Constructing accurate and parsimonious fuzzy models with distinguishable fuzzy sets based on an entropy measure

Parsimony is very important in system modeling as it is closely related to model interpretability. In this paper, a scheme for constructing accurate and parsimonious fuzzy models by generating distinguishable fuzzy sets is proposed, in which the distinguishability of input space partitioning is measured by a so-called ''local'' entropy. By maximizing this entropy measure the optimal number of merged fuzzy sets with good distinguishability can be obtained, which leads to a parsimonious input space partitioning while preserving the information of the original fuzzy sets as much as possible. Different from the existing merging algorithms, the proposed scheme takes into account the information provided by input-output samples to optimize input space partitioning. Furthermore, this scheme possesses the ability to seek a balance between the global approximation ability and distinguishability of input space partitioning in constructing Takagi-Sugeno (TS) fuzzy models. Experimental results have shown that this scheme is able to produce accurate and parsimonious fuzzy models with distinguishable fuzzy sets.

[1]  Giovanna Castellano,et al.  Generation of interpretable fuzzy granules by a double-clustering technique , 2002 .

[2]  Serge Guillaume,et al.  Designing fuzzy inference systems from data: An interpretability-oriented review , 2001, IEEE Trans. Fuzzy Syst..

[3]  T. Kavli ASMO—Dan algorithm for adaptive spline modelling of observation data , 1993 .

[4]  H. Carter Fuzzy Sets and Systems — Theory and Applications , 1982 .

[5]  Michio Sugeno,et al.  Fuzzy identification of systems and its applications to modeling and control , 1985, IEEE Transactions on Systems, Man, and Cybernetics.

[6]  Rami Zwick,et al.  Measures of similarity among fuzzy concepts: A comparative analysis , 1987, Int. J. Approx. Reason..

[7]  Andrzej Bargiela,et al.  Similarity vs. Possibility in measuring Fuzzy Sets Distinguishability , 2004 .

[8]  Jujang Lee,et al.  Adaptive network-based fuzzy inference system with pruning , 2003, SICE 2003 Annual Conference (IEEE Cat. No.03TH8734).

[9]  Andrew K. C. Wong,et al.  A new method for gray-level picture thresholding using the entropy of the histogram , 1985, Comput. Vis. Graph. Image Process..

[10]  John Yen,et al.  Improving the interpretability of TSK fuzzy models by combining global learning and local learning , 1998, IEEE Trans. Fuzzy Syst..

[11]  János Abonyi,et al.  Learning fuzzy classification rules from labeled data , 2003, Inf. Sci..

[12]  Xizhao Wang,et al.  On the optimization of fuzzy decision trees , 2000, Fuzzy Sets Syst..

[13]  F. Klawonn,et al.  A new approach to fuzzy partitioning , 2001, Proceedings Joint 9th IFSA World Congress and 20th NAFIPS International Conference (Cat. No. 01TH8569).

[14]  James C. Bezdek,et al.  Pattern Recognition with Fuzzy Objective Function Algorithms , 1981, Advanced Applications in Pattern Recognition.

[15]  Bernhard Sendhoff,et al.  Extracting Interpretable Fuzzy Rules from RBF Networks , 2003, Neural Processing Letters.

[16]  Magne Setnes,et al.  Rule-based modeling: precision and transparency , 1998, IEEE Trans. Syst. Man Cybern. Part C.

[17]  Shang-Ming Zhou,et al.  Improving the interpretability of Takagi-Sugeno fuzzy model by using linguistic modifiers and a multiple objective learning scheme , 2004, 2004 IEEE International Joint Conference on Neural Networks (IEEE Cat. No.04CH37541).

[18]  Li-Xin Wang,et al.  Adaptive fuzzy systems and control , 1994 .

[19]  Isak Gath,et al.  Unsupervised Optimal Fuzzy Clustering , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[20]  Jyh-Shing Roger Jang,et al.  ANFIS: adaptive-network-based fuzzy inference system , 1993, IEEE Trans. Syst. Man Cybern..

[21]  Joos Vandewalle,et al.  Constructing fuzzy models with linguistic integrity from numerical data-AFRELI algorithm , 2000, IEEE Trans. Fuzzy Syst..

[22]  Christopher J. Harris,et al.  Fuzzy local linearization and local basis function expansion in nonlinear system modeling , 1999, IEEE Trans. Syst. Man Cybern. Part B.

[23]  Uzay Kaymak,et al.  Similarity measures in fuzzy rule base simplification , 1998, IEEE Trans. Syst. Man Cybern. Part B.

[24]  Christopher J. Harris,et al.  A hybrid learning scheme combining EM and MASMOD algorithms for fuzzy local linearization modeling , 2001, IEEE Trans. Neural Networks.

[25]  Donald B. Rubin,et al.  Max-imum Likelihood from Incomplete Data , 1972 .

[26]  Takeshi Furuhashi,et al.  Conciseness of Fuzzy Models , 2003 .

[27]  Takeshi Furuhashi,et al.  On interpretability of fuzzy models based on conciseness measure , 2001, 10th IEEE International Conference on Fuzzy Systems. (Cat. No.01CH37297).

[28]  J. Casillas Interpretability issues in fuzzy modeling , 2003 .

[29]  Kim-Fung Man,et al.  Agent-based evolutionary approach for interpretable rule-based knowledge extraction , 2005, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews).

[30]  Ashok K. Goel Adaptive Modeling , 2002 .

[31]  José Valente de Oliveira,et al.  Semantic constraints for membership function optimization , 1999, IEEE Trans. Syst. Man Cybern. Part A.

[32]  Xia Hong,et al.  Adaptive Modelling, Estimation and Fusion from Data: A Neurofuzzy Approach , 2002, Advanced information processing.