多變量函數型典型相關分析

Abstract

現今工業進入奈米的世界，維持產品良率變得更加重要，製程更精密的同時，產品品質更是嚴格。隨著奈米製程的進步，如何在數以百計的變數之中，找到關鍵變因以預防產品缺陷便是生產線最關切的課題。由於資訊科技的進步，資料的儲存漸漸以能夠清楚看到變數隨時間變化的剖面資料(profile data)為大宗，此類型的變數可視為函數型變數。本論文以函數資料分析為基礎，提出了一個能夠同時考慮兩群函數型變數之間相關性分析的統計方法。現有的相關性分析僅能在兩個函數型變數之間找相關性，本論文將其推廣至更高維度，不僅適合用來分析剖面資料，還能同時分析數個函數型變數，找到與品質具有影響力的變因。此方法可以應用在很多領域，本論文將之應用於果蠅死亡曲線的相關性分析，探討母果蠅死亡曲線與公果蠅死亡曲線、公果蠅數量比例及果蠅密度三個函數型變數的相關性。

Nanotechnology has opened a new world in manufacturing processes. These more sophisticated processes are naturally accompanied with more stringent quality standards. Thus, maintaining or even improving the product yield has become an important issue in the manufacturing industry. With the rapid development of nanometer processes, how to find those key factors that could affect the process quality among possibly thousands of variables in order to increase the yield is one of the most important issues concerned in the production line. With advance computer technology, profile data of a functional variable showing the change of the variable over time, gradually has become a main type of data. However, for profile data, the existing “functional canonical correlation analysis” methods developed in the literature can only analyze the association between two functional variables. In this thesis, we propose a statistical method called “multivariate functional canonical correlation analysis” to explore the association between two groups of multiple functional variables based on functional data analysis. In addition, the proposed method can be easily modified to explore the correlation between multiple functional variables and one or more response variables. With that, the method has a potential to find among many functional variables the quality-related variables. A simulation study demonstrates the effectiveness of the method. The proposed method is applicable in many areas. As an illustrative example, the method is applied to a real dataset of medflies to investigate the association between the mortality of the female medflies and the mortality of the male medflies along with another two functional variables.