Hilbert space on probability density functions with Aitchison geometry

Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densitiesby generalizing the Aitchison geometry for compositions in the simplex into the set probability densities

Distributions on the simplex

The simplex, the sample space of compositional data, can be structured as a real Euclidean space. This fact allows to work with the coefficients with respect to an orthonormal basis. Over these coefficients we apply standard real analysis, inparticular, we define two different laws of probability trought the density function and we study their main properties

When a data set can be considered compositional?

Traditionally, compositional data has been identified with closed data, and the simplex has been considered as the natural sample space of this kind of data. In our opinion, the emphasis on the constrained nature ofcompositional data has contributed to mask its real nature. More crucial than the constraining property of compositional data is the scale-invariant property of this kind of data. Indeed, when we are considering only few parts of a full composition we are not working with constrained data but our data are still compositional. We believe that it is necessary to give a more precisedefinition of composition. This is the aim of this oral contribution

Spatial Analysis of Cell Composition Data

A version of Matheron’s discrete Gaussian model is applied to cell composition data.The examples are for map patterns of felsic metavolcanics in two different areas. Q-Qplots of the model for cell values representing proportion of 10 km x 10 km cell areaunderlain by this rock type are approximately linear, and the line of best fit can be usedto estimate the parameters of the model. It is also shown that felsic metavolcanics in theAbitibi area of the Canadian Shield can be modeled as a fractal

Conditional compositional biplots: theory and application

The biplot has proved to be a powerful descriptive and analytical tool in many areasof applications of statistics. For compositional data the necessary theoreticaladaptation has been provided, with illustrative applications, by Aitchison (1990) andAitchison and Greenacre (2002). These papers were restricted to the interpretation ofsimple compositional data sets. In many situations the problem has to be described insome form of conditional modelling. For example, in a clinical trial where interest isin how patients’ steroid metabolite compositions may change as a result of differenttreatment regimes, interest is in relating the compositions after treatment to thecompositions before treatment and the nature of the treatments applied. To study thisthrough a biplot technique requires the development of some form of conditionalcompositional biplot. This is the purpose of this paper. We choose as a motivatingapplication an analysis of the 1992 US President ial Election, where interest may be inhow the three-part composition, the percentage division among the three candidates -Bush, Clinton and Perot - of the presidential vote in each state, depends on the ethniccomposition and on the urban-rural composition of the state. The methodology ofconditional compositional biplots is first developed and a detailed interpretation of the1992 US Presidential Election provided. We use a second application involving theconditional variability of tektite mineral compositions with respect to major oxidecompositions to demonstrate some hazards of simplistic interpretation of biplots.Finally we conjecture on further possible applications of conditional compositionalbiplots

Analyzing shapes as compositions of distances

We propose to analyze shapes as “compositions” of distances in Aitchison geometry asan alternate and complementary tool to classical shape analysis, especially when sizeis non-informative.Shapes are typically described by the location of user-chosen landmarks. Howeverthe shape – considered as invariant under scaling, translation, mirroring and rotation– does not uniquely define the location of landmarks. A simple approach is to usedistances of landmarks instead of the locations of landmarks them self. Distances arepositive numbers defined up to joint scaling, a mathematical structure quite similar tocompositions. The shape fixes only ratios of distances. Perturbations correspond torelative changes of the size of subshapes and of aspect ratios. The power transformincreases the expression of the shape by increasing distance ratios. In analogy to thesubcompositional consistency, results should not depend too much on the choice ofdistances, because different subsets of the pairwise distances of landmarks uniquelydefine the shape.Various compositional analysis tools can be applied to sets of distances directly or afterminor modifications concerning the singularity of the covariance matrix and yield resultswith direct interpretations in terms of shape changes. The remaining problem isthat not all sets of distances correspond to a valid shape. Nevertheless interpolated orpredicted shapes can be backtransformated by multidimensional scaling (when all pairwisedistances are used) or free geodetic adjustment (when sufficiently many distancesare used)

Comparing correspondence analysis and the log-ratio alternative for representing categorical data

We compare correspondance análisis to the logratio approach based on compositional data. We also compare correspondance análisis and an alternative approach using Hellinger distance, for representing categorical data in a contingency table. We propose a coefficient which globally measures the similarity between these approaches. This coefficient can be decomposed into several components, one component for each principal dimension, indicating the contribution of the dimensions to the difference between the two representations. These three methods of representation can produce quite similar results. One illustrative example is given

CoDa-dendrogram: A new exploratory too

The use of orthonormal coordinates in the simplex and, particularly, balance coordinates, has suggested the use of a dendrogram for the exploratory analysis of compositional data. The dendrogram is based on a sequential binary partition of a compositional vector into groups of parts. At each step of a partition, one group of parts isdivided into two new groups, and a balancing axis in the simplex between both groupsis defined. The set of balancing axes constitutes an orthonormal basis, and the projections of the sample on them are orthogonal coordinates. They can be represented in adendrogram-like graph showing: (a) the way of grouping parts of the compositional vector; (b) the explanatory role of each subcomposition generated in the partition process;(c) the decomposition of the total variance into balance components associated witheach binary partition; (d) a box-plot of each balance. This representation is useful tohelp the interpretation of balance coordinates; to identify which are the most explanatory coordinates; and to describe the whole sample in a single diagram independentlyof the number of parts of the sample

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

Modelling structural zeros in compositional data

This analysis was stimulated by the real data analysis problem of householdexpenditure data. The full dataset contains expenditure data for a sample of 1224 households. The expenditure is broken down at 2 hierarchical levels: 9 major levels (e.g. housing, food, utilities etc.) and 92 minor levels. There are also 5 factors and 5 covariates at the household level. Not surprisingly, there are a small number of zeros at the major level, but many zeros at the minor level. The question is how best to model the zeros. Clearly, models that tryto add a small amount to the zero terms are not appropriate in general as at least some of the zeros are clearly structural, e.g. alcohol/tobacco for households that are teetotal. The key question then is how to build suitable conditional models. For example, is the sub-composition of spendingexcluding alcohol/tobacco similar for teetotal and non-teetotal households?In other words, we are looking for sub-compositional independence. Also, what determines whether a household is teetotal? Can we assume that it is independent of the composition? In general, whether teetotal will clearly depend on the household level variables, so we need to be able to model this dependence. The other tricky question is that with zeros on more than onecomponent, we need to be able to model dependence and independence of zeros on the different components. Lastly, while some zeros are structural, others may not be, for example, for expenditure on durables, it may be chance as to whether a particular household spends money on durableswithin the sample period. This would clearly be distinguishable if we had longitudinal data, but may still be distinguishable by looking at the distribution, on the assumption that random zeros will usually be for situations where any non-zero expenditure is not small.While this analysis is based on around economic data, the ideas carry over tomany other situations, including geological data, where minerals may be missing for structural reasons (similar to alcohol), or missing because they occur only in random regions which may be missed in a sample (similar to the durables)

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

Compositional data analysis with R

Examples of compositional data. The simplex, a suitable sample space for compositional data and Aitchison's geometry. R, a free language and environment for statistical computing and graphics

El Creixement urbá a la zona volcànica de la Garrotxa

Descripció del creixement urbà dels municipis que s’inclouen dins la zona del Parc Natural de la Zona Volcànica de la Garrotxa i de com aquest influeix en el parc. Esment del Campus europeu de medi ambient que es celebrà a Olot al 1991

La Ciutat, el paisatge i la nació: Una lectura de l'obra de Pere Coromines

Estudi de l’obra de Pere Coromines i del seu pensament polític a través d’aquesta. Se centra en el tractament que fa l’escriptor de temes com ara la nació, la ciutat (Barcelona), la naturalesa, el paisatge, l’Empordà, la Mediterrània. En paraules de l’autor, “en l'obra de Coromines es detecta una tensió entre la temptació de remuntar-se a un paisatge primigeni, natural, no contaminat, i la reivindicació del paper de la Ciutat en la conformació de la Catalunya nacional”

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