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239
Interval Type2 Fuzzy Logic Systems Made Simple,” in
 IEEE Trans. on Fuzzy Systems
, 2006
"... Abstract—To date, because of the computational complexity of using a general type2 fuzzy set (T2 FS) in a T2 fuzzy logic system (FLS), most people only use an interval T2 FS, the result being an interval T2 FLS (IT2 FLS). Unfortunately, there is a heavy educational burden even to using an IT2 FLS. ..."
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Cited by 68 (14 self)
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Abstract—To date, because of the computational complexity of using a general type2 fuzzy set (T2 FS) in a T2 fuzzy logic system (FLS), most people only use an interval T2 FS, the result being an interval T2 FLS (IT2 FLS). Unfortunately, there is a heavy educational burden even to using an IT2 FLS. This burden has to do with first having to learn general T2 FS mathematics, and then specializing it to an IT2 FSs. In retrospect, we believe that requiring a person to use T2 FS mathematics represents a barrier to the use of an IT2 FLS. In this paper, we demonstrate that it is unnecessary to take the route from general T2 FS to IT2 FS, and that all of the results that are needed to implement an IT2 FLS can be obtained using T1 FS mathematics. As such, this paper is a novel tutorial that makes an IT2 FLS much more accessible to all readers of this journal. We can now develop an IT2 FLS in a much more straightforward way. Index Terms—Fuzzy logic system, interval type2 fuzzy sets, type2 fuzzy logic system, type2 fuzzy sets. I.
Learning fuzzy rules and approximate reasoning in fuzzy neural networks and hybrid system
 Fuzzy Sets Syst
, 1996
"... Abstract The paper considers both knowledge acquisition and knowledge interpretation tasks as tightly connected and continuously interacting processes in a contemporary knowledge engineering system. Fuzzy rules are used here as a framework for knowledge representation. An algorithm REFuNN for fuzzy ..."
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Cited by 52 (23 self)
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Abstract The paper considers both knowledge acquisition and knowledge interpretation tasks as tightly connected and continuously interacting processes in a contemporary knowledge engineering system. Fuzzy rules are used here as a framework for knowledge representation. An algorithm REFuNN for fuzzy rules extraction from adaptive fuzzy neural networks (FuNN) is proposed. A case study of Iris classification is chosen to illustrate the algorithm. Interpretation of fuzzy rules is possible by using fuzzy neural networks or by using standard fuzzy inference methods. Both approaches are compared in the paper based on the case example. A hybrid environment FuzzyCOPE which facilitates neural network simulation, fuzzy rules extraction from fuzzy neural networks and fuzzy rules interpretation by using different methods for approximate reasoning is briefly described.
A FuzzyGenetic Approach to Breast Cancer Diagnosis
, 1999
"... The automatic diagnosis of breast cancer is an important, realworld medical problem. In this paper we focus on the Wisconsin breast cancer diagnosis (WBCD) problem, combining two methodologiesfuzzy systems and evolutionary algorithmsso as to automatically produce diagnostic systems. We find t ..."
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Cited by 40 (7 self)
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The automatic diagnosis of breast cancer is an important, realworld medical problem. In this paper we focus on the Wisconsin breast cancer diagnosis (WBCD) problem, combining two methodologiesfuzzy systems and evolutionary algorithmsso as to automatically produce diagnostic systems. We find that our fuzzygenetic approach produces systems exhibiting two prime characteristics: first, they attain high classification performance (the best shown to date), with the possibility of attributing a confidence measure to the output diagnosis; second, the resulting systems involve a few simple rules, and are therefore (human) interpretable. 1999 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy systems; Genetic algorithms; Breast cancer diagnosis www.elsevier.com/locate/artmed 1.
Selforganized fuzzy system generation from training examples
 IEEE Trans. Fuzzy Syst
, 2000
"... Abstract—In the synthesis of a fuzzy system two steps are generally employed: the identification of a structure and the optimization of the parameters defining it. This paper presents a methodology to automatically perform these two steps in conjunction using a threephase approach to construct a fu ..."
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Cited by 33 (10 self)
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Abstract—In the synthesis of a fuzzy system two steps are generally employed: the identification of a structure and the optimization of the parameters defining it. This paper presents a methodology to automatically perform these two steps in conjunction using a threephase approach to construct a fuzzy system from numerical data. Phase 1 outlines the membership functions and system rules for a specific structure, starting from a very simple initial topology. Phase 2 decides a new and more suitable topology with the information received from the previous step; it determines for which variable the number of fuzzy sets used to discretize the domain must be increased and where these new fuzzy sets should be located. This, in turn, decides in a dynamic way in which part of the input space the number of fuzzy rules should be increased. Phase 3 selects from the different structures obtained to construct a fuzzy system the one providing the best compromise between the accuracy of the approximation and the complexity of the rule set. The accuracy and complexity of the fuzzy system derived by the proposed selforganized fuzzy rule generation procedure (SOFRG) are studied for the problem of function approximation. Simulation results are compared with other methodologies such as artificial neural networks, neurofuzzy systems, and genetic algorithms. Index Terms—Function approximation, fuzzy system design, generation of membership functions and rules. I.
Modified GathGeva Fuzzy Clustering for Identification of TakagiSugeno Fuzzy Models
 IEEE Transactions on Systems, Man, and Cybernetics
, 2001
"... The construction of interpretable TakagiSugeno (TS) fuzzy models by means of clustering is addressed. First, it is shown how the antecedent fuzzy sets and the corresponding consequent parameters of the TS model can be derived from clusters obtained by the GathGeva algorithm. To preserve the part ..."
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Cited by 33 (6 self)
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The construction of interpretable TakagiSugeno (TS) fuzzy models by means of clustering is addressed. First, it is shown how the antecedent fuzzy sets and the corresponding consequent parameters of the TS model can be derived from clusters obtained by the GathGeva algorithm. To preserve the partitioning of the antecedent space, linearly transformed input variables can be used in the model. This may, however, complicate the interpretation of the rules. To form an easily interpretable model that does not use the transformed input variables, a new clustering algorithm is proposed, based on the Expectation Maximization (EM) identification of Gaussian mixture models. This new technique is applied to two wellknown benchmark problems: the MPG (miles per gallon) prediction and a simulated secondorder nonlinear process. The obtained results are compared with results from the literature.
Flexible neurofuzzy systems
 IEEE TRANS. NEURAL NETW
, 2003
"... In this paper, we derive new neurofuzzy structures called flexible neurofuzzy inference systems or FLEXNFIS. Based on the input–output data, we learn not only the parameters of the membership functions but also the type of the systems (Mamdani or logical). Moreover, we introduce: 1) softness to f ..."
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Cited by 29 (6 self)
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In this paper, we derive new neurofuzzy structures called flexible neurofuzzy inference systems or FLEXNFIS. Based on the input–output data, we learn not only the parameters of the membership functions but also the type of the systems (Mamdani or logical). Moreover, we introduce: 1) softness to fuzzy implication operators, to aggregation of rules and to connectives of antecedents; 2) certainty weights to aggregation of rules and to connectives of antecedents; and 3) parameterized families of Tnorms and Snorms to fuzzy implication operators, to aggregation of rules and to connectives of antecedents. Our approach introduces more flexibility to the structure and design of neurofuzzy systems. Through computer simulations, we show that Mamdanitype systems are more suitable to approximation problems, whereas logicaltype systems may be preferred for classification problems.
Support Vector Learning for Fuzzy RuleBased Classification Systems
, 2003
"... To design a fuzzy rulebased classi cation system (fuzzy classi er) with good generalization abilityina high dimensional feature space has been an active research topic for a long time. As a powerful machine learning approach for pattern recognition problems, support vector machine (SVM) is known to ..."
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Cited by 29 (1 self)
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To design a fuzzy rulebased classi cation system (fuzzy classi er) with good generalization abilityina high dimensional feature space has been an active research topic for a long time. As a powerful machine learning approach for pattern recognition problems, support vector machine (SVM) is known to have good generalization ability. More importantly, an SVM can work very well on a high (or even in nite) dimensional feature space. This paper investigates the connection between fuzzy classi ers and kernel machines, establishes a link between fuzzy rules and kernels, and proposes a learning algorithm for fuzzy classi ers. We rst show that a fuzzy classi er implicitly de nes a translation invariant kernel under the assumption that all membership functions associated with the same input variable are generated from location transformation of a reference function. Fuzzy inference on the IFpart of a fuzzy rule can be viewed as evaluating the kernel function. The kernel function is then proven to be a Mercer kernel if the reference functions meet certain spectral requirement. The corresponding fuzzy classi er is named positive de  nite fuzzy classi er (PDFC). A PDFC can be built from the given training samples based on a support vector learning approach with the IFpart fuzzy rules given by the support vectors. Since the learning process minimizes an upper bound on the expected risk (expected prediction error) instead of the empirical risk (training error), the resulting PDFC usually has good generalization. Moreover, because of the sparsity properties of the SVMs, the number of fuzzy rules is irrelevant to the dimension of input space. In this sense, we avoid the "curse of dimensionality." Finally, PDFCs with dierent reference functions are constructed using the su...
Fuzzy CoCo: A CooperativeCoevolutionary Approach to Fuzzy Modeling
, 2001
"... Coevolutionary algorithms have received increased attention in the past few years within the domain of evolutionary computation. In this paper, we combine the search power of coevolutionary computation with the expressive power of fuzzy systems, introducing a novel algorithm, Fuzzy CoCo: Fuzzy Coope ..."
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Cited by 27 (7 self)
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Coevolutionary algorithms have received increased attention in the past few years within the domain of evolutionary computation. In this paper, we combine the search power of coevolutionary computation with the expressive power of fuzzy systems, introducing a novel algorithm, Fuzzy CoCo: Fuzzy Cooperative Coevolution. We demonstrate the efficacy of Fuzzy CoCo by applying it to a hard, realworld problembreast cancer diagnosisobtaining the best results to date while expending less computational effort than formerly. Analyzing our results, we derive guidelines for setting the algorithm's parameters given a (hard) problem to solve. We hope Fuzzy CoCo proves to be a powerful tool in the fuzzy modeler's toolkit.
Input Selection for ANFIS Learning
 IN PROCEEDINGS OF THE IEEE INTERNATIONAL CONFERENCE ON FUZZY SYSTEMS
, 1996
"... We present a quick and straightfoward way of input selection for neurofuzzy modeling using ANFIS. The method is tested on two realworld problems: the nonlinear regression problem of automobile MPG (miles per gallon) prediction, and the nonlinear system identification using the Box and Jenkins gas ..."
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Cited by 27 (1 self)
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We present a quick and straightfoward way of input selection for neurofuzzy modeling using ANFIS. The method is tested on two realworld problems: the nonlinear regression problem of automobile MPG (miles per gallon) prediction, and the nonlinear system identification using the Box and Jenkins gas furnace data [1].
Visualizing Fuzzy Points in Parallel Coordinates
, 1999
"... The ability to visualize data often leads to new insights. Data that is more than three dimensional must be visualized as a series of projections or transformed into some other representation which usually causes a loss of details. Parallel coordinates allows one to visualize data in two dimensions ..."
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Cited by 27 (5 self)
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The ability to visualize data often leads to new insights. Data that is more than three dimensional must be visualized as a series of projections or transformed into some other representation which usually causes a loss of details. Parallel coordinates allows one to visualize data in two dimensions without a loss of information. In this paper, we discuss the use of parallel coordinates to visualize fuzzy data. Fuzzy data may consist of fuzzy rules, which can be viewed as cutting a swath through an ndimensional space. Fuzzy clusters may also be considered fuzzy data in a similar way. Examples are given from three domains. The examples show that parallel coordinates can be used to nd extraneous fuzzy rules, separate fuzzy clusters as well as validate previous ndings about data sets.