# On the Monotonicity Property of the TSK Fuzzy Inference System: The Necessity of the Sufficient Conditions and the Monotonicity Test

## Abstract

The sufficient conditions for satisfying the monotonicity property of the Takagi–Sugeno–Kang (TSK) fuzzy inference system (FIS) have shown to be useful in many different applications. However, the related sufficient and necessary conditions are still unknown. As such, even when the sufficient conditions are violated, the TSK FIS model may still able to satisfy the monotonicity property. Therefore, the monotonicity test is used as an approximated method to determine the validness of the monotonicity property. To the best of our knowledge, the use of the monotonicity test in FIS is new. In this paper, we focus on single-input zero-order TSK FIS with Gaussian fuzzy membership functions. An algorithm to test the monotonicity property, either accepting or rejecting an TSK FIS model of being monotone, is devised and analyzed. The relationship between TSK FIS and its capability of satisfying the monotonicity property, along with the sufficient conditions and the outcome of the monotonicity test, is established through a Monte Carlo simulation. The Monte Carlo simulation is a useful necessity test for the sufficient conditions. We define a necessity measure of the sufficient conditions (NMSC) as the probability that a randomly generated monotone TSK FIS model (evaluated based on the monotonicity test algorithm) satisfies the sufficient conditions. We empirically show that the NMSC score reduces with increasing number of fuzzy rules. In addition, an application of the TSK FIS model to failure modes and effects analysis is demonstrated. As compared with the sufficient conditions, a better FIS-based model with a lower error measure can be obtained using the monotonicity test. The outcome indicates the effectiveness of the monotonicity test for designing low-dimensional TSK FIS models with large numbers of fuzzy rules.

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## References

1. 1.

Wang, L.-X., Mendel, J.M.: Fuzzy basis functions, universal approximation, and orthogonal least-squares learning. IEEE Trans. Neural Netw. 3(5), 807–813 (1992)

2. 2.

Wang, L.-X.: Fuzzy systems are universal approximators. In: Proceedings of IEEE International Conference on Fuzzy Systems (1992)

3. 3.

Kosko, B.: Fuzzy systems as universal approximators. IEEE Trans. Comput. 43(11), 1329–1333 (1994)

4. 4.

Ying, H.: A sufficient condition on a general class of interval type-2 Takagi-Sugeno fuzzy systems with linear rule consequent as universal approximators. J. Intell. Fuzzy Syst. 29(3), 1219–1227 (2015)

5. 5.

Er, M.J., Mandal, S.: A survey of adaptive fuzzy controllers: nonlinearities and classifications. IEEE Trans. Fuzzy Syst. 24(5), 1095–1107 (2016)

6. 6.

Seki, H., Ishii, H., Mizumoto, M.: On the monotonicity of fuzzy inference methods related to T–S inference method. IEEE Trans. Fuzzy Syst. 18(3), 629–634 (2010)

7. 7.

Kim, J., Lee, J.S.: Sufficient conditions for monotonically constrained functional-type SIRMs connected fuzzy systems. In: Proceedings of IEEE International Conference on Fuzzy Systems, pp. 1–6 (2010)

8. 8.

Van Broekhoven, E., De Baets, B.: Only smooth rule bases can generate monotone Mamdani-Assilian models under center-of-gravity defuzzification. IEEE Trans. Fuzzy Syst. 17(5), 1157–1174 (2009)

9. 9.

Won, J.M., Park, S.Y., Lee, J.S.: Parameter conditions for monotonic Takagi-Sugeno-Kang fuzzy system. Fuzzy Sets Syst. 132(2), 135–146 (2002)

10. 10.

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(6), 1455–1463 (2016)

11. 11.

Li, C.D., Yi, J.Q., Zhang, G.Q.: On the monotonicity of interval type-2 fuzzy logic systems. IEEE Trans. Fuzzy Syst. 22(5), 1197–1212 (2013)

12. 12.

Jee, T.L., Tay, K.M., Lim, C.P.: A new two-stage fuzzy inference system-based approach to prioritize failures in failure mode and effect analysis. IEEE Trans. Reliab. 64(3), 869–877 (2015)

13. 13.

Kouikoglou, V.S., Phillis, Y.A.: On the monotonicity of hierarchical sum-product fuzzy systems. Fuzzy Sets Syst. 160(24), 3530–3538 (2009)

14. 14.

Lindskog, P., Ljung, L.: Ensuring monotonic gain characteristics in estimated models by fuzzy model structures. Automatica 36(2), 311–317 (2000)

15. 15.

Kerk, Y.W., Tay, K.M., Lim, C.P.: An analytical interval fuzzy inference system for risk evaluation and prioritization in failure mode and effect analysis. IEEE Syst. J. 11(3), 1589–1600 (2017)

16. 16.

Husek, P.: On monotonicity of Takagi-Sugeno fuzzy systems with ellipsoidal regions. IEEE Trans. Fuzzy Syst. 24(6), 1673–1678 (2016)

17. 17.

Mandal, S., Jayaram, B.: Monotonicity of SISO fuzzy relational inference with an implicative rule base. IEEE Trans. Fuzzy Syst. 24(6), 1475–1487 (2016)

18. 18.

Won, J.M., Karray, F.: Towards necessity of parametric conditions for monotonic fuzzy systems. IEEE Trans. Fuzzy Syst. 22(2), 465–468 (2014)

19. 19.

Kim, J.W., Won, J.M., Koo, K.M., Lee, J.S.: Monotonic fuzzy systems as universal approximators for monotonic functions. Intell. Autom. Soft Comput. 18(1), 13–31 (2012)

20. 20.

Teh, C.Y., Tay, K.M., Lim, C.P.: Monotone data samples do not always produce monotone fuzzy if-then rules: learning with ad hoc and system identification methods. In: FUZZ-IEEE, pp. 1–7 (2017)

21. 21.

Tay, K.M., Lim, C.P.: On monotonic sufficient conditions of fuzzy inference systems and their applications. Int. J. Uncertain. Fuzz. 19(5), 731–757 (2011)

22. 22.

Tay, K.M., Lim, C.P., Teh, C.Y., Lau, S.H.: A monotonicity index for the monotone fuzzy modeling problem. In: FUZZ-IEEE, pp. 456–463 (2012)

23. 23.

Goldreich, O., Goldwasser, S., Lehman, E., Ron, D., Samorodnitsky, A.: Testing monotonicity. Combinatorial 20, 301–337 (2000)

24. 24.

Ergun, F., Kannan, S., Kumar, S. Ravi, Rubinfeld, R., Viswanathan, M.: Spot-checkers. In: Proceedings of STOC, pp. 259–268 (1988)

25. 25.

Fischer, E., Lehman, E., Newman, I., Raskhodnikova, S., Rubinfeld, R., Samorodnitsky, A.: Monotonicity testing over general post domains. In: Proceedings of STOC, pp. 474–483 (2002)

26. 26.

Dodis, Y., Goldreich, O., Lehman, E., Raskhodnikova, S., Ron, D., Samorodnitsky, A.: Improved testing algorithms for monotonicity. In: Proceedings of RANDOM, pp. 97–108 (1999)

27. 27.

Stamatis, D.H.: Failure Mode and Effect Analysis: FMEA from Theory to Execution. American Society for Quality Control, Milwaukee (2003)

28. 28.

Tay, K.M., Lim, C.P.: On the use of fuzzy inference techniques in assessment models: part II: industrial applications. Fuzzy Optim. Decis. Mak. 7(3), 283–302 (2008)

29. 29.

Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)

30. 30.

Daniels, H., Velikova, M.: Monotone and partially monotone neural network. IEEE Trans. Neural Netw. 21(6), 906–917 (2010)

31. 31.

Fishman, G.S.: Monte Carlo: Concept, Algorithms, and Applications. Springer, Berlin (2003)

32. 32.

Tooranloo, H.S., Ayatollah, A.S.: Pathology the internet banking service quality using Failure Mode and Effect Analysis in interval-valued intuitionistic fuzzy environment. Int. J. Fuzzy Syst. 19(1), 109–123 (2017)

33. 33.

Hoffman, H.: Evolutionary algorithms for fuzzy control system design. Proc. IEEE 89(9), 1318–1333 (2001)

34. 34.

Jang, J.S.R., Sun, C.T., Mizutani, E.: Neural-Fuzzy and Soft Computing. Prentice Hall, Upper Saddle River (1997)

35. 35.

Lin, C.T., George Lee, C.S.: Neural-Fuzzy Systems: A Neural-Fuzzy Synergirm to Intelligent Systems. Prentice-Hall, Upper Saddle River (1996)

36. 36.

Deb, K., Agrawal, R.B.: Simulated binary crossover for continuous search space. Complex Syst. 9(3), 115–148 (1995)

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Correspondence to Chin Ying Teh.

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Teh, C.Y., Tay, K.M. & Lim, C.P. On the Monotonicity Property of the TSK Fuzzy Inference System: The Necessity of the Sufficient Conditions and the Monotonicity Test. Int. J. Fuzzy Syst. 20, 1915–1924 (2018). https://doi.org/10.1007/s40815-018-0509-0

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### Keywords

• Takagi–Sugeno–Kang fuzzy inference system
• Monotonicity property
• Monotonicity test
• Sufficient conditions
• Failure mode and effects analysis
• Fuzzy occurrence model