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33
Designing fuzzy inference systems from data: an interpretabilityoriented review
 IEEE TRANS. FUZZY SYSTEMS
, 2001
"... Fuzzy inference systems (FIS) are widely used for process simulation or control. They can be designed either from expert knowledge or from data. For complex systems, FIS based on expert knowledge only may suffer from a loss of accuracy. This is the main incentive for using fuzzy rules inferred from ..."
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Cited by 88 (16 self)
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Fuzzy inference systems (FIS) are widely used for process simulation or control. They can be designed either from expert knowledge or from data. For complex systems, FIS based on expert knowledge only may suffer from a loss of accuracy. This is the main incentive for using fuzzy rules inferred from data. Designing a FIS from data can be decomposed into two main phases: automatic rule generation and system optimization. Rule generation leads to a basic system with a given space partitioning and the corresponding set of rules. System optimization can be done at various levels. Variable selection can be an overall selection or it can be managed rule by rule. Rule base optimization aims to select the most useful rules and to optimize rule conclusions. Space partitioning can be improved by adding or removing fuzzy sets and by tuning membership function parameters. Structure optimization is of a major importance: selecting variables, reducing the rule base and optimizing the number of fuzzy sets. Over the years, many methods have become available for designing FIS from data. Their efficiency is usually characterized by a numerical performance index. However, for humancomputer cooperation another criterion is needed: the rule interpretability. An implicit assumption states that fuzzy rules are by nature easy to be interpreted. This could be wrong when dealing with complex multivariable systems or when the generated partitioning is meaningless for experts. This paper analyzes the main methods for automatic rule generation and structure optimization. They are grouped into several families and compared according to the rule interpretability criterion. For this purpose, three conditions for a set of rules to be interpretable are defined.
Multiobjective hierarchical genetic algorithm for interpretable fuzzy rulebased knowledge extraction
 FUZZY SETS AND SYSTEMS
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Generating an interpretable family of fuzzy partitions
 IEEE Transactions on Fuzzy Systems
, 2004
"... Abstract—In this paper, we propose a new method to construct fuzzy partitions from data. The procedure generates a hierarchy including best partitions of all sizes from n to two fuzzy sets. The maximum size is determined according to the data distribution and corresponds to the finest resolution lev ..."
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Cited by 30 (9 self)
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Abstract—In this paper, we propose a new method to construct fuzzy partitions from data. The procedure generates a hierarchy including best partitions of all sizes from n to two fuzzy sets. The maximum size is determined according to the data distribution and corresponds to the finest resolution level. We use an ascending method for which a merging criterion is needed. This criterion is based on the definition of a special metric distance suitable for fuzzy partitioning, and the merging is done under semantic constraints. The distance we define does not handle the point coordinates, but directly their membership degrees to the fuzzy sets of the partition. This leads to the introduction of the notions of internal and external distances. The hierarchical fuzzy partitioning is carried independently over each dimension, and, to demonstrate the partition potential, they are used to build fuzzy inference system using a simple selection mechanism. Due to the merging technique, all the fuzzy sets in the various partitions are interpretable as linguistic labels. The tradeoff between accuracy and interpretability constitutes the most promising aspect in our approach. Well known data sets are investigated and the results are compared with those obtained by other authors using different techniques. The method is also applied to real world agricultural data, the results are analyzed and weighed against those achieved by other methods, such as fuzzy clustering or discriminant analysis. Index Terms—Distance, fuzzy partitioning, interpretability, learning, rule induction. I.
Multiobjective evolutionary optimization of the size, shape, and position parameters of radial basis function networks for function approximation
 IEEE Transactions on Neural Networks
"... Abstract—This paper presents a multiobjective evolutionary algorithm to optimize radial basis function neural networks (RBFNNs) in order to approach target functions from a set of inputoutput pairs. The procedure allows the application of heuristics to improve the solution of the problem at hand by ..."
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Cited by 27 (7 self)
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Abstract—This paper presents a multiobjective evolutionary algorithm to optimize radial basis function neural networks (RBFNNs) in order to approach target functions from a set of inputoutput pairs. The procedure allows the application of heuristics to improve the solution of the problem at hand by including some new genetic operators in the evolutionary process. These new operators are based on two wellknown matrix transformations: singular value decomposition (SVD) and orthogonal least squares (OLS), which have been used to define new mutation operators that produce local or global modifications in the radial basis functions (RBFs) of the networks (the individuals in the population in the evolutionary procedure). After analyzing the efficiency of the different operators, we have shown that the global mutation operators yield an improved procedure to adjust the parameters of the RBFNNs. Index Terms—Evolutionary computation, neural networks, radial basis functions (RBFs), orthogonal transformations, heuristics.
Improving Clustering Technique for Functional Approximation Problem Using Fuzzy Logic: ICFA algorithm
 Lecture Notes in Computer Science
, 2005
"... Abstract—To date, clustering techniques have always been oriented to solve classification and pattern recognition problems. However, some authors have applied them unchanged to construct initial models for function approximators. Nevertheless, classification and function approximation problems prese ..."
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Cited by 23 (14 self)
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Abstract—To date, clustering techniques have always been oriented to solve classification and pattern recognition problems. However, some authors have applied them unchanged to construct initial models for function approximators. Nevertheless, classification and function approximation problems present quite different objectives. Therefore it is necessary to design new clustering algorithms specialized in the problem of function approximation. This paper presents a new clustering technique, specially designed for function approximation problems, which improves the performance of the approximator system obtained, compared with other models derived from traditional classification oriented clustering algorithms and input–output clustering techniques. Index Terms—Clustering techniques, function approximation, model initialization. I.
AgentBased Evolutionary Approach for Interpretable RuleBased Knowledge Extraction
 IEEE Trans. Syst., Man, Cybern. C
, 2005
"... Abstract—An agentbased evolutionary approach is proposed to extract interpretable rulebased knowledge. In the multiagent system, each fuzzy set agent autonomously determines its own fuzzy sets information, such as the number and distribution of the fuzzy sets. It can further consider the interpret ..."
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Cited by 11 (2 self)
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Abstract—An agentbased evolutionary approach is proposed to extract interpretable rulebased knowledge. In the multiagent system, each fuzzy set agent autonomously determines its own fuzzy sets information, such as the number and distribution of the fuzzy sets. It can further consider the interpretability of fuzzy systems with the aid of hierarchical chromosome formulation and interpretabilitybased regulation method. Based on the obtained fuzzy sets, the Pittsburghstyle approach is applied to extract fuzzy rules that take both the accuracy and interpretability of fuzzy systems into consideration. In addition, the fuzzy set agents can cooperate with each other to exchange their fuzzy sets information and generate offspring agents. The parent agents and their offspring compete with each other through the arbitrator agent based on the criteria associated with the accuracy and interpretability to allow them to remain competitive enough to move into the next population. The performance with emphasis upon both the accuracy and interpretability based on the agentbased evolutionary approach is studied through some benchmark problems reported in the literature. Simulation results show that the proposed approach can achieve a good tradeoff between the accuracy and interpretability of fuzzy systems. Index Terms—Hierarchical chromosome formulation, interpretability and accuracy, multiagent system, multiobjective decision making. I.
Interpretability improvements to find the balance interpretabilityaccuracy in fuzzy modeling: an overview
 in Interpretability Issues in Fuzzy Modeling
, 2003
"... Abstract. System modeling with fuzzy rulebased systems (FRBSs), i.e. fuzzy modeling (FM), usually comes with two contradictory requirements in the obtained model: the interpretability, capability to express the behavior of the real system in an understandable way, and the accuracy, capability to fa ..."
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Cited by 10 (4 self)
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Abstract. System modeling with fuzzy rulebased systems (FRBSs), i.e. fuzzy modeling (FM), usually comes with two contradictory requirements in the obtained model: the interpretability, capability to express the behavior of the real system in an understandable way, and the accuracy, capability to faithfully represent the real system. While linguistic FM (mainly developed by linguistic FRBSs) is focused on the interpretability, precise FM (mainly developed by TakagiSugenoKang FRBSs) is focused on the accuracy. Since both criteria are of vital importance in system modeling, the balance between them has started to pay attention in the fuzzy community in the last few years. The chapter analyzes mechanisms to find this balance by improving the accuracy in linguistic FM: deriving the membership functions, improving the fuzzy rule set derivation, or extending the model structure. 1
Building Fuzzy Graphs: Features and Taxonomy of Learning for NonGridOriented Fuzzy RuleBased Systems
, 2001
"... The use of Mamdanitype fuzzy rulebased systems (FRBSs) allows us to deal with the modeling of systems building a linguistic model clearly interpretable by human beings. However, the accuracy obtained is not sometimes as good as desired. This fact relates to the restriction imposed when using lingu ..."
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Cited by 8 (4 self)
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The use of Mamdanitype fuzzy rulebased systems (FRBSs) allows us to deal with the modeling of systems building a linguistic model clearly interpretable by human beings. However, the accuracy obtained is not sometimes as good as desired. This fact relates to the restriction imposed when using linguistic variables, which forces the membership functions considered in each fuzzy linguistic rule to belong to a common set of them, i.e., to use a global grid. To solve this problem, in the last few years a new variant has been proposed working directly with fuzzy variables in the fuzzy rules instead of linguistic terms, thus ignoring the said restriction. Therefore, these systems, which are totally equivalent to fuzzy graphs (defined by Zadeh as granular representations of functional dependencies and relations), do not consider a global grid and could be named nongridoriented (NGO) FRBSs. Of course, the main objective of these models is the accuracy of the system instead its interpretability. Until now, NGO FRBSs have been little considered and developed in the literature. However, and due to their good accuracy, their use is increasing thus making necessary a wide analysis on the features and associated learning methods in the NGO domain. This contribution aims at analyzing the structure and framework of NGO FRBSs, as well as making a taxonomy of learning methods considering the constrains imposed on the fuzzy sets in the generation process. Some automatic learning techniques and methods proposed in the literature to build these fuzzy graphs will be also reviewed and analyzed when solving several applications of different nature.
Structure identification in complete rulebased fuzzy systems
 IEEE Trans. Fuzzy
, 2002
"... Abstract—The identification of a model is one of the key issues in the field of fuzzy system modeling and function approximation theory. There are numerous approaches to the issue of parameter optimization within a fixed fuzzy system structure but no reliable method to obtain the optimal topology of ..."
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Cited by 7 (3 self)
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Abstract—The identification of a model is one of the key issues in the field of fuzzy system modeling and function approximation theory. There are numerous approaches to the issue of parameter optimization within a fixed fuzzy system structure but no reliable method to obtain the optimal topology of the fuzzy system from a set of input–output data. This paper presents a reliable method to obtain the structure of a complete rulebased fuzzy system for a specific approximation accuracy of the training data, i.e., it can decide which input variables must be taken into account in the fuzzy system and how many membership functions (MFs) are needed in every selected input variable in order to reach the approximation target with the minimum number of parameters. Index Terms—Complete rulebased fuzzy systems, function approximation, fuzzy modeling, system identification. I.
Fuzzy Logic and Neurofuzzy Systems: A Systematic Introduction
 International Journal of Artificial Intelligence and Expert Systems
, 2011
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