Symbolic Cluster Analysis for Distribution Valued Dissimilarity

The symbolic data analysis is a new trend in multivariate descriptive statistics whose main purpose consists in analyzing and processing set-valued random variables. Such variables are derived by summarizing large datasets and abstracting information in aggregated form. Some typical examples of symbolic datasets are those encoded by means of interval-valued variables or modal variables. Unlike classical data, symbolic data can be structured and can contain internal variation. The aim of this paper is to extend the formal framework of symbolic data analysis for allowing fuzzy-valued variables to deal with. Some related approaches based on granular computing are also proposed or discussed.

’symbolic Data Analysis’ (SDA) provides tools for analyzing ’symbolic’ data, i.e., data matrices X = (xkj) where the entries xkj are intervals, sets of categories, or frequency distributions instead of ‘single values’ (a real number, a category) as in the classical case. There exists a large number of empirical algorithms that generalize classical data analysis methods (PCA, clustering, factor analysis, etc.) to the ‘symbolic’ case. In this context, various optimization problems are formulated (optimum class centers, optimum clustering, optimum scaling,…). This paper presents some cases related to dissimilarities and class centers where explicit solutions are possible. We can integrate these results in the context of an appropriate κ-means clustering algorithm. Moreover, and as a first step to probabilistically based results in SDA, we consider the definition and determination of set-valued class ‘centers’ in SDA and relate them to theorems on the ‘approximation of distributions by sets’.

Symbolic data analysis (SDA) is an emerging area of statistics concerned with understanding and modelling data that takes distributional form (i.e. symbols), such as random lists, intervals and histograms. It was developed under the premise that the statistical unit of interest is the symbol, and that inference is required at this level. Here we consider a different perspective, which opens a new research direction in the field of SDA. We assume that, as with a standard statistical analysis, inference is required at the level of individual-level data. However, the individual-level data are unobserved, and are aggregated into observed symbols—group-based distributional-valued summaries—prior to the analysis. We introduce a novel general method for constructing likelihood functions for symbolic data based on a desired probability model for the underlying measurement-level data, while only observing the distributional summaries. This approach opens the door for new classes of symbol design and construction, in addition to developing SDA as a viable tool to enable and improve upon classical data analyses, particularly for very large and complex datasets. We illustrate this new direction for SDA research through several real and simulated data analyses, including a study of novel classes of multivariate symbol construction techniques.

In many domains of human activities it is now quite common to record huge sets of data in large data bases. It becomes a task of first importance to summarize these data in terms of their underlying concepts in order to extract new knowledge from them. These concepts can only be described by more complex type of data which we call symbolic data as they contain internal variation and they are structured. In this context, we have a rapidly increasing need to extend standard data analysis methods (exploratory, graphical representations, clustering, factorial analysis, discrimination,…) to these symbolic data.

Chapter 40 - Categorization in Symbolic Data Analysis

Dynamic time series smoothing for symbolic interval data applied to neuroscience

Pierre Bourdieu belongs to the most often cited social scientists worldwide, whose books and papers are translated into many languages. His most famous book is probably La Distinction (in French, 1979; English version Distinction, 1984), which had already been translated into 10 languages by 2002, the year he died (cf. Delsaut and Rivière, 2002), including German, Spanish, Russian, and Japanese. Within the social sciences, the book is appraised as one of the most important contributions to stratification research. Outside of France, it is less known that an essential part of Pierre Bourdieu’s work is closely related to his interest in statistics and methodology, and here especially in the work of his friend Jean-Paul Benzécri.

New interval clustering technology for symbolic data analysis (SDA) on smart phones is shown to be beneficial for mobile computing devices for smart data analysis in this paper. A new interval clustering method that combined the rough set with interval possibilistic fuzzy C-means (IPFCM) algorithm under Euclidean distance is proposed and implemented on smart phones. Symbolic clustering algorithms (SCAs) have been widely used for pattern recognition, data mining, artificial intelligence, etc. In general, the SCA is unsupervised classification that is divided into groups according to symbolic data sets. However, the traditional interval fuzzy C-means (IFCM) clustering method still has noisy and data overlapping problems associated with these symbolic interval data. Hence, a new rough set with the interval possibilistic fuzzy C-means (RIPFCM) clustering algorithm with Euclidean distance was proposed to address the symbolic interval data (SID). That is, the proposed method can perform better than the traditional IFCM clustering algorithm for SID clustering in noisy environments and with data overlapping problems. The new RIPFCM algorithm under the Euclidean distance method was proposed to deal with SID on new applications in smart phones. Consequently, this method shows the expansion of the smart phone’s computing power and its future application in new SDA.

In symbolic data analysis, we can consider more complex dissimilarity data existing between objects. Dissimilarity between objects may be described in various ways, including using a single value, an interval, a histogram, and a distribution. Analysis of such data may be carried out using multidimensional scaling. For histogram-valued dissimilarity data, Groenen and Winsberg proposed the “Hist- Scal” algorithm, which focuses on the percentiles of histogram dissimilarities [3]. For the I-Scal algorithm [2], the solution is guaranteed to improved after every iteration. However, for the Hist-Scal algorithm, this improvement cannot be guaranteed since iterative majorization is used in combination with the weighted monotone regression. In this paper, we propose a new improved algorithm applicable to percentile dissimilarities, called “the hyperbox model Percen-Scal” algorithm. Since the algorithm is based on iterative majorization, an improvement in the solution is guaranteed after every iteration of the algorithm.We apply both the hyperbox model Percen-Scal and the Hist-Scal algorithms to artificial data and compare the results obtained from both methods. We then discuss the validity of the results produced using the hyperbox model Percen-Scal algorithm.

Numerical induction from data is one of the statistical data analysis tasks, which uses a tabular model, with almost exclusively numerical features, as data representation formalism. The output representations are different: from functions to probability distributions, from surface equations to tables of indexes. One approach to extend the classical data analysis techniques to symbolic objects is the Symbolic Data Analysis: the input and the output of classical techniques are expressed in a symbolic way, so guaranteeing the comprehensibility of both the observations and the results, while the processing techniques, although appropriately adapted, maintain the efficiency of the classical statistical inferential models. Also in the field of Machine Learning several methods have been proposed to extend some inductive approaches from statistical data analysis to data represented as attribute-value couples. Sometimes these approaches transform ideas and principles coming from numerical induction to handle propositional calculus descriptions, otherwise they combine different techniques in order to treat numerical-continuous data and algebraic-symbolic data differently. The aim here is to improve the efficiency and to preserve the expressive power of the representations during the learning process, and to save the accuracy and flexibility of the numerical techniques during the recognition phase. This kind of integration is more and more complex when using first-order computational learning models, which are useful for handling object descriptions in structured domains, when not only the properties of objects but also the relations between different objects must be considered. The necessity arises from integrating different computational strategies, different knowledge representations and different processing methods in a naive combination of classifiers or, more meaningfully, in a real integration within a unique theoretical framework. When building machine learning systems increasing attention is given to handling symbolic and numerical information inside the same system. In the paper, we face the problem of handling both numerical and symbolic data in first-order models, distinguishing the phase of model generation from examples, and the phase of model recognition by means of a flexible probabilistic subsumption test.

This paper deals with the analysis of datasets, where the subjects are described by the estimated means of a p-dimensional variable. Classical statistical methods of data analysis do not treat measurements affected by intrinsic variability, as in the case of estimates, so that the heterogeneity induced among subjects by this condition is not taken into account. In this paper a way to solve the problem is suggested in the context of symbolic data analysis, whose specific aim is to handle data tables where single valued measurements are substituted by complex data structures like frequency distributions, intervals, and sets of values. A principal component analysis is carried out according to this proposal, with a significant improvement in the treatment of information.

The development of Symbolic Data Analysis comes from the need both to process more general data than classical techniques of Data Analysis do and to develop methods that yield easily interpretable results. In this paper we show how we may enlarge the domain of the data at the input and obtain an “explained” output of a clustering method by adopting notions of Symbolic Data Analysis. We start by recalling the definitions and properties of symbolic objects (Diday (1987b), Diday and Brito (1989)). We shall consider objects that take one and only one value per variable, objects that may present more than one value per variable, and objects such that the definition of a variable depends on the value taken by another one. We then compare notions defined on symbolic objects to similar notions present in the literature (Wille (1982), Ganter (1984), Duquenne (1986), Guenoche (1989)) and show how the former extend the latter. We then recall pyramidal clustering and the main properties of pyramids (Diday (1986)). Pyramids are halfway between hierarchies and lattices: they generalize the former by allowing the presence of non-disjoint clusters, however a pyramid does not present crossing in its graphical representation like lattices do. This intermediate situation led us to adopt pyramids to structure symbolic objects: they allow the definition of a structure on the objects representing inheritance without losing “too much” information, and they have a readable graphical representation. We present an algorithm of “symbolic pyramidal clustering”. This algorithm may apply to a data set of some kind of symbolic objects considering even the case of dependence between variables. As output it yields a pyramid whose clusters are represented by symbolic objects meeting a given property. The inheritance structure between the clusters will then allow for the generation of rules.

It is, by now, well recognized that real data are in intervals and not in points. Unfortunately, classical statistical theory is not capable of handling data in intervals. Here, methodology for drawing univariate and bivariate histograms and computation of descriptive statistics, like sample mean and sample variance for Symbolic interval-valued data is discussed. It is hoped that researchers would start employing this type of ‘Symbolic data analysis’ to their datasets.

패턴 인식, 데이터 분석, 침입 탐지, 이미지 처리, 바이오 인포매틱스 등과 같은 수많은 분야에서 클러스터링 분석이 사용되고 있다. 기존의 많은 연구들은 수치 데이터에만 기반을 두고 있다. 그러나 구간 데이터, 히스토그램, 심지어는 함수들을 값으로 갖는 변수들을 다루는 심볼릭 데이터 분석이 부상하고 있다. 본 논문에서는 이런 심볼릭 데이터들을 클러스터링하기 위하여 비대칭적 유사도를 제안한다. 또한 평균 유사도 값(ASV)에 기반한 클러스터링 방법도 개발한다. 제안하는 클러스터링의 결과는 기존 방법들과 다르며, 매우 고무적인 결과를 보여준다. Clustering analysis has been widely used in numerous applications like pattern recognition, data analysis, intrusion detection, image processing, bioinformatics and so on. Much of previous work has been based on the numeric data only. However, symbolic data analysis has emerged to deal with variables that can have intervals, histograms, and even functions as values. In this paper, we propose a non symmetric proximity based clustering approach for symbolic objects. A method for clustering symbolic patterns based on the average similarity value(ASV) is explored. The results of the proposed clustering method differ from those of the existing methods and the results are very encouraging.

Symbolic data analysis aims at extending classical data analysis to data representing classes of individuals instead of single individuals. A major problem in symbolic data analysis is discrimination, that is the generation of data representing classes. Such data can be expressed as classification rules, which are learned from training examples. The paper addresses the problem of learning classification rules from examples described by both numeric and symbolic attributes, so that the discretization of the continuous-valued attributes is performed during the learning process. The proposed technique has been embedded into a classification learning system, named INDUBI/CSL, and tested on several data sets.KeywordsSymbolic data analysisDiscriminationdiscretization of continuous-valued attributes