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

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 case study of how precision could be conveyed if the multivariate nature of data has to be taken into account. In the official release of the Swiss earnings structure survey, the total salary is broken down into several wage components. We follow Aitchison's approach for the analysis of compositional data, which is based on logratios of components. We first present diferent multivariate analyses of the compositional data whereby the wage components are broken down by economic activity classes. Then we propose a number of ways to assess precision

I-SS: integrated supervision systems approach based on interactive components

Process supervision is the activity focused on monitoring the process operation in order to deduce conditions to maintain the normality including when faults are present Depending on the number/distribution/heterogeneity of variables, behaviour situations, sub-processes, etc. from processes, human operators and engineers do not easily manipulate the information. This leads to the necessity of automation of supervision activities. Nevertheless, the difficulty to deal with the information complicates the design and development of software applications. We present an approach called "integrated supervision systems". It proposes multiple supervisors coordination to supervise multiple sub-processes whose interactions permit one to supervise the global process

Diagnosing Patients Combining Principal Components Analysis and Case Based Reasoning

This paper addresses the application of a PCA analysis on categorical data prior to diagnose a patients data set using a Case-Based Reasoning (CBR) system. The particularity is that the standard PCA techniques are designed to deal with numerical attributes, but our medical data set contains many categorical data and alternative methods as RS-PCA are required. Thus, we propose to hybridize RS-PCA (Regular Simplex PCA) and a simple CBR. Results show how the hybrid system produces similar results when diagnosing a medical data set, that the ones obtained when using the original attributes. These results are quite promising since they allow to diagnose with less computation effort and memory storage

Application of the european customer satisfaction index to postal services. Structural equation models versus partial least squars

Customer satisfaction and retention are key issues for organizations in today’s competitive market place. As such, much research and revenue has been invested in developing accurate ways of assessing consumer satisfaction at both the macro (national) and micro (organizational) level, facilitating comparisons in performance both within and between industries. Since the instigation of the national customer satisfaction indices (CSI), partial least squares (PLS) has been used to estimate the CSI models in preference to structural equation models (SEM) because they do not rely on strict assumptions about the data. However, this choice was based upon some misconceptions about the use of SEM’s and does not take into consideration more recent advances in SEM, including estimation methods that are robust to non-normality and missing data. In this paper, both SEM and PLS approaches were compared by evaluating perceptions of the Isle of Man Post Office Products and Customer service using a CSI format. The new robust SEM procedures were found to be advantageous over PLS. Product quality was found to be the only driver of customer satisfaction, while image and satisfaction were the only predictors of loyalty, thus arguing for the specificity of postal services

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 is divided into two new groups, and a balancing axis in the simplex between both groups is 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 a dendrogram-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 with each binary partition; (d) a box-plot of each balance. This representation is useful to help the interpretation of balance coordinates; to identify which are the most explanatory coordinates; and to describe the whole sample in a single diagram independently of the number of parts of the sample

Modelling structural zeros in compositional data

This analysis was stimulated by the real data analysis problem of household expenditure 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 try to 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 spending excluding 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 one component, 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 durables within 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 to many 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)

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). (...)

An R Library for Compositional Data Analysis in Archaeometry

Compositional data naturally arises from the scientific analysis of the chemical composition of archaeological material such as ceramic and glass artefacts. Data of this type can be explored using a variety of techniques, from standard multivariate methods such as principal components analysis and cluster analysis, to methods based upon the use of log-ratios. The general aim is to identify groups of chemically similar artefacts that could potentially be used to answer questions of provenance. This paper will demonstrate work in progress on the development of a documented library of methods, implemented using the statistical package R, for the analysis of compositional data. R is an open source package that makes available very powerful statistical facilities at no cost. We aim to show how, with the aid of statistical software such as R, traditional exploratory multivariate analysis can easily be used alongside, or in combination with, specialist techniques of compositional data analysis. The library has been developed from a core of basic R functionality, together with purpose-written routines arising from our own research (for example that reported at CoDaWork'03). In addition, we have included other appropriate publicly available techniques and libraries that have been implemented in R by other authors. Available functions range from standard multivariate techniques through to various approaches to log-ratio analysis and zero replacement. We also discuss and demonstrate a small selection of relatively new techniques that have hitherto been little-used in archaeometric applications involving compositional data. The application of the library to the analysis of data arising in archaeometry will be demonstrated; results from different 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 and positive 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 working in coordinates, and the object-oriented programming paradigm of R. In this way, called functions automatically select the most appropriate type of analysis as a function of the geometry. The graphical capabilities include ternary diagrams and tetrahedrons, various compositional plots (boxplots, barplots, piecharts) and extensive graphical tools for principal components. Afterwards, ortion and proportion lines, straight lines and ellipses in all geometries can be added to plots. The package is accompanied by a hands-on-introduction, documentation for every function, demos of the graphical capabilities and plenty of usage examples. It allows direct and parallel computation in all four vector spaces and provides the beginner with a copy-and-paste style of data analysis, while letting advanced users keep the functionality and customizability they demand of R, as well as all necessary tools to add own analysis routines. A complete example is included in the appendix

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

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 10-12 variables and approximates completely compositional data if the main component, sil- ica, is included. We suggested that what has been termed `crude' principal component analysis (PCA) of standardized data often identi ed interpretable pattern in the data more 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 may not be structure carrying, can dominate an analysis and obscure pattern associated with variables present at higher absolute levels. We investigate this further using sub- compositional data relating to archaeological glasses found on Israeli sites. A simple model 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 components associated with the sand source. A `crude' PCA of standardized data shows two clear compositional groups that can be interpreted in terms of di erent recipes being used at di erent periods, re ected in absolute di erences in the composition. LRA analysis can be 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 LRA are di erently interpreted as showing that the source of sand used to make the glass di ered. These results are complementary. One relates to the recipe used. The other relates to the composition (and presumed sources) of one of the ingredients. It seems to be axiomatic in some expositions of LRA that statistical analysis of compositional data should focus on relative variation via the use of ratios. Our analysis suggests that absolute 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 made generally 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 is studied from an energetic point of view and the squares of wind components can be considered as compositional variables. To do so, each component has to be divided by the module of the corresponding vector. In this work the theoretical analysis of the components of the wind as compositional data is presented and also the conclusions that can be obtained from the point of view of the practical applications as well as those that can be derived from the application of this technique in different conditions of weather

Grain size analyses in tin-lead glazes based on 2d-sections

A problem in the archaeometric classification of Catalan Renaissance pottery is the fact, that the clay supply of the pottery workshops was centrally organized by guilds, and therefore usually all potters of a single production centre produced chemically similar ceramics. However, analysing the glazes of the ware usually a large number of inclusions in the glaze is found, which reveal technological differences between single workshops. These inclusions have been used by the potters in order to opacify the transparent glaze and to achieve a white background for further decoration. In order to distinguish different technological preparation procedures of the single workshops, at a Scanning Electron Microscope the chemical composition of those inclusions as well as their size in the two-dimensional cut is recorded. Based on the latter, a frequency distribution of the apparent diameters is estimated for each sample and type of inclusion. Following an approach by S.D. Wicksell (1925), it is principally possible to transform the distributions of the apparent 2D-diameters back to those of the true three-dimensional bodies. The applicability of this approach and its practical problems are examined using different ways of kernel density estimation and Monte-Carlo tests of the methodology. Finally, it is tested in how far the obtained frequency distributions can be used to classify the pottery