Thierry Denux

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.

Reasoning with imprecise belief structures

This paper extends the theory of belief functions by introducing new concepts and techniques, allowing to model the situation in which the beliefs held by a rational agent may only be expressed (or are only known) with some imprecision. Central to our approach is the concept of interval-valued belief structure (IBS), defined as a set of belief structures verifying certain constraints. Starting from this definition, many other concepts of Evidence Theory (including belief and plausibility functions, pignistic probabilities, combination rules and uncertainty measures) are generalized to cope with imprecision in the belief numbers attached to each hypothesis. An application of this new framework to the classification of patterns with partially known feature values is demonstrated.

Fault diagnosis in railway track circuits using Dempster-Shafer classifier fusion

This paper addresses the problem of fault detection and isolation in railway track circuits. A track circuit can be considered as a large-scale system composed of a series of trimming capacitors located between a transmitter and a receiver. A defective capacitor affects not only its own inspection data (short circuit current) but also the measurements related to all capacitors located downstream (between the defective capacitor and the receiver). Here, the global fault detection and isolation problem is broken down into several local pattern recognition problems, each dedicated to one capacitor. The outputs from local neural network or decision tree classifiers are expressed using the Dempster-Shafer theory and combined to make a final decision on the detection and localization of a fault in the system. Experiments with simulated data show that correct detection rates over 99% and correct localization rates over 92% can be achieved using this approach, which represents a major improvement over the state of the art reference method.

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.

Likelihood-based belief function: Justification and some extensions to low-quality data

This note is a rejoinder to comments by Dubois and Moral about my paper “Likelihood-based belief function: justification and some extensions to low-quality data” published in this issue. The main comments concern (1) the axiomatic justification for defining a consonant belief function in the parameter space from the likelihood function and (2) the Bayesian treatment of statistical inference from uncertain observations, when uncertainty is quantified by belief functions. Both issues are discussed in this note, in response to the discussants’ comments.

Extending stochastic ordering to belief functions on the real line

In this paper, the concept of stochastic ordering is extended to belief functions on the real line defined by random closed intervals. In this context, the usual stochastic ordering is shown to break down into four distinct ordering relations, called credal orderings, which correspond to the four basic ordering structures between intervals. These orderings are characterized in terms of lower and upper expectations. We then derive the expressions of the least committed (least informative) belief function credally less (respectively, greater) than or equal to a given belief function. In each case, the solution is a consonant belief function that can be described by a possibility distribution. A simple application to reliability analysis is used as an example throughout the paper.

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.

Combining statistical and expert evidence using belief functions: Application to centennial sea level estimation taking into account climate change

Estimation of extreme sea levels for high return periods is of prime importance in hydrological design and flood risk assessment. Common practice consists of inferring design levels from historical observations and assuming the distribution of extreme values to be stationary. However, in recent years, there has been a growing awareness of the necessity to integrate the effects of climate change in environmental analysis. In this paper, we present a methodology based on belief functions to combine statistical judgements with expert evidence in order to predict the future centennial sea level at a particular location, taking into account climate change. Likelihood-based belief functions derived from statistical observations are combined with random intervals encoding expert assessments of the 21st century sea level rise. Monte Carlo simulations allow us to compute belief and plausibility degrees for various hypotheses about the design parameter.

CECM: Constrained evidential C-means algorithm

In clustering applications, prior knowledge about cluster membership is sometimes available. To integrate such auxiliary information, constraint-based (or semi-supervised) methods have been proposed in the hard or fuzzy clustering frameworks. This approach is extended to evidential clustering, in which the membership of objects to clusters is described by belief functions. A variant of the Evidential C-means (ECM) algorithm taking into account pairwise constraints is proposed. These constraints are translated into the belief function framework and integrated in the cost function. Experiments with synthetic and real data sets demonstrate the interest of the method. In particular, an application to medical image segmentation is presented.

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.