Conceptual knowledge discovery and data analysis

In this paper, we discuss Conceptual Knowledge Discovery in Databases (CKDD) in its connection with Data Analysis. Our approach is based on Formal Concept Analysis, a mathematical theory which has been developed and proven useful during the last 20 years. Formal Concept Analysis has led to a theory of conceptual information systems which has been applied by using the management system TOSCANA in a wide range of domains. In this paper, we use such an application in database marketing to demonstrate how methods and procedures of CKDD can be applied in Data Analysis. In particular, we show the interplay and integration of data mining and data analysis techniques based on Formal Concept Analysis. The main concern of this paper is to explain how the transition from data to knowledge can be supported by a TOSCANA system. To clarify the transition steps we discuss their correspondence to the five levels of knowledge representation established by R. Brachman and to the steps of empirically grounded theory building proposed by A. Strauss and J. Corbin.

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

Possible solution of some essential zero problems in compositional data analysis

One of the tantalising remaining problems in compositional data analysis lies in how to deal with data sets in which there are components which are essential zeros. By an essential zero we mean a component which is truly zero, not something recorded as zero simply because the experimental design or the measuring instrument has not been sufficiently sensitive to detect a trace of the part. Such essential zeros occur in many compositional situations, such as household budget patterns, time budgets, palaeontological zonation studies, ecological abundance studies. Devices such as nonzero replacement and amalgamation are almost invariably ad hoc and unsuccessful in such situations. From consideration of such examples it seems sensible to build up a model in two stages, the first determining where the zeros will occur and the second how the unit available is distributed among the non-zero parts. In this paper we suggest two such models, an independent binomial conditional logistic normal model and a hierarchical dependent binomial conditional logistic normal model. The compositional data in such modelling consist of an incidence matrix and a conditional compositional matrix. Interesting statistical problems arise, such as the question of estimability of parameters, the nature of the computational process for the estimation of both the incidence and compositional parameters caused by the complexity of the subcompositional structure, the formation of meaningful hypotheses, and the devising of suitable testing methodology within a lattice of such essential zero-compositional hypotheses. The methodology is illustrated by application to both simulated and real compositional data

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

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

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). (...)

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

Monitoring procedures in environmental geochemistry and compositional data analysis theory

First discussion on compositional data analysis is attributable to Karl Pearson, in 1897. However, notwithstanding the recent developments on algebraic structure of the simplex, more than twenty years after Aitchison’s idea of log-transformations of closed data, scientific literature is again full of statistical treatments of this type of data by using traditional methodologies. This is particularly true in environmental geochemistry where besides the problem of the closure, the spatial structure (dependence) of the data have to be considered. In this work we propose the use of log-contrast values, obtained by a simplicial principal component analysis, as LQGLFDWRUV of given environmental conditions. The investigation of the log-constrast frequency distributions allows pointing out the statistical laws able to generate the values and to govern their variability. The changes, if compared, for example, with the mean values of the random variables assumed as models, or other reference parameters, allow defining monitors to be used to assess the extent of possible environmental contamination. Case study on running and ground waters from Chiavenna Valley (Northern Italy) by using Na+, K+, Ca2+, Mg2+, HCO3-, SO4 2- and Cl- concentrations will be illustrated

Compositional Data Analysis with R

R from http://www.r-project.org/ is ‘GNU S’ – a language and environment for statistical computing and graphics. The environment in which many classical and modern statistical techniques have been implemented, but many are supplied as packages. There are 8 standard packages and many more are 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 is a MixeR package for compositional data analysis that provides support for operations on compositions: perturbation and power multiplication, subcomposition with or without residuals, 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 of subsets 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 simple and 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 as evidence 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 these conclusions 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 the marked regional dimension in its distribution across the country. Amongst these dissenters it is generally agreed that non-employment is concentrated amongst older males 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 the problem. 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 of the non-employment rate and the share of those not employed through reason of sickness or 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 major drawbacks however is its susceptibility to response biases. A known strategy to control these biases has been the use of so-called ipsative items. Ipsative items are items that require the respondent to make between-scale comparisons within each item. The selected option determines to which scale the weight of the answer is attributed. Consequently in questionnaires only consisting of ipsative items every respondent is allotted an equal amount, i.e. the total score, that each can distribute differently over the scales. Therefore this type of response format yields data that can be considered compositional from its inception. Methodological oriented psychologists have heavily criticized this type of item format, since the resulting data is also marked by the associated unfavourable statistical properties. Nevertheless, clinicians have kept using these questionnaires to their satisfaction. This investigation therefore aims to evaluate both positions and addresses the similarities and differences between the two data collection methods. The ultimate 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 three psychological questionnaires, which were administered to 502 first-year students in psychology according to a balanced within-subjects design. Previous research only compared the direct ipsative scale scores with the derived ipsative scale scores. The use of compositional data analysis techniques also enables one to compare derived normative score ratios with direct normative score ratios. The addition of the second comparison not only offers the advantage of a better-balanced research strategy. In principle it also allows for parametric testing in the evaluation

Validation of order rank scales based on compositional data analysis: a proposal

Usually, psychometricians apply classical factorial analysis to evaluate construct validity of order rank scales. Nevertheless, these scales have particular characteristics that must be taken into account: total scores and rank are highly relevant

Compositional data analysis of vote shares in the 2001 Australian Federal election

First application of compositional data analysis techniques to Australian election data

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

In any discipline, where uncertainty and variability are present, it is important to have principles which are accepted as inviolate and which should therefore drive statistical modelling, 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 and from these a sensible, meaningful methodology has been developed for the statistical analysis of compositional data, the application of inappropriate and/or meaningless methods persists in many areas of application. This paper identifies at least ten common fallacies and confusions in compositional data analysis with illustrative examples and provides readers with necessary, and hopefully sufficient, arguments to persuade the culprits why and how they should amend their ways