Previous |  Up |  Next

Article

Title: On continuity of the entropy-based differently implicational algorithm (English)
Author: Tang, Yiming
Author: Pedrycz, Witold
Language: English
Journal: Kybernetika
ISSN: 0023-5954 (print)
ISSN: 1805-949X (online)
Volume: 55
Issue: 2
Year: 2019
Pages: 307-336
Summary lang: English
.
Category: math
.
Summary: Aiming at the previously-proposed entropy-based differently implicational algorithm of fuzzy inference, this study analyzes its continuity. To begin with, for the FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens) problems, the continuous as well as uniformly continuous properties of the entropy-based differently implicational algorithm are demonstrated for the Tchebyshev and Hamming metrics, in which the R-implications derived from left-continuous t-norms are employed. Furthermore, four numerical fuzzy inference examples are provided, and it is found that the entropy-based differently implicational algorithm can obtain more reasonable solution in contrast with the fuzzy entropy full implication algorithm. Finally, in the entropy-based differently implicational algorithm, we point out that the first fuzzy implication reflects the effect of rule base, and that the second fuzzy implication embodies the inference mechanism. (English)
Keyword: fuzzy inference
Keyword: fuzzy entropy
Keyword: compositional rule of inference
Keyword: continuity
MSC: 03B52
MSC: 94D05
idZBL: Zbl 07144940
idMR: MR4014589
DOI: 10.14736/kyb-2019-2-0307
.
Date available: 2019-09-30T15:04:09Z
Last updated: 2020-04-02
Stable URL: http://hdl.handle.net/10338.dmlcz/147839
.
Reference: [1] Baczyński, M., Jayaram, B.: (S,N)- and R-implications: A state-of-the-art survey..Fuzzy Set Syst. 159 (2008), 1836-1859. MR 2428086, 10.1016/j.fss.2007.11.015
Reference: [2] Baczyński, M., Jayaram, B.: Fuzzy implications (Studies in Fuzziness and Soft Computing, Vol. 231..Springer, Berlin Heidelberg 2008. MR 2428086
Reference: [3] Chaudhuria, B. B., Rosenfeldb, A.: A modified Hausdorff distance between fuzzy sets..Inform. Sci. 118 (1999), 159-171. MR 1723219, 10.1016/s0020-0255(99)00037-7
Reference: [4] Dai, S. S., Pei, D. W., Wang, S. M.: Perturbation of fuzzy sets and fuzzy reasoning based on normalized Minkowski distances..Fuzzy Set Syst. 189 (2012), 63-73. MR 2871353, 10.1016/j.fss.2011.07.012
Reference: [5] Dai, S. S., Pei, D. W., Guo, D. H.: Robustness analysis of full implication inference method..Int. J. Approx. Reason. 54 (2013), 653-666. MR 3041100, 10.1016/j.ijar.2012.11.007
Reference: [6] Liu, F., Zhang, W. G., Fu, J. H.: A new method of obtaining the priority weights from an interval fuzzy preference relation..Inform. Sci. 185 (2012), 32-42. 10.1016/j.ins.2011.09.019
Reference: [7] Liu, F., Zhang, W. G., Wang, Z. X.: A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making.Eur. J. Oper. Res. 218 (2012), 747-754. MR 2881747, 10.1016/j.ejor.2011.11.042
Reference: [8] Fodor, J. C.: Contrapositive symmetry of fuzzy implications..Fuzzy Set Syst. 69 (1995), 141-156. MR 1317882, 10.1016/0165-0114(94)00210-x
Reference: [9] Fodor, J., Roubens, M.: Fuzzy Preference Modeling and Multicriteria Decision Support..Kluwer Academic Publishers, Dordrecht, 1994. 10.1007/978-94-017-1648-2
Reference: [10] Gottwald, S.: A Treatise on Many-Valued Logics..Research Studies, Studies in Logic and Computation 9, Baldock 2001. Zbl 1048.03002, MR 1856623
Reference: [11] Guo, F. F., Chen, T. Y., Xia, Z. Q.: Triple I methods for fuzzy reasoning based on maximum fuzzy entropy principle..Fuzzy Syst. Math. 17 (2003), 55-59. MR 2026787
Reference: [12] Hong, D. H., Hwang, S. Y.: A note on the value similarity of fuzzy systems variable..Fuzzy Set Syst. 66 (1994), 383-386. MR 1300296, 10.1016/0165-0114(94)90107-4
Reference: [13] Jayaram, B.: On the law of importation $(x\wedge y) \rightarrow z \equiv (x\rightarrow (y\rightarrow z))$ in fuzzy logic..IEEE Trans. Fuzzy Syst. 16 (2008), 130-144. 10.1109/tfuzz.2007.895969
Reference: [14] Jaynes, E. T.: Where do we stand on maximum entropy?.In: The Maximum Entropy Formalism (R. .D. Levine and M. Tribus, eds.), MIT Press, Cambridge 1978, pp. 15-118. MR 0521743
Reference: [15] Jaynes, E. T.: On the rationale of maximum-entropy methods..Proc. IEEE, 70 (1982), 939-952. 10.1109/proc.1982.12425
Reference: [16] Jenei, S.: Continuity on Zadeh's compositional rule of inference..Fuzzy Set Syst. 104 (1999), 333-339. MR 1688064, 10.1016/s0165-0114(97)00198-x
Reference: [17] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms..Kluwer Academic Publishers, Dordrecht 2000. Zbl 1087.20041, MR 1790096, 10.1007/978-94-015-9540-7
Reference: [18] Li, H. X.: Probability representations of fuzzy systems..Sci. China Ser. F-Inf. Sci. 49 (2006), 339-363. MR 2250341, 10.1007/s11432-006-0339-9
Reference: [19] Li, H., Lee, E. S.: Interpolation representations of fuzzy logic systems..Comput. Math. Appl. 45 (2003), 1683-1693. MR 1993238, 10.1016/s0898-1221(03)00147-0
Reference: [20] Luo, M. X., Liu, B.: Robustness of interval-valued fuzzy inference triple I algorithms based on normalized Minkowski distance..J. Log. Algebr. Methods 86 (2017), 298-307. MR 3575372, 10.1016/j.jlamp.2016.09.006
Reference: [21] Mas, M., Monserrat, M., Torrens, J., Trillas, E.: A survey on fuzzy implication functions..IEEE Trans. Fuzzy Syst. 15 (2007), 1107-1121. 10.1109/tfuzz.2007.896304
Reference: [22] Novák, V., Perfilieva, I., Močkoř, J.: Mathematical Principles of Fuzzy Logic..Kluwer Academic Publishes, Boston, Dordrecht 1999. Zbl 0940.03028, MR 1733839, 10.1007/978-1-4615-5217-8
Reference: [23] Pang, L. M., Tay, K. M., Lim, C. P.: Monotone fuzzy rule relabeling for the zero-order TSK fuzzy inference system..IEEE Trans. Fuzzy Syst. 24 (2016), 1455-1463. 10.1109/tfuzz.2016.2540059
Reference: [24] Pedrycz, W.: Granular Computing: Analysis and Design of Intelligent Systems..CRC Press/Francis and Taylor, Boca Raton 2013.
Reference: [25] Pedrycz, W.: From fuzzy data analysis and fuzzy regression to granular fuzzy data analysis..Fuzzy Set Syst. 274 (2015), 12-17. MR 3355341, 10.1201/9781315216737
Reference: [26] Pedrycz, W., Wang, X. M.: Designing fuzzy sets with the use of the parametric principle of justifiable granularity..IEEE Trans. Fuzzy Syst. 24 (2016), 489-496. 10.1109/tfuzz.2015.2453393
Reference: [27] Pei, D. W.: $R_{0}$ implication: characteristics and applications..Fuzzy Set Syst. 131 (2002), 297-302. MR 1939842, 10.1016/s0165-0114(02)00053-2
Reference: [28] Pei, D. W.: Unified full implication algorithms of fuzzy reasoning..Inform. Sci. 178 (2008), 520-530. MR 2363234, 10.1016/j.ins.2007.09.003
Reference: [29] Rosenfeld, A.: Distances between fuzzy sets..Pattern Recogn. Lett. 3 (1985), 229-233. 10.1016/0167-8655(85)90002-9
Reference: [30] Sarkoci, P., Šabo, M.: Information boundedness principle in fuzzy inference process..Kybernetika 38 (2002), 327-338. MR 1944313
Reference: [31] Szmidt, E., Kacprzyk, J.: Entropy for intuitionistic fuzzy sets..Fuzzy Set Syst. 118 (2001), 467-477. Zbl 1045.94007, MR 1809394, 10.1016/s0165-0114(98)00402-3
Reference: [32] Tang, Y. M.: Differently implicational hierarchical inference algorithm under interval-valued fuzzy environment..In: Proc. 2015 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2015), Istanbul, pp. 1-8. MR 3324890, 10.1109/fuzz-ieee.2015.7337944
Reference: [33] Tang, Y. M., Liu, X. P.: Differently implicational universal triple I method of (1, 2, 2) type..Comput. Math. Appl. 59 (2010), 1965-1984. MR 2595972, 10.1016/j.camwa.2009.11.016
Reference: [34] Tang, Y. M., Ren, F. J.: Universal triple I method for fuzzy reasoning and fuzzy controller..Iran. J. Fuzzy Syst. 10 (2013), 1-24. MR 3154637, 10.1109/fskd.2013.6816179
Reference: [35] Tang, Y. M., Ren, F. J.: Variable differently implicational algorithm of fuzzy inference..J. Intell. Fuzzy Syst. 28 (2015), 1885-1897. MR 3324890, 10.3233/IFS-141476
Reference: [36] Tang, Y. M., Ren, F. J.: Fuzzy systems based on universal triple I method and their response functions..Int. J. Inf. Tech. Decis. 16 (2017), 443-471. 10.1142/s0219622014500746
Reference: [37] Tang, Y. M., Ren, F. J., Chen, Y. X.: Differently implicational $\alpha$-universal triple I restriction method of (1, 2, 2) type..J. Syst. Eng. Electron. 23 (2012), 560-573. 10.1109/jsee.2012.00070
Reference: [38] Tang, Y. M., Yang, X. Z.: Symmetric implicational method of fuzzy reasoning..Int. J. Approx. Reason. 54 (2013), 1034-1048. MR 3081298, 10.1016/j.ijar.2013.04.012
Reference: [39] Tang, Y. M., Yang, X. Z., Yue, F.: Universal triple I method with maximum fuzzy entropy employing R-implications..In: Proc. the 10th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD 2013), pp. 125-129. 10.1109/fskd.2013.6816179
Reference: [40] Wang, L. X.: A Course in Fuzzy Systems and Control..Prentice-Hall, Englewood Cliffs, NJ, 1997.
Reference: [41] Wang, G. J.: On the logic foundation of fuzzy reasoning..Inform. Sci. 117 (1999), 47-88. MR 1705095, 10.1016/s0020-0255(98)10103-2
Reference: [42] Wang, G. J., Fu, L.: Unified forms of triple I method..Comput. Math. Appl. 49 (2005), 923-932. MR 2135223, 10.1016/j.camwa.2004.01.019
Reference: [43] Wang, G. J., Zhou, H. J.: Introduction to Mathematical Logic and Resolution Principle..Co-published by Science Press and Alpha International Science Ltd., 2009.
Reference: [44] Yang, X. Y., Yu, F. S., Pedrycz, W.: Long-term forecasting of time series based on linear fuzzy information granules and fuzzy inference system..Int. J. Approx. Reasoning 81 (2017), 1-27. MR 3589730, 10.1016/j.ijar.2016.10.010
Reference: [45] Zadeh, L. A.: Outline of a new approach to the analysis of complex systems and decision processes..IEEE Trans. Syst. Man Cyber. 3 (1973), 28-44. MR 0309582, 10.1109/tsmc.1973.5408575
.

Files

Files Size Format View
Kybernetika_55-2019-2_6.pdf 526.9Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo