A flexible soft nonlinear quantile-based regression model
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Hesamian, Gholamreza; Johannssen, Arne; Chukhrova, Nataliya Article — Published Version A flexible soft nonlinear quantile-based regression model Fuzzy Optimization and Decision Making Provided in Cooperation with: Springer Nature Suggested Citation: Hesamian, Gholamreza; Johannssen, Arne; Chukhrova, Nataliya (2025) : A flexible soft nonlinear quantile-based regression model, Fuzzy Optimization and Decision Making, ISSN 1573-2908, Springer US, New York, NY, Vol. 24, Iss. 1, pp. 129-153, https://doi.org/10.1007/s10700-025-09441-5 This Version is available at: https://hdl.handle.net/10419/323315 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/
Vol.:(0123456789) Fuzzy Optimization and Decision Making (2025) 24:129–153 https://doi.org/10.1007/s10700-025-09441-5 A flexible soft nonlinear quantile‑based regression model GholamrezaHesamian1· ArneJohannssen2· NataliyaChukhrova3 Accepted: 25 January 2025 / Published online: 6 March 2025 © The Author(s) 2025 Abstract There are several models for soft regression analysis in the literature, but relatively few are based on quantiles, and these models are limited to the linear case. As quantile-based regression models offer a series of benefits (like robustness and handling of asymmetric distributions) but have not been considered in the nonlinear case, we present the first soft nonlinear quantile-based regression model in this paper. Considering nonlinearity instead of limiting to linearity in the modeling brings numerous advantages such as a higher flexibility, more accurate predictions, a better model fit and an improved explainability/interpretability of the model. In particular, we embed fuzzy quantiles into nonlinear regression analysis with crisp predictor variables and fuzzy responses. We propose a new method for parameter estimation by implementing a three-stage technique on the basis of the center and the spreads. In the framework of this procedure, we utilize kernel-fitting, a least quantile loss function, least absolute errors, and generalized cross-validation criteria to estimate the model parameters. We perform comprehensive comparative analysis with other soft nonlinear regression models that have demonstrated superiority in previous studies. The results reveal that the proposed nonlinear quantile-based regression technique leads to better outcomes compared to the competitors. Keywords Cross-validation· Explainability· Fuzzy quantiles· Fuzzy regression· Kernel-fitting· Least absolute errors· Robustness * Arne Johannssen [email protected] Gholamreza Hesamian [email protected] Nataliya Chukhrova nataliya.chukhrov[email protected] 1 Department ofStatistics, Payame Noor University, Tehran19395-3697, Iran 2 Faculty ofBusiness Studies, Harz University ofApplied Sciences, 38855Wernigerode, Germany 3 Department ofHealth Economics, Faculty ofMedicine Mannheim, University ofHeidelberg, 68167Mannheim, Germany
130 G.Hesamian et al. 1 Introduction Regression analysis can be employed to predict outcomes, investigate relationships, build models, and test hypotheses, making these models a versatile and widely-used statistical technique. In many situations, however, the underlying data is imprecise, which causes serious problems in common regression analysis as data is assumed to be crisp. In real-life applications, data is often imprecise due to errors (e.g., measurement and sampling errors), missing observations, uncertainty, ambiguity or variability that reduce the accuracy and reliability of the data. Therefore, it is crucial to consider these aspects when analyzing the data to ensure the validity and reliability of the conclusions. For this reason, soft/fuzzy regression analysis has been introduced to handle imprecise data and related issues. This field is the most active research area within fuzzy statistics and there are numerous proposals to address fuzziness in the framework of regression models. The most important fields are possibilistic regression, least squares / least absolutes regression and regression models based on machine learning techniques, see the systematic review on this topic provided by Chukhrova and Johannssen (2019). Among various regression techniques, quantile-based regression models can be seen as generalization of least squares estimation for conditional mean models to predict conditional quantile functions. Quantile-based regression models offer various benefits over common mean-based regression models, such as: • Quantile-based regression models are less sensitive to outliers in the data, because they estimate the conditional distribution of the response variable rather than just the conditional mean. • By estimating the conditional distribution of the response variable, quantilebased regression models can capture changes in the relationship between the variables across different parts of the distribution, which is why they allow for a more flexible analysis of the relationship between the variables. • The coefficients estimated by quantile-based regression models have a clear interpretation, as they represent the effect of the predictor variable on the corresponding quantile of the response variable. • When the distribution of the response variable is asymmetric, quantile-based regression models can provide a more accurate and nuanced analysis of the relationship between the variables. Recently, quantile regression models have found their way into the literature of fuzzy regression analysis. First, Arefi (2019) introduced a quantile linear regression model based on fuzzy responses and fuzzy predictors by extending the conventional quantile-based loss function to estimate the fuzzy coefficients. Second, Hesamian and Akbari (2019) proposed a fuzzy semi-parametric quantile-based linear regression model, also by considering fuzzy responses and fuzzy predictors. They used both the common semi-parametric method and the quantile regression model for the estimation of the unknown model parameters. Third, Chachi and Chaji (2021) presented a parameter estimation method for a fuzzy
131 A flexible soft nonlinear quantile-based regression model linear regression model with fuzzy responses and crisp predictors through mathematical programming based on a weighted aggregation of ordered residuals. However, the above-mentioned approaches cannot include nonlinearities in the modeling, as they are limited to the inflexible linear case. In many situations, we do not know much about the underlying nature of the relationship within the set {yi,xi} , where xi =(x i1 ,x i2 ,…,x ip ) ⊤ . For such cases, it is assumed that there is a nonlinear relationship between yi and xi , i.e., yi=f(xi)+𝜖i , i=1, 2, …,n . There are several popular parametric techniques such as polynomials, B-splines, Gaussian functions, wavelet bases or radial basis functions to approximate f. In the context of fuzzy nonlinear regression models with crisp predictor and fuzzy response variables, several fuzzy parametric (Asadolahi etal., 2021; D’Urso & Gastaldi, 2002; De Hierro etal., 2016; Ferraro etal., 2010; Jiang etal., 2013; Hesamian & Akbari, 2020a) and nonparametric (Cheng & Lee, 1999; Hong & Hwang, 2003; Wang etal., 2007) nonlinear regression models have been proposed. As the known fuzzy quantile-based regression models are limited to the linear case, we propose a flexible fuzzy nonlinear quantile-based regression model in this paper. When conducting nonlinear regression analysis, the approach developed by Balasundaram and Meena (2019) is one of the most effective techniques. The main property of this approach is that it provides a simple and effective estimator for the regression function f that is always within the range of the response variable. In this paper, we choose this way of parameter estimation due to its inherent simplicity compared to more sophisticated nonlinear techniques. In particular, the proposed fuzzy nonlinear quantile-based regression model offers several advantages and is based on the following innovative methodology: • It is a combined method that unites the benefits of quantile-based regression and fuzzy nonlinear regression analysis. Thus, it can handle asymmetries in the underlying distribution, is more robust and more flexible, leads in general to a higher model fit, has a higher predictive performance, and it provides an improved explainability and hence a better interpretability of the results. • A kernel-based nonlinear regression model is considered and the concept of fuzzy quantile functions of an LR fuzzy random variable is employed. In this way, a new nonlinear regression technique based on fuzzy quantiles with crisp predictors and LR fuzzy responses is developed. • A three-stage procedure involving a nonlinear quantileand kernel-based regression model and two nonlinear kernel-based regression models is suggested for parameter estimation. Within this three-stage estimation procedure, some hybrid algorithms are implemented using generalized cross-validation, least absolute errors as well as quantile loss function. The paper is structured as follows. In Sect.2, necessary basics on fuzzy numbers are briefly given and the concept of a fuzzy quantile is introduced. In Sect.3, the new methodology is presented to develop the fuzzy nonlinear quantile-based regression model. Afterward, in Sect. 4 an algorithm is discussed to estimate the model parameters. Section 5 presents four application studies where the
132 G.Hesamian et al. effectiveness of the new regression model is compared with various established fuzzy regression approaches. Finally, the paper concludes in Sect.6. 2 Fuzzy numbers andfuzzy quantile function In this section some basics are briefly discussed that we need to develop the nonlinear quantile-based regression model with crisp predictors and fuzzy responses. 2.1 Essentials onfuzzy numbers A fuzzy set A is characterized by a membership function 𝜇 A(x) that assigns membership degrees between zero and one to the elements x∈𝕏 , i.e., it holds 0≤𝜇 A(x)≤1 . A fuzzy number (FN) A is a fuzzy set, which is convex and normalized on ℝ , and 𝜇 A(x) is an upper semi-continuous function. Within the class of FNs so called LeftRight FNs (LR-FNs) A=(aL ,a,aU) LR , aL<a<aU , with are most prominent as they provide a promising possibility to model uncertainty in a simple and practical way. In (2.1), the continuous functions L(.) and R(.) characterize the left and right spreads of the LR-FN and it holds L(0)=R(0)=1 , L(1)=R(1)=0 . Let FLR(ℝ) denote the set of all LR-FNs, and LL-FNs stands for symmetric LR-FNs. The handling of LR-FNs is especially easy when considering a so called triangular fuzzy number (TFN) characterized by Let A=(a L ,a,a U )LR and B=(b L ,b,b U )LR be two LR-FNs, then the following operations are defined: • Addition: � A⊕� B=(a L +b L ,a+b,a U +b U )LR . • Scalar multiplication: The squared distance between two LR-FNs A and B is defined by (2.1) 𝜇 � A(x)= ⎧ ⎪ ⎨ ⎪ ⎩ L � a−x a−aL � if aL≤x≤a , R � x−a aU−a � if a<x<aU (2.2) 𝜇 A(x)= ⎧ ⎪ ⎨ ⎪ ⎩ x−aL a−aLif aL≤x≤a, aU−x a+aUif a≤x≤aU, 0 if x−[aL,aU] . (2.3) 𝜆⊗ � A= { (𝜆a L ,𝜆a;𝜆a U )LR if 𝜆> 0, (𝜆aU,𝜆a;𝜆aL) RL if 𝜆< 0.
133 A flexible soft nonlinear quantile-based regression model with c1 = ∫1 0 L−1(𝛼)d 𝛼 and c2 = ∫1 0 R−1(𝛼)d𝛼. There are several approaches to rank fuzzy quantities based on a real-valued criterion or a preference degree. Here, a simple ranking criterion is suggested to compare two LR-FNs. Definition 2.1 Let A= (aL , a ,a U ) LR and B= ( b L ,b,b U ) LR , then it holds A⪰ B if aL≥bL ,a≥b and aU≥bU . Further, the partial order “ ⪰ ” meets the properties below for three LR-FNs A , B and C : (1) A⪰ A . (2) If A⪰ B and B⪰ A then A= B . (3) If A⪰ B and B⪰ C then A⪰ C . Finally, there are different ways in expressing an LR-FN, e.g., A=(a L ,a,a U )LR is equivalent to A=(a;la,ra)LR with la=a−aL and ra=aU−a . 2.2 Fuzzy quantile function First, we define LR fuzzy random variables (LR-FRVs). Definition 2.2 Let (Ω,A,P) be a common probability space. Then the fuzzyvalued mapping Y∶Ω →F LR( ℝ ) is referred to as LR-FRV if the mappings YL ,Y,YU∶Ω → ℝ are random variables (RVs) with Y=(Y L ,Y,Y U )LR and P(YL<Y<YU)=1 . LR-FRVs Y1 and Y2 are i.i.d. if (Y1,Y2) , ( Y L 1 ,Y L 2) and ( Y U 1 ,Y U 2) are i.i.d. RVs. In addition, Y1,…, Yn is an LR fuzzy random sample if all the Yi ’s are i.i.d. LR-FRVs. An observed LR fuzzy random sample is denoted by y1,…,yn . Two types of LR-FRVs that are commonly employed are (1) Y = ((1−aY 1 )Y,Y,(1+aY 2 )Y) LR for a positive RV Y with aY 1 ,a Y 2 ∈(0, 1 ] and (2) Y =(Y−bY 1 ,Y,Y+bY 2 ) LR , where zero is part of the support of Y and bY 1 ,b Y 2 > 0 . In this paper we employ the notion of a fuzzy cumulative distribution function (c.d.f.) inspired by Hesamian and Chachi (2015): Definition 2.3 Let YL , Y and YU be common RVs with c.d.f. FYL , FY , and FYU , respectively, where P(YL<Y<YU)=1 . The fuzzy c.d.f. of Y at y=(yL ,y,yU)LR is then defined in the way D 2( A, B)=( | a−b |2 +c1 | (a−a L )−(b−b L ) |2 +c2 | (a U −a)−(b U −b) |2 ) 3, (2.5) F Y (y)=(F Y U(yL),F Y (y),F Y L(yU)) LR,
134 G.Hesamian et al. where FY U(y L )=P(Y U≤ y L) , FY(y)=P(Y≤y) and FY L(y U )=P(Y L≤ y U) . Definition 2.4 Let YL , Y and YU be common RVs with c.d.f. FYL , FY , and FYU , respectively, where P(YL<Y<YU)=1 . The fuzzy quantile function of Y at fuzzy quantile level 𝜏 =( 𝜏 L, 𝜏 , 𝜏 U)LR ∈ F LR(0, 1) is defined by with According to the definition of “ ⪰ ” (see Definition 2.1), we can state some properties for the fuzzy quantile of an LR-FRV: Lemma 2.1 Let Y =(YL ,Y,YU) LR then it holds: (1) Q Y ( F Y (y)) = y for any y =(y L ,y,y U ) LR (2) F Y ( Q Y (𝜏 )) ⪰ 𝜏 for any 𝜏 =(𝜏 L ,𝜏,𝜏 U ) L (3) � QI(Y)⊕ � b (𝜏)= � b⊕I(QY(𝜏 )) for any RV Y and level 𝜏∈(0, 1) with b =(bL ,b,bU) LR Proof According to Definitions 2.3 and 2.4, we have which proves (1). To prove (2), first, note that In addition, it holds According to Definition 2.1, it can be concluded that F Y ( Q Y (𝜏 )) ⪰ 𝜏 . Moreover, to prove (3), according to arithmetic operations of LR-FNs, we find that ◻ (2.6) Q Y (𝜏 )=(F−1 Y U(𝜏L),F−1 Y (𝜏),F−1 Y L(𝜏U)) LR, (2.7) F −1 YU(𝜏 L )=inf{y∶FYU(y)≥𝜏 L } , F−1 Y(𝜏)=inf{y∶FY(y)≥𝜏}, F −1 Y L(𝜏U)=inf{y∶F Y L(y)≥𝜏U} . (2.8) Q Y ( F Y (y))=(F−1 YU (F Y U(yL)),F−1 Y (F Y (y)),F−1 YL (F Y L(yU))) LR =(yL,y,yU) LR =y , (2.9) F Y ( Q Y ( t))=(F Y U(F−1 Y U(𝜏L)),F Y (F−1 Y (𝜏)),F Y L(F−1 Y L(𝜏U))) LR. (2.10) FY U(F −1 Y U(𝜏 L )) ≥𝜏 L ,F Y (F −1 Y (𝜏)) ≥𝜏,F Y L(F −1 Y L(𝜏 U )) ≥𝜏 U. (2.11) � Q Y⊕� b(𝜏)=(F−1 Y+bL(𝜏),F−1 Y+b(𝜏),F−1 Y+bU(𝜏))LR =(bL+QY(𝜏),QY(𝜏),bU+QY(𝜏)) LR = � b ⊕ I(QY( 𝜏 )).
135 A flexible soft nonlinear quantile-based regression model Following Lemma 2.1, it is obvious that the fuzzy quantile extends the conventional properties of quantile functions. This is true because we used the ranking criterion ⪰ in this study. Example 2.1 Consider an FRV Y=(Y L ,Y,Y U )LR , where YL∼exp( 𝜆 L) , Y∼exp(𝜆) , YU∼exp( 𝜆 U) with 𝜆L<𝜆<𝜆 U . First note that Therefore, according to Definition 2.4, we have with Example 2.2 Let � Y=�𝜇 ⊕ I(𝜖) where 𝜇 =(𝜇L ,𝜇,𝜇U)LR and 𝜖∼N(0, 𝜎2) . According to Lemma 2.1 (3), it holds � Q� Y (𝜏)=�𝜇 ⊕ I(Z 1−𝜏 𝜎)=(𝜇L+Z 1−𝜏 𝜎,𝜇+Z 1−𝜏 𝜎,𝜇U + Z1−𝜏 𝜎 )LR . Here, Z1−𝜏 is the (1−𝜏) -quantile of the standard normal distribution. 3 The fuzzy nonlinear quantile‑based regression model Let {( x i , y i )} i=1,…,n be a fuzzy data set, where x i =(x i1 ,…,x im ) ⊤ (fixed values) and yi ∈F LR (ℝ ) are observed values of the FRVs. Inspired by Balasundaram and Meena (2019), we consider the following fuzzy (kernel-based) nonlinear regression model where 1. Kh(xi,A)=(Kh(xi,x1),…,Kh(xi,xn)) with bandwidth h and kernel function Kh(., .) , 2. Yi =(YL i ,Y i ,YU i ) LR , 3. w=( w L , w , w U)LR with wL =(w L 1 ,…,w L n ) ⊤ , w=(w1,…,wn)⊤ and wU =(w U 1 ,…,w U n ) ⊤ . 4. 𝜖 i=𝜇 +I(𝜖i) indicates an LR fuzzy error term, where 𝜖1,…,𝜖n are i.i.d. normally distributed RVs with zero mean and variance 𝜎2 , and it holds 𝜇 =(𝜇L ,𝜇,𝜇U)LR . According to Definition 2.4, the proposed model (3.1) can be written in the form (2.12) FY (y)=1−e−𝜆y,F Y L(y)=1−e−𝜆 L y,F Y U(y)=1−e−𝜆 U y . (2.13) Q Y (𝜏 )=(F − 1 Y U(𝜏L),F − 1 Y (𝜏),F − 1 Y L(𝜏U)) LR, (2.14) F−1 YU(𝜏L)=inf{y∶FYU(y)≥𝜏L}= −1 𝜆Ulog(1−𝜏L), F−1 Y(𝜏)=inf{y∶FY(y)≥𝜏}=−1 𝜆log(1−𝜏), F −1 YL(𝜏U)=inf{y∶FYL(y)≥𝜏U}=−1 𝜆 Llog(1−𝜏U) . (3.1) � Yi=( K h( x i,A)⊗ w )⊕ �𝜖 i,i=1, …,n,
136 G.Hesamian et al. Now, the fuzzy quantile of an FRV introduced in Sect.2.2 is utilized for obtaining the fuzzy quantile of the model (3.2). Following Lemma 2.1, the fuzzy quantile of Yi can be evaluated at the exact quantile level 𝜏 in the way where � b (𝜏)=�𝜇 ⊕ I(Z 1−𝜏 𝜎)=(𝜇L+Z 1−𝜏 𝜎,𝜇+Z 1−𝜏 𝜎,𝜇U+Z 1−𝜏 𝜎) LR . Since K(xi,A) are positive quantities, the right-hand side of (3.3) is an LR-FN. Based on the second notation of LR-FNs (see Sect.2.1, last paragraph), the fuzzy quantile of Yi can be rewritten as where w=(w1,…,wn)⊤ , l w=(l w1 ,…,l wn ) ⊤ , r w=(r w1 ,…,r wn ) ⊤ . Considering the equality of two LR-FNs in (3.4), three separate (nonlinear) regression models can be stated in the following way: (L) l QY i =K hl (x i ,A)l w +lb , i=1, …,n , (C) QYi(𝜏)=Kh(xi,A)w+b(𝜏) , i=1, …,n , (R) r QY i =K hr (x i ,A)r w +rb , i=1, …,n . The parameters w =( � w 1 ,…, � w n ) ⊤ , 𝜏 , h and b (𝜏)=(b(𝜏);l b ,r b ) LR can be estimated by considering three ordinary nonlinear regression models L, C, R. To achieve this, we decompose the training patterns {(x i ,y i =(y i ;l y i,r y i) LR )} into three data sets (l y i,x i ) , (yi,xi) , and (r y i,x i ) , i=1, …,n . Thus, the center and the spreads of the fuzzy coefficients w =(w 1 ,…,w n ) ⊤ , 𝜏 , h and b(𝜏) can be separately estimated in each step. All regression coefficients should be simultaneously estimated in each step. Note that L and R provide two nonlinear regression models, while C is a nonlinear quantilebased regression model. 4 Parameter estimation In order to estimate the model parameters, the following steps can be implemented for the nonlinear regression models L, C, R defined in Sect.3. 4.1 Estimating theparameters ofmodel C As for parameter estimation related to the regression model QYi(𝜏)=Kh(xi,A)w+b(𝜏) (model C) using the data (yi,xi) , i=1, …,n , three target functions are required to (3.2) � Yi = ( (K h (x i ,A)⊗ w)⊕ �𝜇 ) ⊕I(𝜖 i ) . (3.3) � Q � Yi (𝜏)= ( (Kh(xi,A)⊗ w)⊕ �𝜇 ) ⊕I(Z1−𝜏𝜎)) = (K(xi,A)⊗ w)⊕ � b(𝜏) , (3.4) Q Y i (𝜏)=(Kh(xi,A)w+b(𝜏);K(xi,A)lw+lb,K(xi,A)rw+rb)LR ,
143 A flexible soft nonlinear quantile-based regression model lower value of RMSE for the introduced model in this paper. Thus, the predictive performance of the newly proposed fuzzy nonlinear quantile-based regression model is better compared to the other models, and therefore it is superior to all the competitors for the investigated data set in this example. Example 5.2 This example is based on a fuzzy data set that is related to the development of a sophisticated software system to run a virtual mall, where the programmers worked in parallel with a team of seven beta testers (D’Urso & Gastaldi, 2002; Hesamian etal., 2024). During the 30-months software development, the team of beta testers have written comprehensive monthly reports. In any of these reports, the team provided an overall average score in the form of a symmetric TFN. In the following, we use the fuzzy univariate nonlinear regression model for the above data set, where K(xi,A)=(K(xi,x1),…,K(xi,xn)) and The optimal values of h, hl , 𝜏 , b ( 𝜏 )=( b ( 𝜏 ); l b)T , and w can be determined by means of the following two stages: (L) l QY i =K hl (x i ,A)l w +lb , i=1, …, 30 , (5.5) � Yi =(K h (x i ,A)⊗ w)⊕ �𝜖 i ,i=1, … , 30, (5.6) K h(xi,xj)= √ 1 2𝜋 e−0.5 (xi−xj)2 h . Table 1 Values of yi =(y i ;l yi ) LL , S (y i , y i) and D2(y i , y i) for the proposed method in Example 5.1 No yi =(y i ;l yi ) LL S ( yi , yi) D 2( yi , yi) 1 (12.30;1.845)LL 1 0 2 (20.90;3.135)LL 1 0 3 (39;5.85)LL 1 0 4 (47.90;0.185)LL 1 0 5 (5.60;0.84)LL 1 0 6 (25.90;3.885)LL 1 0 7 (37.30;5.595)LL 1 0 8 (21.90;3.285)LL 1 0 9 (18.10;2.715)LL 1 0 10 (21;3.15)LL 1 0 11 (34.90;5.23)LL 1 0.00005 12 (57.20;8.58)LL 1 0 13 (0.69;1.0389)LL 0.527 1.74 14 (25.90;3.885)LL 1 0 15 (54.90;8.235)LL 1 0
144 G.Hesamian et al. (C) Q Y i(𝜏)=K h (x i ,A)w+b(𝜏) , i=1, …, 30 . That is, (1) the optimal values of hl , l wh l and lb can be evaluated via steps L1–L2 based on the observed values (l y i,x i ) , i=1, …, 30 , and (2) the unknown model components h, wh and b(𝜏) can be estimated based on steps C1–C4 by using the observed data set (yi,xi) , i=1, …, 30 . The values of y i =(y i ;l yi ) LL , D 2(y i , y i) , and S (y i , y i) are given in Table3. In this example, we compare our method with the fuzzy regression models by De Hierro etal. (2016), D’Urso and Gastaldi (2002), Table 2 Comparative analysis in Example 5.1 Method Model components MSM RMSE D’Urso and Gastaldi (2002) y (x)=( f(x);l f(x) ) LL f ( x )=−120 +38.76 x1 −2.415 x2 +3.26 x3 0.63 5.36 l f(x)=10.5 +6.25x 1 +1.71x 2 −4.46x3 Kula and Apaydin (2008) 𝛽0 = (−120.36;24.26) LL , 𝛽1 =(32.95;7.21) LL 𝛽2 = (−4.01;1.22) LL , 𝛽3 =(2.62;2.64) LL 0.618 5.75 Wang etal. (2007) h=8 0.639 5.239 Hesamian and Akbari (2020b) h=2 , k=8 0.702 4.55 Hong and Hwang (2003) w0 = (−145.32;18.21) LL , w1 =(29.51;9.29) LL w2 = (−3.21;1.44) LL , w3 =(3.41;3.24) LL 0.65 5.15 Asadolahi etal. (2021) w0=( 11.80;1.30 )LL , w1= (− 1.04;0.0006 )LL w2=(2.61;0.33)LL , w3 =(3.74;2.60) LL w4 =(6.43;0) LL , w5 =(0.08;0) LL w6=( 3.79;0 )LL , w7=( 4.46;0.20 )LL 0.734 3.59 w8 =(6.21;0.0001) LL , w9 =(1.21;0) LL w10 =(2.54;0) LL , w11 =(6.39;0.95) LL w12 =( 6.37;3.36 )LL , w13 = (− 1.14;0 )LL w14 =(2.39;0.69) LL , w15 =(7.32;0.52) LL Proposed model b=( 26.68;0.10 )LL, w1=( 20.17;0.12 )LL, w2 = (−23.79;0.784) LL, w3 =(19.24;0.54) LL w4 =(27.42;1.63) LL, w5 = (−25.67;0.26) LL, w6= (− 2.91;0.50 )LL, w7=( 11.49;0.17 )LL 0.927 1.899 w8 = (−39.73;0.99) LL, w9 =(7.57;0.44) LL, w10 = (−6.89;0.52) LL, w11 =(3.84;0.43) LL w12 =( 49.72;4.37 )LL, w13 = (− 40.40;0.13 )LL, w14 =(12.51;0.72) LL , w15 =(26.48;4.73) LL 𝜏 =0.7 , h=4 , hl = 1
145 A flexible soft nonlinear quantile-based regression model Kula and Apaydin (2008), Chachi etal. (2016), Wang etal. (2007), Ferraro etal. (2010), Hesamian and Akbari (2020a, 2020b), Hong and Hwang (2003), and Asadolahi etal. (2021). In Table4 the results of our comparative analysis including estimated model parameters and performance measures can be seen. Analyzing the results in Table4, we find the highest value of MSM and the lowest value of RMSE for our proposed regression model ( MSM =0.82 , RMSE =1.46 ). Thus, the new model provides a higher degree of similarity on the one hand and a lower prediction error on the other hand in comparison to the fuzzy regression models under consideration, confirming the superiority of the nonlinear quantile-based regression model over the other approaches. Table 3 Values of yi =(y i ;l yi ) LL , S (y i , y i) and D2(y i , y i) for the presented model in Example 5.2 No yi =(y i ;l yi ) LL S ( yi , yi) D 2( yi , yi) 1 (6.0293;1.99721)T 0.798339 0.0851244 2 (5;3.4)T 1 0 3 (8;5.3)T 1 0 4 (13;2.4)T 0.0140845 9.48 5 (19;5.331)T 0.0257109 38.8636 6 (19;5.6)T 1 0 7 (20;4.3)T 1 0 8 (25.3461;3.01)T 0.245816 5.53222 9 (25;2.2)T 1 0 10 (26.9619;4.35)T 0.982134 0.00228494 11 (26;5.1)T 1 0 12 (23.8419;3.0245)T 0.54447 1.8018 13 (5.11;1.2843)T 0.813722 0.0234222 14 (8;2.1)T 1 0 15 (10;5.02393)T 0.570272 2.10891 16 (10;4.1)T 1 0 17 (13;4.498)T 0.597673 1.5616 18 (13;3.5)T 1 0 19 (20;6.4)T 1 0 20 (28;7.7)T 1 0 21 (29;5.2)T 1 0 22 (30.0157;3.85790)T 0.89707 0.0653973 23 (35;5.2)T 1 0 24 (44;6.3)T 1 0 25 (44.3;6.82091)T 0.81555 0.506267 26 (43.0991;5.15)T 0.962597 0.0106541 27 (47;4.2)T 1 0 28 (48;5.3)T 1 0 29 (49.0162;5.13680)T 0.693062 1.88994 30 (48.36;5.3896)T 0.795188 0.413663
146 G.Hesamian et al. Table 4 Comparative analysis in Example 5.2 Method Model components MSM RMSE De Hierro etal. (2016) y ( x )=( f ( x ); s ( x )) T f ( x )=−14.419 +13.516 x −1.724 x 2−0.081 x 3−0.001 x4 0.39 9.87 s(x)=0.581 +2.29x−0.26x2+0.01x3−0.0001x4 D’Urso and Gastaldi (2002) y (x)=( f(x);l f(x) ) T f(x)= 5.31 + 2.52 x− 0.18 x 2 + 0.005 x3 0.31 12.10 l f(x)=5.96 −0.03x Kula and Apaydin (2008) 𝛽0 =(3.18;1.12) T , 𝛽1 =(1.5;0.4) T 𝛽2=( 0.19;0.13 )T 0.31 12.27 Chachi etal. (2016) 𝛽0 =(2.45;0.52) T , 𝛽1 =(0.66;0.17) T 𝛽2=( 0.000001;0 )T 0.28 13.73 Wang etal. (2007) h=0.08 0.41 10.49 Ferraro etal. (2010) y (x)=(y(x);l y(x) ) T y(x)=−18.21 +14.74x−1.66x2+0.072x3−0.001x4 l y ( x )=−0.17 +2.29x−0.17x 2 +0.004x 3 −1.96 ×10 −5 x 4 0.43 10.23 Hesamian and Akbari (2020a) 𝛽0 =(0.019;1.56) T , 𝛽1 =(1.68;0.232) T 𝛽2=( 0.307;0 )T , b1= (− 1.024;0 )T b2 =(1.85;0) T , b3 = (−1.198;0.00001) T 0.42 9.82 K=3 , t1=5, t2=12, t3=16 , 𝜆=1 Hesamian and Akbari (2020b) h=5 , k=1 0.40 10.208 Hong and Hwang (2003) w0= (− 14.41;0.58 )T , w1=( 13.51;2.29 )T w2 = (−1.72;0.26) T , w3 =(0.08;0.01) T 0.46 8.24 Asadolahi etal. (2021) w0 =(10.72;0.93) T , w1 = (−2.72;0) T w2= (− 2.42;0 )T , w3= (− 1.59;0.132 )T w4 = (−0.89;0.882) T , w5 = (−0.1;1.35) T w6 =(0.245;1.480) T , w7 =(2.64;1.32) T w8=( 3.55;1 )T , w9=( 4.19;0.665 )9T w10 =(4.83;0.402) T , w11 =(3.06;0.242) T w12 =(1.07;0.170) T , w13 =(0.33;0.148, 0) T w14 = (− 2.84;0.158 )T , w15 = (− 2.94;0.211 )T 0.55 7.47 w16 = (−3.30;0.333) T , w17 = (−1.78;0.529) T w18 =(0.94;0.764) T , w19 =(1.46;0.954) T w20 =( 2.11;0.991 )T , w21 =( 2.67;0.807 )T w22 =(3.89;0.437) T , w23 =(2.23;0.001) T w24 =(2.11;0) T , w25 =(2.82;0) T
147 A flexible soft nonlinear quantile-based regression model Example 5.3 In this third example we consider a real data set with 21 observations that aims to analyze the relationship between the atmospheric concentration of carbon monoxide ( y ) and three meteorological predictor variables in Rome: humidity ( x1 ), rain ( x2 ), wind speed ( x3 ) (Hesamian etal., 2024). The response variable has been reported in the form of LR-FNs with L(x)=exp(−x) , R( x )=1∕(1+ x 2) , while the predictor variables have crisp values. Wang etal. (2007) employed the following fuzzy multiple regression model to capture the relation between y and x1,x2,x3 : Here, we investigate the above relationship by implementing the following model: The results of the comparative analysis are summarized in Table 5. We consider 13 competitors of various kinds in order to obtain a representative and comprehensive comparison: Choi and Yoon (2010), Zeng etal. (2017), Taheri and Kelkinnama (5.7) � yi=�w0⊕ 3 ⨁ j=1 (�wij ⊗xij)⊕ �𝜀 i,i=1, … , 21 (5.8) � Yi =(K h (x i ,A)⊗ w)⊕ �𝜖 i ,i=1, … , 21 Table 4 (continued) Method Model components MSM RMSE w26 =( 3.56;0 )T , w27 =( 5.51;0.178 )T w28 =(7.25;0.686) T , w29 =(8.92;1.136) T w30 =(9.51;1.418) T Proposed model b (𝜏 )=(17.280;0, 0) T ,w 1 = (−11.3173;0.58, 7.01) T, w2= (− 7.1110;1.33, 0 )T , w3= (− 6.5733;0.71, 4.48 )T, w4 = (−4.0607;5.14, 4.45) T , w5 = (−7.4598;5.45, 2.73) T, w6 =(3.8772;0, 4.07) T , w 7 = (−4.9458;3.87, 0.91) T w8=( 4.7932;6.21, 2.07 )T , w9= (− 2.8439;0, 2.36 )T, w10 =(10.0855;2.42, 5.71) T , w11 = (−8.9716;5.00, 0) T w12 =(21.2993;0.01, 4.66) T , w 13 = (−28.1973;1.18, 0) T, w14 =( 5.1703;0.91, 0 )T , w15 = (− 10.9986;5.58, 4.07 )T, w16 = (−3.8492;0, 0) T , w17 = (−3.7330;3.02, 4.12) T, 0.82 1.46 w18 = (−5.7769;0, 5.86) T , w 19 = (−1.5642;4.25, 0) T w20 =( 6.6130;0.93, 6.11 )T , w21 =( 1.9484;6.53, 3.15 )T, w22 =(4.8021;0, 4.22) T , w23 =(3.3954;0.63, 0) T w24 =(14.2488;0, 2.17) T , w 25 =(12.2567;4.26, 3.72) T, w26 =( 6.5624;1.97, 0 )T , w27 =( 15.0507;0.26, 7.96 )T, w28 =(11.9609;1.17, 0) T , w29 =(11.2484;3.44, 1.57) T w30 =(21.4031;2.51, 2.50) T ,𝜏 =0.7, h= h l = 0.7
148 G.Hesamian et al. Table 5 Comparative analysis in Example 5.3 Method Model components MSM RMSE Choi and Yoon (2010) w0 =(0.769;0.361, 0.905) T , w1 =(0.048;0.012, 0.002) T w2= (− 0.644;1.101, 0.775 )T , w3 = (−0.988;0.520, 0.232) T 0.54 11.10 Zeng etal. (2017) w0=( 1.53;0.147, 1.50 )T , w1=( 0.037;0.024, 0.002 )T w2 =(0.753;0.189, 0.234) T , w3= (− 1.025;0.009, 0.005 )T 0.58 9.025 Taheri and Kelkinnama (2012) w0 =(1.34;1.12, 0.87) T , w1 =(0.142;0.023, 0) T w2= (− 2.750;0, 0 )T , w3= (− 1.038;0, 0 )T 0.50 11.69 Kula and Apaydin (2008) w0=( 0.535;0.361, 0.014 )T , w1=( 0.048;0.018, 0.014 )T w2= (−0.835;0.398, 0)T , w3 = (−0.904;0, 0.020) T 0.56 9.87 Choi and Buckley (2008) w0=( 2.99;0.101, 0.020 )T , w1=( 0.029;0.023, 0.020 )T w2= (− 1.278;0, 0 )T , w3= (− 1.14;0, 0 )T 0.55 10.36 Wang etal. (2007) h=5.25 0.58 9.03 Jiang etal. (2013) w0 =(0.73;0.33, 0.90) T , w1 =(0.69;0, 0) T w2 = (−0.49;0.83, 0.51) T , w3 = (−1.03, 0.70, 0.079) T , h=0.5 0.51 12.93 Atalay etal. (2015) w0 =(0.39;0.60, 0.41) T , w1 =(0.106;0, 0.008) T w2= (− 0.59;0.52, 0.005 )T , w3= (− 1.049;0, 0.016 )T 0.60 8.63 h= 0.5, k1= k2=1 Icen and Demirhan (2016) w0 =(1.37;0.172, 0.012) T , w1 =(0.037;0, 0.012) T w2= (−0.75;0.143, 0.193)T , w3 = (−1.02;0, 0.011) T 0.57 9.53 Hesamian and Akbari (2020b) h=1 , k=0.5 0.53 10.73 Arefi (2019) w0=( 0.341;1.034, 0.821 )T , w1=( 0.043;0.121, 0.142 )T w2= (− 0.721;0.052, 0.072 )T , w3 = (−0.008;0.010, 0.012) T , 𝜏 =0.5 0.56 8.54 D’Urso and Gastaldi (2002) y(x)=(f(x);l f(x) ,r f(x) ) T f(x)=0.420152 +0.038745x1−0.82130129x2−1.02135975x3 lf ( x ) =0.05 +0.0286x 1 −0.200x 2 −0.05219x3 rf(x)=−0.384175858 +0.0329x 1 −0.0225264x 2 −0.322588x3 0.502 11.01 Asadolahi etal. (2021) w0=( 3.605;0, 0 )T , w1= (− 4.010;0, 0 )T w2 = (− 2.1610;0, 0 ) T , w3 =(0.9986;0, 0) T w4 =(3.2599;1.60, 1.55) T , w5 =(0.9369;1.67, 1.80) T w6= (− 0.7129;0.165, 0 )T , w7= (− 1.4941;0.71, 0 )T w8 =(3.9898;0.90, 0.74) T , w9 =(2.2226;0.49, 1.08) 9T w10 =(0.6054;0, 0.07) T , w11 = (−0.3640;0.42, 0.65) T 0.64 7.23 w12 =( 1.1437;1.53, 2.17 )T , w13 =( 0.3731;1.60, 0.14 )T
149 A flexible soft nonlinear quantile-based regression model (2012), Kula and Apaydin (2008), Choi and Buckley (2008), Wang etal. (2007), Jiang etal. (2013), Atalay etal. (2015), Icen and Demirhan (2016), Hesamian and Akbari (2020b), Arefi (2019), D’Urso and Gastaldi (2002), and Asadolahi et al. (2021). In this example we observe the same clear picture as in the previous examples: the highest value of MSM and the lowest value of RMSE can be found for our newly proposed fuzzy regression model. That is, considering the degree of similarity as well as the predictive performance of all the compared models, we have a clear best method, and this is the fuzzy nonlinear quantile-based regression model with crisp predictor and fuzzy response variables. Example 5.4 In this example, we analyze a large data set of Shanghai housing price data (Hesamian etal., 2024). The fuzzy dependent variable represents the acceptable purchase price, expressed by TFNs y i =(y i ;l y i,r y i) T . Additionally, there are six crisp explanatory variables: housing size ( x1 ), mortgage interest rate ( x2 ), real estate tax ( x3 ), down payment ratio ( x4 ), annual household income ( x5 ), family population ( x6 ). According to the proposed method, the fuzzy predicted values can be determined via yi =(y i ;l yi ,r yi ) T , i=1, 2, …, 147 , where Table 5 (continued) Method Model components MSM RMSE w14 = (−2.3193;0.43, 0.41) T , w15 = (−0.9893;0, 0) T w16 = (−0.5327;0, 0) T , w17 = (−1.9571;0, 0) T w18 = (−1.2844;1.50, 0.93) T , w19 = (−1.2687;0, 0) T w20 = (− 0.9986;0, 0 )T , w21 =( 0.8715;0, 0 )T Proposed model b ( 𝜏 )=(2.6461;0, 0)T, w 1= (−1.68;0, 0)T, w 2=(0.0769;0.53, 0.30)T, w3=( 0.7219;0.53, 0.76 )T , w4=( 3.2956;0, 1.39 )T, w5=( 1.0787;1.42, 1.42 )T , w6= (− 1.1128;1.80, 0 )T, w7 = (−0.1529;0.79, 0.24) T , w 8 =(0.6294;0.21, 1.39) T, w9 =(0.7415;1.41, 0.98) T , w 10 =(1.3161;0.97, 1.50) T, 0.87 4.67 w11 =( 0.0930;1.35, 0.98 )T , w12 =( 1.1248;1.62, 2.05 )T, w13 = (−0.6108;2.42, 0.84) T , w 14 = (−1.5312;0, 0) T, w15 =( 0.9069;0, 0.26 )T , w16 = (− 3.0121;0, 0 )T, w17 = (−0.3858;0.35, 0.57) T , w 18 = (−0.4612;1.24, 0.68) T, w19 = (−2.0747;0.18, 0) T , w 20 =(0.3412;0, 0) T, w21 =( 1.4886;1.49, 0.06 )T 𝜏 = 0.7, h= hl= hr=0.7
150 G.Hesamian et al. (L) l y i=K hl(x i ,A)l w +l b , (C) y i =K h (x i ,A) w+ b(𝜏 ) , (R) r y i=K h r(x i ,A)r w +r b with xi =(x i1 ,x i2 ,…,x i6 ) ⊤ . The parameters 𝜏 , hl , h , hr , l w = ( l �w1 ,l �w2 ,…,l �w147 ) ⊤ , w = (� w 1 , � w 2 ,…, � w 147 ) ⊤ , r w = ( r �w1 ,r �w2 ,…,r �w147 ) ⊤ , l b , b(𝜏 ) and r b can be estimated via Table 6 Estimated coefficients and performance measures for the considered fuzzy regression models in Example 5.4 Method Model components MSM RMSE Zeng etal. (2017) w0 = (−570.59;5.236, 7.20) T , w1 =(8.243;1.773, 0.982) T w2= (− 7.238;0.3058, 0.8429 )T , w3= (− 52.663;3.528, 3.762 )T 0.61 30.55 w4 = (−0.127;0.432, 0.472) T , w5 =(11.856;1.89, 1.3621) T w6 = (−3.275;0.0893, 0.657) T Choi and Buckley (2008) w0= (− 496.6;127.25, 13048 )T , w1=( 7.26;1.25, 0.968 )T w2 = (−4.256;0.625, 1.025) T , w3 = (−45.628;17.236, 11.25) T 0.47 35.94 w4 = (−0.471;1.528, 0.568) T , w5 =(7.25;0.528, 1.856) T w6= (− 2.369;1.745, 0.658 )T Asadolahi etal. (2021) Step 1: C=30 , 𝛾 =4 , 𝜖 =3 , 𝜇 =0.01 Step 2: C=20 , 𝛾 =1 , 𝜖 =2 , 𝜇 =0.02 0.68 47.26 Step 3: C=30 , 𝛾 =2 , 𝜖 =1 , 𝜇 =0.01 Khammar etal. (2021) w0 = ((−570.63;0;0) T , w1 =(7.61;0.47;0) T w2 = (−5.78;0;0) T , w3 = (−54.47;0;0.04) T w4= ((− 0.7;0;0 )T , w5=( 14.05;3.60;3.16 )T 0.71 37.94 w6 = (−2.10;0;0) T , h=26.4 D’Urso and Gastaldi (2002) y (x)=(f(x);l f(x) ,r f(x) ) T f(x)=−572 +8.467x1−6.925x2−50.227x3 −0.4328x4+9.862x5−3.106x6 lf(x)=−134 +4.58x1−1.725x2+36.025x3 0.61 30.74 −2.324x4 + 1.253x5 − 22.35x6 rf(x)=−90.3 +1.115x1+1.28x2+11.925x3 +0.12x4+2.325x5+2.358x6 Wang etal. (2007) h=5 0.54 32.24 Cheng and Lee (1999) h=3.5 0.50 34.76 Proposed model 𝜏 = 0.55, b(𝜏 )=(− 571;0, 0 )T , h= 4.4, hl= 2.1, hr=2.2 0.84 24.67
151 A flexible soft nonlinear quantile-based regression model three distinct optimization algorithms L, C and R proposed in Sect.4. The results can be found in Table 6. As there are numerous estimated parameters { l w , w,r w} , they are not included in this table. Considering the performance measures we get MSM =0.84 and RMSE =24.67 . We compare the results of our model with the respective results of several other models including (Zeng et al., 2017; Choi & Buckley, 2008; Asadolahi etal., 2021; Khammar etal., 2021; D’Urso & Gastaldi, 2002; Wang etal., 2007), and Cheng and Lee (1999). The results of all the models are presented in Table6. Upon interpreting the values, it is evident that the proposed model achieves better results. Consistent with the previous examples, these findings reaffirm the superiority of the proposed quantile-based model over various fuzzy regression models for the considered data set. 6 Conclusions The proposed method in this paper is the first quantile-based soft regression model that can embed nonlinearities in the modeling while the previous approaches in the literature are restricted to the inflexible linear case. The introduced model is based on quantiles of fuzzy random variables and embeds them into fuzzy nonlinear regression modeling. To this aim, the quantile function of a fuzzy random variable was employed and its properties were investigated in relation to LR fuzzy numbers. To estimate the regression coefficients, three ordinary nonlinear regression models were obtained based on the center, left, and right spreads of the introduced quantile regression model. In each stage, some of the model parameters are estimated by implementing hybrid optimization algorithms, kernel-based fitting, quantile loss function, and generalized cross-validation criteria. We have conducted comprehensive comparative analysis by performing three case studies based on real data sets and by including numerous competitors, i.e., fuzzy regression models of various kinds that have been proven to be superior in previous studies. In the framework of these comparisons it has been shown that both the degree of similarity and the predictive power of the proposed model is better compared to all the competitors. Beyond the innovative methodology, the proposed model has various applicability and management implications, which we briefly discuss in the following. There are mainly four applicability implications, namely the robustness to data asymmetry and outliers (i.e., ideal for financial risk analysis, housing price predictions and medical studies), flexibility in real-world data modeling (well-suited for complex systems with uncertainty, e.g., in environmental studies and supply chains), broad domain applicability (like in agriculture, climate studies and health research), and improved predictive performance (i.e., reliable for high-stakes decision-making). As for management implications, we would like to highlight five, namely enhanced decision-making (identifies risks and opportunities across data quantiles, supporting strategic choices), operational efficiency (efficient for big data, aiding industries like retail and logistics), resilient planning (accounts for uncertainty, helping leaders in volatile contexts like market forecasting), policy implications (aids targeted interventions and resource allocation for policymakers), and cross-disciplinary synergy
152 G.Hesamian et al. (encourages collaboration between data scientists and decision-makers for better outcomes). Finally, as the introduced quantile-based regression model is focused on LR fuzzy numbers, it would be a promising future path to transfer the proposed method to other types of fuzzy numbers. Extensions of the model to other types of fuzzy quantities such as Pythagorean fuzzy sets, Z-fuzzy clouds, and hesitant fuzzy linguistic term sets are other potential topics for future studies. Acknowledgements The authors would like to thank the editor and both anonymous reviewers for their valuable feedback and suggestions, which were important and helpful to improve the paper. Funding Open Access funding enabled and organized by Projekt DEAL. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/ licenses/by/4.0/. References Akbari, M.G. & G. Hesamian (2022). A fuzzy linear regression model with autoregressive fuzzy errors based on exact predictors and fuzzy responses. Computational and Applied Mathematics, 41, Article number: 284. Arefi, M. (2019). Quantile fuzzy regression based on fuzzy outputs and fuzzy parameters. Soft Computing, 24, 311–320. Asadolahi, M., Akbari, M. G., Hesamian, G., & Arefi, M. (2021). A robust support vector regression with exact predictors and fuzzy responses. International Journal of Approximate Reasoning, 132, 206–225. Atalay, K. D., Eraslan, E., & Cinar, M. O. (2015). A hybrid algorithm based on fuzzy linear regression analysis by quadratic programming for time estimation: An experimental study in manufacturing industry. Journal of Manufacturing Systems, 36, 182–188. Balasundaram, S., & Meena, Y. (2019). Robust support vector regression in primal with asymmetric huber Loss. Neural Processing Letters, 49, 1399–1431. Chachi, J., & Chaji, A. (2021). An OWA-based approach to quantile fuzzy regression. Computers and Industrial Engineering, 159, 107498. Chachi, J., Taheri, S. M., & Pazhand, H. R. (2016). Suspended load estimation using L1 -fuzzy regression, L2 -fuzzy regression and MARS-fuzzy regression models. Hydrological Sciences Journal, 61, 1489–1502. Cheng, C. B., & Lee, E. S. (1999). Non-parametric Fuzzy Regression K -NN and Kernel Smoothing Techniques. Computers and Mathematics with Applications, 38, 239–251. Choi, S. H., & Buckley, J. J. (2008). Fuzzy regression using least absolute deviation estimators. Soft Computing, 12, 257–263. Choi, S. H., & Yoon, J. H. (2010). General fuzzy regression using least squares method. International Journal of Systems Science, 41, 477–485. Chukhrova, N., & Johannssen, A. (2019). Fuzzy regression analysis: systematic review and bibliography. Applied Soft Computing, 84, 105708. D’Urso, P., & Gastaldi, T. (2002). An orderwise polynomial regression procedure for fuzzy data. Fuzzy Sets and Systems, 130, 1–19.