On assessing the \(H\) value in fuzzy linear regression.

*(English)*Zbl 0791.62073Summary: There are certain circumstances under which the application of statistical regression is not appropriate or even feasible because it makes rigid assumptions about the statistical properties of the model. Fuzzy regression, a nonparametric method, can be quite useful in estimating the relationships among variables where the available data are very limited and imprecise, and variables are interacting in an uncertain, qualitative, and fuzzy way. Thus, it may have considerable practical applications in many management and engineering problems.

In this paper, the relationship among the \(H\) value, membership function shape, and spreads of fuzzy parameters in fuzzy linear regression is determined, and the sensitivity of the spread with respect to the \(H\) value and membership function shape is examined. The spread of a fuzzy parameter increases as a higher value of \(H\) and/or a decreasingly concave or increasingly convex membership function is employed. By utilizing the relationship among the \(H\) value, membership function, and spreads of the fuzzy parameters, a systematic approach to assessing a proper \(H\) parameter value is also developed. The approach developed and illustrated enables a decision maker’s beliefs regarding the shape and range of the possibility distribution of the model to be reflected more systematically, and consequently should yield more reliable and realistic results from fuzzy regression. The resulting regressing equations could, for example, also be used as constraints in a fuzzy mathematical optimization model, such as in quality function deployment.

In this paper, the relationship among the \(H\) value, membership function shape, and spreads of fuzzy parameters in fuzzy linear regression is determined, and the sensitivity of the spread with respect to the \(H\) value and membership function shape is examined. The spread of a fuzzy parameter increases as a higher value of \(H\) and/or a decreasingly concave or increasingly convex membership function is employed. By utilizing the relationship among the \(H\) value, membership function, and spreads of the fuzzy parameters, a systematic approach to assessing a proper \(H\) parameter value is also developed. The approach developed and illustrated enables a decision maker’s beliefs regarding the shape and range of the possibility distribution of the model to be reflected more systematically, and consequently should yield more reliable and realistic results from fuzzy regression. The resulting regressing equations could, for example, also be used as constraints in a fuzzy mathematical optimization model, such as in quality function deployment.

##### MSC:

62J05 | Linear regression; mixed models |

62J99 | Linear inference, regression |

62G99 | Nonparametric inference |

##### Keywords:

\(H\) value; membership function shape; spreads of fuzzy parameters; fuzzy linear regression; possibility distribution
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\textit{H. Moskowitz} and \textit{K. Kim}, Fuzzy Sets Syst. 58, No. 3, 303--327 (1993; Zbl 0791.62073)

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