- 學位論文
基於多值m 測度之模糊積分迴歸模式之實證研究
An experimental study of fuzzy integral regression model based on polyvalent m-measure
摘要
當欲進行綜合評價之多種屬性間具潛在交互作用時，傳統可加性測度分析方 法雖計算方便，常功效不彰。例如某工程若甲乙兩人分別獨自工作，每日分別可 完成工程之 1 , 1 20 30 ，若兩人合作一起工作，則每日共同完成之工程不一定是 1 1 1 20 30 12 + = ，要看兩人之合作是否良好，而有增多或減少之情形，此時可考慮 採用模糊測度與模糊積分，常用之模糊測度，有Sugeno(1974)之λ 測度，及Zadeh (1978)之P 測度，該二模糊測度均只有唯一之解，且λ 測度不恆存在直接解 (closed form )，P 測度雖恆存在次可加性測度，但靈敏度不足，劉湘川(2006a,b) 先後提出基於P 測度之改進模糊測度，二值m 測度，及多值m 測度則能兼顧前二 者之優點，有多於一個之靈敏直接解，特別是多值m 測度可有無限多之靈敏直接 解可供選擇，藉以求取Choquet 積分值及Segeno 積分值，則可得甚多選擇之整 合計分多重決策之可行解法，當有效標應變數存在時，劉湘川並提出模糊積分迴 歸模式，藉以可選擇最適模糊測度解，惟僅有理論之探析，尚缺實物驗證，本研 究除簡介模糊測度、模糊積分與模糊積分迴歸模式之主要概念與發展外，特別採 用學理上廣為接受之五折交互驗證法 ( 5-fold cross validation )，以均方誤 差(MSE)及平方根均方誤差(RMSE)為比較準則， 以485 位國中畢業學生之理化、 生物、地球科學三門學科畢業成績，及其國中畢業基本能力測驗之自然科成績， 藉以評估三種分別基於λ 測度、 P 測度與多值m 測度之Choquet 積分迴歸模式， 與脊迴歸模式及複線性迴歸模式之預測效力，其中，三種Choquet 積分迴歸模式 之各學科基本測度均採用同一組學分比：0.5:0.25:0.25 ，實驗結果證實；基於 多值m 測度之Choquet 積分迴歸模式有最佳表現，其餘四種模式依次為基於λ 測 度之Choquet 積分迴歸模式，脊迴歸模式，複線性迴歸模式，及基於P 測度之 Choquet 積分迴歸模式。
並列摘要
When interactions among attributes exist in multiple decision-making problems， the performance of the traditional additive scale method is poor. For example , in a project，if two people work alone, the 1/20，1/30th of the project can be completed by them separately every day ，If they work together，the 1/20+1/30=1/12th of the project can be completed by them or not, according to the cooperating situation of them in every day，Non-additive fuzzy measures and fuzzy integral can be applied to improve this situation. Theλ-measure (Sugeno, 1974) and P-measure (Zadeh, 1978) are two well-known fuzzy measures. Hsiang-Chuan Liu(2006a,b) also proposed some improved non-additive fuzzy measures based on P-measure, the two valued m-measure and the polyvalent m-measure can be used, Specially the polyvalent m-measure has infinitely many solutions may be chosen，Choquet integral and Sugeno integral with this proposed generalized m-measure is applied to obtain the aggregation score of the entrance examination of graduate school. When effective dependent variable existence，Hsiang-Chuan Liu suggested to use the fuzzy integral regression model based on the most suitable improved fuzzy measures by only theoretical analyses, In order to lacking of the practical experimental study，in this research, not only the main concept and development of the fuzzy measure, the fuzzy integral and fuzzy integral regression mode are given，but also an educational data experiment is conducted for comparing the performances of the different forecasting models. A real data set with 485 samples from a junior high school in Taiwan including the independent variables, examination scores of three courses, physics and chemistry, biology, and geoscience, and the dependent variable, the score of the Basic Competence Test of junior high school is applied to evaluate the performances of Sugeno and Choquet integral regression models based on the polyvalent m-measure, λ-measure, P-measure, a ridge regression model, and a multiple linear regression model by using 5-fold cross validation method to compute the mean square error (MSE) and the rooted mean square error (RMSE) of the dependent variable, Responding the ratio of the credit hour for three courses, all of the fuzzy measures about the independent variables are assigned the same singleton measures as iii 0.5:0.25:0.25，experimental result confirmation；Fuzzy integral regression model based on polyvalent m-measure has the best performance，other four kind of patterns are in turnλ measure Choquet of based on the integral regression model，ridge regression model，multiple linear regression model，and based on P measure Choquet of integral regression model。