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Weighted Least Squares and Least Median Squares estimation for the fuzzy linear regression analysis

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Abstract

In this paper, we discuss the problem of regression analysis in a fuzzy domain. By considering an iterative Weighted Least Squares estimation approach, we propose a general linear regression model for studying the dependence of a general class of fuzzy response variable, i.e., \(LR_2\) fuzzy variable or trapezoidal fuzzy variable,on a set of crisp or \(LR_2\) fuzzy explanatory variables. We also show some theoretical properties and a suitable generalization of the determination coefficient in order to investigate the goodness of fit of the regression model. Furthermore, we discuss some theoretical issues and an assessment of imprecision of the regression function. Finally, we suggest a robust version of the fuzzy regression model based on the Least Median Squares estimation approach which is able to neutralize and/or smooth the disruptive effects of possible crisp or fuzzy outliers in the estimation process. A simulation study and two empirical applications are presented.

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Notes

  1. \(\mathbf {M}_1,\,\mathbf {M}_2,\,\mathbf {L},\,\text { and } \mathbf {R}\) also contain a vector of ones, related to the intercepts of the model.

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Authors and Affiliations

  1. Dipartimento di Scienze Sociali ed Economiche, Sapienza-Università di Roma, P.za Aldo Moro, 5, 00185 , Rome, Italy

    Pierpaolo D’Urso & Riccardo Massari

Authors
  1. Pierpaolo D’Urso
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  2. Riccardo Massari
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Corresponding author

Correspondence to Pierpaolo D’Urso.

Appendix: Iterative solutions of the \(LR_2\) output–\(LR_2\) inputs regression model

Appendix: Iterative solutions of the \(LR_2\) output–\(LR_2\) inputs regression model

By substituting in (7) the expressions (5a)–(5d), and by putting the first partial derivatives with respect to each coefficient equal to zero, the following iterative solutions are obtained.

$$\begin{aligned} {\varvec{\alpha }}_1&= (2-2\lambda \gamma _1+\lambda ^2\gamma _1^2+\rho ^2 \delta _1^2)^{-1}({\mathbf {M}}^{\prime }_1\mathbf {WM}_1)^{-1}{\mathbf {M}}^{\prime }_1\mathbf {W}\nonumber \\&\cdot \{2\mathbf {m}_1-(\mathbf {M}_2{\varvec{\alpha }}_2+\mathbf {L}{\varvec{\alpha }}_l+\mathbf {R}{\varvec{\alpha }}_r)(2-2\lambda \gamma _1+\lambda ^2\gamma _1^2+\rho ^2 \delta _1^2)\nonumber \\&-\lambda [\mathbf {m}_1 \gamma _1+\mathbf {l}-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \gamma _2- \mathbf {1}\gamma _0] \nonumber \\&+\lambda ^2 \gamma _1[\mathbf {l}-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \gamma _2-\mathbf {1} \gamma _0]\nonumber \\&+ \rho \delta _1[\mathbf {m}_2-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r)] \nonumber \\&+ \rho ^2 \delta _1[\mathbf {r}-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \delta _2 -\mathbf {1} \delta _0]\}\end{aligned}$$
(20a)
$$\begin{aligned} {\varvec{\alpha }}_2&= (2-2\lambda \gamma _1+\lambda ^2\gamma _1^2+\rho ^2 \delta _1^2)^{-1}({\mathbf {M}}^{\prime }_2\mathbf {WM}_2)^{-1}{\mathbf {M}}^{\prime }_2\mathbf {W}\nonumber \\&\cdot \{2\mathbf {m}_1-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {L}{\varvec{\alpha }}_l+\mathbf {R}{\varvec{\alpha }}_r)(2-2\lambda \gamma _1+\lambda ^2\gamma _1^2+\rho ^2 \delta _1^2)\nonumber \\&-\lambda [\mathbf {m}_1 \gamma _1+\mathbf {l}-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \gamma _2- \mathbf {1}\gamma _0] \nonumber \\&+\lambda ^2 \gamma _1[\mathbf {l}-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \gamma _2-\mathbf {1} \gamma _0]\nonumber \\&+ \rho \delta _1[\mathbf {m}_2-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r)] \nonumber \\&+ \rho ^2 \delta _1[\mathbf {r}-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \delta _2 -\mathbf {1} \delta _0]\}\end{aligned}$$
(20b)
$$\begin{aligned} {\varvec{\alpha }}_l&= (2-2\lambda \gamma _1+\lambda ^2\gamma _1^2+\rho ^2 \delta _1^2)^{-1}({\mathbf {L}}^{\prime }\mathbf {WL})^{-1}{\mathbf {L}}^{\prime }\mathbf {W}\nonumber \\&\cdot \{2\mathbf {m}_1-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2+\mathbf {R}{\varvec{\alpha }}_r)(2-2\lambda \gamma _1+\lambda ^2\gamma _1^2+\rho ^2 \delta _1^2)\nonumber \\&-\lambda [\mathbf {m}_1 \gamma _1+\mathbf {l}-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \gamma _2- \mathbf {1}\gamma _0] \nonumber \\&+\lambda ^2 \gamma _1[\mathbf {l}-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \gamma _2-\mathbf {1} \gamma _0]\nonumber \\&+ \rho \delta _1[\mathbf {m}_2-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r)] \nonumber \\&+ \rho ^2 \delta _1[\mathbf {r}-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \delta _2 -\mathbf {1} \delta _0]\}\end{aligned}$$
(20c)
$$\begin{aligned} {\varvec{\alpha }}_r&= (2-2\lambda \gamma _1+\lambda ^2\gamma _1^2+\rho ^2 \delta _1^2)^{-1}({\mathbf {R}}^{\prime }\mathbf {WR})^{-1}{\mathbf {R}}^{\prime }\mathbf {W}\nonumber \\&\cdot \{2\mathbf {m}_1-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2+\mathbf {L}{\varvec{\alpha }}_l)(2-2\lambda \gamma _1+\lambda ^2\gamma _1^2+\rho ^2 \delta _1^2)\nonumber \\&-\lambda [\mathbf {m}_1 \gamma _1+\mathbf {l}-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \gamma _2- \mathbf {1}\gamma _0] \nonumber \\&+\lambda ^2 \gamma _1[\mathbf {l}-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \gamma _2-\mathbf {1} \gamma _0]\nonumber \\&+ \rho \delta _1[\mathbf {m}_2-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r)] \nonumber \\&+ \rho ^2 \delta _1[\mathbf {r}-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \delta _2 -\mathbf {1} \delta _0]\}\end{aligned}$$
(20d)
$$\begin{aligned} {\varvec{\beta }}_1&= (2+\lambda ^2 \gamma _2+2\rho \delta _2+\rho ^2 \delta _2^2)^{-1}({\mathbf {M}}^{\prime }_1\mathbf {WM}_1)^{-1}{\mathbf {M}}^{\prime }_1\mathbf {W}\nonumber \\&\cdot \{2\mathbf {m}_2-(\mathbf {M}_2{\varvec{\beta }}_2+\mathbf {L}{\varvec{\beta }}_l+\mathbf {R}{\varvec{\beta }}_r)(2+\lambda ^2 \gamma _2+2\rho \delta _2+\rho ^2 \delta _2^2)\nonumber \\&-\lambda \gamma _2[\mathbf {m}_1-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)] \nonumber \\&+\lambda ^2 \gamma _2[\mathbf {l}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r) \gamma _1-\mathbf {1} \gamma _0]\nonumber \\&+ \rho [\mathbf {m}_2\delta _2+\mathbf {r}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)\delta _1-\mathbf {1}\delta _0] \nonumber \\&+ \rho ^2 \delta _2[\mathbf {r}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r) \delta _1 -\mathbf {1} \delta _0]\}\end{aligned}$$
(20e)
$$\begin{aligned} {\varvec{\beta }}_2&= (2+\lambda ^2 \gamma _2+2\rho \delta _2+\rho ^2 \delta _2^2)^{-1}({\mathbf {M}}^{\prime }_2\mathbf {WM}_2)^{-1}{\mathbf {M}}^{\prime }_2\mathbf {W}\nonumber \\&\cdot \{2\mathbf {m}_2-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {L}{\varvec{\beta }}_l+\mathbf {R}{\varvec{\beta }}_r)(2+\lambda ^2 \gamma _2+2\rho \delta _2+\rho ^2 \delta _2^2)\nonumber \\&-\lambda \gamma _2[\mathbf {m}_1-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)] \nonumber \\&+\lambda ^2 \gamma _2[\mathbf {l}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r) \gamma _1-\mathbf {1} \gamma _0]\nonumber \\&+ \rho [\mathbf {m}_2\delta _2+\mathbf {r}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)\delta _1-\mathbf {1}\delta _0] \nonumber \\&+ \rho ^2 \delta _2[\mathbf {r}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r) \delta _1 -\mathbf {1} \delta _0]\}\end{aligned}$$
(20f)
$$\begin{aligned} {\varvec{\beta }}_l&= (2+\lambda ^2 \gamma _2+2\rho \delta _2+\rho ^2 \delta _2^2)^{-1}({\mathbf {L}}^{\prime }\mathbf {WL})^{-1}{\mathbf {L}}^{\prime }\mathbf {W}\nonumber \\&\cdot \{2\mathbf {m}_2-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2+\mathbf {R}{\varvec{\beta }}_r)(2+\lambda ^2 \gamma _2+2\rho \delta _2+\rho ^2 \delta _2^2)\nonumber \\&-\lambda \gamma _2[\mathbf {m}_1-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)] \nonumber \\&+\lambda ^2 \gamma _2[\mathbf {l}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r) \gamma _1-\mathbf {1} \gamma _0]\nonumber \\&+ \rho [\mathbf {m}_2\delta _2+\mathbf {r}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)\delta _1-\mathbf {1}\delta _0] \nonumber \\&+ \rho ^2 \delta _2[\mathbf {r}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r) \delta _1 -\mathbf {1} \delta _0]\}\end{aligned}$$
(20g)
$$\begin{aligned} {\varvec{\beta }}_r&= (2+\lambda ^2 \gamma _2+2\rho \delta _2+\rho ^2 \delta _2^2)^{-1}({\mathbf {R}}^{\prime }\mathbf {WR})^{-1}{\mathbf {R}}^{\prime }\mathbf {W}\nonumber \\&\cdot \{2\mathbf {m}_2-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2+\mathbf {L}{\varvec{\beta }}_l)(2+\lambda ^2 \gamma _2+2\rho \delta _2+\rho ^2 \delta _2^2)\nonumber \\&-\lambda \gamma _2[\mathbf {m}_1-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)] \nonumber \\&+\lambda ^2 \gamma _2[\mathbf {l}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r) \gamma _1-\mathbf {1} \gamma _0]\nonumber \\&+ \rho [\mathbf {m}_2\delta _2+\mathbf {r}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)\delta _1-\mathbf {1}\delta _0] \nonumber \\&+ \rho ^2 \delta _2[\mathbf {r}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r) \delta _1 -\mathbf {1} \delta _0]\}\end{aligned}$$
(20h)
$$\begin{aligned} \gamma _1&= \lambda ^{-1}[(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)^{\prime } \mathbf {W} (\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)]^{-1}\nonumber \\&(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)^{\prime }\mathbf {W} \{(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)-\mathbf {m}_1\nonumber \\&+\lambda [\mathbf {l}-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \gamma _2 -\mathbf {1}\gamma _0]\}\end{aligned}$$
(20i)
$$\begin{aligned} \gamma _2&= \lambda ^{-1}[(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r)^{\prime } \mathbf {W} (\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r)]^{-1}\nonumber \\&(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r)^{\prime }\mathbf {W} \{(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)-\mathbf {m}_1\nonumber \\&+\lambda [\mathbf {l}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r) \gamma _1 -\mathbf {1}\gamma _0]\}\end{aligned}$$
(20j)
$$\begin{aligned} \gamma _0&= \lambda ^{-1}({\mathbf {1}}^{\prime }\mathbf {W 1})^{-1}{\mathbf {1}}^{\prime }\mathbf {W}\{(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)-\mathbf {m}_1\nonumber \\&+\lambda [\mathbf {l}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r) \gamma _1 - (\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \gamma _2]\}\nonumber \\ \end{aligned}$$
(20k)
$$\begin{aligned} \delta _1&= \rho ^{-1}[(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)^{\prime } \mathbf {W} (\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)]^{-1}\nonumber \\&(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)^{\prime }\mathbf {W} \{\mathbf {m}_2-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r)\nonumber \\&+\rho [\mathbf {r}-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \delta _2 -\mathbf {1}\delta _0]\}\end{aligned}$$
(20l)
$$\begin{aligned} \delta _2&= \rho ^{-1}[(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r)^{\prime } \mathbf {W} (\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r)]^{-1}\nonumber \\&(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r)^{\prime }\mathbf {W} \{\mathbf {m}_2-(\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r)\nonumber \\&+\rho [\mathbf {r}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r) \delta _1 -\mathbf {1}\delta _0]\}\end{aligned}$$
(20m)
$$\begin{aligned} \delta _0&= \rho ^{-1}({\mathbf {1}}^{\prime }\mathbf {W 1})^{-1}{\mathbf {1}}^{\prime }\mathbf {W}\{\mathbf {m}_2-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r)\nonumber \\&+\rho [\mathbf {r}-(\mathbf {M}_1{\varvec{\alpha }}_1+\mathbf {M}_2{\varvec{\alpha }}_2 +\mathbf {L}{\varvec{\alpha }}_l + \mathbf {R}{\varvec{\alpha }}_r) \delta _1 - (\mathbf {M}_1{\varvec{\beta }}_1+\mathbf {M}_2{\varvec{\beta }}_2 +\mathbf {L}{\varvec{\beta }}_l + \mathbf {R}{\varvec{\beta }}_r) \delta _2]\}.\nonumber \\ \end{aligned}$$
(20n)

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D’Urso, P., Massari, R. Weighted Least Squares and Least Median Squares estimation for the fuzzy linear regression analysis. METRON 71, 279–306 (2013). https://doi.org/10.1007/s40300-013-0025-9

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