Non if-then fuzzy models

IEEE Transactions on Systems, Man and Cybernetics, Part B (Cybernetics), 2000

This paper gives some known theoretical results about fuzzy rule-based classifiers and offers a few new ones. The ability of Takagi-Sugeno-Kang (TSK) fuzzy classifiers to match exactly and to approximate classification boundaries is discussed. The lemma by Klawonn and Klement about the exact match of a classification boundary in 2 is extended from monotonous to arbitrary functions. Equivalence between fuzzy rule-based and nonfuzzy classifiers (1-nn and Parzen) is outlined. We specify the conditions under which a class of fuzzy TSK classifiers turn into lookup tables. It is shown that if the rule base consists of all possible rules (all combinations of linguistic labels on the input features), the fuzzy TSK model is a lookup classifier with hyperbox cells, regardless of the type (shape) of the membership functions used. The question "why fuzzy?" is addressed in the light of these results.

Studies in Fuzziness and Soft Computing

The series "Studies in Fuzziness and Soft Computing" contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valued logic, and related fields. The publications within "Studies in Fuzziness and Soft Computing" are primarily monographs and edited volumes. They cover significant recent developments in the field, both of a foundational and applicable character. An important feature of the series is its short publication time and worldwide distribution. This permits a rapid and broad dissemination of research results.

Journal of Mathematical Analysis and Applications, 1982

A general framework for a theory is presented that encompasses both statistical uncertainty. which falls within the province of probability theory, and nonstatistical uncertamty. which relates to the concept of a fuzzy set and possibility theory [L. A. Zadeh, J. FUZZJ Sers I (1978). 3-281. The concept of a fuzzy integral ts used to define the expected value of a random vartable. Properties of the fuzzy expectation are stated and a mean-value theorem for the fuzzy integral is proved. Comparisons between the fuzzy and the Lebesgue integral are presented. After a new concept of dependence IS formulated, various convergence concepts are defined and their relationshtps are studied by using a Chebyshevlike inequality for the fuzzy Integral. The possibility of using this theory m Bayestan estimation with fuzzy prior mformation IS explored.

The aim of this work is to compare between what seems to be entirely different two highly developing "fuzzy probability" theories. The first theory had been developed firstly by statisticians and the other separately by physicists. We start by indicating the needs to develop such theories and what helped to develop each, then we will establish the basis of the two theories and illustrate that each indeed extends classical probability theory. Moreover, we will try to see whether or not any of the two theory can be embedded into the other.

Springer eBooks, 2005

Preface vii 1 Pattern Recognition 1 1.1 Fuzzy models for pattern recognition 1 1.2 Why fuzzy pattern recognition? 1.3 Overview of the volume 8 1.4 Comments and bibliography 2 Cluster Analysis for Object Data 2.1 Cluster analysis 2.2 Batch point-prototype clustering models A. The c-means models B. Semi-supervised clustering models 23 C. Probabilistic Clustering D. Remarks on HCM/FCM/PCM E. The Reformulation Theorem 2.3 Non point-prototype clustering models A. The Gustafson-Kessel (GK) Model B. Linear manifolds as prototypes C. Spherical Prototypes D. Elliptical Prototypes E. Quadric Prototypes F. Norm induced shell prototypes G. Regression models as prototypes H. Clustering for robust parametric estimation 2.4 Cluster Validity A. Direct Measures B. Davies-Bouldin Index C. Dunn's index D. Indirect measures for fuzzy clusters E. Standardizing and normalizing indirect indices F. Indirect measures for non-point prototype models G. Fuzzification of statistical indices 2.5 Feature Analysis 2.6 Comments and bibliography vi FUZZY PATTERN RECOGNITION 3 Cluster Analysis for Relational Data 3.1 Relational Data A. Crisp Relations B. Fuzzy Relations 3.2 Object Data to Relational Data 3.3 Hierarchical Methods 3.4 Clustering by decomposition of fuzzy relations 3.5 Relational clustering with objective functions A. The Fuzzy Non Metric (FNM) model B. The Assignment-Prototype (AP) Model C. The relational fuzzy c-means (RFCM) model D. The non-Euclidean RFCM (NERFCM) model 3.6 Cluster validity for relational models 3.7 Comments and bibliography 4 Classifier Design 4.1 Classifier design for object data 4.2 Prototype classifiers A. The nearest prototype classifier B. Multiple prototype designs 4.3 Methods of prototype generation A. Competitive learning networks B. Prototype relabeling C. Sequential hard c-means (SHCM) D. Learning vector quantization (LVQ) E. Some soft versions of LVQ F. Case Study : LVQ and GLVQ-F 1-nmp designs G. The soft competition scheme (SCS) H. Fuzzy learning vector quantization (FLVQ) 1. The relationship between c-Means and CL schemes J. The mountain "clustering" method (MCM) 4.4 Nearest neighbor classifiers 4.5 The Fuzzy Integral 4.6 Fuzzy Rule-Based Classifiers A. Crisp decision trees B. Rules from crisp decision trees C. Crisp decision tree design D. Fuzzy system models and function approximation E. The Chang -Pavlidis fuzzy decision tree F. Fuzzy relatives of 1D3 G. Rule-based approximation based on clustering H. Heuristic rule extraction I. Generation of fuzzy labels for training data 4.7 Neural-like architectures for classification A. Biological and mathematical neuron models B. Neural network models C. Fuzzy Neurons D. Fuzzy aggregation networks E. Rule extraction with fuzzy aggregation networks Contents vii 4.8 Adaptive resonance models A. The ARTl algorithm B. Fuzzy relatives of ART C. Radial basis function networks 4.9 Fusion techniques A. Data level fusion B. Feature level fusion C. Classifier fusion 4.10 Syntactic pattern recognition A. Language-based methods B. Relation-based methods 4.11 Comments and bibliography 5 Image Processing and Computer 'Vision 5.

Information and Control, 1973

The problem of making decisions to classify the objects of a certain universe into two or more suitable classes has been considered in the setting of fuzzy sets theory. A measure of the total amount of uncertainty that arises in making decisions has been proposed in the general case. This quantity reduces to the "entropy" of a fuzzy set in the case of two classes. Other quantities which play a relevant role in this theory are the "energy" and the "effective power" of a fuzzy set, defined as N N Z-,J, and * ES,, t=1 l~l respectively, where w is a nonnegative weight function and ¢ a nonnegative constant. If go = constant and ~ q= 0, the energy is proportional to the effective power and, therefore, to the "power" of the fuzzy set. The maximum of the uncertainty has been calculated in some cases of interest, keeping constant the total energy and effective power. In particular the Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein distributions are derived. Some applications to decision theory are considered in the case of both deterministic and probabilistic decisions. Finally, the analogies that exist between the previous concepts and the thermodynamic ones are discussed.

Lecture Notes in Computer Science, 1997

We examine the principle capabilities and limits of fuzzy classi ers that are based on a nite set of fuzzy if{then rules like they are used for fuzzy controllers, except that the conclusion of a rule speci es a discrete class instead of a (fuzzy) real output value. Our results show that in the two{dimensional case, for classi cation problems whose solutions can only be solved approximately by crisp classi cation 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 su cient.

Automatic Control and Computer Sciences, Vol. 49, , 2015

In a fuzzy classifier, the maping “inputs–output” is described by the linguistic 〈If–then〉 rules, the antecedents in which contain the fuzzy terms “low,” “average,” “high,” and so on. To increase the correctness, the fuzzy classifier is learned by using experimental data. The problems with equal and different costs of various classification errors are discussed. A new criterion is offered for problems with undistinguishable types of errors, in addition to the two known criteria. A new one implies that the distance between the desired and real fuzzy results of classification for the cases of a wrong decision is weighted by the penalty factor. The learning criteria are generalized for problems of classification with the cost matrix. The conducted computer experiments on the wine recognition and heart disease diagnostics problems show that the best quality parameters of tuning fuzzy classifiers are achieved by a new learning criterion.

International journal of …, 2002

When designing rule-based models and classifiers, some precision is sacrificed to obtain linguistic interpretability. Understandable models are not expected to outperform black boxes, but usually fuzzy learning algorithms are statistically validated by contrasting them with black-box models. Unless performance of both approaches is equivalent, it is difficult to judge whether the fuzzy one is doing its best, because the precision gap between the best understandable model and the best black-box model is not known.

2004

When designing any type of fuzzy rule based system, considerable effort is placed in identifying the correct number of fuzzy sets and the fine tuning of the corresponding membership functions. Once a rule base has been formulated a fuzzy inference strategy must be applied in order to combine grades of membership. Considerable time and effort is spent trying to determine the number of fuzzy sets for a given system while substantially less time is invested in obtaining the most suitable inference strategy. This paper investigates a number of theoretical proven fuzzy inference strategies in order to assess the impact of these strategies on the performance of a fuzzy rule based classifier system. A fuzzy inference framework is proposed, which allows the investigation of five pure theoretical fuzzy inference operators in two real world applications. An additional two novel fuzzy-neural strategies are proposed and a comparative study is undertaken. The results show that the selection of the most suitable inference strategy for a given domain can lead to a significant improvement in performance.