, and also the projection approach. To compute statistics in the presence of outliers, robust statistics are adapted to the characteristics of compositional data, based on the minimum covariance determinant approach. The outlier classification is based on four different models of outlier occur- rence and Monte-Carlo-based tests for their characterization. Furthermore the package provides special plots helping to understand the nature of outliers in the dataset. Keywords: coda-dendrogram, lost values, MAR, missing data, MCD estimator, robustness, rounded zeros

A unified approach for representing rows and columns in contingency tables

By using suitable parameters, we present a uni¯ed aproach for describing four methods for representing categorical data in a contingency table. These methods include: correspondence analysis (CA), the alternative approach using Hellinger distance (HD), the log-ratio (LR) alternative, which is appropriate for compositional data, and the so-called non-symmetrical correspondence analysis (NSCA). We then make an appropriate comparison among these four methods and some illustrative examples are given. Some approaches based on cumulative frequencies are also linked and studied using matrices. Key words: Correspondence analysis, Hellinger distance, Non-symmetrical correspondence analysis, log-ratio analysis, Taguchi inertia

A new distribution on the simplex containing the Dirichlet family

The Dirichlet family owes its privileged status within simplex distributions to easyness of interpretation and good mathematical properties. In particular, we recall fundamental properties for the analysis of compositional data such as closure under amalgamation and subcomposition. From a probabilistic point of view, it is characterised (uniquely) by a variety of independence relationships which makes it indisputably the reference model for expressing the non trivial idea of substantial independence for compositions. Indeed, its well known inadequacy as a general model for compositional data stems from such an independence structure together with the poorness of its parametrisation. In this paper a new class of distributions (called Flexible Dirichlet) capable of handling various dependence structures and containing the Dirichlet as a special case is presented. The new model exhibits a considerably richer parametrisation which, for example, allows to model the means and (part of) the variance-covariance matrix separately. Moreover, such a model preserves some good mathematical properties of the Dirichlet, i.e. closure under amalgamation and subcomposition with new parameters simply related to the parent composition parameters. Furthermore, the joint and conditional distributions of subcompositions and relative totals can be expressed as simple mixtures of two Flexible Dirichlet distributions. The basis generating the Flexible Dirichlet, though keeping compositional invariance, shows a dependence structure which allows various forms of partitional dependence to be contemplated by the model (e.g. non-neutrality, subcompositional dependence and subcompositional non-invariance), independence cases being identified by suitable parameter configurations. In particular, within this model substantial independence among subsets of components of the composition naturally occurs when the subsets have a Dirichlet distribution

Statistical evidences of cyclic changes in volcanic gas chemistry composition by inverse modelling

The identification of compositional changes in fumarolic gases of active and quiescent volcanoes is one of the most important targets in monitoring programs. From a general point of view, many systematic (often cyclic) and random processes control the chemistry of gas discharges, making difficult to produce a convincing mathematical-statistical modelling. Changes in the chemical composition of volcanic gases sampled at Vulcano Island (Aeolian Arc, Sicily, Italy) from eight different fumaroles located in the northern sector of the summit crater (La Fossa) have been analysed by considering their dependence from time in the period 2000-2007. Each intermediate chemical composition has been considered as potentially derived from the contribution of the two temporal extremes represented by the 2000 and 2007 samples, respectively, by using inverse modelling methodologies for compositional data. Data pertaining to fumaroles F5 and F27, located on the rim and in the inner part of La Fossa crater, respectively, have been used to achieve the proposed aim. The statistical approach has allowed us to highlight the presence of random and not random fluctuations, features useful to understand how the volcanic system works, opening new perspectives in sampling strategies and in the evaluation of the natural risk related to a quiescent volcano

Elemental analysis of New Zealand Nephrite Jade (pounamu) for forensic purposes

Pounamu (NZ jade), or nephrite, is a protected mineral in its natural form following the transfer of ownership back to Ngai Tahu under the Ngai Tahu (Pounamu Vesting) Act 1997. Any theft of nephrite is prosecutable under the Crimes Act 1961. Scientific evidence is essential in cases where origin is disputed. A robust method for discrimination of this material through the use of elemental analysis and compositional data analysis is required. Initial studies have characterised the variability within a given nephrite source. This has included investigation of both in situ outcrops and alluvial material. Methods for the discrimination of two geographically close nephrite sources are being developed. Key Words: forensic, jade, nephrite, laser ablation, inductively coupled plasma mass spectrometry, multivariate analysis, elemental analysis, compositional data analysis

Hardy-Weinberg Equilibrium and the Ternary Plot

The Hardy-Weinberg law, formulated about 100 years ago, states that under certain assumptions, the three genotypes AA, AB and BB at a bi-allelic locus are expected to occur in the proportions p2, 2pq, and q2 respectively, where p is the allele frequency of A, and q = 1-p. There are many statistical tests being used to check whether empirical marker data obeys the Hardy-Weinberg principle. Among these are the classical xi-square test (with or without continuity correction), the likelihood ratio test, Fisher's Exact test, and exact tests in combination with Monte Carlo and Markov Chain algorithms. Tests for Hardy-Weinberg equilibrium (HWE) are numerical in nature, requiring the computation of a test statistic and a p-value. There is however, ample space for the use of graphics in HWE tests, in particular for the ternary plot. Nowadays, many genetical studies are using genetical markers known as Single Nucleotide Polymorphisms (SNPs). SNP data comes in the form of counts, but from the counts one typically computes genotype frequencies and allele frequencies. These frequencies satisfy the unit-sum constraint, and their analysis therefore falls within the realm of compositional data analysis (Aitchison, 1986). SNPs are usually bi-allelic, which implies that the genotype frequencies can be adequately represented in a ternary plot. Compositions that are in exact HWE describe a parabola in the ternary plot. Compositions for which HWE cannot be rejected in a statistical test are typically “close" to the parabola, whereas compositions that differ significantly from HWE are “far". By rewriting the statistics used to test for HWE in terms of heterozygote frequencies, acceptance regions for HWE can be obtained that can be depicted in the ternary plot. This way, compositions can be tested for HWE purely on the basis of their position in the ternary plot (Graffelman & Morales, 2008). This leads to nice graphical representations where large numbers of SNPs can be tested for HWE in a single graph. Several examples of graphical tests for HWE (implemented in R software), will be shown, using SNP data from different human populations