A Homogeneous Set-Theoretical Frame for Clustering Fuzzy Relational Data

Our purpose is to provide a set-theoretical frame to clustering fuzzy relational data basically based on cardinality of the fuzzy subsets that represent objects and their complementaries, without applying any crisp property. From this perspective we define a family of fuzzy similarity indexes which includes a set of fuzzy indexes introduced by Tolias et al, and we analyze under which conditions it is defined a fuzzy proximity relation. Following an original idea due to S. Miyamoto we evaluate the similarity between objects and features by means the same mathematical procedure. Joining these concepts and methods we establish an algorithm to clustering fuzzy relational data. Finally, we present an example to make clear all the process

Application of compositional data analysis to geochemical data of marine sediments

In an earlier investigation (Burger et al., 2000) five sediment cores near the Rodrigues Triple Junction in the Indian Ocean were studied applying classical statistical methods (fuzzy c-means clustering, linear mixing model, principal component analysis) for the extraction of endmembers and evaluating the spatial and temporal variation of geochemical signals. Three main factors of sedimentation were expected by the marine geologists: a volcano-genetic, a hydro-hydrothermal and an ultra-basic factor. The display of fuzzy membership values and/or factor scores versus depth provided consistent results for two factors only; the ultra-basic component could not be identified. The reason for this may be that only traditional statistical methods were applied, i.e. the untransformed components were used and the cosine-theta coefficient as similarity measure. During the last decade considerable progress in compositional data analysis was made and many case studies were published using new tools for exploratory analysis of these data. 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 visual interpretation of the factor scores would lead to a revision of earlier results and to answers 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 biplot scattergram, extracting compositional factors, ilr-transformation of the components and visualization of the factor scores in a spatial context: The compositional factors will be plotted versus depth (time) of the core samples in order to facilitate the identification of the expected sources of the sedimentary process. Kew words: compositional data analysis, biplot, deep sea sediments

Discovering similarities in the time use patterns of the Spanish Autonomous Communities by Fuzzy techniques

In 2000 the European Statistical Office published the guidelines for developing the Harmonized European Time Use Surveys system. Under such a unified framework, the first Time Use Survey of national scope was conducted in Spain during 2002– 03. The aim of these surveys is to understand human behavior and the lifestyle of people. Time allocation data are of compositional nature in origin, that is, they are subject to non-negativity and constant-sum constraints. Thus, standard multivariate techniques cannot be directly applied to analyze them. The goal of this work is to identify homogeneous Spanish Autonomous Communities with regard to the typical activity pattern of their respective populations. To this end, fuzzy clustering approach is followed. Rather than the hard partitioning of classical clustering, where objects are allocated to only a single group, fuzzy method identify overlapping groups of objects by allowing them to belong to more than one group. Concretely, the probabilistic fuzzy c-means algorithm is conveniently adapted to deal with the Spanish Time Use Survey microdata. As a result, a map distinguishing Autonomous Communities with similar activity pattern is drawn. Key words: Time use data, Fuzzy clustering; FCM; simplex space; Aitchison distance

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

La relación entre el aprendizaje léxico y el desarrollo gramatical

El presente trabajo se centra en estudiar la relación que existe entre el desarrollo de léxico y el de la morfosintaxis. Concretamente pretendemos explorar el tipo de vocabulario que mejor predice el desarrollo de la morfología verbal y el de la complejidad gramatical, así como establecer el tipo de relación entre desarrollo léxico y desarrollo morfosintáctico. La muestra comprende 517 niños de edades comprendidas entre los 18 meses y los 30 meses. Los datos se han recogido a partir de la adaptación al catalán del instrumento MacArthur-Bates Communicative Development Inventories (CDI). Los resultados muestran que el mejor predictor del desarrollo morfológico y gramatical es el vocabulario de clase cerrada, conjuntamente con el vocabulario general. Por otra parte, se observa una relación predominantemente lineal entre el desarrollo del léxico y el desarrollo morfosintáctico

Clustering compositional data trajectories

Our essay aims at studying suitable statistical methods for the clustering of compositional data in situations where observations are constituted by trajectories of compositional data, that is, by sequences of composition measurements along a domain. Observed trajectories are known as “functional data” and several methods have been proposed for their analysis. In particular, methods for clustering functional data, known as Functional Cluster Analysis (FCA), have been applied by practitioners and scientists in many fields. To our knowledge, FCA techniques have not been extended to cope with the problem of clustering compositional data trajectories. In order to extend FCA techniques to the analysis of compositional data, FCA clustering techniques have to be adapted by using a suitable compositional algebra. The present work centres on the following question: given a sample of compositional data trajectories, how can we formulate a segmentation procedure giving homogeneous classes? To address this problem we follow the steps described below. First of all we adapt the well-known spline smoothing techniques in order to cope with the smoothing of compositional data trajectories. In fact, an observed curve can be thought of as the sum of a smooth part plus some noise due to measurement errors. Spline smoothing techniques are used to isolate the smooth part of the trajectory: clustering algorithms are then applied to these smooth curves. The second step consists in building suitable metrics for measuring the dissimilarity between trajectories: we propose a metric that accounts for difference in both shape and level, and a metric accounting for differences in shape only. A simulation study is performed in order to evaluate the proposed methodologies, using both hierarchical and partitional clustering algorithm. The quality of the obtained results is assessed by means of several indices