以兩階段法建構直覺式模糊迴歸模式

迴歸分析是一種統計學上分析數據的方法，建立解釋變數與反應變數之間的關聯性。由於現實環境中，所獲得的資料並非都是明確的數值(crisp value)，有些資料本身是模糊的(fuzzy)，致使傳統的分析方法難以使用。為了處理這類型的資料，Zadeh學者在1965年提出模糊理論(Fuzzy set theory)，將不確定性表達於資料型態上，許多學者遂將迴歸分析方法擴展至模糊環境中，做更廣泛的應用。此外，直覺式模糊集合(Intuitionistic fuzzy sets, IFS)為模糊理論之延伸，在歸屬度函數外，加入非歸屬度函數來衡量，因此直覺式模糊集合能表達更多資訊，較貼近於現實環境。本研究根據直覺式模糊集合與模糊迴歸理論，建立直覺式模糊迴歸模型，並利用兩階段法進行求解。本研究採用迴歸係數為明確值、解釋變數與反應變數為直覺式模糊數(Intuitionistic fuzzy number, IFN)的迴歸估計式。由於所求解的迴歸係數為明確數值，可以獲得解釋變數與反應變數之間的確切關係，提供決策者更明確的決策資訊。本研究在前置階段會先取得直覺式模糊資料。然後是求解階段，採用兩階段法，第一階段先對原始直覺式模糊資料做解模糊化(defuzzified)，轉換成一組明確資料，接著將該資料做傳統最小平方法，求得一組明確的迴歸係數估計值；第二階段，在最小化總誤差量的目標下，根據距離測度建立一個數學規劃(mathematical programming)模型，並使用Arefi and Taheri學者在2015年所提出之範例，在反應變數觀察值與估計值差距最小的目標下，求解直覺式模糊調整變數(Intuitionistic fuzzy adjustment variable)，以減少模糊估計誤差。模式建構完成後，進行誤差值的分析評估，相較於舊有文獻之方法，本研究模型的誤差衡量指標有更理想的表現；而在模型的穩健度上，本研究之模式透過交叉驗證(cross-validation)後所得的結果，可以驗證明此方法之合理性。

關鍵字

直覺式模糊集合；模糊迴歸；兩階段法；解模糊化； αβ-截集

並列摘要

Regression analysis is one of the most widely used decision-making tools. It allows decision-makers to determine the relationship between input variables and output variables. In the complex real-world environment, data may be uncertain, written in linguistic terms, or based on personal subjective attitudes. Therefore, fuzzy set theory was developed to deal with this datatype. To express the essence of uncertainty better, scholars proposed the concept of intuitionistic fuzzy sets (IFS) as a generalization of fuzzy set theory. In addition to positive information, it also includes negative information. In this study, the regression coefficients are crisp values, and the input variables and output variable are intuitionistic fuzzy numbers (IFN). IFN multiplied with each other will lead to an over-increase in the spread of IFN. In other words, the fuzziness of the numbers will be over-increased. Differing from previous studies on intuitionistic fuzzy regression (IFR), this study proposes a two-stage approach to construct IFR model based on the distance concept. In the first stage, the fuzzy observations are defuzzified so that the classical least-squares method can be applied to find a crisp regression line. In the second stage, the adjustment variable of the model, which represents the fuzziness of the data, is determined to give the model the best explanatory power and the smaller estimation error between the observed and estimated values. The predictive ability of the obtained models is evaluated using similarity and distance measures. The results indicate that the model proposed in this study has better performance than those in previous studies. As for the robustness of the model, the cross-validation of the model also proves that the rationality of this method is sufficient. Furthermore, this study demonstrates the applicability of the proposed two-stage approach in handling a problem with asymmetric intuitionistic fuzzy numbers.