Constructing a Fuzzy Controller from Data Frank Klawonn and Rudolf Kruse 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 exists that are exactly working in this spirit: 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. The Relation between Inference and Interpolation in the Framework ofFuzzy Systems Frank Klawonn and Vilem Novak This papers aims at clarifying the meaning of different interpretations of the Max-Min or, more generally, the Max-t-norm rule in fuzzy systems. It turns out that basically two distinct approaches play an important role in fuzzy logic and its applications: fuzzy interpolation on the basis of an imprecisely known function and logical inference in the presence of fuzzy information. Fuzzy Control on the Basis of Equality Relations - with an Example from Idle Speed Control F. Klawonn, J. Gebhardt, R. Kruse The way engineers use fuzzy control in real world applications is often not coherent with an understanding of the control rules as logical statements or implications. In most cases fuzzy control can be seen as an interpolation of a partially specified control function in a vague environment, which reflects the indistinguishability of measurements or control values. In this paper we show that equality relations turn out to be the natural way to represent such vague environments and we develop suitable interpolation methods to obtain a control function. \par As a special case of our approach we obtain Mamdani's model and can justify the inference mechanism in this model and the use of triangular membership functions not only for the reason of simplified computations, and we can explain why typical fuzzy partitions are preferred. We also obtain a criterion for reasonable defuzzification strategies. The fuzzy control methodology introduced in this paper has been applied successfully in a case study of engine idle speed control for the Volkswagen Golf GTI. Similarity in Fuzzy Reasoning Frank Klawonn, J. L. Castro Fuzzy set theory is based on a fuzzification' of the predicate $\in$ (element of), the concept of membership degrees is considered as fundamental. In this paper we elucidate the connection between indistinguishability modelled by fuzzy equivalence relations and fuzzy sets. We show that the indistinguishability inherent to fuzzy sets can be computed and that this indistinguishability cannot be overcome in approximate reasoning. For our investigations we generalize from the unit interval as the basis for fuzzy sets, to the framework of GL-monoids that can be understood as a generalization of MV-algebras. Residuation is a basic concept in GL-monoids and many proofs can be formulated in a simple and clear way instead of using special properties of the unit interval. Fuzzy Sets and Vague Environments Frank Klawonn In this paper we propose a natural approach to handle imprecise numbers as they arise for example from measurements. Fuzzy sets turn out to be a canonical representation for such imprecise numbers that are induced by taking different tolerance or error bounds into account. Fuzzy sets are induced by scaling factors that describe the magnitude of the imprecision. On the other, the scaling factors can be derived from given fuzzy sets so that we have a correspondence between scaling factors and fuzzy sets. When these concepts are applied to control problems, the max-min rule is rediscovered as an interpolations technique. Viewing fuzzy control as an interpolation technique in vague environments enables us to validate various concepts for the design and tuning of fuzzy controllers and suggests new also new methods based on clear semantics. A Lukasiewicz Logic Based Prolog Frank Klawonn and Rudolf Kruse Prolog is a programming language based on a restricted subset of classical first order predicate logic. In order to overcome some problems of classical logic to handle imperfect human knowledge, we provide a formal framework for a Lukasiewicz logic based Prolog system. The use of Lukasiewicz logic with its connection to Ulam games enables us to deal with partial inconsistencies by interpreting the truth values as relative distance to contradiction. We also present the software tool LULOG which is based on the theoretical results of this paper and can be seen as a Prolog system for many-valued logic. Applications of LULOG to an Ulam game and an example of reasoning with imperfect knowledge are also discussed. Equality Relations as a Basis for Fuzzy Control F. Klawonn, R. Kruse The aim of this paper is to introduce a fuzzy control model with well-founded semantics in order to explain the concepts applied in fuzzy control. Assuming that the domains of the input- and output variables for the process are endowed with equality relations, that reflect the indistinguishability of values lying closely together, the use of triangular and trapezoidal membership functions can be justified and max-$\sqcap$ inference where $\sqcap$ is a t-norm turns out to be a consequence of our model. Distinguishing between a functional and a relational view of the control rules it is possible to explain when defuzzification strategies like MOM or COA are appropriate or lead to undesired results. Techniques and Applications of Control Systems Based on Knowledge-Based Interpolation F. Klawonn, R. Kruse Fuzzy control was established as an alternative control method when it is difficult to develop a suitable mathematical model of the process, but expert knowledge in the form of vague rule is available. Although the principal idea of a fuzzy set as a model of a vague linguistic expression is very appealing, a naive approach to fuzzy sets can cause tedious problems. Without a concrete semantics for fuzzy sets the design of a fuzzy controller can end up in a trial and error experiment with a large number of parameters and options. In this paper we review an approach to fuzzy control that interprets fuzzy sets as vague values in a vague environment. The vague environment is characterised by a scaling function that describes how sensitive the process reacts when a certain value is slightly changed. We also discuss a regression technique based on the concept of vague environments, enabling to construct a fuzzy controller from data. The Role of Similarity in Fuzzy Reasonin F. Klawonn Fuzzy reasoning mechanisms are designed to cope with vague and uncertain knowledge and information. In this paper we demonstrate that from the vagueness inherent in a fuzzy system a canonical indistinguishability of objects can be derived which cannot be overcome by the standard reasoning schemes. We discuss also the consequences for fuzzy logic in the narrow sense. Fuzzy Shell Cluster Analysis F. Klawonn, R. Kruse and H. Timm In this paper we survey the main approaches to fuzzy shell cluster analysis which is simply a generalization of fuzzy cluster analysis to shell like clusters, i.e. clusters that lie in nonlinear subspaces. Therefore we introduce the main principles of fuzzy cluster analysis first. In the following we present some fuzzy shell clustering algorithms. In many applications it is necessary to determine the number of clusters as well as the classification of the data set. Subsequently therefore we review the main ideas of unsupervised fuzzy shell cluster analysis. Finally we present an application of unsupervised fuzzy shell cluster analysis in computer vision. Prolog Extensions to Many--Valued Logics F. Klawonn The aim of this paper is to show that a restriction of a logical language to clauses like Horn clauses, as they are used in Prolog, applied to [0,1]-valued logics leads to calculi with a sound and complete proof theory. In opposition to other models where generally the set of axioms as well as the deduction schemata are enriched we restrict ourselves to a simple modification of the deduction rules of classical logic without adding new axioms. In our model the truth values from the unit interval can be interpreted in a probabilistic sense, so that a value between 0 and 1 is not just intuitively interpreted as a degree of truth'. Learning the Rule Base of a Fuzzy Controller by a Genetic Algorithm J. Hopf and F. Klawonn For the design of a fuzzy controller it is necessary to choose, besides other parameters, suitable membership functions for the linguistic terms and to determine a rule base. This paper deals with the problem of finding a good rule base - the basis of a fuzzy controller. Consulting experts still is the usual but time-consuming and therefore rather expensive method. Besides, after having designed the controller, one cannot be sure that the rule base will lead to near optimal control. This paper shows how to reduce significantly the period of development (and the costs) of fuzzy controllers with the help of genetic algorithms and, above all, how to engender a rule base which is very close to an optimum solution. The example of the inverted pendulum is used to demonstrate how a genetic algorithm can be designed for an automatic construction of a rule base. So this paper does not deal with the tuning of an existing fuzzy controller but with the genetic (re-)production of rules, even without the need for experts. Thus, a program is engendered, consisting of simple IF...THEN instructions. Context Sensitive Fuzzy Clustering Annete Keller, Frank Klawonn We introduce an objective function-based fuzzy clustering technique that incorporates linear combinations of attributes in the distance function. The main application field of our method is image processing where a a comparison pixel by pixel is usually not adequate, but the environment of a pixel or a groupd of pixels characterize important properties of an image or parts of it. In addition, our approach can be seen as a generalization of other fuzzy clustering techniques like the axes-parallel version of the Gustafson-Kessel algorithm. Fuzzy Clustering Based on Modified Distance Measures Frank Klawonn, Annete Keller The well-known fuzzy c-means algorithm is an objective function based fuzzy clustering technique that extends the classical k-means method to fuzzy partitions. By replacing the Euclidean distance in the objective function other cluster shapes than the simple (hyper-)spheres of the fuzzy c-means algorithm can be detected, for instance ellipsoids, lines or shells of circles and ellipses. We propose a modified distance function that is based on the scalar product and allows to detect a new kind of cluster shape and also lines and (hyper-)planes. Fuzzy Clustering with Weighting of Data Variables Annete Keller, Frank Klawonn We introduce an objective function-based fuzzy clustering technique that assigns one influence parameter to each single data variable for each cluster. Our method is not only suited to detect structures or groups in unevenly over the structure's single domains distributed data, but gives also information about the influence of individual variables on the detected groups. In addition, our approach can be seen as as generalization of the well-known fuzzy c-means clustering algorithm. Mathematical Analysis of Fuzzy Classifiers Frank Klawonn and Erich-Peter Klement We examine the principle capabilities and limits of fuzzy classifiers that are based on a finite set of fuzzy if-then rules like they are used for fuzzy controllers, except that the conclusion of a rule specifies a discrete class instead of a (fuzzy) real output value. Our results show that in the two-dimensional case, for classification problems whose solutions can only be solved approximately by crisp classification rules, very simple fuzzy rules provide an exact solution. However, in the multi-dimensional case, even for linear separable problems, max-min rules are not sufficient. Fuzzy Clustering and Fuzzy Rules Frank Klawonn and Annette Keller Fuzzy clustering offers various possibilities for learning fuzzy if-then rules from data for classification tasks as well as for function approximation problems like in fuzzy control. In this paper we review approaches for deriving rules from data by fuzzy clustering and discuss some of their common problems. As a consequence, we propose a new method which is specifically tailored for the task of learning rules. Fuzzy Clustering with Evolutionary Algorithms Frank Klawonn Abstract Objective function based fuzzy clustering aims at finding a fuzzy partition by optimizing a function evaluating a (fuzzy) assignment of a given data set to clusters, that are characterized by a set of parameters, the so-called prototypes. The iterative optimization technique usually requires the objective function not only to be differentiable, but prefers also an analytical solution for the equations of necessary conditions for local optima. Evolutionary algorithms are known to be an alternative robust optimization technique which are applicable to quite general forms of objective functions. We investigate the possibility of making use of evolutionary algorithms in fuzzy clustering. Our experiments and theoretical investigations show that the application of evolutionary algorithms to shell clustering, where the clusters are in the form of geometric contours, is not very promising due to the shape of the objective function, whereas they can be helpful in finding solid clusters that are not smooth, for example rectangles or cubes. These types of clusters play an important role for fuzzy rule extraction from data. Fuzzy Cluster Analysis for Identification of Gene Regulating Regions Lars Pickert, Frank Klawonn, Edgar Wingender The main approach of this work is the implementation of a cluster analysis program for identification of regulatory regions in genomes. These regions are important parts of the genetic pool in higher developed organisms. They are composed of several basic elements, so called transcription factor sites, which can be identified by special analysis tools more or less vaguely. The program we have developed is able to search for two-dimensional clusters in the results of such analysis tools to give hints on gene regulatory regions. For this purpose two fuzzy clustering algorithms have been implemented: The fuzzy c-means (FCM) and the Gath and Geva fuzzy clustering algorithm (GG) with two conventional cluster validity methods and one which has been developed especially for this application. All results of the cluster analysis program can be visualized and documented automatically. Neuro-Fuzzy Classification Initialized by Fuzzy Clustering D. Nauck, F. Klawonn In this paper we discuss how a neuro-fuzzy classifier can be initialized by rules generated by fuzzy clustering. The neuro-fuzzy classifier NEFCLASS can learn fuzzy classification rules completely from data. The learning algorithm for fuzzy sets can be constrained in order to obtain interpretable classifiers. However, fuzzy clustering provides more sophisticated rule learning procedures. We show that the learning process of NEFCLASS produces better results, if it is initialized by fuzzy clustering. Derivation of Fuzzy Classification Rules from Multidimensional Data F. Klawonn and R. Kruse A This paper describes techniques for deriving fuzzy classification rules based on special modified fuzzy clustering algorithms. The basic idea is that each fuzzy cluster induces a fuzzy classification rule. The fuzzy sets appearing in a rule associated with a fuzzy cluster are obtained by projecting the cluster to the one-dimensional coordinate spaces. In order to allow clusters of varying shape and size we derive special fuzzy clustering algorithms which are searching for clusters in the form of axes-parallel hyper-ellipsoids. Our method can be applied to classification tasks where the classification of the sample data is known as well as when it is not known. Clustering Methods in Fuzzy Control F. Klawonn, R. Kruse Fuzzy controllers can be interpreted as an interpolation technique on the basis of fuzzy clusters of input/output pairs. It is therefore obvious that fuzzy clustering algorithms are a promising tool for supporting the design of a fuzzy controller when data of the process to be controlled are available. This paper discusses the possibilities and limitations of fuzzy clustering for fuzzy control. Similarity Based Reasoning Frank Klawonn This paper is devoted to the duality between fuzzy sets and equality relations. It comprises various results that allow to interchange from one framework to the other. Finally it is shown that in fuzzy reasoning the inherent similarity characterized by equality relations cannot be avoided. Modifications of Genetic Algorithms for Designing and Optimizing FuzzyControllers J. Kinzel, F. Klawonn, R. Kruse This paper investigates the possibilities for applications of genetic algorithms to tuning and optimizing fuzzy controllers, or even to generate fuzzy controllers automatically. There are various ad-hoc approaches to use genetic algorithms for the design of fuzzy controllers, which already indicated good results. However, there is a need for systematic techniques that take the properties of fuzzy controllers and genetic algorithm into account in order to obtain fast convergence and to be able to tackle more complex control problems. Fuzzy Clustering with Evolutionary Algorithms Frank Klawonn and Annette Keller Objective function based fuzzy clustering aims at finding a fuzzy partition by optimizing a function evaluating a (fuzzy) assignment of a given data set to clusters, that are characterized by a set of parameters, the so-called prototypes. The iterative optimization technique usually requires the objective function not only to be differentiable, but prefers also an analytical solution for the equations of necessary conditions for local optima. Evolutionary algorithms are known to be an alternative robust optimization technique which are applicable to quite general forms of objective functions. We investigate the possibility of making use of evolutionary algorithms in fuzzy clustering. Our experiments and theoretical investigations show that the application of evolutionary algorithms to shell clustering, where the clusters are in the form of geometric contours, is not very promising due to the shape of the objective function, whereas they can be helpful in finding solid clusters that are not smooth, for example rectangles or cubes. These types of clusters play an important role for fuzzy rule extraction from data. Fuzzy Max-Min Classifiers Decide locally on the Basis of Two Attributes Birka von Schmidt, Frank Klawonn Fuzzy classification systems differ from fuzzy controllers in the form of their outputs. For classification problems a decision between a finite number of discrete classes has to be made, whereas in fuzzy control the output domain is usually continuous, i.e. a real interval. In this paper we consider fuzzy classification systems using the max-min inference scheme and classifying an unknown datum on the basis of maximum matching, i.e. assigning it to the class appearing in the consequent of the rule whose premise fits best. We basically show that this inference scheme locally takes only two attributes (variables) into account for the classification decision. Letzte Änderung am 07.09.2006