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題名 模糊隨機變數在線性迴歸模式上的應用
Fuzzy Random Variables and Its Applications in Fuzzy Regression Model
作者 曾能芳
貢獻者 吳柏林<br>鄭宇庭
曾能芳
關鍵詞 集合表徵
模糊隨機變數
模糊迴歸模式
模糊期望值
模糊分配函數
模糊不偏性
set representation
fuzzy random variables
fuzzy regression model
fuzzy expected value
fuzzy distribution function
fuzzy unbiased
日期 2002
上傳時間 10-May-2016 18:56:11 (UTC+8)
摘要 　　傳統迴歸分析是假設觀測值的不確定性來自於隨機現象，本文則應用模糊隨機變數概念於迴歸模式的架構，考慮將隨機現象和模糊認知並列研究。針對樣本模糊數(x<sub>i</sub>, Y<sub>i</sub>)，我們進行模糊迴歸參數估計，並稱此為模糊迴歸模式分析。模糊迴歸參數估計大都採用線性規劃，求出適當區間，將觀測模糊數Y<sub>i</sub>的分佈範圍全部覆蓋。但是此結果並不能充分反映觀測樣本Y<sub>i</sub>的特性。本研究提出一套模糊迴歸參數的估計方法，其結果對觀測樣本的解釋將更為合理，且具有模糊不偏的特性。在分析過程中，我們亦提出一些模糊統計量如模糊期望值、模糊變異數、模糊中位數的定義，以增加對這些參數的模糊理解。最後在本文中也針對台灣景氣指標與經濟成長率作實務分析，說明模糊迴歸模式的適用性。
　　Conventional study on the regression analysis is based on the conception that the uncertainty of observed data comes from the random property. However, in this paper we consider both of the random property and the fuzzy perception to construct the regression model by using of fuzzy random variables. For the fuzzy sample (x<sub>i</sub>,Y<sub>i</sub>), we will process the parameters estimation of the fuzzy regression, and we call this process as fuzzy regression analysis. The parameters estimation for a fuzzy regression model is generally derived by the linear programming scheme. But it`s result usually doesn`t sufficiently reflect the characteristics of the observed samples. Hence in this paper we propose an alternative technique for parameters estimation in constructing the fuzzy regression model. The result will describe the observed data better than the conventional method did, moreover it will have the fuzzy unbiased properties. For the purpose of fuzzy perception on the fuzzy random variables, we also give definitions for certain important fuzzy statistics such as fuzzy expected value, fuzzy variance and fuzzy median. Finally, we give an example about the Taiwan Business Cycle and the Taiwan Economic Growth Rate for illustration.
參考文獻 Agee, W. S. and Turner, R. H. (1979). Application of Robust Regression to Trajectory data Reduction, In Robustness in Statistics (R. L. Launer and G. N. Wilkinson, eds). London: Academic Press.
     Chanas, S. & Florkiewicz, B. (1991). Deriving Expected Values from Probabilities of Fuzzy Subsets. European Journal of Operational Research, Vol. 50, p199-210.
     Hwang, C.M. & Yao, J.S. (1996). Independent Fuzzy Random Variables and their Application. Fuzzy Sets and Systems, Vol. 82, p335-350.
     Korner, R. (1997). On the Variance of Fuzzy Random Variables. Fuzzy Sets and Systems, Vol. 92, p83-93.
     Kruse, R. & Meyer, K. D. (1987). Statictics with Vague Data (Reidel, Dordrecht, Boston).
     Kwakernaak, H. (1978). Fuzzy Random Variables. Part I: Definitions and theorems. Information Sciences, vol 15, p1-15.
     Puri, M. L. (1986). Fuzzy Random Variables. Journal of Mathematical Analysis and Applications, Vol. 114, p409-422.
     Savic, D.A. & Pedrycz, W. (1991). Evaluation of Fuzzy Linear Regression Models. Fuzzy Set and Systems, Vol. 23, p51-63.
     Stojakovic, M. (1992). Fuzzy Conditional Expectation. Fuzzy Sets and Systems, Vol. 52, p53-60.
     Stojakovic, M. (1994) Fuzzy Random Variables, Expectation, and Martingales . Journal of Mathematical Analysis and Applications, Vol. 184, p594-606.
     Tanaka, H. Uejima, S. and Asai, K. (1980). Fuzzy Linear Regression Model. International Congress on Applied Systems Research and Cybernetics, Aculpoco, Mexico.
     Tanaka, H. Uejima, S. and Asai, K. (1982). Linear Regression Analysis with Fuzzy model. IEEE Trans. SystemsMan Cybernet, Vol. SMC12, p903-907.
     Tanaka, H. & Ishibuchi, H. (1993). An architecture of neural networks with interval weights and its application to fuzzy regression analysis. Fuzzy Sets and Systems, Vol. 57, p27-39.
     Toth, H. (1992). Probabilities andFuzzy Events: an Operational Approach. Fuzzy Sets and Systems, Vol. 48, p113-127.
     Wang, G. & Zhang, Y. (1992). The Theory of Fuzzy Stochastic Processes. Fuzzy Sets and Systems, Vol. 51, p161-178.
     Wu, B. and Tseng, N. F. (2002). A New Approach to Fuzzy Regression Models with Application to Business Cycle Analysis. Fuzzy Sets and Systems (will appear).
     Wu, H.C. (1999). Probability density functions of Fuzzy Random Variables. Fuzzy Sets and Systems, Vol. 105, p139-158.
     Wu, H.C. (2000). The Law of Large Numbers for Fuzzy Random Variables. Fuzzy Sets and Systems, Vol. 116, p245-262.
     Yang, M. & Ko, C. (1997). On cluster-wise fuzzy regression analysis. IEEE Trans. Systems Man Cybernet, Vol. 27, 1-13.
     Yun, K.K. (2000). The Strong Law of Large Numbers for Fuzzy Random Variables. Fuzzy Sets and Systems, Vol. 111, p319-323.
     Zadeh, L. A. (1965). Fuzzy Sets. Information and Control, vol 8, p338-353.
     Zadeh, L. A. (1968). Probability Measures of Fuzzy Events. Journal of Mathematical Analysis and Applications, Vol. 23, p421-427.
描述 博士
國立政治大學
統計學系
86354501
資料來源 http://thesis.lib.nccu.edu.tw/record/#A2010000073
資料類型 thesis
dc.contributor.advisor 吳柏林<br>鄭宇庭zh_TW
dc.contributor.author (Authors) 曾能芳zh_TW
dc.creator (作者) 曾能芳zh_TW
dc.date (日期) 2002en_US
dc.date.accessioned 10-May-2016 18:56:11 (UTC+8)-
dc.date.available 10-May-2016 18:56:11 (UTC+8)-
dc.date.issued (上傳時間) 10-May-2016 18:56:11 (UTC+8)-
dc.identifier (Other Identifiers) A2010000073en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/96299-
dc.description (描述) 博士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計學系zh_TW
dc.description (描述) 86354501zh_TW
dc.description.abstract (摘要) 　　傳統迴歸分析是假設觀測值的不確定性來自於隨機現象，本文則應用模糊隨機變數概念於迴歸模式的架構，考慮將隨機現象和模糊認知並列研究。針對樣本模糊數(x<sub>i</sub>, Y<sub>i</sub>)，我們進行模糊迴歸參數估計，並稱此為模糊迴歸模式分析。模糊迴歸參數估計大都採用線性規劃，求出適當區間，將觀測模糊數Y<sub>i</sub>的分佈範圍全部覆蓋。但是此結果並不能充分反映觀測樣本Y<sub>i</sub>的特性。本研究提出一套模糊迴歸參數的估計方法，其結果對觀測樣本的解釋將更為合理，且具有模糊不偏的特性。在分析過程中，我們亦提出一些模糊統計量如模糊期望值、模糊變異數、模糊中位數的定義，以增加對這些參數的模糊理解。最後在本文中也針對台灣景氣指標與經濟成長率作實務分析，說明模糊迴歸模式的適用性。zh_TW
dc.description.abstract (摘要) 　　Conventional study on the regression analysis is based on the conception that the uncertainty of observed data comes from the random property. However, in this paper we consider both of the random property and the fuzzy perception to construct the regression model by using of fuzzy random variables. For the fuzzy sample (x<sub>i</sub>,Y<sub>i</sub>), we will process the parameters estimation of the fuzzy regression, and we call this process as fuzzy regression analysis. The parameters estimation for a fuzzy regression model is generally derived by the linear programming scheme. But it`s result usually doesn`t sufficiently reflect the characteristics of the observed samples. Hence in this paper we propose an alternative technique for parameters estimation in constructing the fuzzy regression model. The result will describe the observed data better than the conventional method did, moreover it will have the fuzzy unbiased properties. For the purpose of fuzzy perception on the fuzzy random variables, we also give definitions for certain important fuzzy statistics such as fuzzy expected value, fuzzy variance and fuzzy median. Finally, we give an example about the Taiwan Business Cycle and the Taiwan Economic Growth Rate for illustration.en_US
dc.description.tableofcontents 謝辭
     摘要-----1
     Abstract-----2
     目錄-----3
     一、前言-----4
     二、模糊集合與運算-----7
     　　2.1　多值邏輯和模糊集合-----7
     　　2.2　模糊集合運算-----9
     三、模糊隨機變數-----13
     　　3.1　模糊隨機變數的引進-----13
     　　3.2　模糊隨機變數的架構與運算-----15
     四、模糊期望值和模糊分配函數-----18
     　　4.1　模糊期望值-----18
     　　4.2　模糊分配函數-----19
     五、模糊線性迴歸式-----22
     　　5.1　傳統模糊迴歸模式-----22
     　　5.2　測量誤差存在於中點的模糊迴歸模式-----24
     　　5.3　測量誤差存在於中點和半徑的模糊迴歸模式-----28
     六、模糊迴歸參數的推估-----33
     　　6.1　測量誤差存在於中點的模糊迴歸模式-----35
     　　6.2　測量誤差存在於中點和半徑的模糊迴歸模式-----37
     七、實用架構下的模糊統計量-----40
     　　7.1　模糊統計量的引進-----40
     　　7.2　模糊迴歸之模糊統計量-----42
     　　7.3　應用實例-----46
     八、結論-----51
     附錄-----52
     　　A1定理5.1證明：-----52
     　　A2定理5.2證明：-----52
     　　A3定理5.3證明：-----53
     　　A4定理5.4證明：-----55
     　　A5定理5.5證明：-----56
     　　A6定理5.6證明：-----57
     　　A7定理6.1證明：-----58
     　　A8定理6.2證明：-----59
     　　A9定理6.3證明：-----60
     參考文獻-----63		zh_TW
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#A2010000073en_US
dc.subject (關鍵詞) 集合表徵zh_TW
dc.subject (關鍵詞) 模糊隨機變數zh_TW
dc.subject (關鍵詞) 模糊迴歸模式zh_TW
dc.subject (關鍵詞) 模糊期望值zh_TW
dc.subject (關鍵詞) 模糊分配函數zh_TW
dc.subject (關鍵詞) 模糊不偏性zh_TW
dc.subject (關鍵詞) set representationen_US
dc.subject (關鍵詞) fuzzy random variablesen_US
dc.subject (關鍵詞) fuzzy regression modelen_US
dc.subject (關鍵詞) fuzzy expected valueen_US
dc.subject (關鍵詞) fuzzy distribution functionen_US
dc.subject (關鍵詞) fuzzy unbiaseden_US
dc.title (題名) 模糊隨機變數在線性迴歸模式上的應用zh_TW
dc.title (題名) Fuzzy Random Variables and Its Applications in Fuzzy Regression Modelen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) Agee, W. S. and Turner, R. H. (1979). Application of Robust Regression to Trajectory data Reduction, In Robustness in Statistics (R. L. Launer and G. N. Wilkinson, eds). London: Academic Press.
     Chanas, S. & Florkiewicz, B. (1991). Deriving Expected Values from Probabilities of Fuzzy Subsets. European Journal of Operational Research, Vol. 50, p199-210.
     Hwang, C.M. & Yao, J.S. (1996). Independent Fuzzy Random Variables and their Application. Fuzzy Sets and Systems, Vol. 82, p335-350.
     Korner, R. (1997). On the Variance of Fuzzy Random Variables. Fuzzy Sets and Systems, Vol. 92, p83-93.
     Kruse, R. & Meyer, K. D. (1987). Statictics with Vague Data (Reidel, Dordrecht, Boston).
     Kwakernaak, H. (1978). Fuzzy Random Variables. Part I: Definitions and theorems. Information Sciences, vol 15, p1-15.
     Puri, M. L. (1986). Fuzzy Random Variables. Journal of Mathematical Analysis and Applications, Vol. 114, p409-422.
     Savic, D.A. & Pedrycz, W. (1991). Evaluation of Fuzzy Linear Regression Models. Fuzzy Set and Systems, Vol. 23, p51-63.
     Stojakovic, M. (1992). Fuzzy Conditional Expectation. Fuzzy Sets and Systems, Vol. 52, p53-60.
     Stojakovic, M. (1994) Fuzzy Random Variables, Expectation, and Martingales . Journal of Mathematical Analysis and Applications, Vol. 184, p594-606.
     Tanaka, H. Uejima, S. and Asai, K. (1980). Fuzzy Linear Regression Model. International Congress on Applied Systems Research and Cybernetics, Aculpoco, Mexico.
     Tanaka, H. Uejima, S. and Asai, K. (1982). Linear Regression Analysis with Fuzzy model. IEEE Trans. SystemsMan Cybernet, Vol. SMC12, p903-907.
     Tanaka, H. & Ishibuchi, H. (1993). An architecture of neural networks with interval weights and its application to fuzzy regression analysis. Fuzzy Sets and Systems, Vol. 57, p27-39.
     Toth, H. (1992). Probabilities andFuzzy Events: an Operational Approach. Fuzzy Sets and Systems, Vol. 48, p113-127.
     Wang, G. & Zhang, Y. (1992). The Theory of Fuzzy Stochastic Processes. Fuzzy Sets and Systems, Vol. 51, p161-178.
     Wu, B. and Tseng, N. F. (2002). A New Approach to Fuzzy Regression Models with Application to Business Cycle Analysis. Fuzzy Sets and Systems (will appear).
     Wu, H.C. (1999). Probability density functions of Fuzzy Random Variables. Fuzzy Sets and Systems, Vol. 105, p139-158.
     Wu, H.C. (2000). The Law of Large Numbers for Fuzzy Random Variables. Fuzzy Sets and Systems, Vol. 116, p245-262.
     Yang, M. & Ko, C. (1997). On cluster-wise fuzzy regression analysis. IEEE Trans. Systems Man Cybernet, Vol. 27, 1-13.
     Yun, K.K. (2000). The Strong Law of Large Numbers for Fuzzy Random Variables. Fuzzy Sets and Systems, Vol. 111, p319-323.
     Zadeh, L. A. (1965). Fuzzy Sets. Information and Control, vol 8, p338-353.
     Zadeh, L. A. (1968). Probability Measures of Fuzzy Events. Journal of Mathematical Analysis and Applications, Vol. 23, p421-427.		zh_TW