當所預測問題的多個自變數之間,如存有潛在的交互作用之時,傳統複線性迴歸方法雖然常被使用,但功效並不顯著。此時可考慮採用非可加性測度模糊積分之迴歸模式。本文旨在探討比較當基本測度與聯合事件測度均為未知時,關於聯合亂度模糊測度(Kojadinovic, 2004 ; Calvo, Kolesarova, Komornikova & Mesiar, 2001 ; Yu, Wen, Xu, & Ester, 2001),複雜度模糊測度(Shieh, Wu,& Liu, 2009),劉湘川(2006, 2008)提出的複相互訊息模糊測度,此三種模糊測度的Choquet積分(Choquet, 1953)迴歸模式,與傳統的複線性迴歸模式,對於群體資料之預測精確度。上述三個模糊測度,分別稱為E測度、C測度和M測度。 本研究使用苗粟縣某中學國中部的自然科畢業成績(包含了理化、生物和地球科學三科)來預測學生參加國中基本學力測驗的自然科成績,利用5-fold交叉驗證的方法,進行實證分析。實驗結果顯示,此三種Choquet積分迴歸模式均優於傳統的複線性迴歸模式,而在三種Choquet積分迴歸模式中,基於M測度的Choquet積分迴歸模式有較佳之結果。
When interactions among independent variables exist in forecasting problems, the performance of the traditional multiple linear regression method is poor. On the contrary, non-additive fuzzy measures and fuzzy integral can be considered. In this study, when the basic event is unknown, three kinds of the Choquet integral (Choquet, 1953) regression models with fuzzy measures based on joint entropy (Kojadinovic, 2004 ; Calvo, Kolesarova, Komornikova & Mesiar, 2001 ; Yu, Wen, Xu, & Ester, 2001), complexity (Shieh, Wu, & Liu, 2009), multiple mutual information (劉湘川, 2006, 2008) and traditional multiple linear regression are compared for grouped data. The above three fuzzy measures are called E-measure, C-measure and M-measure respectively. In this study, a real grouped data set comes from a junior high school including three courses (namely physics and chemistry, biology, and geoscience) for natural science, are used 5-fold cross validation to evaluate the students’ performance based on a Basic Competence Test. Experimental result shows that all of the three Choquet integral regression models are better than the traditional regression model. The Choquet integral regression model based on M-measure has the best performance.