在稀疏模糊規則庫系統中模糊規則通常不完整，在此情形下，系統可能無法完整執行模糊推理以獲得合理之推論結果。為了克服此缺點，我們需要在稀疏模糊規則庫系統中提出模糊內插推理之方法。在本論文中，我們提出兩個新的模糊內插推理方法，其中第一個方法利用模糊集合之斜率比之技術以簡單的計算步驟即可處理複雜的多邊形之模糊集合，其產生之推理結果比目前已存在的方法具有更合理之推理結果。本論文所提之第二個模糊內插推理方法根據粒子群最佳化演算法提出模糊規則之前提變數之權重學習方法，並應用於處理電腦預測問題、多元回歸問題、及時間序列預測問題上。我們並以統計分析的方法將我們所提之新方法與目前已存在之方法作比較。實驗結果顯示本論文中所提之以粒子群最佳化演算法為基礎之加權式模糊內插推理新方法使用學習出之最佳權重比目前已存在之方法具有更小之誤差率。

In sparse fuzzy rule-based systems, the fuzzy rule bases usually are incomplete. In this situation, the systems may not properly perform fuzzy reasoning to get reasonable consequences. In order to overcome the drawback of sparse fuzzy rule-based systems, there is an increasing demand to develop fuzzy interpolative reasoning techniques in sparse fuzzy rule-based systems. We apply the proposed method to deal with the computer activity prediction problem, multivariate regression problems, and time series prediction problem. Based on statistical analysis techniques, the experimental results show that the proposed weighted fuzzy interpolative reasoning method by the use of the optimally learned weights that were obtained by the proposed PSO-based weight-learning algorithm has statistically significantly smaller error rates than the existing methods.

[1] P. Baranyi, L. T. Koczy, and T. D. Gedeon, “A generalized concept for fuzzy rule interpolation,” IEEE Trans. Fuzzy Syst., vol. 12, no. 6, pp. 820-837, 2004.
[2] P. Baranyi, D. Tikk, Y. Yam, and L. T. Kozcy, “A new method for avoiding abnormal conclusion for α-cut based rule interpolation,” in Proc. 1999 IEEE Int. Conf. Fuzzy Syst., pp. 383-388, 1999.
[3] B. Bouchon-Meunier, C. Marsala, and M. Rifqi, “Interpolative reasoning based on graduality,” in Proc. 2000 IEEE Int. Conf. Fuzzy Syst., pp. 483-487, 2000.
[4] G. E. P. Box, G. M. Jenkins, and G. C. Reinsel, Time Series Analysis: Forecasting and Control, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1994.
[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 Trans. Fuzzy Syst. vol. 16, no. 5, pp.1285-1301, 2008.
[6] S. M. Chen and Y. C. Chang, “Weighted fuzzy rule interpolation based on GA-based weight-learning techniques,” IEEE Trans. Fuzzy Syst. vol. 19, no. 4, pp. 729-744, 2011.
[7] 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.
[8] 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.
[9] 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.
[10] S. M. Chen and Y. K. Ko, “Fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on α-cuts and transformation techniques,” IEEE Trans. Fuzzy Syst., vol. 16, no. 6, pp. 1626-1648, 2008.
[11] S. M. Chen, Y. K. Ko, Y. C. Chang, and J. S. Pan, “Weighted fuzzy interpolative reasoning based on weighted increment transformation and weighted ratio transformation techniques,” IEEE Trans. Fuzzy Syst., vol. 17, no. 6, pp. 1412-1427, 2009.
[12] 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,” in Expert Systems with Applications, vol. 39, no. 15, pp. 11961-11969, 2012.
[13] S. M. Chen, W. C. Hsin, S. W. Yang, and Y. C. Chang, “A new method for weighted fuzzy interpolative reasoning based on PSO-based weights-learning techniques,” In Proc. 2012 International Conference on Machine Learning and Cybernetics, Xian, Shaanxi, China, 2012.
[14] 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.
[15] R. S. Cowder, “Predicting the Mackey-Glass time series with cascade-correlation learning,” in Connectionist Models: Proc. 1990 Summer School (edited by D. S. Touretzky, J. L. Elman, T. J. Sejnowski, and G. E. Hinton), California: Morgan Kaufmann Publishers, pp. 117-123, 1990.
[16] J. Demsar, “Statistical comparisons of classifiers over multiple data sets,” J. Mach. Learn. Res., vol. 7, no. 1, pp. 1-30, 2006.
[17] R. C. Eberhart and Y. Shi, “Comparing Inertia Weights and Constriction Factors in Particle Swarm Optimization,” in Proc. 2000 IEEE Int. Conf. Evolutionary Computation, pp. 84-88, 2000.
[18] A. Frank and A. Asuncion, UCI Machine Learning Repository, [Online]. Available: http://archive.ics.uci.edu/ml.
[19] S. Garcia and F. Herrera, “An extension on “Statistical comparisons of classifiers overmultiple data sets” for all pairwise comparisons,” J. Mach. Learn. Res., vol. 9, no. 12, pp. 2677-2694, 2008.
[20] M. A. Hall, Correlation-Based Feature Subset Selection for Machine Learning, Ph.D. Dissertation, Dept. Comput. Sci., Univ. Waikato, Hamilton, New Zealand, 1999.
[21] S. Holm, “A simple sequentially rejective multiple test procedure,” Scand. J. Stat., vol. 6, pp. 65-70, 1979.
[22] W. H. Hsiao, S. M. Chen, and C. H. Lee, “A new interpolative reasoning method in sparse rule-based systems,” Fuzzy Sets Systems, vol. 93, no. 1, pp. 17-22, 1998.
[23] Z. Huang and Q. Shen, “A new fuzzy interpolative reasoning method based on center of gravity,” in Proc. 2003 IEEE Int. Conf. Fuzzy Syst., vol. 1, pp. 25-30, 2003.
[24] Z. Huang and Q. Shen, “Fuzzy interpolative reasoning via scale and move transformations,” IEEE Trans. Fuzzy Syst., vol. 14, no. 2, pp. 340-359, 2006.
[25] Z. Huang and Q. Shen, “Fuzzy interpolation and extrapolation: A practical approach,” IEEE Trans. Fuzzy Syst., vol. 16, no. 1, pp. 13-28, 2008.
[26] R. L. Iman and J. M. Davenport, “Approximations of the critical region of the Friedman statistic,” Commun. Stat., vol. 9, no. 6, pp. 571-595, 1980.
[27] J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proc. I995 ICEC, Perth, Australia, 1995.
[28] R. J. Hyndman. (2010, Sep.), Time Series Data Library, [Online]. Available: http://robjhyndman.com/TSDL.
[29] L. T. Koczy and K. Hirota, “Size reduction by interpolation in fuzzy rule bases,” IEEE Trans. Syst., Man, Cybern., vol. 27, no. 1, pp. 14-25, 1997.
[30] L. T. Koczy and K. Hirota, “Approximate reasoning by linear rule interpolation and general approximation,” Int. J. Approx. Reason., vol. 9, no. 3, pp. 197-225, 1993.
[31] L. T. Koczy and K. Hirota, “Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases,” Inf. Sci., vol. 71, no. 1-2, pp. 169-201, 1993.
[32] L. T. Koczy and S. Kovacs, “Shape of the fuzzy conclusion generated by linear interpolation in trapezoidal fuzzy rule bases,” in Proceedings of the 2nd European Congress on Intelligent Techniques and Soft Computing, Aachen, pp. 1666-1670, 1994.
[33] S. G. Kong and B. Kosko, “Adaptive fuzzy systems for backing up a truck-and-trailer,” IEEE Trans. Neural Netw., vol. 3, no. 5, pp. 211-223, 1992.
[34] 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.
[35] Y. M. Li, D. M. Huang, E. C. C. Tsang, and L. N. Zhang, “Weighted fuzzy interpolative reasoning method,” in Proc. 4th Int. Conf. Mach. Learning Cybern., Guangzhou, China, pp. 18-21, 2005.
[36] A. Lotfi. (2000). Fuzzy inference system toolbox for MATLAB (FISMAT). [Online]. Available: http://www.lotfi.net/research/fismat/index.html.
[37] M. Mizumoto and H.-J. Zimmermann, “Comparison of fuzzy reasoning methods,” Fuzzy Sets Syst., vol. 15, no. 3, pp. 253-283, 1982.
[38] W. Z. Qiao, M. Mizumoto, and S. Y. Yan, “An improvement to Koczy and Hirota’s interpolative reasoning in sparse fuzzy rule bases,” Int. J. Approx. Reason., vol. 15, no. 3, pp. 185-201, 1996.
[39] C. E. Rasmussen, R. M. Neal, G. Hinton, D. Camp, M. Revow, Z. Ghahramani, R. Kustra, and R. Tibshirani, “Data for evaluating learning in valid experiments (delve),” [Online]. Available: http://www.cs.toronto.edu/~delve/.
[40] A. Riid and E. Rustern, “Fuzzy logic control: Truck backer-upper problem revised,” in Proc. IEEE Int. Conf. Fuzzy Systems, vol. 1, pp. 513-516, 2001.
[41] D. Tikk and P. Baranyi, “Comprehensive analysis of a new fuzzy rule interpolation method,” IEEE Trans. Fuzzy Syst., vol. 8, no. 3, pp. 281-296, 2000.
[42] B. Wang, Q. Zhang, W. Liu, and Y. Shi, “Multidimensional fuzzy interpolative reasoning method based on λ-width similarity,” in Proc. 2007 IEEE 8th ACIS Int. Conf. Softw. Eng., Artif. Intell., Netw., Parallel/Distrib. Comput., Qingdao, China, vol. 1, pp. 776-781, 2007.
[43] L. X. Wang and J. M. Mendel, “Generating fuzzy rules by learning from examples,” IEEE Trans. Syst., Man, Cybern., vol. 22, no. 6, pp. 1414-1426, 1992.
[44] I. H. Witten and E. Frank, Data Mining: Practical Machine Learning Tools and Techniques with Java Implementations. San Mateo, CA: Morgan Kaufmann, 1999.
[45] 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 Trans. Fuzzy Syst., vol. 13, no. 6, pp. 809-819, 2005.
[46] Y. Yam, P. Baranyi, D. Tikk, and L. T. Koczy, “Eliminating the abnormality problem of α-cut based interpolation,” in Proc. 8th IFSAWorld Congress, 1999, vol. 2, pp. 726-766.
[47] Y. Yam and L. T. Koczy, “Representing membership functions as points in high dimensional spaces for fuzzy interpolation and extrapolation,” IEEE Trans. Fuzzy Syst., vol. 8, no. 6, pp. 761-772, 2000.
[48] Y. Yam, M. L. Wong, and P. Baranyi, “Interpolation with function space representation of membership functions,” IEEE Trans. Fuzzy Syst., vol. 14, no. 3, pp. 398-411, 2006.
[49] L. A. Zadeh, “Fuzzy sets,” Inform. Control, vol. 8, pp. 338-353, 1965.