A tale of two logits, compositional data analysis and zero observations

The application of compositional data analysis through log ratio trans-formations corresponds to a multinomial logit model for the shares themselves.This model is characterized by the property of Independence of Irrelevant Alter-natives (IIA). IIA states that the odds ratio in this case the ratio of shares is invariant to the addition or deletion of outcomes to the problem. It is exactlythis invariance of the ratio that underlies the commonly used zero replacementprocedure in compositional data analysis. In this paper we investigate using thenested logit model that does not embody IIA and an associated zero replacementprocedure and compare its performance with that of the more usual approach ofusing the multinomial logit model. Our comparisons exploit a data set that com-bines voting data by electoral division with corresponding census data for eachdivision for the 2001 Federal election in Australia

Weighted Logratio Biplots, Correspondence Analysis and Spectral Maps

Starting with logratio biplots for compositional data, which are based on the principle of subcompositional coherence, and then adding weights, as in correspondence analysis, we rediscover Lewi's spectral map and many connections to analyses of two-way tables of non-negative data. Thanks to the weighting, the method also achieves the property of distributional equivalence

An R Library for Compositional Data Analysis in Archaeometry

Compositional data naturally arises from the scientific analysis of the chemicalcomposition of archaeological material such as ceramic and glass artefacts. Data of thistype can be explored using a variety of techniques, from standard multivariate methodssuch as principal components analysis and cluster analysis, to methods based upon theuse of log-ratios. The general aim is to identify groups of chemically similar artefactsthat could potentially be used to answer questions of provenance.This paper will demonstrate work in progress on the development of a documentedlibrary of methods, implemented using the statistical package R, for the analysis ofcompositional data. R is an open source package that makes available very powerfulstatistical facilities at no cost. We aim to show how, with the aid of statistical softwaresuch as R, traditional exploratory multivariate analysis can easily be used alongside, orin combination with, specialist techniques of compositional data analysis.The library has been developed from a core of basic R functionality, together withpurpose-written routines arising from our own research (for example that reported atCoDaWork'03). In addition, we have included other appropriate publicly availabletechniques and libraries that have been implemented in R by other authors. Availablefunctions range from standard multivariate techniques through to various approaches tolog-ratio analysis and zero replacement. We also discuss and demonstrate a smallselection of relatively new techniques that have hitherto been little-used inarchaeometric applications involving compositional data. The application of the libraryto the analysis of data arising in archaeometry will be demonstrated; results fromdifferent analyses will be compared; and the utility of the various methods discussed

A compositional data analysis package for R providing multiple approaches

”compositions” is a new R-package for the analysis of compositional and positive data.It contains four classes corresponding to the four different types of compositional andpositive geometry (including the Aitchison geometry). It provides means for computation,plotting and high-level multivariate statistical analysis in all four geometries.These geometries are treated in an fully analogous way, based on the principle of workingin coordinates, and the object-oriented programming paradigm of R. In this way,called functions automatically select the most appropriate type of analysis as a functionof the geometry. The graphical capabilities include ternary diagrams and tetrahedrons,various compositional plots (boxplots, barplots, piecharts) and extensive graphical toolsfor principal components. Afterwards, ortion and proportion lines, straight lines andellipses in all geometries can be added to plots. The package is accompanied by ahands-on-introduction, documentation for every function, demos of the graphical capabilitiesand plenty of usage examples. It allows direct and parallel computation inall four vector spaces and provides the beginner with a copy-and-paste style of dataanalysis, while letting advanced users keep the functionality and customizability theydemand of R, as well as all necessary tools to add own analysis routines. A completeexample is included in the appendix

Compositional Data Analysis with R

R from http://www.r-project.org/ is ‘GNU S’ – a language and environment for statistical computingand graphics. The environment in which many classical and modern statistical techniques havebeen implemented, but many are supplied as packages. There are 8 standard packages and many moreare available through the cran family of Internet sites http://cran.r-project.org .We started to develop a library of functions in R to support the analysis of mixtures and our goal isa MixeR package for compositional data analysis that provides support foroperations on compositions: perturbation and power multiplication, subcomposition with or withoutresiduals, centering of the data, computing Aitchison’s, Euclidean, Bhattacharyya distances,compositional Kullback-Leibler divergence etc.graphical presentation of compositions in ternary diagrams and tetrahedrons with additional features:barycenter, geometric mean of the data set, the percentiles lines, marking and coloring ofsubsets of the data set, theirs geometric means, notation of individual data in the set . . .dealing with zeros and missing values in compositional data sets with R procedures for simpleand multiplicative replacement strategy,the time series analysis of compositional data.We’ll present the current status of MixeR development and illustrate its use on selected data sets

New Features of CoDaPack. An Userfriendly Compositional Data Package

The statistical analysis of compositional data is commonly used in geological studies.As is well-known, compositions should be treated using logratios of parts, which aredifficult to use correctly in standard statistical packages. In this paper we describe thenew features of our freeware package, named CoDaPack, which implements most of thebasic statistical methods suitable for compositional data. An example using real data ispresented to illustrate the use of the package

Convex linear combination processes for compositions

Aitchison and Bacon-Shone (1999) considered convex linear combinations ofcompositions. In other words, they investigated compositions of compositions, wherethe mixing composition follows a logistic Normal distribution (or a perturbationprocess) and the compositions being mixed follow a logistic Normal distribution. Inthis paper, I investigate the extension to situations where the mixing compositionvaries with a number of dimensions. Examples would be where the mixingproportions vary with time or distance or a combination of the two. Practicalsituations include a river where the mixing proportions vary along the river, or acrossa lake and possibly with a time trend. This is illustrated with a dataset similar to thatused in the Aitchison and Bacon-Shone paper, which looked at how pollution in aloch depended on the pollution in the three rivers that feed the loch. Here, I explicitlymodel the variation in the linear combination across the loch, assuming that the meanof the logistic Normal distribution depends on the river flows and relative distancefrom the source origins

Meaning of the λ parameter of skew–normal and log–skew normal distributions in fluid geochemistry

The literature related to skew–normal distributions has grown rapidly in recent yearsbut at the moment few applications concern the description of natural phenomena withthis type of probability models, as well as the interpretation of their parameters. Theskew–normal distributions family represents an extension of the normal family to whicha parameter (λ) has been added to regulate the skewness. The development of this theoreticalfield has followed the general tendency in Statistics towards more flexible methodsto represent features of the data, as adequately as possible, and to reduce unrealisticassumptions as the normality that underlies most methods of univariate and multivariateanalysis. In this paper an investigation on the shape of the frequency distribution of thelogratio ln(Cl−/Na+) whose components are related to waters composition for 26 wells,has been performed. Samples have been collected around the active center of Vulcanoisland (Aeolian archipelago, southern Italy) from 1977 up to now at time intervals ofabout six months. Data of the logratio have been tentatively modeled by evaluating theperformance of the skew–normal model for each well. Values of the λ parameter havebeen compared by considering temperature and spatial position of the sampling points.Preliminary results indicate that changes in λ values can be related to the nature ofenvironmental processes affecting the data

Thermodynamics and log–contrast analysis in fluid geochemistry

There are two principal chemical concepts that are important for studying the naturalenvironment. The first one is thermodynamics, which describes whether a system is atequilibrium or can spontaneously change by chemical reactions. The second main conceptis how fast chemical reactions (kinetics or rate of chemical change) take place wheneverthey start. In this work we examine a natural system in which both thermodynamics andkinetic factors are important in determining the abundance of NH+4 , NO−2 and NO−3 insuperficial waters. Samples were collected in the Arno Basin (Tuscany, Italy), a system inwhich natural and antrophic effects both contribute to highly modify the chemical compositionof water. Thermodynamical modelling based on the reduction-oxidation reactionsinvolving the passage NH+4 -& NO−2 -& NO−3 in equilibrium conditions has allowed todetermine 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 expressesthe concentration of H+ in solution, redox potential is used to express the tendency of anenvironment to receive or supply electrons. In this context, oxic environments, as thoseof river systems, are said to have a high redox potential because O2 is available as anelectron acceptor.Principles of thermodynamics and chemical kinetics allow to obtain a model that oftendoes not completely describe the reality of natural systems. Chemical reactions may indeedfail to achieve equilibrium because the products escape from the site of the rectionor because reactions involving the trasformation are very slow, so that non-equilibriumconditions exist for long periods. Moreover, reaction rates can be sensitive to poorly understoodcatalytic effects or to surface effects, while variables as concentration (a largenumber of chemical species can coexist and interact concurrently), temperature and pressurecan have large gradients in natural systems. By taking into account this, data of 91water samples have been modelled by using statistical methodologies for compositionaldata. The application of log–contrast analysis has allowed to obtain statistical parametersto be correlated with the calculated Eh values. In this way, natural conditions in whichchemical equilibrium is hypothesised, as well as underlying fast reactions, are comparedwith those described by a stochastic approach

Compositional analysis of archaeological glasses

At CoDaWork'03 we presented work on the analysis of archaeological glass composi-tional data. Such data typically consist of geochemical compositions involving 10-12variables and approximates completely compositional data if the main component, sil-ica, is included. We suggested that what has been termed `crude' principal componentanalysis (PCA) of standardized data often identi ed interpretable pattern in the datamore readily than analyses based on log-ratio transformed data (LRA). The funda-mental problem is that, in LRA, minor oxides with high relative variation, that maynot be structure carrying, can dominate an analysis and obscure pattern associatedwith variables present at higher absolute levels. We investigate this further using sub-compositional data relating to archaeological glasses found on Israeli sites. A simplemodel for glass-making is that it is based on a `recipe' consisting of two `ingredients',sand and a source of soda. Our analysis focuses on the sub-composition of componentsassociated with the sand source. A `crude' PCA of standardized data shows two clearcompositional groups that can be interpreted in terms of di erent recipes being used atdi erent periods, reected in absolute di erences in the composition. LRA analysis canbe undertaken either by normalizing the data or de ning a `residual'. In either case,after some `tuning', these groups are recovered. The results from the normalized LRAare di erently interpreted as showing that the source of sand used to make the glassdi ered. These results are complementary. One relates to the recipe used. The otherrelates to the composition (and presumed sources) of one of the ingredients. It seemsto be axiomatic in some expositions of LRA that statistical analysis of compositionaldata should focus on relative variation via the use of ratios. Our analysis suggests thatabsolute di erences can also be informative

Interpretation of wind components as compositional variables

The classical statistical study of the wind speed in the atmospheric surface layer is madegenerally from the analysis of the three habitual components that perform the wind data,that is, the component W-E, the component S-N and the vertical component,considering these components independent.When the goal of the study of these data is the Aeolian energy, so is when wind isstudied from an energetic point of view and the squares of wind components can beconsidered as compositional variables. To do so, each component has to be divided bythe module of the corresponding vector.In this work the theoretical analysis of the components of the wind as compositionaldata is presented and also the conclusions that can be obtained from the point of view ofthe practical applications as well as those that can be derived from the application ofthis technique in different conditions of weather

Assessing the Precision of Compositional Data in a Stratified Double Stage Cluster Sample: Application to the Swiss Earnings Structure Survey

Precision of released figures is not only an important quality feature of official statistics,it is also essential for a good understanding of the data. In this paper we show a casestudy of how precision could be conveyed if the multivariate nature of data has to betaken into account. In the official release of the Swiss earnings structure survey, the totalsalary is broken down into several wage components. We follow Aitchison's approachfor the analysis of compositional data, which is based on logratios of components. Wefirst present diferent multivariate analyses of the compositional data whereby the wagecomponents are broken down by economic activity classes. Then we propose a numberof ways to assess precision

Interactions between grain size and composition of sediments: two examples

Two contrasting case studies of sediment and detrital mineral composition are investigated in order to outline interactions between chemical composition and grain size. Modern glacial sediments exhibit a strong dependence of the two parameters due to the preferential enrichment of mafic minerals, especially biotite, in the fine-grained fractions. On the other hand, the composition of detrital heavy minerals (here: rutile) appears to be not systematically related to grain-size, but is strongly controlled by location, i.e. the petrology of the source rocks of detrital grains. This supports the use of rutile as a well-suited tracer mineral for provenance studies. The results further suggest that (i) interpretations derived from whole-rock sediment geochemistry should be flanked by grain-size observations, and (ii) a more sound statistical evaluation of these interactions require the development of new tailor-made statistical tools to deal with such so-called two-way compositions

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 differencesbetween them based on their origin and also on commercial criteria. A particularanalysis based on their vertical zonality is also given for samples from Solnechnoedeposit. All the statistical analysis are made on the subcomposition formed by seventrace elements in cassiterite (In, Sc, Be, W, Nb, Ti and V) using the Aitchison’methodology of analysis of compositional data

A compositional approach to stable isotope data analysis

Isotopic data are currently becoming an important source of information regardingsources, evolution and mixing processes of water in hydrogeologic systems. However, itis not clear how to treat with statistics the geochemical data and the isotopic datatogether. We propose to introduce the isotopic information as new parts, and applycompositional data analysis with the resulting increased composition. Results areequivalent to downscale the classical isotopic delta variables, because they are alreadyrelative (as needed in the compositional framework) and isotopic variations are almostalways very small. This methodology is illustrated and tested with the study of theLlobregat River Basin (Barcelona, NE Spain), where it is shown that, though verysmall, isotopic variations comp lement geochemical principal components, and help inthe better identification of pollution sources