Theil-Sen Estimators for fuzzy regression model

Document Type : Research Paper

Authors

Behbahan Khatam Alanbia university of technology

Abstract

Both traditional and fuzzy regression analyses have demonstrated the significant characteristics of the least-squares methodology as a method for parameter estimation.} The presence of outliers in the sample and/or minor variations in the dataset might impact the behaviour and characteristics of the least-squares estimators (LSE)‎. ‎In contrast‎, ‎robust approaches provide estimators of the parameters that are resilient to the aforementioned unfavourable effects‎. ‎This study aims to expand upon the Theil-Sen estimator in fuzzy regression analysis‎, ‎with the objective of obtaining consistent findings even when outliers are present‎. ‎\rd{ We demonstrate the effectiveness of the suggested technique through simulation experiments and real-world examples‎, ‎comparing it to commonly used fuzzy regression models‎. ‎The applicative examples are based on hydrology and atmospheric environment datasets‎. ‎We also show the sensitivity analysis of the estimated parameters using a Monte-Carlo simulation study‎, ‎demonstrating the effectiveness of the suggested estimators in comparison to other established approaches in the field of fuzzy regression analysis‎. ‎The results showed that the Theil-Sen estimator (TSE) is very effective in cases where there are outliers‎, ‎and the calculation error is smaller compared to other methods.

Keywords

Main Subjects

References

[1] M. G. Akbari, G. Hesamian, A partial-robust-ridge-based regression model with fuzzy predictors-responses, Journal
of Computational and Applied Mathematics, 351 (2019), 290-301. https://doi.org/10.1016/j.cam.2018.11.006
[2] M. G. Amiri, A. R. Zarei, J. Abedi-Koupai, S. Eslamian, The performance of fuzzy regression method for estimating
of reference evapotranspiration under controlled environment, International Journal of Hydrology Science and
Technology, 9(1) (2019), 28-38. https://doi.org/10.1504/IJHST.2019.096791
[3] M. Arefi, Quantile fuzzy regression based on fuzzy outputs and fuzzy parameters, Soft Computing, 24(1) (2020),
311-320. https://doi.org/10.1007/s00500-019-04424-2
[4] H. T. S. U. K. Asai, S. Tanaka, K. Uegima, Linear regression analysis with fuzzy model, IEEE Transactions Systems,
Man, and Cybernetics, 12 (1982), 903-907. https://doi.org/10.1109/TSMC.1982.4308925
[5] E. Bas, Robust fuzzy regression functions approaches, Information Sciences, 613 (2022), 419-434.
https://doi.org/10.1016/j.ins.2022.09.047
[6] J. Chachi, A weighted least squares fuzzy regression for crisp input-fuzzy output data, IEEE Transactions on Fuzzy
Systems, 27(4) (2017), 739-748. https://doi.org/10.1109/TFUZZ.2018.2868554
[7] J. Chachi, M. Roozbeh, A fuzzy robust regression approach applied to bedload transport data, Communications in
Statistics-Simulation and Computation, 46(3) (2017), 1703-1714. https://doi.org/10.1080/03610918.2015.1010002
[8] J. Chachi, S. M. Taheri, P. D’Urso, Fuzzy regression analysis based on M-estimates, Expert Systems with Applications,
187 (2022), 115891. https://doi.org/10.1016/j.eswa.2021.115891
[9] J. Chachi, S. M. Taheri, S. Fattahi, S. A. Hosseini Ravandi, Two robust fuzzy regression models and their applications
in predicting imperfections of cotton yarn, Journal of Textiles and Polymers, 4(2) (2016), 60-68.
[10] Y. H. O. Chang, B. M. Ayyub, Fuzzy regression methods-a comparative assessment, Fuzzy Sets and Systems,
119(2) (2001) 187-203. https://doi.org/10.1016/S0165-0114(99)00091-3
[11] G. Charizanos, H. Demirhan, D. ˙I¸cen, A Monte Carlo fuzzy logistic regression framework against imbalance and
separation, Information Sciences, 655 (2024), 119893. https://doi.org/10.1016/j.ins.2023.119893
[12] Y. S. Chen, Outliers detection and confidence interval modification in fuzzy regression, Fuzzy Sets and Systems,
119(2) (2001), 259-272. https://doi.org/10.1016/S0165-0114(99)00049-4
[13] H. Chervenkov, K. Slavov, Theil-sen estimator for the parameters of the generalized extreme value distributions:
Demonstration for meteorological applications, Comptes rendus de I’Acad´emie bulgare des Sciences, 70(12) (2017),
1701-1708.
[14] H. Chervenkov, K. Slavov, Theil-Sen estimator vs. ordinary least squares-trend analysis for selected
ETCCDI climate indices, Comptes Rendus de L’Acad´emie Bulgare des Sciences, 72 (2019), 47-54.
https://doi.org/10.7546/CRABS.2019.01.06
[15] N. Chukhrova, A. Johannssen, Fuzzy regression analysis: Systematic review and bibliography, Applied Soft Computing,
84 (2019), 105708. https://doi.org/10.1016/j.asoc.2019.105708
[16] R. Coppi, P. D’Urso, P. Giordani, A. Santoro, Least squares estimation of a linear regression model with LR fuzzy response, Computational Statistics and Data Analysis, 51 (2006), 267-286. https://doi.org/10.1016/j.csda.2006.04.036
[17] X. Dang, H. Peng, X. Wang, H. Zhang, Theil-sen estimators in a multiple linear regression model, Olemiss Edu,
The University of Mississippi, 2008.
[18] P. D’Urso, R. Massari, Weighted least squares and least median squares estimation for the fuzzy linear regression
analysis, Metron, 71 (2013), 279-306. https://doi.org/10.1007/s40300-013-0025-9
[19] P. D’Urso, R. Massari, A. Santoro, Robust fuzzy regression analysis, Information Sciences, 181(19) (2011), 4154-
4174. https://doi.org/10.1016/j.ins.2011.04.031
[20] E. Egrioglu, E. Bas, Robust intuitionistic fuzzy regression functions approaches, Information Sciences, 638 (2023),
118992. https://doi.org/10.1016/j.ins.2023.118992
[21] A. H. El-Shaarawi, W. W. Piegorsch, Encyclopedia of environmetrics, Volume 1, John Wiley and Sons, p. 19, ISBN
978-0-471-89997-6, 2001.
[22] R. Fernandes, S. G. Leblanc, Parametric (modified least squares) and non-parametric (Theil-Sen) linear regressions
for predicting biophysical parameters in the presence of measurement errors, Remote Sensing of Environment, 95(3)
(2005), 303-316. https://doi.org/10.1016/j.rse.2005.01.005
[23] M. B. Ferraro, P. Giordani, A proposal of robust regression for random fuzzy sets. In synergies of soft computing and
statistics for intelligent data analysis, Springer Berlin Heidelberg, 2013. https://doi.org/10.1007/978-3-642-33042-
1 13
[24] G. Hesamian, M. G. Akbari, A robust multiple regression model based on fuzzy random variables, Journal of
Computational and Applied Mathematics, 388 (2021), 113270. https://doi.org/10.1016/j.cam.2020.113270
[25] P. J. Huber, Robust statistics, John Wiley and Sons, 2004.
[26] W. L. Hung, M. S. Yang, An omission approach for detecting outliers in fuzzy regression models, Fuzzy Sets and
Systems, 157(23) (2006), 3109-3122. https://doi.org/10.1016/j.fss.2006.08.004
[27] R. Jiang, Y. Xin, Z. Chen, Y. Zhang, A medical big data access control model based on fuzzy trust prediction and
regression analysis, Applied Soft Computing, 117 (2022), 108423. https://doi.org/10.1016/j.asoc.2022.108423
[28] L. Kokkinen, Studying social determinants of health using fuzzy-set qualitative comparative analysis: A worked
example, Social Science and Medicine, 309 (2022), 115241. https://doi.org/10.1016/j.socscimed.2022.115241
[29] K. S. Kula, A. Apaydin, Fuzzy robust regression analysis based on the ranking of fuzzy sets, International
Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 16(05) (2008), 663-681.
https://doi.org/10.1142/S0218488508005558
[30] J. M. Leski, M. Kotas, On robust fuzzy c-regression models, Fuzzy Sets and Systems, 279 (2015), 112-129.
https://doi.org/10.1016/j.fss.2014.12.004
[31] Y. Li, X. He, X. Liu, Fuzzy multiple linear least squares regression analysis, Fuzzy Sets and Systems, 459 (2023),
118-143. https://doi.org/10.1016/j.fss.2022.06.012
[32] F. Maturo, S. Hoˇskov´a-Mayerov´a, Fuzzy regression models and alternative operations for economic and social
sciences, Recent Trends in Social Systems: Quantitative Theories and Quantitative Models, (2017), 235-247.
https://doi.org/10.1007/978-3-319-40585-8 21
[33] Z. Mei, T. Zhao, X. Xie, Hierarchical fuzzy regression tree: A new gradient boosting approach to design a TSK
fuzzy model, Information Sciences, 652 (2024), 119740. https://doi.org/10.1016/j.ins.2023.119740
[34] E. Nasrabadi, S. M. Hashemi, Robust fuzzy regression analysis using neural networks, International
Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 16(04) (2008), 579-598.
https://doi.org/10.1142/S021848850800542X
[35] E. Nasrabadi, S. M. Hashemi, M. Ghatee, An LP-based approach to outliers detection in fuzzy regression
analysis, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 15(04) (2007), 441-456.
https://doi.org/10.1142/S0218488507004789
[36] G. Peters, Fuzzy linear regression with fuzzy intervals, Fuzzy Sets and Systems, 63(1) (1994), 45-55.
https://doi.org/10.1016/0165-0114(94)90144-9
[37] P. J. Rousseeuw, A. M. Leroy, Robust regression and outlier detection, John Wiley and Sons, 2005.
[38] M. Salman, X. Long, G. Wang, D. Zha, Paris climate agreement and global environmental efficiency:
New evidence from fuzzy regression discontinuity design, Energy Policy, 168 (2022), 113128.
https://doi.org/10.1016/j.enpol.2022.113128
[39] H. Theil, A rank-invariant method of linear and polynomial regression analysis, Proceedings of the Royal Netherlands
Academy of Sciences, 53 (1950), 368-392 (Part I), 521-525 (Part II), 1397-1412 (Part III).
[40] S. Varga, Robust estimations in classical regression models versus robust estimations in fuzzy regression models,
Kybernetika, 43(4) (2007), 503-508.
[41] H. Von Storch, F. W. Zwiers, Statistical analysis in climate research, Cambridge University Press, 2022.
[42] R. Xu, C. Li, Multidimensional least-squares fitting with a fuzzy model, Fuzzy Sets and Systems, 119 (2001),
215-223. https://doi.org/10.1016/S0165-0114(98)00350-9
[43] T. Zaman, K. Alaku¸s, Integrating Jackknife into the Theil-Sen estimator in multiple linear regression model,
Revstat-Statistical Journal, 21(1) (2023), 97-114. https://doi.org/10.57805/revstat.v21i1.398
[44] Y. A. Zelenkov, E. V. Lashkevich, Fuzzy regression model of the impact of technology on living standards, Business
Informatics, 14(3) (2023), 67-81. https://doi.org/10.17323/2587-814X.2020.3.67.81
[45] J. Zhou, H. Zhang, Y. Gu, A. A. Pantelous, Affordable levels of house prices using fuzzy linear regression analysis:
the case of Shanghai, Soft Computing, 22 (2018), 5407-5418. https://doi.org/10.1007/s00500-018-3090-4
Iranian Journal of Fuzzy Systems
Volume 21, Issue 3
May and June 2024
Pages 177-192

Files

Share

How to cite

Statistics