Fuzzy modeling using the similarity-based approximate reasoning system

Chachi, Jalal; Jalalvand, Mehdi

doi:10.22103/jmmr.2024.22970.1587

Fuzzy modeling using the similarity-based approximate reasoning system

Document Type : Research Paper

Authors

Jalal Chachi ¹

Mehdi Jalalvand ²

¹ Faculty of Mathematical Sciences and Computer, Department of Statistics, Shahid Chamran University of Ahvaz

² Faculty of Mathematical Sciences and Computer, Department of Mathematics, Shahid Chamran University of Ahvaz

10.22103/jmmr.2024.22970.1587

Abstract

Just as we humans use many different types of inferential procedures to help us understand things or to make decisions, there are many different fuzzy logic inferential procedures, including similarity-based approaches. Similarity measures can be seen not only as a general notion but also as a particular family of fuzzy relations which play crucial roles for the motivation and the whole design of similarity-based reasoning. In the context of similarity-based reasoning, several issues merit concern. One is the representation of implication relation and two is the composition of a fuzzy implication relation with an observed system fact. The others are continuity and robustness of these systems which are the soul that must be inherited in the newly setup frameworks. Therefore, the purpose of this study is to introduce a new similarity-based approximate reasoning system which is based on introducing a new class of similarity measure on the space of $LR$-fuzzy numbers. Therefore, first, a new class of similarity measures is introduced between fuzzy sets. The similarity measure is needed in order to activate rules which are in terms of linguistic variables. Second, it is proved that the proposed measures satisfy the properties of the axiomatic definition as well as the other properties by a theorem. Next, we validate the effectiveness of the proposed similarity measure in a bidirectional approximate reasoning system in order to provide a nonlinear mapping of fuzzy input data into fuzzy output data. Finally, using existing experimental data from Uniaxial Compressive Strength (UCS) testing, the fuzzy inference system constitutive model is produced to describe the influence of joint geometry (joint location, trace length and orientation) on the UCS of rock. The numerical results will show that the proposed model based on similarity-based approximate reasoning systems has better performance compared with the Mamdani fuzzy inference systems and the multivariate regression.

Keywords

Approximate reasoning system

Fuzzy modeling

Similarity measure

Statistical learning

Subjects

Fuzzy Mathematics and Soft Computing

[1] Arkes, J. (2023). Regression analysis: a practical introduction. Routledge. https://doi.org/10.4324/9781351011099

[2] Atanasov, A. B., Zavatone-Veth, J. A., & Pehlevan, C. (2024). Scaling and renormalization in high-dimensional regression. arXiv preprint arXiv:2405.00592. https://doi.org/10.48550/arXiv.2405.00592

[3] Bas, E. (2022). Robust fuzzy regression functions approaches. Information Sciences, 613, 419-434. https://doi.org/10.1016/j.ins.2022.09.047

[4] Bas, E., & Egrioglu, E. (2022). A fuzzy regression functions approach based on gustafson-kessel clustering algorithm. Information Sciences, 592, 206-214. https://doi.org/10.1016/j.ins.2022.01.057

[5] Berk, R. A. (2020). Statistical learning as a regression problem. In Statistical learning from a regression perspective (pp. 1{72). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-40189-4 1

[6] Cao, J., Zhou, T., Zhi, S., Lam, S., Ren, G., Zhang, Y., Wang, Y. Dong, Y., & Cai, J. (2024). Fuzzy inference system with interpretable fuzzy rules: Advancing explainable arti cial intelligence for disease diagnosis-a comprehensive review. Information Sciences, 662, 120212. https://doi.org/10.1016/j.ins.2024.120212

[7] Chachi, J., & Chaji, A. (2021). An owa-based approach to quantile fuzzy regression. Computers & Industrial Engineering, 159, 107498. https://doi.org/10.1016/j.cie.2021.107498

[8] Chachi, J., Kazemifard, A., & Jalalvand, M. (2021). A multi-attribute assessment of fuzzy regression models. Iranian Journal of Fuzzy Systems, 18, 131-148. https://doi.org/10.22111/ijfs.2021.6181

[9] Chachi, J., & Taheri, S. M. (2013). A uni ed approach to similarity measures between intuitionistic fuzzy sets. International Journal of Intelligent Systems, 28, 669-685. https://doi.org/10.1002/int.21596

[10] Chachi, J., Taheri, S. M., & D'Urso, P. (2022). Fuzzy regression analysis based on m-estimates. Expert Systems with Applications, 187, 115891. https://doi.org/10.1016/j.eswa.2021.115891

[11] Chukhrova, N., & Johannssen, A. (2019). Fuzzy regression analysis: Systematic review and bibliography. Applied Soft Computing, 84, 105708. https://doi.org/10.1016/j.asoc.2019.105708

[12] Couso, I., Garrido, L., & Sanchez, L. (2013). Similarity and dissimilarity measures between fuzzy sets: a formal relational study. Information Sciences, 229, 122-141. https://doi.org/10.1016/j.ins.2012.11.012

[13] Cox, J. A., Wu, Y., & Davies, A. M. A. (2024). Does animacy a ect visual statistical learning? revisiting the e ects of selective attention and animacy on visual statistical learning. Quarterly Journal of Experimental Psychology, 77, 492-510.
https://doi.org/10.1177/17470218231173883

[14] Diamond, P. (1988). Fuzzy least squares. Information Sciences, 46, 141-157. https://doi.org/10.1016/0020-0255(88)90047-3

[15] D'Urso, P., & Chachi, J. (2022). Owa fuzzy regression. International journal of approximate reasoning, 142, 430-450. https://doi.org/10.1016/j.ijar.2021.12.009

[16] D'Urso, P., Chachi, J., Kazemifard, A., & De Giovanni, L. (2024). Owa-based multicriteria decision making based on fuzzy methods. Annals of Operations Research, 1-35. https://doi.org/10.1007/s10479-024-05926-5

[17] Dvorak, A., Jayaram, B., & Stepnicka, M. (2021). Similarity-based reasoning from the perspective of extensionality. In 19th world congress of the inter-national fuzzy systems association (ifsa), 12th conference of the european society for fuzzy logic and technology (eusat), and 11th international summer school on aggregation operators (agop) (pp.
322-329). https://doi.org/10.2991/asum.k.210827.043

[18] Feng, Y., Fan, J., & Suykens, J. A. (2020). A statistical learning approach to modal regression. Journal of Machine Learning Research, 21(2), 1-35. https://doi.org/10.48550/arXiv.1702.05960

[19] Fiedler, C., Herty, M., & Trimpe, S. (2024). On kernel-based statistical learning theory in the mean eld limit. Advances in Neural Information Processing Systems, 36, arXiv:2310.18074. https://doi.org/10.48550/arXiv.2310.18074

[20] Frost, R., Armstrong, B. C., & Christiansen, M. H. (2019). Statistical learning research: A critical review and possible new directions. Psychological Bulletin, 145(12), 1128. https://doi.org/10.1037/bul0000210

[21] Gorgin, S., Karvandi, M. S., Moghari, S., Fallah, M. K., & Lee, J.-A. (2024). A hardware realization framework for fuzzy inference system optimization. Electronics, 13(4), 690. https://doi.org/10.3390/electronics13040690

[22] Hesamian, G. (2017). Measuring similarity and ordering based on interval type-2 fuzzy numbers. IEEE Transactions on Fuzzy Systems, 25(4), 788-798. https://doi.org/10.1109/TFUZZ.2016.2578342

[23] Hesamian, G., & Chachi, J. (2017). On similarity measures for fuzzy sets with applications to pattern recognition, decision making, clustering, and approximate reasoning. Journal of Uncertain Systems, 11(1), 35-48. https://www.academia.edu/7732486

[24] Hothorn, T. (2023). Cran task view: Machine learning & statistical learning. Comprehensive R Archive Network (CRAN), Version 2023-07-20. https://CRAN.R-project.org/view=MachineLearning

[25] Imam, M., Adam, S., Dev, S., & Nesa, N. (2024). Air quality monitoring using statistical learning models for sustainable environment. Intelligent Systems with Applications, 200333. https://doi.org/10.1016/j.iswa.2024.200333

[26] Irani, J., Pise, N., & Phatak, M. (2016). Clustering techniques and the similarity measures used in clustering: A survey. International journal of computer applications, 134(7), 9-14. https://doi.org/10.5120/ijca2016907841

[27] James, G., Witten, D., Hastie, T., Tibshirani, R., & Taylor, J. (2023). Statistical learning. In An introduction to statistical learning: with applications in python (pp. 15-67). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-031-38747-0

[28] Kazemifard, A., & Chachi, J. (2022). Madm approach to analyse the performance of fuzzy regression models. Journal of Ambient Intelligence and Humanized Computing, 13, 4019-4031. https://doi.org/10.1007/s12652-021-03394-4

[29] Klement, E. P., Mesiar, R., & Pap, E. (2000). Triangular norms. Dordrecht, Netherland: Springer. https://doi.org/10.1007/978-94-015-9540-7

[30] Klir, G., & Yuan, B. (2002). Fuzzy sets and fuzzy logic: Theory and applications. Taiwan: Prentice-Hall. https://books.google.com/books?id=AOhQAAAAMAAJ

[31] Li, F., & Shen, Q. (2024). Fuzzy rule-based inference: Advances and applications in reasoning with approximate knowledge interpolation. https://doi.org/10.1007/978-981-97-0491-0[32] MacDonell, S. G. (2011). The impact of

sampling and rule set size on generated fuzzy inference system predictive accuracy: analysis of a software engineering data set. In International conference on engineering applications of neural networks (pp. 360-369).
https://doi.org/10.1007/978-3-642-23960-1 43

[33] Mazandarani, M., & Li, X. (2020). Fractional fuzzy inference system: The new generation of fuzzy inference systems. IEEE Access, 8, 126066-126082. https://doi.org/10.1109/ACCESS.2020.3008064

[34] Miloudi, S., Wang, Y., & Ding, W. (2021). An improved similarity-based clustering algorithm for multi-database mining. Entropy, 23(5), 553. https://doi.org/10.20944/preprints202104.0256.v1

[35] Mouzouris, G., & Mendel, J. (1997). Nonsingleton fuzzy logic systems: theory and application. IEEE Transactions on Fuzzy Systems, 5(1), 56-71. https://doi.org/10.1109/91.554447

[36] Nguyen, A.-T., Taniguchi, T., Eciolaza, L., Campos, V., Palhares, R., & Sugeno, M. (2019). Fuzzy control systems: Past, present and future. IEEE Computational Intelli-gence Magazine, 14(1), 56-68. https://doi.org/10.1109/MCI.2018.2881644[37] Ojha, V., Abraham, A., & Snasel, V. (2019). Heuristic design of fuzzy inference systems: A review of three decades of research. Engineering Applications of Arti cial Intelligence, 85, 845-864. https://doi.org/10.1016/j.engappai.2019.08.010

[38] Pattanayak, R. M., Behera, H., & Panigrahi, S. (2021). A novel probabilistic intuitionistic fuzzy set based model for high order fuzzy time series forecasting. Engineering Applications of Arti cial Intelligence, 99, 104136.
https://doi.org/10.1016/j.engappai.2020.104136

[39] Scholkopf, B., & Smola, A. J. (2018). Learning with kernels: support vector machines, regularization, optimization, and beyond. MIT press. https://doi.org/10.7551/mitpress/4175.001.0001

[40] Sharifani, K., & Amini, M. (2023). Machine learning and deep learning: A review of methods and applications. World Information Technology and Engineering Journal, 10(7), 3897-3904. https://ssrn.com/abstract=4458723

[41] Tak, N. (2020). Type-1 possibilistic fuzzy forecasting functions. Journal of Computational and Applied Mathematics, 370, 112653. https://doi.org/10.1016/j.cam.2019.112653

[42] Tanaka, H., Uegima, S., & Asai, K. (1982). Linear regression analysis with fuzzy model. IEEE Transactions on Systems, Man, and Cybernetics, 12, 903-907. http://dx.doi.org/10.1109/TSMC.1982.4308925

[43] Utt, Z., Volya, D., & Mishra, P. (2024). Quantum measurement classi cation using statistical learning. ACM Transactions on Quantum Computing, 5, 1-16. https://doi.org/10.1145/3644823

[44] Volosencu, C. (2024). Adaptive neuro-fuzzy inference system as a universal estimator. IntechOpen. https://books.google.com/books?id=NtAHEQAAQBAJ

[45] Wagner, C., Pourabdollah, A., McCulloch, J., John, R., & Garibaldi, J. M. (2016). A similarity-based inference engine for non-singleton fuzzy logic systems. In 2016 ieee international conference on fuzzy systems (fuzz-ieee) (pp. 316-323).
https://doi.org/10.1109/FUZZ-IEEE.2016.7737703

[46] Wasantha, P. L. P., Ranjith, P. G., & Viete, D. R. (2012). Constitutive models describing the in uence of the geometry of partially-spanning joints on jointed rock mass strength: Regression and fuzzy logic analysis of experimental data. Expert Systems with Applications, 39(9), 7663-7672. https://doi.org/10.1016/j.eswa.2012.01.025

[47] Welch, W. J. (1982). Algorithmic complexity: three np-hard problems in computational statistics. Journal of Statistical Computation and Simulation, 15, 17-25. https://doi.org/10.1080/00949658208810560

[48] Xu, Y., & Zeevi, A. (2024). Towards optimal problem dependent generalization error bounds in statistical learning theory. Mathematics of Operations Research. https://doi.org/10.1287/moor.2021.0076

[49] Yang, M.-S., & Shih, H.-M. (2001). Cluster analysis based on fuzzy relations. Fuzzy Sets and Systems, 120(2), 197-212. https://doi.org/10.1016/S0165-0114(99)00146-3

[50] Zadeh, L. A. (1965). Fuzzy sets. Informtion Control, 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X

[52] Zeng, W., Liu, Y., Cui, H., Ma, R., & Xu, Z. (2022). Interval possibilistic c-means algorithm and its application in image segmentation. Information Sciences, 612, 465-480. https://doi.org/10.1016/j.ins.2022.08.082

[53] Zhang, T. (2023). Mathematical analysis of machine learning algorithms. Cambridge
University Press. https://doi.org/10.1017/9781009093057