Weighted fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on piecewise fuzzy entropies of fuzzy sets

In this paper, we propose a new method for weighted fuzzy interpolative reasoning in sparse fuzzy rule-based systems based on piecewise fuzzy entropies of fuzzy sets. First, the proposed method uses the representative values of antecedent fuzzy sets, the representative values of observation fuzzy sets, and the representative values of consequence fuzzy sets of fuzzy rules to get the characteristic points of the fuzzy interpolative result represented by a fuzzy set. Then, it calculates the piecewise fuzzy entropies between any two characteristic points of the antecedent fuzzy sets, the piecewise fuzzy entropies between any two characteristic points of the observation fuzzy sets, and the piecewise fuzzy entropies between any two characteristic points of the consequence fuzzy sets of the fuzzy rules, respectively. Then, it calculates the weights of the antecedent fuzzy sets of each fuzzy rule, respectively, and calculates the weight of each fuzzy rule. Then, it calculates the piecewise fuzzy entropies between any two characteristic points of the fuzzy interpolative result. Finally, it uses the secant method to calculate the degree of membership of each obtained characteristic point of the fuzzy interpolative result. The experimental results show that the proposed method outperforms the existing methods for dealing with the multivariate regression problems, the Mackey-Glass chaotic time series prediction problem, and the time series prediction problems.

[1]  Qiang Shen,et al.  Closed form fuzzy interpolation , 2013, Fuzzy Sets Syst..

[2]  Shyi-Ming Chen,et al.  A new method for weighted fuzzy interpolative reasoning based on weights-learning techniques , 2010, International Conference on Fuzzy Systems.

[3]  Qiang Shen,et al.  Backward fuzzy interpolation and extrapolation with multiple multi-antecedent rules , 2012, 2012 IEEE International Conference on Fuzzy Systems.

[4]  Churn-Jung Liau,et al.  Fuzzy Interpolative Reasoning for Sparse Fuzzy-Rule-Based Systems Based on the Areas of Fuzzy Sets , 2008, IEEE Transactions on Fuzzy Systems.

[5]  László T. Kóczy,et al.  Size reduction by interpolation in fuzzy rule bases , 1997, IEEE Trans. Syst. Man Cybern. Part B.

[6]  Settimo Termini,et al.  A Definition of a Nonprobabilistic Entropy in the Setting of Fuzzy Sets Theory , 1972, Inf. Control..

[7]  Jerry M. Mendel,et al.  Generating fuzzy rules by learning from examples , 1992, IEEE Trans. Syst. Man Cybern..

[8]  Der-Chiang Li,et al.  The data complexity index to construct an efficient cross-validation method , 2010, Decis. Support Syst..

[9]  C. Marsala,et al.  Interpolative reasoning based on graduality , 2000, Ninth IEEE International Conference on Fuzzy Systems. FUZZ- IEEE 2000 (Cat. No.00CH37063).

[10]  Qiang Shen,et al.  Adaptive fuzzy interpolation with prioritized component candidates , 2011, 2011 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2011).

[11]  Shyi-Ming Chen,et al.  Fuzzy interpolative reasoning based on the ratio of fuzziness of rough-fuzzy sets , 2015, Inf. Sci..

[12]  Shyi-Ming Chen,et al.  A new interpolative reasoning method in sparse rule-based systems , 1998, Fuzzy Sets Syst..

[13]  Shyi-Ming Chen,et al.  Weighted fuzzy interpolative reasoning systems based on interval type-2 fuzzy sets , 2013, Inf. Sci..

[14]  S. Holm.  A Simple Sequentially Rejective Multiple Test Procedure , 1979 .

[15]  László T. Kóczy,et al.  Approximate reasoning by linear rule interpolation and general approximation , 1993, Int. J. Approx. Reason..

[16]  Qiang Shen,et al.  Adaptive Fuzzy Interpolation , 2011, IEEE Transactions on Fuzzy Systems.

[17]  Qiang Shen,et al.  Fuzzy interpolative reasoning via scale and move transformations , 2006, IEEE Transactions on Fuzzy Systems.

[18]  Shyi-Ming Chen,et al.  Weighted fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on transformation techniques , 2008, 2008 International Conference on Machine Learning and Cybernetics.

[19]  Qiang Shen,et al.  Fuzzy Interpolation and Extrapolation: A Practical Approach , 2008, IEEE Transactions on Fuzzy Systems.

[20]  Paulo Cortez,et al.  Using sensitivity analysis and visualization techniques to open black box data mining models , 2013, Inf. Sci..

[21]  Shyi-Ming Chen,et al.  A new method for fuzzy interpolative reasoning based on ranking values of polygonal fuzzy sets and automatically generated weights of fuzzy rules , 2015, 2015 International Conference on Machine Learning and Cybernetics (ICMLC).

[22]  László T. Kóczy,et al.  Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases , 1993, Inf. Sci..

[23]  Qiang Shen,et al.  Towards adaptive interpolative reasoning , 2009, 2009 IEEE International Conference on Fuzzy Systems.

[24]  Péter Baranyi,et al.  Comprehensive analysis of a new fuzzy rule interpolation method , 2000, IEEE Trans. Fuzzy Syst..

[25]  Yan Shi,et al.  Multidimensional Fuzzy Interpolative Reasoning Method Based on \lambda-Width Similarity , 2007 .

[26]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1971 .

[27]  L. Zadeh.  Probability measures of Fuzzy events , 1968 .

[28]  Qiang Shen,et al.  Generalisation of Scale and Move Transformation-Based Fuzzy Interpolation , 2011, J. Adv. Comput. Intell. Intell. Informatics.

[29]  Shyi-Ming Chen,et al.  A new method for weighted fuzzy interpolative reasoningbased onpiecewise fuzzy entropies of fuzzy sets , 2015, 2015 International Conference on Machine Learning and Cybernetics (ICMLC).

[30]  R. Iman,et al.  Approximations of the critical region of the fbietkan statistic , 1980 .

[31]  Claude E. Shannon,et al.  The mathematical theory of communication , 1950 .

[32]  E.C.C. Tsang,et al.  Weighted fuzzy interpolative reasoning method , 2005, 2005 International Conference on Machine Learning and Cybernetics.

[33]  David J. Sheskin,et al.  Handbook of Parametric and Nonparametric Statistical Procedures , 1997 .

[34]  Yan Shi,et al.  An improvement to Kóczy and Hirota's interpolative reasoning in sparse fuzzy rule bases , 1996, Int. J. Approx. Reason..

[35]  Shyi-Ming Chen,et al.  Fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on the slopes of fuzzy sets , 2012, Expert Syst. Appl..

[36]  Qiang Shen,et al.  Adaptive fuzzy interpolation with uncertain observations and rule base , 2011, 2011 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2011).

[37]  Shyi-Ming Chen,et al.  Weighted Fuzzy Rule Interpolation Based on GA-Based Weight-Learning Techniques , 2011, IEEE Transactions on Fuzzy Systems.

[38]  László T. Kóczy,et al.  Fuzzy rule interpolation for multidimensional input spaces with applications: a case study , 2005, IEEE Transactions on Fuzzy Systems.

[39]  Chengyuan Chen,et al.  A new method for rule interpolation inspired by rough-fuzzy sets , 2012, 2012 IEEE International Conference on Fuzzy Systems.

[40]  D. Tikk,et al.  A new method for avoiding abnormal conclusion for /spl alpha/-cut based rule interpolation , 1999, FUZZ-IEEE'99. 1999 IEEE International Fuzzy Systems. Conference Proceedings (Cat. No.99CH36315).

[41]  Shyi-Ming Chen,et al.  Fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on the ranking values of fuzzy sets , 2008, Expert Syst. Appl..

[42]  László T. Kóczy,et al.  Representing membership functions as points in high-dimensional spaces for fuzzy interpolation and extrapolation , 2000, IEEE Trans. Fuzzy Syst..

[43]  Qiang Shen,et al.  A new fuzzy interpolative reasoning method based on center of gravity , 2003, The 12th IEEE International Conference on Fuzzy Systems, 2003. FUZZ '03..

[44]  László T. Kóczy,et al.  A generalized concept for fuzzy rule interpolation , 2004, IEEE Transactions on Fuzzy Systems.

[45]  Qiang Shen,et al.  Adaptive fuzzy interpolation and extrapolation with multiple-antecedent rules , 2010, International Conference on Fuzzy Systems.

[46]  Shyi-Ming Chen,et al.  Fuzzy Interpolative Reasoning for Sparse Fuzzy Rule-Based Systems Based on ${\bm \alpha}$-Cuts and Transformations Techniques , 2008, IEEE Transactions on Fuzzy Systems.

[47]  Shyi-Ming Chen,et al.  Fuzzy rule interpolation based on principle membership functions and uncertainty grade functions of interval type-2 fuzzy sets , 2011, Expert Syst. Appl..

[48]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[49]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[50]  Shyi-Ming Chen,et al.  Fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on interval type-2 fuzzy sets , 2011, Expert Syst. Appl..

[51]  Yeung Yam,et al.  Interpolation with function space representation of membership functions , 2006, IEEE Transactions on Fuzzy Systems.

[52]  Shyi-Ming Chen,et al.  A new method for multiple fuzzy rules interpolation with weighted antecedent variables , 2008, 2008 IEEE International Conference on Systems, Man and Cybernetics.

[53]  Fakhri Karray,et al.  Fuzzy entropy: a brief survey , 2001, 10th IEEE International Conference on Fuzzy Systems. (Cat. No.01CH37297).

[54]  D.S. Yeung,et al.  A fuzzy interpolative reasoning method , 2004, Proceedings of 2004 International Conference on Machine Learning and Cybernetics (IEEE Cat. No.04EX826).

[55]  M. Govindarajan,et al.  Performance Optimization of Data Mining Application Using Radial Basis Function Classifier , 2009 .

[56]  Jeng-Shyang Pan,et al.  Fuzzy Rules Interpolation for Sparse Fuzzy Rule-Based Systems Based on Interval Type-2 Gaussian Fuzzy Sets and Genetic Algorithms , 2013, IEEE Transactions on Fuzzy Systems.

[57]  Shyi-Ming Chen,et al.  Multi-variable fuzzy forecasting based on fuzzy clustering and fuzzy rule interpolation techniques , 2010, Inf. Sci..

[58]  Sankar K. Pal,et al.  Roughness of a Fuzzy Set , 1996, Inf. Sci..

[59]  Hiok Chai Quek,et al.  Scale and move transformation-based fuzzy rule interpolation with interval type-2 fuzzy sets , 2013, 2013 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE).

[60]  Hiok Chai Quek,et al.  Backward fuzzy rule interpolation with multiple missing values , 2013, 2013 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE).

[61]  Francisco Herrera,et al.  Advanced nonparametric tests for multiple comparisons in the design of experiments in computational intelligence and data mining: Experimental analysis of power , 2010, Inf. Sci..

[62]  Qiang Shen,et al.  Scale and move transformation-based fuzzy interpolative reasoning: a revisit , 2004, 2004 IEEE International Conference on Fuzzy Systems (IEEE Cat. No.04CH37542).

[63]  Shyi-Ming Chen,et al.  Fuzzy rule interpolation based on the ratio of fuzziness of interval type-2 fuzzy sets , 2011, Expert Syst. Appl..

[64]  Steven C. Chapra,et al.  Numerical Methods for Engineers , 1986 .

[65]  M. Friedman.  The Use of Ranks to Avoid the Assumption of Normality Implicit in the Analysis of Variance , 1937 .