Rudolf Kruse

Symbolic and Quantitative Approaches to Reasoning with Uncertainty

In recent years it has become apparent that an important part of the theory of artificial intelligence is concerned with reasoning on the basis of uncertain, incomplete, or inconsistent information. A variety of formalisms have been developed, including nonmonotonic logic, fuzzy sets, possibility theory, belief functions, and dynamic models of reasoning such as belief revision and Bayesian networks. Several European research projects have been formed in the area and the first European conference was held in 1991. This volume contains the papers accepted for presentation at ECSQARU-93, the European Conference on Symbolicand Quantitative Approaches to Reasoning and Uncertainty, held at the University of Granada, Spain, November 8-10, 1993.

A neuro-fuzzy method to learn fuzzy classification rules from data

Abstract Neuro-fuzzy systems have recently gained a lot of interest in research and application. Neuro-fuzzy models as we understand them are fuzzy systems that use local learning strategies to learn fuzzy sets and fuzzy rules. Neuro-fuzzy techniques have been developed to support the development of e.g. fuzzy controllers and fuzzy classifiers. In this paper we discuss a learning method for fuzzy classification rules. The learning algorithm is a simple heuristics that is able to derive fuzzy rules from a set of training data very quickly, and tunes them by modifying parameters of membership functions. Our approach is based on NEFCLASS, a neuro-fuzzy model for pattern classification. We also discuss some results obtained by our software implementation of NEFCLASS, which is freely available on the Internet.

Introduction to Neural Networks

(Artificial) neural networks are information processing systems, whose structure and operation principles are inspired by the nervous system and the brain of animals and humans. They consist of a large number of fairly simple units, the so-called neurons, which are working in parallel. These neurons communicate by sending information in the form of activation signals, along directed connections, to each other.

Obtaining interpretable fuzzy classification rules from medical data

For many application problems classifiers can be used to support a decision making process. In some domains-in areas like medicine especially-it is preferable not to use black box approaches. The user should be able to understand the classifier and to evaluate its results. Fuzzy rule based classifiers are especially suitable, because they consist of simple linguistically interpretable rules and do not have some of the drawbacks of symbolic or crisp rule based classifiers. Classifiers must often be created from data by a learning process, because there is not enough expert knowledge to determine their parameters completely. A simple and convenient way to learn fuzzy classifiers from data is provided by neuro-fuzzy approaches. In this paper we discuss extensions to the learning algorithms of neuro-fuzzy classification (NEFCLASS), a neuro-fuzzy approach for data analysis that we have presented before. We present interactive strategies for pruning rules and variables from a trained classifier to enhance its readability, and demonstrate our approach on a small example.

NEFCLASSmdash;a neuro-fuzzy approach for the classification of data

In this paper we present NEFCLASS, a neuro-fuzzy system for the classification of data. This approach is based on our generic model of a fuzzy perceptron which can be used to derive fuzzy neural networks or neural fuzzy systems for specific domains. The presented model derives fuzzy rules from data to classify patterns into a number of (crisp) classes. NEFCLASS uses a supervised learning algorithm based on fuzzy error backpropagation that is used in other derivations of the fuzzy perceptron. Introduction Combinations of neural networks and fuzzy systems are very popular (for an overview see [4, 6]), but most of the approaches are not easy to compare because they use very different architectures, activation functions, propagation and learning algorithms, etc. In [s] we presented a fuzzy perceptron as a generic model of multilayer fuzzy neural networks. It can be used as a common base for neuro-fuzzy architectures in order to ease the comparision of different approaches. By applying additional constraints to the definition of the fuzzy perceptron one can e.g. obtain a structure that can be interpreted as a usual fuzzy controller, and easily create a neuro-fuzzy controller this way [3, 8, 91. In thii paper we present an approach to neuro-fuzzy data analysis. The goal is to derive fuzzy rules from a set of data that can be separated in different crisp classes, i.e. at this moment we do not consider data where the patterns belong to overlapping or fuzzy categories. The fuzziness involved is due to an imperfect or incomplete measurement of features thus rendering it difficult to assign a pattern to the correct category. “Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commerical advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission.” @ 19% ACM O-89791-658-1 95 COO2 3.50 The fuzzy rules describing the data are of the form: if ~~ is g1 and zz is ~2 and . . . end zn is p, then the pattern (21,22, . . . ,z,) belongs to class i, where~l,... , pn are fuzzy sets. The task of the NEFCLASS model is to discover these rules and to learn the shape of the membership functions. We will first briefly present the fuzzy perceptron model in section II, and in section III we show how the NEFCLASS model is derived from it. We also present the supervised learning algorithm. In the fourth section we discuss the learning results we got by applying NEFCLASS to the SRIS data set, and we compare the results to other approaches. The Fuzzy Perceptron A fuzzy perceptron has the architecture of an usual multilayer perceptron, but the weights are modelled as fuzzy sets and the activation, output, and propagation functions are changed accordingly. The intention of this model is to be interpretable in form of linguistic rules and to be able to use prior rule based knowledge, so the learning has not to start from scratch. In [s] we suggested a generic model for fuzzy neural networks based on a 3-layer fuzzy perceptron. By using it to derive neural fuzzy systems for special domains, it would be possible to evaluate these diierent neuro-fuzzy approaches by means of the same underlying model. The fuzzy perceptron was used to derive the NEFCON model [3, 8, 91 for neuro-fuzzy controll applications, and it is now used to define the NEFCLASS model discussed in this paper. We will therefore shortly present the definition of the generic fuzzy perceptron. Definition 1 A %-layer fuzzy perceptron is D S-layer feedforward neural network (U, W, NET, A, 0, ex) with the following specifications: (i) I! = U U; is a non-empty set of units (neurons) igM andM={1,2,3} is theindezsetofu. Foralli,jE M, U; # B and U; n Uj = t? with i # j holds. U1 is called input layer, Us rule layer (hidden layer), and Ii3 output layer. (ii) The structure of the network (connections) is defined as W : U x U --+ T(R), such that there are only connections W(u, w) with u E Cr,, u E U,+I : E all fuzzy subsets of L). {1,2)) (T(R) is the set of

Neuro-fuzzy systems for function approximation

Abstract We present a neuro-fuzzy architecture for function approximation based on supervised learning. The learning algorithm is able to determine the structure and the parameters of a fuzzy system. The approach is an extension to our already published NEFCON and NEFCLASS models which are used for control or classification purposes. The proposed extended model, which we call NEFPROX, is more general and can be used for any application based on function approximation.

Induction of Association Rules: Apriori Implementation

We describe an implementation of the well-known apriori algorithm for the induction of association rules [Agrawal et al. (1993), Agrawal et al. (1996)] that is based on the concept of a prefix tree. While the idea to use this type of data structure is not new, there are several ways to organize the nodes of such a tree, to encode the items, and to organize the transactions, which may be used in order to minimize the time needed to find the frequent itemsets as well as to reduce the amount of memory needed to store the counters. Consequently, our emphasis is less on concepts, but on implementation issues, which, however, can make a considerable difference in applications.

Constructing a fuzzy controller from data

Abstract Fuzzy control at the executive level can be interpreted as an approximation technique for a control function based on typical, imprecisely specified input-output tuples that are represented by fuzzy sets. The imprecision is characterized by similarity relations that are induced by transformations of the canonical distance function between real numbers. Taking this interpretation of fuzzy controllers into account, in order to derive a fuzzy controller from observed data typical input-output tuples have to be identified. In addition, a concept of similarity based on a transformations of the canonical distance is needed in order to characterize the typical input-output tuples by suitable fuzzy sets. A variety of fuzzy clustering algorithms that are exactly working in this spirit exists: They identify prototypes and assign fuzzy sets to the prototypes on the basis of a suitable transformed distance. In this paper we discuss how such fuzzy clustering techniques can be applied to construct a fuzzy controller from data and introduce special clustering algorithms that are tailored for this problem.

An extension to possibilistic fuzzy cluster analysis

We explore an approach to possibilistic fuzzy clustering that avoids a severe drawback of the conventional approach, namely that the objective function is truly minimized only if all cluster centers are identical. Our approach is based on the idea that this undesired property can be avoided if we introduce a mutual repulsion of the clusters, so that they are forced away from each other. We develop this approach for the possibilistic fuzzy c-means algorithm and the possibilistic Gustafson‐Kessel algorithm. In our experiments we found that in this way we can combine the partitioning property of the probabilistic fuzzy c-means algorithm with the advantages of a possibilistic approach w.r.t. the interpretation of the membership degrees.

Fusion: General concepts and characteristics

The problem of combining pieces of information issued from several sources can be encountered in various fields of application. This paper aims at presenting the different aspects of information fusion in different domains, such as databases, regulations, preferences, sensor fusion, etc., at a quite general level. We first present different types of information encountered in fusion problems, and different aims of the fusion process. Then we focus on representation issues which are relevant when discussing fusion problems. An important issue is then addressed, the handling of conflicting information. We briefly review different domains where fusion is involved, and describe how the fusion problems are stated in each domain. Since the term fusion can have different, more or less broad, meanings, we specify later some terminology with respect to related problems, that might be included in a broad meaning of fusion. Finally we briefly discuss the difficult aspects of validation and evaluation. © 2001 John Wiley & Sons, Inc.