A general approach to fuzzy regression models based on different loss functions

Abstract

In this paper, a new general approach is presented to fit a fuzzy regression model when the response variable and the parameters of model are as fuzzy numbers. In this approach, for estimating the parameters of fuzzy regression model, a new definition of objective function is introduced based on the different loss functions and under the averages of differences between the $\alpha$-cuts of errors. The application of the proposed approach is studied using a simulated data set and some real data sets in the presence of different types of outliers.

1 Introduction

The fuzzy regression analysis is based on some regression models between a response variable and some explanatory variables when some quantities are as imprecise. Also, among the observed data, we may encounter with some outliers. In such situations, we need to introduce a regression model which supports these limitations. An approach for solving these limitations is using of the robust fuzzy regression models.

In the following, we review some works on robust fuzzy regression models. Arefi (2020) studied a robust fuzzy regression based on a generalized quantile loss function under the fuzzy outputs and fuzzy parameters. Chang and Lee (1994) presented fuzzy least absolute deviations regression based on the ranking of fuzzy numbers. An approach to fit a robust least squares fuzzy regression based on a kernel function is investigated by Khammar et al. (2020). Oussalah and De Schutter (2002) proposed a fuzzy regression model with the combination of least trimmed squares (LTS) and least median squares (LMS). They studied the performance of the proposed model when the data are contaminated by outliers. Sanli and Apaydin (2004) investigated a robust estimation procedure for fuzzy linear regression model with fuzzy input–output data based on the least median squares method. A robust approach to model a fuzzy linear regression based on M-estimators is studied by Shon (2005). Choi and Buckley (2008) utilized the least absolute deviations (LAD) method for estimating the parameters of a fuzzy regression model and investigated the performance of the proposed model under the fuzzy outlier data. D’Urso et al. (2011) and D’Urso and Massari (2013) proposed a robust fuzzy linear regression model with crisp inputs and fuzzy outputs based on the least median squares-weighted least squares (LMS-WLS) estimation procedure. Some approaches to fit the fuzzy regression models based the least absolutes method are investigated by Chachi and Taheri (2016), Taheri and Kelkinnama (2012), and Zeng et al. (2016). Based on the least trimmed squares estimation, Chachi and Roozbeh (2017) proposed a estimation procedure for determining the coefficients of a fuzzy regression model with crisp input–fuzzy output data. A weighted least-squares fuzzy regression model under crisp input–fuzzy output data and fuzzy coefficients is provided by Chachi (2019). For considering some other approaches of fuzzy regression models, see Chen and Hsueh (2007, 2009), Nasrabadi and Hashemi (2008), Hao and Chiang (2008), Kula et al. (2012), Mosleh et al. (2010), Arefi and Taheri (2015), Lopez et al. (2016), Hesamian and Akbari (2019), and Rapaic et al. (2019).

This paper is organized as follows: In Sect. 2, some preliminary concepts about fuzzy sets and fuzzy numbers are provided. In Sect. 3, a new general approach based on the different loss functions is investigated to fit the fuzzy regression models when the response variable and the parameters of models are as fuzzy numbers. In this section, we present some indices of goodness of fit to evaluate the proposed fuzzy regression models. Also, the cross-validation method is provided to examine the predictive ability of the proposed fuzzy regression models. Some numerical examples to assess the effectiveness of the proposed method are presented in Sect. 4. Finally, in Sect. 5, some concluding remarks are provided.

2 Preliminary concepts

In this section, we recall some notations and preliminary concepts on fuzzy sets (see Zimmermann 2001).

Let $\Omega$ be an universal set. A fuzzy set $\tilde{N}$ of $\Omega$ is defined by the membership function $\tilde{N}:\Omega \rightarrow [0, 1]$. The $\alpha$-cut of $\tilde{N}$ is as $\tilde{N}[\alpha ]=\{x\in R:\tilde{N}(x)\ge \alpha \}$ for $0<\alpha \le 1$.

Definition 1

A fuzzy set $\tilde{N}$ on $\Omega$ is called a fuzzy number, if

• (i) $\widetilde{N}(x)=1$ for some $x\in \Omega$,

• (ii) $\widetilde{N}[\alpha ]$ is a closed bounded interval for $0<\alpha \le 1$.

Definition 2

Let $\tilde{N}$ be a fuzzy number, then a LR fuzzy number is defined by the following membership function

$$\begin{aligned}\tilde{N}(x)= \left\{ \begin{array}{ll} L\big (\frac{m-x}{l}\big ) &{} x\le m, \\ R\big (\frac{x-m}{r}\big ) &{} x >m, \end{array}\right. \end{aligned}$$

where $l, r\ge 0$ and L(.) and R(.) are the strictly decreasing functions as $L, R :R^+ \rightarrow [0,1]$. It is denoted by $\tilde{N}=(m, l,r)_{LR}$.

Remark 1

In a LR fuzzy number $\tilde{N}$, if $L(x) = R(x)$, then $\tilde{N}$ is called the LL fuzzy number and is denoted as $\tilde{N}=(m,l,r)_{LL}$. For $L(x) = R(x) = 1-x$ for all $x\in [0,1]$, $\tilde{N}$ is called a triangular fuzzy number and is denoted by $\tilde{N}=(m,l,r)_T$. Also, for $l=r$, $\tilde{N}$ is a symmetric triangular fuzzy number as $\tilde{N}=(m,l)_T$.

Proposition 1

Assume that $\tilde{A}=(m_a, l_a,r_a)_{LR}$ and $\tilde{B}=(m_b, l_b,r_b)_{LR}$ are two LR fuzzy numbers and $\lambda \in R-\{0\}$. Some of the arithmetic operations based on extension principle (Zimmermann 2001) are given as follows

$$\begin{aligned} \lambda \otimes \tilde{A}= & {} \left\{ \begin{array}{ll} (\lambda m_a,\lambda l_a,\lambda r_a)_{LR} &{} \lambda >0,\\ (\lambda m_a,-\lambda r_a,-\lambda l_a)_{RL} &{} \lambda <0, \end{array}\right. \\ \tilde{A}\oplus \tilde{B}= & {} (m_a+m_b, l_a+l_b, r_a+r_b)_{LR}. \end{aligned}$$

3 Methodology

In this section, we introduce a new approach in fuzzy regression theory based on the concept of $\alpha$-cuts for the crisp inputs and the fuzzy output when the parameters of model are as fuzzy quantities.

3.1 Fuzzy regression model

In the following, we introduce fuzzy regression model under crisp input–fuzzy output variables. Our aim is to fit an optimal fuzzy linear regression model to this data set. Suppose that the fuzzy linear regression model based on crisp input–fuzzy output variables and the fuzzy parameters is given as follows

$$\begin{aligned} \tilde{Y}_i=\tilde{\beta _0}\oplus (\tilde{\beta _1}\otimes x_{i1})\oplus (\tilde{\beta _2}\otimes x_{i2})\oplus \cdots \oplus (\tilde{\beta _p}\otimes x_{ip}), \end{aligned}$$

where, $\tilde{\beta _j}=(\beta _j, \gamma _j)_{LL}$, $j=0,\ldots ,p$, and $\widetilde{Y}_i=(y_i, s_{i})_{LL}$, $i=1,\ldots ,n$. The estimated fuzzy response variables are obtained as follows:

$$\begin{aligned} \nonumber \hat{\tilde{Y_i}}= & {} \tilde{\beta _0}\oplus (\tilde{\beta _1}\otimes x_{i1})\oplus (\tilde{\beta _1}\otimes x_{i2})\oplus \cdots \oplus (\tilde{\beta _p}\otimes x_{ip})\\= & {} \bigg (\sum _{j=0}^{p}\beta _jx_{ij}, \sum _{j=0}^{p}\gamma _j x_{ij}\bigg )_{LL},~~i=1,\ldots ,n, \end{aligned}$$

(1)

where $x_{i0}=1$, $i=1,\ldots ,n$.

3.2 Objective function

In the following, we first introduce some differences between the $\alpha$-cuts of fuzzy numbers, and then the objective functions of fuzzy regression models are provided based on loss function on these differences as follows.

Definition 1

Let $\tilde{A}$ and $\tilde{B}$ be two fuzzy numbers with $\alpha$-cuts $\tilde{A}[\alpha ]=[\tilde{A}^L[\alpha ], \tilde{A}^R[\alpha ]]$ and $\tilde{B}[\alpha ]=[\tilde{B}^L[\alpha ], \tilde{B}^R[\alpha ]]$. The averages of differences between the lower and upper $\alpha$-cuts of $\tilde{A}$ and $\tilde{B}$ are defined as

$$\begin{aligned}&DL(\tilde{A},\tilde{B})=\int _{0}^{1}(\tilde{A}^L[\alpha ]-\tilde{B}^L[\alpha ])d\alpha ,\\&DR(\tilde{A},\tilde{B})=\int _{0}^{1}(\tilde{A}^R[\alpha ]-\tilde{B}^R[\alpha ])d\alpha . \end{aligned}$$

Remark 2

Let $\tilde{A}$, $\tilde{B}$ and $\tilde{C}$ be three fuzzy numbers. Then, DL(., .) and DR(., .) in Definition 1 satisfy the following properties:

• (i) $DL(\tilde{A},\tilde{A})=0$ and $DR(\tilde{A},\tilde{A})=0$,

• (ii) $DL(\tilde{A},\tilde{B})=-DL(\tilde{B},\tilde{A})$ and $DR(\tilde{A},\tilde{B})=-DR(\tilde{B},\tilde{A})$,

• (iii) $DL(\tilde{A},\tilde{B})+DL(\tilde{B},\tilde{C})=DL(\tilde{A},\tilde{C})$ and $DR(\tilde{A},\tilde{B})+DR(\tilde{B},\tilde{C})=DR(\tilde{A},\tilde{C})$.

Remark 3

In a special case, if $\tilde{A}=(m_a, l_a,r_a)_{LR}$ and $\tilde{B}=(m_b, l_b,r_b)_{LR}$ are two LR fuzzy numbers, then

$$\begin{aligned}&DL(\tilde{A},\tilde{B})=(m_a-m_b)-\lambda (l_a-l_b),\\&DR(\tilde{A},\tilde{B})=(m_a-m_b)+\rho (r_a-r_b), \end{aligned}$$

where $\lambda =\int _{0}^{1}L^{-1}(\alpha )d\alpha$ and $\rho =\int _{0}^{1}R^{-1}(\alpha )d\alpha$. Furthermore, if $\tilde{A}=(m_a, l_a,r_a)_{T}$ and $\tilde{B}=(m_b, l_b,r_b)_{T}$ are two triangular fuzzy numbers, then

$$\begin{aligned}&DL(\tilde{A},\tilde{B})=(m_a-m_b)-\frac{1}{2}(l_a-l_b),\\&DR(\tilde{A},\tilde{B})=(m_a-m_b)+\frac{1}{2}(r_a-r_b), \end{aligned}$$

Remark 4

If the LR fuzzy numbers $\tilde{A}=(m_a, l_a,r_a)_{LR}$ and $\tilde{B}=(m_b, l_b,r_b)_{LR}$ are reduced to the crisp numbers (i.e. $l_a=r_a=l_b=r_b=0$), then $DL(\tilde{A},\tilde{B})=DR(\tilde{A},\tilde{B})=m_a-m_b$.

Definition 3

Suppose that $\tilde{Y}_i$ and $\hat{\widetilde{Y}}_i$ are the observed fuzzy response variable and the estimated fuzzy response variable, respectively. The objective function based on Definition 1 is defined as

$$\begin{aligned} O=\frac{1}{2}\sum _{i=1}^{n}\Big [\psi (DL(\tilde{Y}_i, \hat{\tilde{Y}}_i))+\psi ((DR(\tilde{Y_i}, \hat{\tilde{Y_i}}))\Big ], \end{aligned}$$

(2)

where $\psi (.)$ is a loss function.

In the objective function O, the relation $\frac{1}{2}\big (\psi (DL(\tilde{Y_i},\hat{\tilde{Y_i}}))+\psi ((DR(\tilde{Y_i},\hat{\tilde{Y_i}}))\big )$ can be considered as the loss value of error between $\tilde{Y}_i$ and $\hat{\tilde{Y}}_i$. Note that based on the different loss functions $\psi (.)$, we can obtain the different fuzzy regression models. Some loss functions are listed as follows:

$$\begin{aligned}&(1) \ \ \text {Squared error loss function}: \ \ \ \psi ^1(e)=e^2,\\&(2) \ \ \text {Absolute error loss function}: \ \ \ \psi ^2(e)=|e|,\\&(3) \ \ \text {Quantile loss function }\mathrm{(Koenker 2005)}: \\&\quad \psi ^3_\tau (e)=|e||\tau -I(e<0)|, \ \ 0<\tau \le 1,\\&(4) \ \ \text {Huber loss function }\mathrm{(Huber 1981)}: \\&\quad \psi ^4_{c}(e)= \left\{ \begin{array}{ll} \frac{1}{2}e^2, &{} |e|\le c,\\ c|e|-\frac{1}{2}c^2, &{} |e|> c \end{array}\right. , c>0. \end{aligned}$$

Remark 5

In the loss function $\psi ^3_\tau$, $\tau$ is the quantile level and for $\tau =0.5$, $\psi ^3_\tau$ is reduced to the loss function $\psi ^2$. Also, in the loss function $\psi ^4_c$, based on some of indices of goodness of fit, we choose the value of c in interval [0, 2] (Huber 1981 says that the good value of c is in interval [1, 2]. So, it is taken as $c=1.5$).

From the relation (2), the objective functions based on the above loss functions are provided as follows:

$$\begin{aligned} (1) \ \ O_{\psi ^1}&=\frac{1}{2}\sum _{i=1}^{n}\bigg ((DL(\tilde{Y_i}, \hat{\tilde{Y_i}}))^2+(DR(\tilde{Y_i}, \hat{\tilde{Y_i}}))^2\bigg ).\\ (2) \ \ O_{\psi ^2}&=\frac{1}{2}\sum _{i=1}^{n}\bigg (|DL(\tilde{Y_i}, \hat{\tilde{Y_i}})|+|DR(\tilde{Y_i}, \hat{\tilde{Y_i}})|\bigg ).\\ (3) \ \ O_{\psi ^3_{\tau }}&=\frac{1}{2}\sum _{i=1}^{n}\bigg [\Big (|DL(\tilde{Y_i}, \hat{\tilde{Y_i}})|\times |\tau -I(DL(\tilde{Y_i}, \hat{\tilde{Y_i}})< 0)|\Big )\\&\quad +\Big (|DR(\tilde{Y_i}, \hat{\tilde{Y_i}})|\times |\tau -I(DR(\tilde{Y_i}, \hat{\tilde{Y_i}})< 0)|\Big )\bigg ].\\ (4) \ \ O_{\psi ^4_c}&=\frac{1}{2}\sum _{i=1}^{n}\bigg [\Big ((DL(\tilde{Y_i}, \hat{\tilde{Y_i}}))^2/2-\big ((DL(\tilde{Y_i}, \hat{\tilde{Y_i}}))^2/2\\&\quad -c|DL(\tilde{Y_i}, \hat{\tilde{Y_i}})|+c^2/2\big )I(|DL(\tilde{Y_i}, \hat{\tilde{Y_i}})|> c)\Big ) \nonumber \\&\quad + \Big ((DR(\tilde{Y_i}, \hat{\tilde{Y_i}}))^2/2-\big ((DR(\tilde{Y_i}, \hat{\tilde{Y_i}}))^2/2\\&\quad -c|DR(\tilde{Y_i}, \hat{\tilde{Y_i}})|+c^2/2\big )I(|DR(\tilde{Y_i}, \hat{\tilde{Y_i}})|> c)\Big )\bigg ]. \end{aligned}$$

By minimizing these objective functions under the parameters of model, we can obtain the optimal fuzzy regression models.

3.3 Estimation of model parameters

Based on the above objective functions, we can calculate the estimations of parameters of models as follows (assume that $x_{ij}\ge 0$).

(A) Squared error loss function: Based on relation (1), the objective function $O_{\psi ^1}$ is rewritten as

$$\begin{aligned} O_{\psi ^1}= & {} \frac{1}{2}\sum _{i=1}^{n}\left[ \left[ \left( y_i-\sum _{j=0}^{p}\beta _jx_{ij}\right) -\lambda \left( s_{i}-\sum _{j=0}^{p}\gamma _jx_{ij}\right) \right] ^2 \right. \\&\left. +\left[ \left( y_i-\sum _{j=0}^{p}\beta _jx_{ij}\right) +\lambda \left( s_{i}-\sum _{j=0}^{p}\gamma _jx_{ij}\right) \right] ^2\right] \\= & {} \left( \mathbf{Y}-\mathbf{X}{\beta }\right) ' \left( \mathbf{Y}-\mathbf{X}{\beta }\right) +\lambda ^2\left( \mathbf{S}-\mathbf{X}{\gamma }\right) ' \left( \mathbf{S}-\mathbf{X}{\gamma }\right) . \end{aligned}$$

where

$$\begin{aligned}&\mathbf{X}=\left[ \begin{array}{cccc} 1 &{} x_{11} &{} \ldots &{} x_{1p}\\ 1 &{} x_{21} &{} \ldots &{} x_{2p}\\ \vdots &{}\vdots &{}\vdots &{}\vdots \\ 1 &{} x_{n1}&{} \ldots &{} x_{np} \end{array}\right] ,~~ \mathbf{Y}=\left[ \begin{array}{c} y_1\\ y_2\\ \vdots \\ y_n \end{array}\right] ,~~ \mathbf{S}=\left[ \begin{array}{c} s_{1}\\ s_{2}\\ \vdots \\ s_{n} \end{array}\right] ,\\&{\beta }=\left[ \begin{array}{c} \beta _0\\ \beta _1\\ \vdots \\ \beta _p \end{array}\right] ,~~ {\gamma }=\left[ \begin{array}{c} \gamma _0\\ \gamma _1\\ \vdots \\ \gamma _p \end{array}\right] .\end{aligned}$$

By differentiating of $O_{\psi ^1}$ with respect to $\beta _j$ and $\gamma _j$, the estimations of parameters are obtained as follows:

$$\begin{aligned} \nonumber&{\hat{\beta }}=(\mathbf{X^{'}}{} \mathbf{X})^{-1}{} \mathbf{X^{'}}{} \mathbf{Y},\\&\hat{{\gamma }}=(\mathbf{X^{'}}{} \mathbf{X})^{-1}{} \mathbf{X^{'}}{} \mathbf{S}. \end{aligned}$$

Remark 6

If we encounter with some negative spreads, we can again run the fuzzy regression model with considering such parameters as crisp (i.e. the related spreads are zero). Thus, the centers are obtained as before, but the spreads are calculated as follows

$$\begin{aligned}\hat{\gamma }^{*}=(\mathbf{X^{*^{'}}}\mathbf{X^{*}})^{-1}{} \mathbf{X^{*^{'}}}{} \mathbf{S},\end{aligned}$$

where $\mathbf{X^{*}}$ is like $\mathbf{X}$ in which the columns corresponding to crisp parameters are removed.

(B) Absolute error loss function and quantile loss function: For minimizing the objective functions $O_{\psi ^2}$ and $O_{\psi ^3}$, we can translate it into the linear programming problems as follows. Assume that $DL(\tilde{Y_i},\hat{\tilde{Y_i}})=d^{L+}_i-d^{L-}_i$ and $DR(\tilde{Y_i}, \hat{\tilde{Y_i}})=d^{R+}_i-d^{R-}_i$ with $d^{L+}_i=\max (DL(\tilde{Y_i},\hat{\tilde{Y_i}}),0)$, $d^{L-}_i=-\min (DL(\tilde{Y_i},\hat{\tilde{Y_i}}),0)$, $d^{R+}_i=\max (DR(\tilde{Y_i},\hat{\tilde{Y_i}}),0)$, and $d^{R-}_i=-\min (DR(\tilde{Y_i},\hat{\tilde{Y_i}}),0)$, then the objective functions are presented as

$$\begin{aligned} O_{\psi ^2}&=\frac{1}{2}\sum _{i=1}^{n}\Big [|DL(\tilde{Y_i}, \hat{\tilde{Y_i}})|+|DR(\tilde{Y_i}, \hat{\tilde{Y_i}})|\Big ]\\&=\frac{1}{2}\sum _{i=1}^{n}\Big [(d^{L+}_i+d^{L-}_i)+(d^{R+}_i+d^{R-}_i)\Big ],\\ O_{\psi ^3_{\tau }}&=\frac{1}{2}\sum _{i=1}^{n}\bigg [\Big (|DL(\tilde{Y_i}, \hat{\tilde{Y_i}})|\times |\tau -I(DL(\tilde{Y_i}, \hat{\tilde{Y_i}})< 0)|\Big )\\&\quad +\Big (|DR(\tilde{Y_i}, \hat{\tilde{Y_i}})|\times |\tau -I(DR(\tilde{Y_i}, \hat{\tilde{Y_i}})< 0)|\Big )\bigg ]\\&=\frac{1}{2}\sum _{i=1}^{n}\Big [\tau (d^{L+}_i+d^{R+}_i)+(1-\tau )(d^{L-}_i+d^{R-}_i)\Big ]. \end{aligned}$$

Hence, the linear programming problems are given as follows:

$$\begin{aligned}&\text {min} \ \ O_{\psi ^2}=\frac{1}{2}\sum _{i=1}^{n}\Big [d^{L+}_i+d^{L-}_i+d^{R+}_i+d^{R-}_i\Big ] \\&\text {s.t.}\\&\quad d^{L+}_i-d^{L-}_i=DL(\tilde{Y_i}, \hat{\tilde{Y_i}}) =(y_i-\mathop {\sum }\nolimits _{j=0}^{p}\beta _jx_{ij})\\&\qquad -\lambda (s_{i}-\mathop {\sum }\nolimits _{j=0}^{p}\gamma _jx_{ij}),\\&\quad d^{R+}_i-d^{R-}_i=DR(\tilde{Y_i}, \hat{\tilde{Y_i}}) =(y_i-\mathop {\sum }\nolimits _{j=0}^{p}\beta _jx_{ij})\\&\qquad +\lambda (s_{i}-\mathop {\sum }\nolimits _{j=0}^{p}\gamma _jx_{ij}),\\&\quad d^{L+}_i\ge 0,~~ d^{L-}_i\ge 0,~~ d^{R+}_i\ge 0,~~ d^{R-}_i\ge 0,~~ i=1,2,\ldots ,n,\\&\quad \beta _j\in R,~~ \gamma _j\ge 0, \quad j=0,\ldots ,p. \end{aligned}$$

and

$$\begin{aligned}&\text {min} \ \ O_{\psi ^3_{\tau }}=\frac{1}{2}\sum _{i=1}^{n}\Big [\tau (d^{L+}_i+d^{R+}_i)+(1-\tau )(d^{L-}_i+d^{R-}_i)\Big ] \\&\text {s.t.}\\&\quad d^{L+}_i-d^{L-}_i=DL(\tilde{Y_i}, \hat{\tilde{Y_i}}) =(y_i-\mathop {\sum }\nolimits _{j=0}^{p}\beta _jx_{ij})\\&\qquad -\lambda (s_{i}-\mathop {\sum }\nolimits _{j=0}^{p}\gamma _jx_{ij}),\\&\quad d^{R+}_i-d^{R-}_i=DR(\tilde{Y_i}, \hat{\tilde{Y_i}}) =(y_i-\mathop {\sum }\nolimits _{j=0}^{p}\beta _jx_{ij})\\&\qquad +\lambda (s_{i}-\mathop {\sum }\nolimits _{j=0}^{p}\gamma _jx_{ij}),\\&\quad d^{L+}_i\ge 0,~~ d^{L-}_i\ge 0,~~ d^{R+}_i\ge 0,~~ d^{R-}_i\ge 0,~~ i=1,2,\ldots ,n,\\&\quad \beta _j\in R,~~ \gamma _j\ge 0, \ \ j=0,\ldots ,p. \end{aligned}$$

Table 1 Data set in Example 1

Table 2 Comparison study between different models in Example 1

Remark 7

Note that for solving the above linear programming problems, we can use “LINGO” software (Schrage 2006) or “Mathematica” software (Wolfram 2015).

(C) Huber loss function: It is simple to verify that the Huber loss function can be equivalently rewritten as

$$\begin{aligned} \psi ^4_c(e)=\frac{1}{2}\big [e^2-(e-c)^2_+-(-e-c)^2_+\big ]. \end{aligned}$$

Hence, the objective functions $O_{\psi ^4_c}$ based on matrix forms are obtained as follows

$$\begin{aligned} O_{\psi ^4_c}&=\frac{1}{4}\bigg [ \big (\mathbf{Y}-\mathbf{X}\beta -\lambda (S-\mathbf{X}\gamma )\big )^{'}\big (\mathbf{Y}-{} \mathbf{X}\beta -\lambda (\mathbf{S}-\mathbf{X}\gamma )\big )\nonumber \\&\quad -\big (\mathbf{Y}-\mathbf{X}\beta -\lambda (\mathbf{S}-\mathbf{X}\gamma )-c\mathbf{L}\big )_{+}^{'}\nonumber \\&\quad \big (\mathbf{Y}-\mathbf{X}\beta -\lambda (\mathbf{S}-\mathbf{X}\gamma )-c\mathbf{L}\big )_{+}\nonumber \\&\quad -\big (\mathbf{X}\beta -\mathbf{Y}-\lambda (\mathbf{X}\gamma -\mathbf{S})-c\mathbf{L}\big )_{+}^{'}\nonumber \\&\quad \big (\mathbf{X}\beta -\mathbf{Y}-\lambda (\mathbf{X}\gamma -\mathbf{S})-c\mathbf{L}\big )_{+}\nonumber \\&\quad + \big (\mathbf{Y}-\mathbf{X}\beta +\lambda (S-\mathbf{X}\gamma )\big )^{'}\big (\mathbf{Y}-\mathbf{X}\beta +\lambda (\mathbf{S}-\mathbf{X}\gamma )\big )\nonumber \\&\quad -\big (\mathbf{Y}-\mathbf{X}\beta +\lambda (\mathbf{S}-\mathbf{X}\gamma )-c\mathbf{L}\big )_{+}^{'} \nonumber \\&\quad \big (\mathbf{Y}-\mathbf{X}\beta +\lambda (\mathbf{S}-\mathbf{X}\gamma )-c\mathbf{L}\big )_{+}\nonumber \\&\quad -\big (\mathbf{X}\beta -\mathbf{Y}+\lambda (\mathbf{X}\gamma -\mathbf{S})-c\mathbf{L}\big )_{+}^{'} \nonumber \\&\quad \big (\mathbf{X}\beta -\mathbf{Y}+\lambda (\mathbf{X}\gamma -\mathbf{S})-c\mathbf{L}\big )_{+} \bigg ], \end{aligned}$$

(3)

where $\mathbf{L}=(1, 1,\ldots ,1)'$. By differentiating (3) with respect to the vectors $\beta$ and $\gamma$, we have

$$\begin{aligned} \frac{\partial O_{\psi ^4_c}}{\partial \beta }&=-\mathbf{X^{'}}{} \mathbf{Y}+\mathbf{X^{'}}{} \mathbf{X}\beta \\&\quad +\frac{1}{2}{} \mathbf{X^{'}}[H_1(\beta , \gamma )+H_2(\beta , \gamma )]=0,\\ \frac{\partial O_{\psi ^4_c}}{\partial \gamma }&=-\lambda ^2\mathbf{X^{'}}{} \mathbf{S}+\lambda ^2\mathbf{X^{'}}{} \mathbf{X}\gamma \\&\quad -\frac{1}{2}\lambda \mathbf{X^{'}}[H_1(\beta , \gamma )-H_2(\beta , \gamma )]=0, \end{aligned}$$

where

$$\begin{aligned} H_1(\beta , \gamma )&=(\mathbf{Y}-\mathbf{X}\beta -\lambda (\mathbf{S}-\mathbf{X}\gamma )-c\mathbf{L})_{+}\\&\quad -(\mathbf{X}\beta -\mathbf{Y}-\lambda (\mathbf{X}\gamma -\mathbf{S})-c\mathbf{L})_{+},\\ H_2(\beta , \gamma )&=(\mathbf{Y}-\mathbf{X}\beta +\lambda (\mathbf{S}-\mathbf{X}\gamma )-c\mathbf{L})_{+}\\&\quad -(\mathbf{X}\beta -\mathbf{Y}+\lambda (\mathbf{X}\gamma -\mathbf{S})-c\mathbf{L})_{+}. \end{aligned}$$

Now, the estimations of parameters are obtained by the following simple iterative algorithm:

$$\begin{aligned}&\beta ^{m+1}=(\mathbf{X^{'}}{} \mathbf{X})^{-1}{} \mathbf{X^{'}}\big [\mathbf{Y}-\frac{1}{2}\big (H_1(\beta ^m, \gamma ^m)+H_2(\beta ^m, \gamma ^m)\big )\big ],\\&\gamma ^{m+1}=(\mathbf{X^{'}}{} \mathbf{X})^{-1}{} \mathbf{X^{'}}\big [\mathbf{S}+\frac{1}{2\lambda }\big (H_1(\beta ^m, \gamma ^m)-H_2(\beta ^m, \gamma ^m)\big )\big ], \\&\quad m=0, 1,\ldots . \end{aligned}$$

Table 3 Values of response variable, corresponding predictions by cross-validation method, and indices of goodness of fit of Mohammadi and Taheri’s data set in Example 1

3.4 Goodness of fit of the model

In order to evaluate the proposed fuzzy regression models, we introduce some indices of goodness of fit. Also, using cross-validation method, the performance and predictive ability of proposed regression models are evaluated (see Geisser 1993; Wasserman 2006)

3.4.1 Goodness of fit

Definition 4

(Pappis and Karacapilidis 1993) Suppose that $\tilde{Y}_i$ and $\hat{\tilde{Y}}_i$ are the observed fuzzy response variable and the estimated fuzzy response variable, respectively. The mean of similarity measures (MSM) is defined as

$$\begin{aligned}MSM=\frac{1}{n}\sum _{i=1}^{n}S_{PK}(\hat{\tilde{Y_i}},\tilde{Y_i}),\end{aligned}$$

where $S_{PK}(\hat{\tilde{Y_i}},\tilde{Y_i})=\frac{\text {Card}(\hat{\tilde{Y_i}}\cap \tilde{Y_i})}{\text {Card}(\hat{\tilde{Y_i}}\cup \tilde{Y_i})}$ and $\text {Card}(\tilde{A})$ denotes the cardinal number of $\tilde{A}$ as $\text {Card}(\tilde{A})=\int _{R}^{}\tilde{A}(x)dx$.

Definition 5

(Chen and Dang 2008) Suppose that $\tilde{Y}_i$ and $\hat{\tilde{Y}}_i$ are the observed fuzzy response variable and the estimated fuzzy response variable, respectively. The index for goodness of fit of regression model is defined as

$$\begin{aligned} \bar{G}_1=\frac{1}{n}\sum _{i=1}^{n}S_{CD}(\hat{\tilde{Y_i}},\tilde{Y_i}), \end{aligned}$$

where $S_{CD}(\hat{\tilde{Y_i}},\tilde{Y_i})=\frac{1}{1+E(\hat{\tilde{Y_i}},\tilde{Y_i})}$ and $E(\hat{\tilde{Y_i}},\tilde{Y_i})=\frac{\int _{S_{\tilde{Y_i}}\cup S_{\hat{\tilde{Y_i}}}}\mid \hat{\tilde{Y_i}}(y)-\tilde{Y_i}(y)\mid dy}{\int _{S_{\tilde{Y_i}}}\tilde{Y_i}(y)dy}$. Also, $\hat{\tilde{Y_i}}(y)$ and $\tilde{Y_i}(y)$ are the membership functions of $\hat{\tilde{Y_i}}$ and $\tilde{Y_i}$ with the supports $S_{\hat{\tilde{Y_i}}}$ and $S_{\tilde{Y_i}}$, respectively.

Definition 6

Suppose that $\tilde{Y}_i$ and $\hat{\tilde{Y}}_i$ are the observed fuzzy response variable and the estimated fuzzy response variable, respectively. Based on the objective function $O_{\psi ^2}$, the index of goodness of fit of the model is defined as

$$\begin{aligned}\bar{G}_2=\frac{1}{n}\sum _{i=1}^{n}S_{O}(\hat{\tilde{Y_i}},\tilde{Y_i}),\end{aligned}$$

where $S_{O}(\hat{\tilde{Y_i}},\tilde{Y_i})=\frac{1}{1+O_{\psi ^2}(\hat{\tilde{Y_i}},\tilde{Y_i})}$.

Remark 8

The indices MSM, $\bar{G}_1$ and $\bar{G}_2$ are on the interval [0, 1]. The optimal model is the model with maximum values of MSM, $\bar{G}_1$ or $\bar{G}_2$.

Table 4 Data set in Example 2

Table 5 Different outliers in Example 2

3.4.2 Cross-validation

The cross-validation method is a well-known method for evaluating the performance and predictive ability of regression models. For doing this purpose, we divide the data set with size n into two sets. The first set contains the training data of size $n-1$, which we can use to develop the regression model, and the second set is the testing data of size $k=1$, which is used to evaluate the predictive ability of the presented regression model.

Table 6 Performance of fuzzy regression models in Example 2

Fig. 1

Comparison between fuzzy regression models based on the different loss functions and outliers in Example 2

In this method, we consider n steps. In each step, the ith observation is first deleted from the data set for $i=1,2,\ldots ,n$, and then, the fuzzy regression model is obtained based on the remaining observations (training data set). In final, the value of the ith response variable is predicted based on the proposed fuzzy regression model and is denoted as $\hat{\tilde{Y_i}}^{(-i)}$. To evaluate the performance of fuzzy regression model, we calculate the indices of goodness of fit MSM, $\bar{G}_1$ and $\bar{G}_2$ between $\tilde{Y_i}$ and $\hat{\tilde{Y_i}}^{(-i)}$.

4 Numerical examples

In this section, we present some numerical and simulation examples to illustrate our proposed approach.

Example 1

One of the classical problems in soil science is the measurement of physical, chemical, and/or biological soil properties. The problem results from the difficulty, time, and cost of direct measurements. Mohammadi and Taheri (2004) provided a data set that it includes some soil properties such as the cation exchange capacity (CEC), the sand content percentage (SAND), and the organic matter content (OM) (see Table 1). Now, we wish to model a relationship between CEC (as the response variable) and SAND and OM (as the explanatory variables) by the regression model:

$$\begin{aligned} \tilde{Y}_i=\tilde{\beta _0}\oplus (\tilde{\beta _1}\otimes x_{i1})\oplus (\tilde{\beta _2}\otimes x_{i2}),~~ i=1,\ldots ,24, \end{aligned}$$

where $\tilde{\beta }_j=(\beta _j,\gamma _j)_T$, $j=0,1,2$. But due to some impreciseness in related experimental environment, the response variable $\tilde{Y}_i$ is reported as a symmetric triangular fuzzy number $(y_i, s_i)_T$, in which the spreads are proportional to centers. Here, we consider the sensitivity analysis of model by fuzzifying for different values of spreads as $s_i=wy_i$ with $w =0.05,0.10,0.20$. Note that Arefi (2020) recently presented a quantile fuzzy linear regression model on this data set. Since Arefi’s approach (AR) provided a optimal fuzzy regression model on this data set, we compare our work with this approach (the results are listed in Table 2). Results show that our proposed models based on the quantile loss function with $\tau = 0.5$ and Huber loss function with $c=0.5$ have better performance than the fuzzy model given by Arefi (2020) (for each $w=0.05, 0.1, 0.2$).

Now, to evaluate the performance and predictive ability of optimal regression model based on the quantile loss function with $\tau = 0.5$, we apply the cross-validation method on data set in Table 1 (with $s_i = 0.2y_i$). The results are given in Table 3. Based on the values of goodness-of-fit indices (see Table 3), we can suggest the proposed fuzzy regression model has a suitable performance for predicting the response variables.

Example 2

Consider the data set in Table 4 (Tanaka and Lee 1998). To study the effect of outliers on the our proposed models, we have listed some of different types of outliers in Table 5. Assume that the original fuzzy regression model is as

$$\begin{aligned} \tilde{Y}_i=\tilde{\beta _0}\oplus (\tilde{\beta _1}\otimes x_{i1})\oplus (\tilde{\beta _2}\otimes x^2_{i1})\oplus (\tilde{\beta }_3\otimes x^3_{i1}), ~~~~~i=1,\ldots ,8. \end{aligned}$$

Table 6 lists the optimal fitted fuzzy regression models in the presence of outlier data. In original model, the models proposed based on the absolute error loss function and quantile loss functions (at $\tau =0.75$) have better performance than other models. In the presence of outlier data, the fitted fuzzy regression model based on the quantile loss function at $\tau =0.75$ is robust because the obtained fuzzy regression models based on the quantile loss function at $\tau =0.75$ are approximately similar in all cases (see Fig. 1).

Now, we apply the cross-validation method on data set in Table 4 based on the optimal fuzzy regression model with the quantile loss function at $\tau =0.75$. The results are given in Table 7. The values of goodness-of-fit indices show that the predictive ability of the proposed regression model is suitable.

Example 3

(Tanaka et al. 1982) Consider the data set in Table 8. In this data set, the observations of independent variable are crisp and the observations of dependent variable are presented as the symmetric triangular fuzzy numbers. This data set has been considered by many numbers of researchers. Here, we compare our approach with some other approaches introduced by Diamond (1988) (DM), Chen and Hsueh (2007) $(CH_2)$, Chen and Hsueh (2009) $(CH_1)$, Mosleh et al. (2010) (ME), Taheri and Kelkinnama (2012) (TK), Roldan et al. (2012) (RE), Zeng et al. (2016) (ZE), and Lopez et al. (2016) (HE). Results are given in Table 9. By using the indices of goodness of fit, we can suggest the following cases:

• In between the fuzzy linear regression models based on the least squared errors, the models (SLF), (DM), and $(CH_1)$ have the approximately similar results.

• In between the fuzzy linear regression models based on the least absolute errors, the model (ALF) has better performance than the models $(CH_2)$, (TK), and (ZE).

• The fuzzy linear regression model based on the quantile loss function (QLF) at $\tau =0.6$ provides a model with better performance than other models considered in this example.

Table 7 Values of response variable, corresponding predictions by cross-validation method, and indices of goodness of fit on Tanaka and Lee’s data set in Example 2

Table 8 Data set in Example 3

Table 9 Comparison study between different models in Example 3

Table 10 Performance of fuzzy regression model in Example 4

Example 4

(Simulation study) In this example, we want to design a data set that empirically assesses the robust performance of the proposed fuzzy regression model based on the quantile loss function. Based on a sample of size $n=100$ simulated under a fuzzy regression model, we obtain a data set $(x_i,\tilde{Y_i})$, $i=1,\ldots ,100$. Also, we add three different percentages of simulated outliers as $p.out=\{0.05, 0.10, 0.20\}$ in this the data set, and we want to evaluate the proposed models. Note that we assume that the data set is simulated based on the following model

Fig. 2

Fuzzy regression model under original data and based on quantile loss function with $\tau =0.5$ in Example 4

Fig. 3

Fuzzy regression models in the presence of outliers and based on quantile loss function with $\tau =0.5$ in Example 4

$$\begin{aligned}&\tilde{Y}_i=(13,4)_T\oplus ((7, 1)_T\otimes \sin (\frac{3.14}{24}x_i))\oplus ((0.33, 0.01)_T\\&\qquad \otimes x_i)\oplus \varepsilon _i, i=1,\ldots ,n, \end{aligned}$$

where $x_i$ and $\varepsilon _i$ are generated from $U(-40, 50)$ and N(0, 1.5), respectively. The outlier observations are generated according to two different types: Type I is the outliers in the centers of $\tilde{Y}_i$, and Type II is the outliers in the spreads of $\tilde{Y}_i$. We generate these two types of outliers from the bivariate Gaussian distribution $N_2\left( \left[ \begin{array}{ll} \mu _x\\ \mu _y\\ \end{array} \right] , \left[ \begin{array}{ll} \sigma _X^2 \ \ \ \rho \sigma _X\sigma _Y\\ \rho \sigma _X\sigma _Y\ \ \ \sigma _Y^2\\ \end{array} \right] \right)$ and uniform distribution, respectively. In Type I, we generate observations $(x_i, y_i)$ from two bivariate Gaussian distributions $N_2\left( \left[ \begin{array}{ll} \bar{x}+\varepsilon _i\\ max(y_i)+1.5s_y\\ \end{array} \right] , \left[ \begin{array}{ll} 20 \ \ \ 0\\ 0\ \ \ \ 5\\ \end{array} \right] \right)$ and $N_2\left( \left[ \begin{array}{ll} \bar{x}+\varepsilon _i\\ min(y_i)-1.5s_y\\ \end{array} \right] , \left[ \begin{array}{ll} 20 \ \ \ 0\\ 0\ \ \ \ 5\\ \end{array} \right] \right)$, where $\varepsilon _i$ is generated from $U(-20, 40)$ and $s_y=\sqrt{\frac{1}{n}\sum _{i=1}^{100}(y_i-\overline{y})^2}$. In Type II, we generate the spreads of triangular fuzzy response observations $(y_i, s_{i})_T$ based on U(9, 13). The results of the fitted fuzzy regression models based on the quantile loss function with $\tau =0.25,0.5,0.75$ are given in Table 10. Based on the index of MSM, the model based on the quantile loss function with $\tau =0.5$ has the best results. See Figs. 2 and 3 for the performance of the best fuzzy regression models (using quantile loss function at $\tau =0.5$). They show the robustness performances from our proposed model in original data set and in the models with the presence of outliers. Also, we can consider the similar results based on the results given in Tables 10 and 11. Based on these tables, we have

Table 11 Mean values of the estimated parameters in Example 4

• The MSM in the original data and the MSM without outliers (in the models with outliers) are approximately similar (see Table 10).

• The values of the estimated parameters with original data and the mean values of the estimated parameters in the presence of outliers are approximately similar (see Table 11).

5 Conclusion

In this paper, we present a new approach to fit the fuzzy regression models based on some loss functions when the response variable and the parameters of model are as fuzzy numbers. Some of certain merits in this approach are as follows:

(1):

It is a general approach for fitting the fuzzy regression models based on the different types of loss functions.

(2):

A new definition of the objective function is presented based on loss function and the average of differences between the $\alpha$-cuts of errors.

(3):

Among the proposed loss functions, the fuzzy regression models based on the quantile loss function and Huber loss function are robust under outlier data.

(4):

To evaluate the goodness of fit of the proposed fuzzy regression models, we introduce three indices.