Fuzzy Logistic Regression Analysis Using the Least Squares Method
Subject Areas : Transactions on Fuzzy Sets and Systems
Zahra Behdani
1
,
Majid Darehmiraki
2
1 - .گروه آمار- دانشکده علوم پایه- دانشگاه صنعتی خاتم الانبیاء
2 - Department of Mathematics, Behbahan Khatam Alanbia University of Technology, Behbahan, Khouzestan, Iran.
Keywords: Least square, Distance measure, Logistic regression.,
Abstract :
One of the most efficient statistical tools for modeling the relationship between a dependent variable and several independent variables is regression. In practice, observations relating to one or more variables, or the relationship between variables, may be vague or non-specific. In such cases, classic regression methods will not have enough capability to model data, and one of the alternative methods is regression in a fuzzy environment. The fuzzy logistic regression model provides a framework in the fuzzy environment to investigate the relationship between a binary response variable and a set of covariates. The purpose of this paper is to attempt to develop a fuzzy model that is based on the idea of the possibility of success. These possibilities are characterized {by several} linguistic phrases, including low, medium, and high, among others. Next, we {use a set of precise explanatory variable observations to model the logarithm transformation of "possibilistic odds." We assume that the model's parameters are triangular fuzzy numbers.} We use the least squares method in fuzzy linear regression to estimate the parameters of the provided model. We compute three types of goodness-of-fit criteria to evaluate the model. Ultimately, we model suspected cases of Systemic Lupus Erythematosus (SLE) disease based on significant risk factors to identify the model's application. We do this due to the widespread use of logistic regression in clinical studies and the prevalence of ambiguous observations in clinical diagnosis. Furthermore, to assess the prevalence of diabetes in the community, we will collect a sample of plasma glucose levels, measured two hours after a meal, from each participant in a clinical survey. The proposed model has the potential to rationally replace an ordinary model in modeling the clinically ambiguous condition, according to the findings.
[1] Zadeh LA. Fuzzy sets. Information and Control. 1965; 8(3): 338-353. DOI: https://doi.org/10.1016/S0019-9958(65)90241-X
[2] Asai HTSUK, Tanaka S, Uegima K. Linear regression analysis with fuzzy model. IEEE Transactions on Systems Man Cybernet. 1982; 12(6): 903-907. DOI: https://doi.org/10.1109/TSMC.1982.4308925
[3] Yager RR. Fuzzy prediction based on regression models. Information Sciences. 1982; 26(1): 45-63. DOI: https://doi.org/10.1016/0020-0255(82)90043-3
[4] Jajuga K. Linear fuzzy regression. Fuzzy Sets and Systems. 1986; 20(3): 343-353. DOI: https://doi.org/10.1016/S0165-0114(86)90045-X
[5] Celmins A. Least squares model fitting to fuzzy vector data. Fuzzy Sets and Systems. 1987; 22(3): 245-269. DOI:
https://doi.org/10.1016/0165-0114(87)90070-4
[6] Diamond P. Least squares fitting of several fuzzy variables. InPreprints of Second IFSA World Congress, Tokyo, Japan, 1987, pp. 329-331.
[7] Pourahmad S, Taghi Ayatollahi SM, Taheri SM. Fuzzy logistic regression: a new possibilistic model and its application in clinical vague status. Iranian Journal of Fuzzy Systems. 2011; 8(1): 1-17. DOI: https://doi.org/10.22111/ijfs.2011.232
[8] Pourahmad S, Ayatollahi SMT, Taheri SM, Agahi ZH. Fuzzy logistic regression based on the least squares approach with application in clinical studies. Computers & Mathematics with Applications. 2011; 62(9): 3353-3365. DOI: https://doi.org/10.1016/j.camwa.2011.08.050
[9] Namdari M, Yoon JH, Abadi A, Taheri SM, Choi SH. Fuzzy logistic regression with least absolute deviations estimators. Soft Computing. 2015; 19: 909-917. DOI: https://doi.org/10.1007/s00500-014- 1418-2
[10] Gao Y, Lu Q. A fuzzy logistic regression model based on the least squares estimation. Computational and Applied Mathematics. 2018; 37: 3562-3579. DOI: https://doi.org/10.1007/s40314-017-0531-0
[11] Mustafa S, Asghar S, Hanif M. Fuzzy logistic regression based on least square approach and trapezoidal membership function. Iranian Journal of Fuzzy Systems. 2018; 15(6): 97-106. DOI: https://doi.org/10.22111/ijfs.2018.4369
[12] Salmani F, Taheri SM, Yoon JH, Abadi A, Alavi Majd H, Abbaszadeh A. Logistic regression for fuzzy covariates: Modeling, inference, and applications. International Journal of Fuzzy Systems. 2017; 19: 1635-1644. DOI: https://doi.org/10.1007/s40815-016-0258-x
[13] Salmani F, Taheri SM, Abadi A. A forward variable selection method for fuzzy logistic regression. International Journal of Fuzzy Systems. 2019; 21: 1259-1269. DOI: https://doi.org/10.1007/s40815-019- 00615-z
[14] Bagley SC, White H, Golomb BA. Logistic regression in the medical literature: Standards for use and reporting with particular attention to one medical domain. Journal of Clinical Epidemiology. 2001; 54(10): 979-985. DOI: https://doi.org/10.1016/S0895-4356(01)00372-9
[15] Klippel JH, Stone JH, Crofford LJ, White PH. Primer on the rheumatic diseases. Springer; 2008.
[16] Li Y, He X, Liu X. Fuzzy multiple linear least squares regression analysis. Fuzzy Sets and Systems. 2023; 459: 118-143. DOI: https://doi.org/10.1016/j.fss.2022.06.012
[17] Yang MS, Ko CH. On a class of fuzzy c-numbers clustering procedures for fuzzy data. Fuzzy Sets and Systems. 1996; 84(1): 49-60. DOI: https://doi.org/10.1016/0165-0114(95)00308-8
[18] Diamond P, Körner R. Extended fuzzy linear models and least squares estimates. Computers & Mathematics with Applications. 1997; 33(9): 15-32. DOI: https://doi.org/10.1016/S0898-1221(97)00063-1
[19] Chachi J, Taheri SM, D’Urso P. Fuzzy regression analysis based on M-estimates. Expert Systems with Applications. 2022; 187: 115891. DOI: https://doi.org/10.1016/j.eswa.2021.115891
[20] D’Urso P, Massari R, Santoro A. Robust fuzzy regression analysis. Information Sciences. 2011; 181(19): 4154-4174. DOI: https://doi.org/10.1016/j.ins.2011.04.031
[21] Takemura K. Fuzzy logistic regression analysis for fuzzy input–output data. InProceedings of the joint 2nd International Conference on Soft Computing and Intelligent Systems and the 5th International Symposium on Advanced Intelligent Systems, 2004. pp. 1-6.