Marie-Hélène Masson

ECM: An evidential version of the fuzzy c-means algorithm

A new clustering method for object data, called ECM (evidential c-means) is introduced, in the theoretical framework of belief functions. It is based on the concept of credal partition, extending those of hard, fuzzy, and possibilistic ones. To derive such a structure, a suitable objective function is minimized using an FCM-like algorithm. A validity index allowing the determination of the proper number of clusters is also proposed. Experiments with synthetic and real data sets show that the proposed algorithm can be considered as a promising tool in the field of exploratory statistics.

EVCLUS: evidential clustering of proximity data

A new relational clustering method is introduced, based on the Dempster-Shafer theory of belief functions (or evidence theory). Given a matrix of dissimilarities between n objects, this method, referred to as evidential clustering (EVCLUS), assigns a basic belief assignment (or mass function) to each object in such a way that the degree of conflict between the masses given to any two objects reflects their dissimilarity. A notion of credal partition is introduced, which subsumes those of hard, fuzzy, and possibilistic partitions, allowing to gain deeper insight into the structure of the data. Experiments with several sets of real data demonstrate the good performances of the proposed method as compared with several state-of-the-art relational clustering techniques.

Nonparametric rank-based statistics and significance tests for fuzzy data

Nonparametric rank-based statistics depending only on linear orderings of the observations are extended to fuzzy data. The approach relies on the definition of a fuzzy partial order based on the necessity index of strict dominance between fuzzy numbers, which is shown to contain, in a well-defined sense, all the ordinal information present in the original data. A concept of fuzzy set of linear extensions of a fuzzy partial order is introduced, allowing the approximate computation of fuzzy statistics alpha-cutwise using a Markov Chain Monte Carlo simulation approach. The usual notions underlying significance tests are also extended, leading to the concepts of fuzzy p-value, and graded rejection of the null hypothesis (quantified by a degree of possibility and a degree of necessity) at a given significance level. This general approach is demonstrated in two special cases: Kendalls rank correlation coefficient, and Wilcoxons two-sample rank sum statistic.

Pairwise classifier combination using belief functions

In the so-called pairwise approach to polychotomous classification, a multiclass problem is solved by combining classifiers trained to discriminate between each pair of classes. In this paper, this approach is revisited in the framework of the Dempster-Shafer theory of belief functions, a non-probabilistic framework for quantifying and manipulating partial knowledge. It is proposed to interpret the output of each pairwise classifiers by a conditional belief function. The problem of classifier combination then amounts to computing the non-conditional belief function which is the most consistent, according to some criterion, with the conditional belief functions provided by the classifiers. Experiments with various datasets demonstrate the good performances of this method as compared to previous approaches to the same problem.

Multidimensional scaling of interval-valued dissimilarity data

Multidimensional scaling is a well-known technique for representing measurements of dissimilarity among objects as points in a p-dimensional space. In this paper, this method is extended to the case where dissimilarities are only known to lie within certain intervals. Each object is then no longer represented as point, but as a region of Rp, in such a way that the minimum and maximum distances between two regions approximate the lower and upper bounds of the dissimilarity interval between the two objects. Experiments with real data demonstrate the ability of this method to represent both the structure and the precision of dissimilarity measurements.

Principal component analysis of fuzzy data using autoassociative neural networks

This paper describes an extension of principal component analysis (PCA) allowing the extraction of a limited number of relevant features from high-dimensional fuzzy data. Our approach exploits the ability of linear autoassociative neural networks to perform information compression in just the same way as PCA, without explicit matrix diagonalization. Fuzzy input values are propagated through the network using fuzzy arithmetics, and the weights are adjusted to minimize a suitable error criterion, the inputs being taken as target outputs. The concept of correlation coefficient is extended to fuzzy numbers, allowing the interpretation of the new features in terms of the original variables. Experiments with artificial and real sensory evaluation data demonstrate the ability of our method to provide concise representations of complex fuzzy data.

Clustering interval-valued proximity data using belief functions

The problem of clustering objects based on interval-valued dissimilarities is tackled in the framework of the Dempster--Shafer theory of belief functions. The proposed method assigns to each object a basic belief assignment (or mass function) defined on the set of clusters, in such a way that the belief and the plausibility that any two objects belong to the same cluster reflect, respectively, the observed lower and upper dissimilarity values. Experiments with synthetic and real data sets demonstrate the ability of the method to detect meaningful clusters, even in the presence of imprecise data and outliers.

RECM: Relational evidential c-means algorithm

A new clustering algorithm for proximity data, called RECM (relational evidential c-means) is presented. This algorithm generates a credal partition, a new clustering structure based on the theory of belief functions, which extends the existing concepts of hard, fuzzy and possibilistic partitions. Two algorithms, EVCLUS (Evidential Clustering) and ECM (evidential c-means) were previously available to derive credal partitions from data. EVCLUS was designed to handle proximity data, whereas ECM is a direct extension of fuzzy clustering algorithms for vectorial data. In this article, the relational version of ECM is introduced. It is compared to EVCLUS using various datasets. It is shown that RECM provides similar results to those given by EVCLUS. However, the optimization procedure of RECM, based on an alternate minimization scheme, is computationally much more efficient than the gradient-based procedure used in EVCLUS.

Decision fusion for postal address recognition using belief functions

Combining the outputs from several postal address readers (PARs) is a promising approach for improving the performances of mailing address recognition systems. In this paper, this problem is solved using the Transferable Belief Model, an uncertain reasoning framework based on Dempster-Shafer belief functions. Applying this framework to postal address recognition implies defining the frame of discernment (or set of possible answers to the problem under study), converting PAR outputs into belief functions (taking into account additional information such as confidence scores when available), combining the resulting belief functions, and making decisions. All these steps are detailed in this paper. Experimental results demonstrate the effectiveness of this approach as compared to simple combination rules.

Multidimensional Scaling Methods for Many-Object Sets: A Review.

Given a set of dissimilarities data between n objects, multidimensional scaling is the problem of reconstructing a geometrical pattern of these objects, using n points, so that between-points distance corresponds to between-objects dissimilarity. Often, the collection of input data requires rating the dissimilarities between all n(n - 1)/2 possible pairs of stimuli. When the number of stimuli is large, say n