在製程中，一種連續收集而來且呈現曲線型態的品質特性資料，我們稱之為函數型資料（Functional Data）。傳統上，函數型資料被視為是一種特殊形式的量變曲線（Profile）資料。然而，不論是使用線性或非線性的模型，其在建構品管圖上的共同特性，就是利用已發展完備的多變量品管方法，應用在監控這些從品質資料中估計出來的模型參數。通常在實務的應用上，這些線性或非線性模型的確切函數型式是很難知道的。並且，當我們從這些估計參數中偵測到了異常訊號時，我們也很難從這些異常的估計參數理解其在曲線上的直接影響。而函數型資料分析方法（Functional Data Analysis）提供了更好的解決方法，本論文的目的就是研究如何應用函數型資料分析方法來改善量變曲線資料品管圖的不足。因此，本篇論文提供了一個利用函數型資料分析方法對函數型資料進行統計製程管制的架構。從模擬研究中我們發現，函數型資料管圖對於製程狀態改變的偵測非常敏感。在本篇論文中我們也示範了如何實作函數型資料管圖在實際的資料。

In manufacturing process, a sequence of measurements of quality characteristic is increasingly taken across some continuum, producing a curve that represents the quality of the item. This curve provides the so-called functional data. Traditionally, functional data is treated as a special type of profile data. Regardless of a linear or nonlinear profile, the common approaches of the control chart are based on the multivariate control chart by monitoring the estimated parameter of the predefined linear or nonlinear model. Usually, the model is difficult to know practically, and it is also difficult to identify the abnormal pattern from the outlying parameter. By using the techniques of functional data analysis, we proposed the functional data control chart which can provide a better solution to these problems. In the Monte Carlo simulations, we show that the functional data control chart is sensitive when the underlying process status is changed. By applying to real example data, the new method exhibits a good performance.

Chapter 1 Introduction ……………….………………………………. 01
1.1 Multivariate Statistical Process Control…………………….……… 02
1.2 Control Chart for Monitoring Profile………………………...…….. 02
1.3 Control Chart for Functional Data ………………………......……. 04
1.4 Literature Review ……………………………….……..….……. 05
Chapter 2 Preprocess Functional Data ……………….………………. 07
2.1 Introduction to Functional Data………………………………… 07
2.2 The Basis System …………………………………….……..….. 09
2.2.1 The B-Spline Basis System……………….……………... 10
2.2.2 The Fourier Basis System……………………...………... 11
2.3 From Discretized Data to Functional Data…………………………. 11
Chapter 3 Functional Data Control Chart………………………..…… 14
3.1 Phase I Analysis………………………………………………… 14.
3.2 The Functional Data Control Chart………………………..………. 11
Chapter 4 Simulation Analysis…………………..………...………….. 25
Chapter 5 Example………………………………………...….........….. 40
Chapter 5 Conclusions and Future Research………………..………... 44
5.1Conclusions……………………………………………...…….…44
5.2 Future Research………………………….………………….….. 44
References…………………………………………………………..... 47

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