Abstract
Uncertain linear regression (ULR) model based on symmetric triangular uncertain set has been studied early. This paper extends the symmetric triangular uncertain coefficients to asymmetric triangular uncertain coefficients and builds two methods for estimating the parameters of ULR model. Our aim is to minimize the differences of the uncertain membership functions between the observed and estimated values. Firstly, we propose a linear programming method, whose principle is to minimize the sum of the absolute values of the differences between left width and right width of two triangular uncertain sets for each index i. Secondly, we develop a new nonlinear programming method by maximizing the overlaps of acreage of the estimated and real triangular uncertain sets in a particular \(h_i\)-cut. Then, a criterion is established to evaluate the performance of the proposed approaches. Finally, we use an example based on industrial water demand data of China to illustrate our proposed approaches which are reasonable and compare the explanatory power of the ULR model and traditional linear regression (TLR) model using the presented evaluation criteria, which shows that the performance of the ULR model is obviously better than the TLR model.
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References
Allahviranloo T, Hosseinzadeh AA, Ghanbari M, Haghi E, Nuraei R (2014) On the new solutions for a fully fuzzy linear system. Soft Comput 18:95–107
Chang YH (2001) Hybrid fuzzy least-squares regression analysis and its reliability measures. Fuzzy Sets Syst 119(2):225–246
Chen L, Hsueh C, Chang C (2013) A two-stage approach for formulating fuzzy regression models. Knowl Based Syst 52:302–310
Coppi R, D’Urso P, Giordani P, Santoro A (2006) Least squares estimation of a linear regression model with LR fuzzy response. Comput Stat Data Anal 51(1):267–286
Diamond P (1988) Fuzzy least squares. Inf Sci 46:141–157
Dunyak JP, Wunsch D (2000) Fuzzy regression by fuzzy number neural networks. Fuzzy Sets Syst 112:371–380
Guo H, Wang X, Gao Z (2017) Uncertain linear regression model and its application. J Intell Manuf 28(3):559–564
Guo H, Wang X, Wang L (2017) The normal uncertain set and its application. http://orsc.edu.cn/online/140910.pdf
Hassanpour H, Maleki HR, Yaghoobi MA (2011) A goal programming approach to fuzzy linear regression with fuzzy input–output data. Soft Comput 15:1569–1580
Kim B, Bishu RR (1998) Evaluation of fuzzy linear regression models by comparing membership functions. Fuzzy Sets Syst 100:343–352
Kim HK, Yoon JH, Li Y (2008) Asymptotic properties of leasts quares estimation with fuzzy observations. Inf Sci 178:439–451
Lee HT, Chen SH (2001) Fuzzy regression model with fuzzy input and output data for manpower forecasting. Fuzzy Sets Syst 24(5):205–213
Li J, Zeng W, Xie J, Yin Q (2016) A new fuzzy regression model based on least absolute deviation. Eng Appl Artif Intell 52:54–64
Liu B (2007) Uncertain theory, 2nd edn. Springer, Berlin
Liu B (2009) Some research problems in uncertain theory. J Uncertain Syst 3(1):3–10
Liu B (2010) Uncertain theory: a branch of mathematics for modeling human uncertainty. Springer, Berlin
Liu B (2010) Uncertain set theory and uncertain inference rule with application to uncertain control. J Uncertain Syst 4(2):83–98
Liu B (2012) Membership functions and operational law of uncertain sets. Fuzzy Optim Decis Mak 11(4):387–410
Liu B (2011) Uncertain logic for modeling human language. J Uncertain Syst 5(1):3–20
Li X, Wang X (2017) A regression model based on uncertain set. Asian Bus Res 2(3):33–45
Nasrabadi MM, Nasrabadi E, Nasrabady AR (2005) Fuzzy linear regression analysis: a multi-objective programming approach. Appl Math Comput 163(1):245–251
Soliman SA, Alammari RA (2002) Fuzzy linear parameter estimation algorithms: a new formulation. Int J Electr Power Energy Syst 119(2):415–420
Tanaka H, Uejima S, Asai K (1982) Linear regression analysis with fuzzy models. IEEE Trans Syst Man Cybern 12(6):903–907
Tanaka H, Lee H (1998) Interval regression analysis by quadratic programming approach. IEEE Trans Fuzzy Syst 6(4):473–481
Wu Q, Law R (2010) Fuzzy support vector regression machine with penalizing Gaussian noises on triangular fuzzy number space. Expert Syst Appl 37:7788–7795
Wang X, Gao Z, Guo H (2012) Delphi method for estimating uncertainty distributions. Inf Int Interdiscip J 15(2):449–460
Wang X, Peng Z (2014) Method of moments for estimating uncertainty distribution. J Uncertain Anal Appl 2(1):1–10
Yao K, Liu B (2018) Uncertain regression analysis: an approach for imprecise observation. Soft Comput 22:5579–5582. https://doi.org/10.1007/s00500-017-2521-y
Yen KK, Ghoshray S, Roig G (1999) A Linear regression fuzzy model using triangular fuzzy number coefficients. Fuzzy Sets Syst 106:167–177 Please check and confirm the author names and initials are correctly identified for reference Yen et al. (1999)
Yeylaghi S, Otadi M, Imankhan N (2017) A new fuzzy regression model based on interval-valued fuzzy neural network and its applications to management. Beni-Suef Univ J Basic Appl Sci 6:106–111
Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1(1):3–28
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This work was supported by the National Natural Science Foundation of China (No. 61873084) and the Foundation of Hebei Education Department (No. ZD2017016).
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Wang, X., Li, H. & Guo, H. A new uncertain regression model and its application. Soft Comput 24, 6297–6305 (2020). https://doi.org/10.1007/s00500-019-03938-z
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DOI: https://doi.org/10.1007/s00500-019-03938-z