Fuzzy interpolative reasoning is a very important research topic for sparse fuzzy rule-based systems. In this thesis, we propose two new adaptive fuzzy interpolative reasoning methods for sparse fuzzy rule-based systems based on polygonal fuzzy sets and interval type-2 fuzzy sets, respectively. In the first method of our thesis, we propose a new adaptive fuzzy interpolative reasoning method based on contradiction measures between polygonal fuzzy sets and novel move and transformation techniques. The proposed adaptive fuzzy interpolative reasoning method performs fuzzy interpolative reasoning using the multiple fuzzy rules with multiple antecedent variables fuzzy interpolative reasoning scheme and solves the contradictions after the fuzzy interpolative reasoning processes based on contradiction measures between polygonal fuzzy sets. The experimental results show that the proposed adaptive fuzzy interpolative reasoning method outperforms the existing methods for fuzzy interpolative reasoning in sparse fuzzy rule-based systems. In the second method of our thesis, we propose a new adaptive weighted fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on representative values and similarity measures of interval type-2 fuzzy sets. The experimental results show that the proposed adaptive weighted fuzzy interpolative reasoning method can overcome the drawbacks of the existing adaptive fuzzy interpolative reasoning methods for sparse fuzzy rule-based systems.

Fuzzy interpolative reasoning is a very important research topic for sparse fuzzy rule-based systems. In this thesis, we propose two new adaptive fuzzy interpolative reasoning methods for sparse fuzzy rule-based systems based on polygonal fuzzy sets and interval type-2 fuzzy sets, respectively. In the first method of our thesis, we propose a new adaptive fuzzy interpolative reasoning method based on contradiction measures between polygonal fuzzy sets and novel move and transformation techniques. The proposed adaptive fuzzy interpolative reasoning method performs fuzzy interpolative reasoning using the multiple fuzzy rules with multiple antecedent variables fuzzy interpolative reasoning scheme and solves the contradictions after the fuzzy interpolative reasoning processes based on contradiction measures between polygonal fuzzy sets. The experimental results show that the proposed adaptive fuzzy interpolative reasoning method outperforms the existing methods for fuzzy interpolative reasoning in sparse fuzzy rule-based systems. In the second method of our thesis, we propose a new adaptive weighted fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on representative values and similarity measures of interval type-2 fuzzy sets. The experimental results show that the proposed adaptive weighted fuzzy interpolative reasoning method can overcome the drawbacks of the existing adaptive fuzzy interpolative reasoning methods for sparse fuzzy rule-based systems.

[1] P. Baranyi, T. D. Gedeon, and L. T. Koczy, “A general interpolation technique in fuzzy rule bases with arbitrary membership functions,” in Proceedings of the 1996 IEEE International Conference on Systems, Man and Cybernetics, Beijing, China, 1996, vol. 1, pp. 510-515.
[2] P. Baranyi, L. T. Koczy, and T. D. Gedeon, “A generalized concept for fuzzy rule interpolation,” IEEE Transactions on Fuzzy Systems, vol. 12, no. 6, pp. 820-832, 2004.
[3] B. Bouchon-Meunier, C. Marsala, and M. Rifqi, “Interpolative reasoning based on graduality,” in Proceedings of the 2000 Ninth IEEE International Conference on Fuzzy Systems, San Antonio, Texas, 2000, vol. 1, pp. 483-487
[4] B. Bouchon-Meunier and L. Valverde, “A fuzzy approach to analogical reasoning,” Soft Computing, vol. 3, no. 3, pp. 141-147, 1999.
[5] Y. C. Chang, S. M. Chen, and C. J. Liau, “Fuzzy interpolative reasoning for sparse fuzzy-rule-based systems based on the areas of fuzzy sets,” IEEE Transactions on Fuzzy Systems, vol. 16, no. 5, 1285-1301, 2008.
[6] S. M. Chen and Y. C. Chang, “Weighted fuzzy interpolative reasoning for sparse fuzzy rule-based systems,” Expert Systems with Applications, vol. 38, no. 8, pp. 9564-9572, 2011.
[7] S. M. Chen and Y. C. Chang, “Fuzzy rule interpolation based on principle membership functions and uncertainty grade functions of interval type-2 fuzzy sets,” Expert Systems with Applications, vol. 38, no. 9, pp. 11573-11580, 2011.
[8] S. M. Chen and Y. C. Chang, “Fuzzy rule interpolation based on the ratio of fuzziness of interval type-2 fuzzy sets,” Expert Systems with Applications, vol. 38, no. 10, pp. 12202-12213, 2011.
[9] S. M. Chen and Y. C. Chang, “Weighted fuzzy rule interpolation based on GA-based weight-learning techniques,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 4, pp. 729-744, 2011.
[10] S. M. Chen, S. I. Adam, “Weighted fuzzy interpolated reasoning based on ranking values of polygonal fuzzy sets and new scale and move transformation techniques,” Information Sciences, vol. 435, pp. 184-202, 2018.
[11] S. M. Chen, Y. C. Chang, Z. J. Chen, and C. L. Chen, “Multiple fuzzy rules interpolation with weighted antecedent variables in sparse fuzzy rule-based systems,” International Journal of Pattern Recognition and Artificial Intelligence, vol. 27, no. 5, pp. 1359002-1 - 1359002-15, 2013.
[12] S. M. Chen, Y. C. Chang, and J. S. Pan, “Fuzzy rules interpolation for sparse fuzzy rule-based systems based on interval type-2 Gaussian fuzzy sets and genetic algorithms,” IEEE Transactions on Fuzzy Systems, vol. 21, no. 3, pp. 412-425, 2013.
[13] S. M. Chen, W. C. Hsin, S. W. Yang, and Y. C. Chang, “Fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on the slopes of fuzzy sets,” Expert Systems with Applications, vol. 39, no. 15, pp. 11961-11969, 2012.
[14] S. M. Chen and Y. K. Ko, “Fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on α-cuts and transformations techniques,” IEEE Transactions on Fuzzy Systems, vol. 16, no. 6, pp. 1626-1648, 2008.
[15] S. M. Chen, Y. K. Ko, Y. C. Chang, and J. S. Pang, “Weighted fuzzy interpolative reasoning based on weighted increment transformation and weighted ratio transformation techniques,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 6, pp. 1412-1427, 2009.
[16] S. M. Chen, A. Munif, G. S. Chen, H. C. Liu, and B. C. Kuo, “Fuzzy risk analysis based on ranking generalized fuzzy numbers with different left heights and right heights,” Expert Systems with Applications, vol. 39, no.7, pp. 6320-6334, 2012.
[17] S. M. Chen, “Aggregating fuzzy opinions under the group decision-making environment”, Cybernetics and Systems, vol. 29, no. 4, pp. 363-376, 1998.
[18] S. M. Chen and L. W. Lee, “Fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on interval type-2 fuzzy sets,” Expert Systems with Applications, vol. 38, no. 8, pp. 9947-9957, 2011.
[19] S. M. Chen, L. W. Lee, and V. R. L. Shen, “Weighted fuzzy interpolative reasoning systems based on interval type-2 fuzzy sets,” Information Sciences, vol. 248, pp. 15-30, 2013.
[20] C. Chen, C. Quek, and Q. Shen, “Scale and move transformation-based fuzzy rule interpolation with interval type-2 fuzzy sets,” in Proceedings of the 2013 IEEE International Conference on Fuzzy Systems, Hyderabad, India, 2013.
[21] H.Y. Wang, and S.M. Chen, “Evaluating students’ answerscripts using fuzzy numbers associated with degrees of confidence,” IEEE Transactions on Fuzzy Systems, vol. 16, no. 2, pp. 403-415, 2008.
[22] C. Chen, N. M. Parthaláin, Y. Li, C. Price, C. Quek, and Q. Shen, “Rough-fuzzy rule interpolation,” Information Sciences, vol. 351, pp. 1-17, 2016
[23] S. M. Chen, S. H. Cheng, and Z. J. Chen, “Fuzzy interpolative reasoning based on the ratio of fuzziness of rough-fuzzy sets,” Information Sciences, vol. 299, pp. 394-411, 2015.
[24] S. H. Cheng, S. M. Chen, and C. L. Chen, “Fuzzy interpolative reasoning based on ranking values of polygonal fuzzy sets and automatically generated weights of fuzzy rules,” Information Sciences, vol. 325, pp. 521-540, 2015.
[25] S. M. Chen and S. I. Adam, “Adaptive fuzzy interpolation based on ranking values of interval type-2 polygonal fuzzy sets,” Information Sciences, vol 435, pp. 320-333, 2018.
[26] S. H. Cheng, S. M. Chen, and C. L. Chen, “Adaptive fuzzy interpolation based on ranking values of polygonal fuzzy sets and similarity measures between polygonal fuzzy sets,” Information Sciences, vol. 342, pp. 176-190, 2016.
[27] S. M. Chen and S. I. Adam, “Adaptive fuzzy interpolation based on general representative values of polygonal fuzzy sets and the shift and modification techniques,” Information Sciences, vol. 414, pp. 147-157, 2017.
[28] S. M. Chen and D. Barman, “Adaptive weighted fuzzy interpolative reasoning based on representative values and similarity measures of interval type-2 fuzzy sets,” Information Sciences, vol. 478, pp. 167-185, 2019.
[29] S. M. Chen and Z. J. Chen, “Weighted fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on piecewise fuzzy entropies of fuzzy sets,” Information Sciences, vol. 329, pp. 503-523, 2016.
[30] D. Ciucci, “Orthopairs and granular computing,” Granular Computing, vol. 1, no. 3, pp. 159-170, 2016.
[31] R. Diao, S. Jin, and Q. Shen, “Antecedent selection in fuzzy rule interpolation using feature selection techniques,” in Proceedings of the 2014 IEEE International Conference on Fuzzy Systems, Beijing, China, 2014, pp. 2206-2213.
[32] D. Dubois and H. Prade, “On fuzzy interpolation,” International Journal of General Systems, vol. 28, no. 2, pp. 103-114, 1999.
[33] D. Dubois and H. Prade, “Bridging gaps between several forms of granular computing,” Granular Computing, vol. 1, no. 2, pp. 115-126, 2016.
[34] J. N. S. Eisenberg, W. Cevallos, K. Ponce, K. Levy, S. J. Bates, J. C. Scott, A. Hubbard, N. Viera, P. Endara, M. Espinel, G. Trueba, L. W. Riley, and J. Trostle. “Environmental change and infectious disease: how new roads affect the transmission of diarrheal pathogens in rural Ecuador,” in Proceedings of the National Academy of Science of United States of America, vol. 103, no. 51, pp. 19465, 2006.
[35] W. H. Hsiao, S. M. Chen, and C. H. Lee, “A new interpolative reasoning method in sparse rule-based systems,” Fuzzy Sets and Systems, vol. 93, no. 1, pp. 17-22, 1998.
[36] Z. H. Huang and Q. Shen, “Fuzzy interpolation reasoning via scale and move transformations,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 2, pp. 340-359, 2006.
[37] Z. H. Huang and Q. Shen, “Fuzzy interpolation and extrapolation: A practical approach,” IEEE Transactions on Fuzzy Systems, vol. 16, no. 1, pp. 13-28, 2008.
[38] Z. H. Huang and Q. Shen, “Preserving piece-wise linearity in fuzzy interpolation,” in Proceedings of the 2009 17th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 2009, pp. 575-580.
[39] D. M. Huang, E. C. C. Tsang, and D. S. Yeung, “A fuzzy interpolative reasoning method,” in Proceedings of 2004 International Conference on Machine Learning and Cybernetics, Shanghai, China, 2004, vol. 3, pp. 1826-1830.
[40] S. Jenei, “Interpolation and extrapolation of fuzzy quantities revisited－An axiomatic approach,” Soft Computing, vol. 5, no. 3, pp. 179-193, 2001.
[41] S. Jenei, E. P. Klement, and R. Konzel “Interpolation and extrapolation of fuzzy quantities－The multiple-dimension a case,” Soft Computing, vol. 6, no. 3, pp. 258-270, 2002.
[42] S. Jin, R. Diao, and Q. Shen, “α-cuts-based backward fuzzy interpolation,” in Proceedings of the 2014 IEEE International Conference on Cognitive Informatics & Cognitive Computing, London, UK, 2014, pp. 211-218.
[43] S. Jin, R. Diao, C. Quek, and Q. Shen, “Backward fuzzy rule interpolation with multiple missing values,” in Proceedings of the 2013 IEEE International Conference on Fuzzy Systems, Hyderabad, India, 2013.
[44] S. Jin, R. Diao, C. Quek, and Q. Shen, “Backward fuzzy rule interpolation,” IEEE Transactions on Fuzzy Systems, vol. 22, no. 6, pp. 1682-1698, 2014.
[45] Z. C. Johanyak and K. Szilveszter, “Fuzzy rule interpolation based on polar cuts,” in Computational Intelligence, Theory and Applications (Edited by B. Reusch), Springer, Germany, pp. 499-511, 2006.
[46] J. de Kleer and B. C. Williams, “Diagnosing multiple faults,” Artificial Intelligence, vol. 32, no. 1, pp. 97-130, 1987.
[47] J. de Kleer, “An assumption-based TMS,” Artificial Intelligence, vol. 28, no. 2, pp. 127-162, 1987.
[48] L. T. Kóczy and K. Hirota, “Approximate reasoning by linear rule interpolation and general approximation,” International Journal Approximate Reasoning, vol. 9, no. 3, pp. 197-225, 1993.
[49] L. T. Kóczy and K. Hirota, “Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases,” Information Sciences, vol. 71, no. 1-2, pp. 169-201, 1993.
[50] L. T. Kóczy and K. Hirota, “Ordering, distance and closeness of fuzzy sets,” Fuzzy Sets and Systems, vol. 59, no. 3, pp. 281-293, 1993.
[51] L. T. Koczy and K. Hirota, “Size reduction by interpolation in fuzzy rules bases,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 27, no. 1, pp. 14-25, 1997.
[52] S. Kovacs, “Adapting the scale and move FRI for the fuzziness interpolation of the double fuzzy point rule representation,” in Proceedings of the 2013 IEEE International Conference on Fuzzy Systems, Hyderabad, India, 2013.
[53] S. Kovacs, “Extending the fuzzy rule interpolation “FIVE” by fuzzy observation,” in Computational Intelligence, Theory and Applications (Edited by B. Reusch), Springer, Germany, pp. 485-497, 2006.
[54] L. W. Lee and S. M. Chen, “Fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on the ranking values of fuzzy sets,” Expert Systems with Applications, vol. 35, no.3, pp. 850-864, 2008.
[55] L. W. Lee and S. M. Chen, “Weighted fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on the ranking values of fuzzy sets,” Expert Systems with Applications, vol. 35, no. 3, pp. 850-864, 2008.
[56] L. Livi and A. Sadeghian, “Granular computing, computational intelligence, and the analysis of non-geometric input spaces,” Granular Computing vol. 1, no. 1, pp. 13-20, 2016.
[57] Q. Liang and J. M. Mendel, “Interval type-2 fuzzy logic systems: Theory and design,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 5, pp. 535-550, 2000.
[58] J. M. Mendel and R. I. John, “Type-2 fuzzy sets made simple,” IEEE Transactions on Fuzzy Systems, vol. 10, no. 2, pp. 117-127, 2002.
[59] J. M. Mendel, R. I. John, and F. L. Liu “Interval type-2 fuzzy logic systems made simple,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 6, pp. 808-821, 2006.
[60] N. Naik, R. Diao, C. Quek, and Q. Shen, “Towards dynamic fuzzy rule interpolation,” in Proceedings of the 2013 IEEE International Conference on Fuzzy Systems, Hyderabad, India, 2013.
[61] N. Naik, R. Diao, and Q. Shen, “Genetic algorithm-aided dynamic fuzzy rule interpolation,” in Proceedings of the 2014 IEEE International Conference on Fuzzy Systems, Beijing, China, 2014, pp. 2198-2205.
[62] C. P. Pappis and N. I. Karacapilidis, “A comparative assessment of measures of similarity of fuzzy values,” Fuzzy Sets and Systems, vol. 56, no. 2, pp. 171-174, 1993.
[63] G. Peters and R. Weber, “DCC: A framework for dynamic granular clustering,” Granular Computing, vol. 1, no. 1, pp. 1-11, 2016.
[64] W. Z. Qiao, M. Mizumoto, and S. Y. Yang, “An improvement to Koczy and Hirota’s interpolative reasoning in sparse fuzzy rule bases,” International Journal of Approximate Reasoning, vol. 15, no. 3, pp. 185-201, 1996.
[65] Q. Shen and L. Yang, “Generalisation of scale and move transformation-based fuzzy interpolation,” Journal of Advanced Computational Intelligence and Intelligent Informatics, vol. 15, no. 3, pp. 288-298, 2011.
[66] A. Skowron, A. Jankowski, and S. Dutta, “Interactive granular computing,” Granular Computing, vol. 1, no. 2, pp. 95-113, 2016.
[67] D. Tikk and P. Baranyi, “Comprehensive analysis of a new fuzzy rule interpolation method,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 3, pp. 281-296, 2000.
[68] D. Tikk, I. Joo, L. Koczy, P. Varlaki, B. Moser, and T. D. Gedeon, “Stability of interpolative fuzzy KH controllers,” Fuzzy Sets and Systems, vol. 125, no. 1, pp.105-119, 2002.
[69] L. Ughetto, D. Dubois, and H. Prade, “Fuzzy interpolation by convex completion of sparse rule bases,” in Proceedings of the Ninth IEEE International Conference on Fuzzy Systems, San Antonio, Texas, 2000, vol. 1, pp. 465-470.
[70] K. W. Wong, D. Tikk, T. D. Gedeon, and L. T. Koczy, “Fuzzy rule interpolation for multidimensional input spaces with applications: A case study,” IEEE Transactions on Fuzzy Systems, vol. 13, no. 6, pp. 809-819, 2005.
[71] D. Wu, J. M. Mendel, “A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets”, Information Sciences, vol. 179, pp. 1169-1192, 2009.
[72] S. Yan, M. Mizumoto, and W. Z. Qiao, “Reasoning conditions on Koczy’s interpolative reasoning method in sparse fuzzy rule bases,” Fuzzy Sets and Systems, vol. 75, no. 1, pp. 63-71, 1995.
[73] L. Yang, C. Chen, N. Jin, X. Fu, and Q. Shen, “Closed form fuzzy interpolation with interval type-2 fuzzy sets,” in Proceedings of the 2014 IEEE International Conference on Fuzzy Systems, Beijing, China, 2014, pp. 2184-2191.
[74] L. Yang and Q. Shen, “Adaptive fuzzy interpolation,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 6, pp. 1107-1126, 2011.
[75] L. Yang and Q. Shen, “Adaptive fuzzy interpolation with prioritized component candidates,” in Proceedings of 2011 IEEE International Conference on Fuzzy Systems, Taipei, Taiwan, 2011, pp. 428-435.
[76] L. Yang and Q. Shen, “Adaptive fuzzy interpolation with uncertain observations and rule base,” in Proceedings of 2011 IEEE International Conference on Fuzzy Systems, Taipei, Taiwan, pp. 471-478.
[77] L. Yang and Q. Shen, “Adaptive fuzzy interpolation and extrapolation with multiple-antecedent rules,” in Proceedings of the 2010 IEEE International Conference on Fuzzy Systems, Barcelona, Spain, 2010.
[78] L. Yang and Q. Shen, “Towards Adaptive Interpolative Reasoning,” in Proceedings of the 2009 IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 2009, pp. 542-549.
[79] L. Yang, F. Chao and Q. Shen, “Generalized adaptive fuzzy rule interpolation,” IEEE Transactions on Fuzzy Systems , vol. 25, no. 4, pp. 839-853, 2017.
[80] L. Yang and Q. Shen, “Closed form fuzzy interpolation,” Fuzzy Sets and Systems, vol. 225, pp. 1-22, 2013.
[81] Y. Yao, “A triarchic theory of granular computing,” Granular Computing, vol. 1, no. 2, pp. 145-157, 2016.
[82] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338-353, 1965.
[83] L.A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning-I”, Information Sciences, vol. 8, pp. 199-249, 1975.