Application of compositional data analysis to geochemical data of marine sediments

In an earlier investigation (Burger et al., 2000) five sediment cores near the RodriguesTriple Junction in the Indian Ocean were studied applying classical statistical methods(fuzzy c-means clustering, linear mixing model, principal component analysis) for theextraction of endmembers and evaluating the spatial and temporal variation ofgeochemical signals. Three main factors of sedimentation were expected by the marinegeologists: a volcano-genetic, a hydro-hydrothermal and an ultra-basic factor. Thedisplay of fuzzy membership values and/or factor scores versus depth providedconsistent results for two factors only; the ultra-basic component could not beidentified. The reason for this may be that only traditional statistical methods wereapplied, i.e. the untransformed components were used and the cosine-theta coefficient assimilarity measure.During the last decade considerable progress in compositional data analysis was madeand many case studies were published using new tools for exploratory analysis of thesedata. Therefore it makes sense to check if the application of suitable data transformations,reduction of the D-part simplex to two or three factors and visualinterpretation of the factor scores would lead to a revision of earlier results and toanswers to open questions . In this paper we follow the lines of a paper of R. Tolosana-Delgado et al. (2005) starting with a problem-oriented interpretation of the biplotscattergram, extracting compositional factors, ilr-transformation of the components andvisualization of the factor scores in a spatial context: The compositional factors will beplotted versus depth (time) of the core samples in order to facilitate the identification ofthe expected sources of the sedimentary process.Kew words: compositional data analysis, biplot, deep sea sediments

Compositional data analysis: where we are and where should we be heading?

We take stock of the present position of compositional data analysis, of what has beenachieved in the last 20 years, and then make suggestions as to what may be sensibleavenues of future research. We take an uncompromisingly applied mathematical view,that the challenge of solving practical problems should motivate our theoreticalresearch; and that any new theory should be thoroughly investigated to see if it mayprovide answers to previously abandoned practical considerations. Indeed a main themeof this lecture will be to demonstrate this applied mathematical approach by a number ofchallenging examples

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

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 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

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

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

The agreement between ipsative and normative questionnaires using compositional data analysis techniques

The main instrument used in psychological measurement is the self-report questionnaire. One of its majordrawbacks however is its susceptibility to response biases. A known strategy to control these biases hasbeen the use of so-called ipsative items. Ipsative items are items that require the respondent to makebetween-scale comparisons within each item. The selected option determines to which scale the weight ofthe answer is attributed. Consequently in questionnaires only consisting of ipsative items everyrespondent is allotted an equal amount, i.e. the total score, that each can distribute differently over thescales. Therefore this type of response format yields data that can be considered compositional from itsinception.Methodological oriented psychologists have heavily criticized this type of item format, since the resultingdata is also marked by the associated unfavourable statistical properties. Nevertheless, clinicians havekept using these questionnaires to their satisfaction. This investigation therefore aims to evaluate bothpositions and addresses the similarities and differences between the two data collection methods. Theultimate objective is to formulate a guideline when to use which type of item format.The comparison is based on data obtained with both an ipsative and normative version of threepsychological questionnaires, which were administered to 502 first-year students in psychology accordingto a balanced within-subjects design. Previous research only compared the direct ipsative scale scoreswith the derived ipsative scale scores. The use of compositional data analysis techniques also enables oneto compare derived normative score ratios with direct normative score ratios. The addition of the secondcomparison not only offers the advantage of a better-balanced research strategy. In principle it also allowsfor parametric testing in the evaluation

CoDaPack. An Excel and Visual Basic based software of compositional data analysis. Current version and discussion form upcoming versions

In the eighties, John Aitchison (1986) developed a new methodological approach for the statistical analysis of compositional data. This new methodology was implemented in Basic routines grouped under the name CODA and later NEWCODA inMatlab (Aitchison, 1997). After that, several other authors have published extensions to this methodology: Marín-Fernández and others (2000), Barceló-Vidal and others (2001), Pawlowsky-Glahn and Egozcue (2001, 2002) and Egozcue and others (2003). (...)

Some last thoughts on compositional data analysis

One of the disadvantages of old age is that there is more past than future: this,however, may be turned into an advantage if the wealth of experience and, hopefully,wisdom gained in the past can be reflected upon and throw some light on possiblefuture trends. To an extent, then, this talk is necessarily personal, certainly nostalgic,but also self critical and inquisitive about our understanding of the discipline ofstatistics. A number of almost philosophical themes will run through the talk: searchfor appropriate modelling in relation to the real problem envisaged, emphasis onsensible balances between simplicity and complexity, the relative roles of theory andpractice, the nature of communication of inferential ideas to the statistical layman, theinter-related roles of teaching, consultation and research. A list of keywords might be:identification of sample space and its mathematical structure, choices betweentransform and stay, the role of parametric modelling, the role of a sample spacemetric, the underused hypothesis lattice, the nature of compositional change,particularly in relation to the modelling of processes. While the main theme will berelevance to compositional data analysis we shall point to substantial implications forgeneral multivariate analysis arising from experience of the development ofcompositional data analysis…

The single principle of compositional data analysis, continuing fallacies, confusionsand misunderstandings and some suggested remedies

In any discipline, where uncertainty and variability are present, it is important to haveprinciples which are accepted as inviolate and which should therefore drive statisticalmodelling, statistical analysis of data and any inferences from such an analysis.Despite the fact that two such principles have existed over the last two decades andfrom these a sensible, meaningful methodology has been developed for the statisticalanalysis of compositional data, the application of inappropriate and/or meaninglessmethods persists in many areas of application. This paper identifies at least tencommon fallacies and confusions in compositional data analysis with illustrativeexamples and provides readers with necessary, and hopefully sufficient, arguments topersuade the culprits why and how they should amend their ways

Compositional data analysis: where we are 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. Indeed a main theme of this lecture will be to demonstrate this applied mathematical approach by a number of challenging examples

Some last thoughts on compositional data analysis

One of the disadvantages of old age is that there is more past than future: this, however, may be turned into an advantage if the wealth of experience and, hopefully, wisdom gained in the past can be reflected upon and throw some light on possible future trends. To an extent, then, this talk is necessarily personal, certainly nostalgic, but also self critical and inquisitive about our understanding of the discipline of statistics. A number of almost philosophical themes will run through the talk: search for appropriate modelling in relation to the real problem envisaged, emphasis on sensible balances between simplicity and complexity, the relative roles of theory and practice, the nature of communication of inferential ideas to the statistical layman, the inter-related roles of teaching, consultation and research. A list of keywords might be: identification of sample space and its mathematical structure, choices between transform and stay, the role of parametric modelling, the role of a sample space metric, the underused hypothesis lattice, the nature of compositional change, particularly in relation to the modelling of processes. While the main theme will be relevance to compositional data analysis we shall point to substantial implications for general multivariate analysis arising from experience of the development of compositional data analysis…

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 exactly this invariance of the ratio that underlies the commonly used zero replacement procedure in compositional data analysis. In this paper we investigate using the nested logit model that does not embody IIA and an associated zero replacement procedure and compare its performance with that of the more usual approach of using the multinomial logit model. Our comparisons exploit a data set that com- bines voting data by electoral division with corresponding census data for each division for the 2001 Federal election in Australia