在稀疏模糊規則庫系統中，模糊內插推論是一個很重要的研究課題。在本論文中，我們根據多邊形模糊集合及多邊形模糊集合之排序值在稀疏模糊規則庫系統中提出兩個新方法以作模糊內插推論。在本論文所提之第一個方法中，我們根據多邊形模糊集合之排序值及自動產生模糊規則之權重值提出一個新的模糊內插推論方法。實驗結果顯示我們所提出之方法可以克服目前已存在之模糊內插推論方法之缺點。在本論文所提之第二個方法中，我們根據多邊形模糊集合之排序值及多邊形模糊集合之間的相似度計算提出一個新的自適性模糊內插推論方法。我們所提出之新的自適性模糊內插推論方法使用多個模糊規則及多個前提變數以執行模糊內插推論，並且根據多邊形模糊集合之間的相似度測量以處理模糊內插推論後所產生之矛盾情況。實驗結果顯示我們所提出之自適性模糊內插推論方法優於目前已存在之方法。

Fuzzy interpolative reasoning is a very important research topic for sparse fuzzy rule-based systems. In this thesis, we propose two new fuzzy interpolative reasoning methods for sparse fuzzy rule-based systems based on polygonal fuzzy sets and the ranking values of polygonal fuzzy sets. In the first method of our thesis, we propose a new fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on ranking values of polygonal fuzzy sets and automatically generated weights of fuzzy rules. The experimental results show that the proposed method can overcome the drawbacks of the existing fuzzy interpolative reasoning methods for fuzzy interpolative reasoning in sparse fuzzy rule-based systems. In the second method of our thesis, we propose a new adaptive fuzzy interpolation method based on ranking values of polygonal fuzzy sets and similarity measures between polygonal fuzzy sets. The proposed adaptive fuzzy interpolation method performs fuzzy interpolative reasoning using multiple fuzzy rules with multiple antecedent variables and solves the contradictions after the fuzzy interpolative reasoning processes based on similarity measures between polygonal fuzzy sets. The experimental results show that the proposed adaptive fuzzy interpolation method outperforms the existing methods for fuzzy interpolative reasoning in sparse fuzzy rule-based systems.

[1]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.
[2]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.
[3]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.
[4]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.
[5]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.
[6]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.
[7]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.
[8]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.
[9]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.
[10]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.
[11]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.
[12]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.
[13]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.
[14]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.
[15]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.
[16]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.
[17]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.
[18]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.
[19]Z. H. Huang and Q. Shen, “Preserving piece-wise linearity in fuzzy interpolation,” in Proceedings of 2009 IEEE International Conference on Fuzzy Systems, Korea, 2009, pp. 575-580.
[20]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.
[21]S. Jenei, “Interpolation and extrapolation of fuzzy quantities revisited－An axiomatic approach,” Soft Computing, vol. 5, no. 3, pp. 179-193, 2001.
[22]S. Jenei, E. P. Klement, and R. Konzel “Interpolation and extrapolation of fuzzy quantities－The multiple-dimension case,” Soft Computing, vol. 6, no. 3-4, pp. 258-270, 2002.
[23]S. Jin, R. Diao, and Q. Shen, “α-cut-based backward fuzzy interpolation,” in Proceedings of the 2014 IEEE International Conference on Cognitive Informatics & Cognitive Computing, London, UK, 2014, pp. 211-218.
[24]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.
[25]Z. C. Johanyak, D. Tikk, S. Kovacs, and K. W. Wong, “Fuzzy rule interpolation MATLAB toolbox-FRI toolbox,” in Proceedings of the 2006 IEEE International Conference on Fuzzy Systems, Vancouver, BC, Canada, 2006, pp. 351-357.
[26]J. de Kleer, “An assumption-based TMS,” Artificial Intelligence, vol. 28, no. 2, pp. 127-162, 1986.
[27]J. de Kleer and B. C. Williams, “Diagnosing multiple faults,” Artificial Intelligence, vol. 32, no. 1, pp. 97-130, 1987.
[28]G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall, Upper Saddle River, NJ, U. S. A, 1995.
[29]L. T. Koczy and K. Hirota, “Approximate reasoning by linear rule interpolation and general approximation,” International Journal Approximate Reasoning, vol. 9, no. 3, pp. 197-225, 1993.
[30]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.
[31]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.
[32]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.
[33]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.
[34]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.
[35]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.
[36]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.
[37]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.
[38]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.
[39]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.
[40]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.
[41]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.
[42]K. W. Wong, D. Tikk, T. D. Gedon, 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.
[43]Y. Yam, P. Baranyi, D. Tikk, and L. T. Koczy, “Eliminating the abnormality problem of alpha-cut based fuzzy interpolation,” in Proceedings of the 8th International Fuzzy Systems Association World Congress, Taipei, Taiwan, 1999, vol. 2, pp. 762-766.
[44]Y. Yam and L. T. Koczy, “Representing membership functions as points in high-dimensional spaces for fuzzy interpolation and extrapolation,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 6, pp. 761-772, 2000.
[45]Y. Yam, M. L. Wong, and P. Baranyi, “Interpolation with function space representation of membership functions,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 3, pp. 398-411, 2006.
[46]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.
[47]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.
[48]L. Yang and Q. Shen, “Adaptive fuzzy interpolation,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 6, pp. 1107-1126, 2011.
[49]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.
[50]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, 2011, pp. 471-478.
[51]L. Yang and Q. Shen, “Closed form fuzzy interpolation,” Fuzzy Sets and Systems, vol. 225, pp. 1-22, 2013.
[52]L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338-353, 1965.