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 in archaeometry

We shall call an n × p data matrix fully-compositional if the rows sum to a constant, and sub-compositional if the variables are a subset of a fully-compositional data set1. Such data occur widely in archaeometry, where it is common to determine the chemical composition of ceramic, glass, metal or other artefacts using techniques such as neutron activation analysis (NAA), inductively coupled plasma spectroscopy (ICPS), X-ray fluorescence analysis (XRF) etc. Interest often centres on whether there are distinct chemical groups within the data and whether, for example, these can be associated with different origins or manufacturing technologies

Compositional data analysis of archaeological glass: problems and possible solutions

Presentation in CODAWORK'03, session 4: Applications to archeometry

Exploration of geological variability and possible processes through the use of compositional data analysis: an example using scottish metamorphosed

Developments in the statistical analysis of compositional data over the last twodecades have made possible a much deeper exploration of the nature of variability,and the possible processes associated with compositional data sets from manydisciplines. In this paper we concentrate on geochemical data sets. First we explainhow hypotheses of compositional variability may be formulated within the naturalsample space, the unit simplex, including useful hypotheses of subcompositionaldiscrimination and specific perturbational change. Then we develop through standardmethodology, such as generalised likelihood ratio tests, statistical tools to allow thesystematic investigation of a complete lattice of such hypotheses. Some of these tests are simple adaptations of existing multivariate tests but others require specialconstruction. We comment on the use of graphical methods in compositional dataanalysis and on the ordination of specimens. The recent development of the conceptof compositional processes is then explained together with the necessary tools for astaying- in-the-simplex approach, namely compositional singular value decompositions. All these statistical techniques are illustrated for a substantial compositional data set, consisting of 209 major-oxide and rare-element compositions of metamorphosed limestones from the Northeast and Central Highlands of Scotland.Finally we point out a number of unresolved problems in the statistical analysis ofcompositional processes

New insights on river water chemistry by using non-centred simplicial principal component analysis: a case study

The use of perturbation and power transformation operations permits the investigation of linear processes in the simplex as in a vectorial space. When the investigated geochemical processes can be constrained by the use of well-known starting point, the eigenvectors of the covariance matrix of a non-centred principalcomponent analysis allow to model compositional changes compared with a reference point.The results obtained for the chemistry of water collected in River Arno (central-northern Italy) have open new perspectives for considering relative changes of the analysed variables and to hypothesise the relative effect of different acting physical-chemical processes, thus posing the basis for a quantitative modelling

Kriging coordinates: what does that mean?

Kriging is an interpolation technique whose optimality criteria are based on normality assumptions either for observed or for transformed data. This is the case of normal, lognormal and multigaussian kriging.When kriging is applied to transformed scores, optimality of obtained estimators becomes a cumbersome concept: back-transformed optimal interpolations in transformed scores are not optimal in the original sample space, and vice-versa. This lack of compatible criteria of optimality induces a variety of problems in both point and block estimates. For instance, lognormal kriging, widely used to interpolate positivevariables, has no straightforward way to build consistent and optimal confidence intervals for estimates.These problems are ultimately linked to the assumed space structure of the data support: for instance, positive values, when modelled with lognormal distributions, are assumed to be embedded in the whole real space, with the usual real space structure and Lebesgue measure

Compositional hypotheses of subcompositional stability and specific perturbation change and their testing

In standard multivariate statistical analysis common hypotheses of interest concern changes in mean vectors and subvectors. In compositional data analysis it is now well established that compositional change is most readily described in terms of the simplicial operation of perturbation and that subcompositions replace the marginal concept of subvectors. To motivate the statistical developments of this paper we present two challenging compositional problems from food production processes.Against this background the relevance of perturbations and subcompositions can beclearly seen. Moreover we can identify a number of hypotheses of interest involvingthe specification of particular perturbations or differences between perturbations and also hypotheses of subcompositional stability. We identify the two problems as being the counterpart of the analysis of paired comparison or split plot experiments and of separate sample comparative experiments in the jargon of standard multivariate analysis. We then develop appropriate estimation and testing procedures for a complete lattice of relevant compositional hypotheses

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

The allometry of non-employment. What can compositional data analysis tell us about labour market performance across the UK's regions?

The low levels of unemployment recorded in the UK in recent years are widely cited asevidence of the country’s improved economic performance, and the apparent convergence of unemployment rates across the country’s regions used to suggest that the longstanding divide in living standards between the relatively prosperous ‘south’ and the more depressed ‘north’ has been substantially narrowed. Dissenters from theseconclusions have drawn attention to the greatly increased extent of non-employment(around a quarter of the UK’s working age population are not in employment) and themarked regional dimension in its distribution across the country. Amongst these dissenters it is generally agreed that non-employment is concentrated amongst oldermales previously employed in the now very much smaller ‘heavy’ industries (e.g. coal,steel, shipbuilding).This paper uses the tools of compositiona l data analysis to provide a much richer picture of non-employment and one which challenges the conventional analysis wisdom about UK labour market performance as well as the dissenters view of the nature of theproblem. It is shown that, associated with the striking ‘north/south’ divide in nonemployment rates, there is a statistically significant relationship between the size of the non-employment rate and the composition of non-employment. Specifically, it is shown that the share of unemployment in non-employment is negatively correlated with the overall non-employment rate: in regions where the non-employment rate is high the share of unemployment is relatively low. So the unemployment rate is not a very reliable indicator of regional disparities in labour market performance. Even more importantly from a policy viewpoint, a significant positive relationship is found between the size ofthe non-employment rate and the share of those not employed through reason of sicknessor disability and it seems (contrary to the dissenters) that this connection is just as strong for women as it is for men

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 ideas in the bayesian analysis of categorical data with application to dose finding clinical trials

Compositional random vectors are fundamental tools in the Bayesian analysis of categorical data.Many of the issues that are discussed with reference to the statistical analysis of compositionaldata have a natural counterpart in the construction of a Bayesian statistical model for categoricaldata.This note builds on the idea of cross-fertilization of the two areas recommended by Aitchison (1986)in his seminal book on compositional data. Particular emphasis is put on the problem of whatparameterization to use

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