本論文探討函數型線性模型 Y (t) = β0(t) + Xβ(t) + ε(t)，其中應變數 Y (t) 為函數型的隨機變數，解釋變數為純量型變數，而 ε(t) 則假設為具同質性和獨立性之誤差。對於係數函數 β(t)，我們假設其為某些未知本質編碼 Φ(t) 的線性組合，亦即 β(t) = ΓΦ(t)，其中 Γ 稱為本質效應。本論文所研究的課題是辨識及估計重要的 Φ(t) 和 Γ。因為滿足 β(t) = ΓΦ(t) 的 Φ(t) 有無窮多組解，我們必須設定 Φ(t) 多滿足一些條件，方能有唯一解。本論文將本質編碼視為投影方向，並將函數型線性模型投影至本質編碼上轉換成常見的單變量線性模型。藉由函數型應變數之投影量能被此單變量線性模型中的解釋變數解釋最多變異之準則，以唯一定義本質編碼，並透過此準則對其做估計。在求得本質編碼之估計 Φˆ (t) 後，可以在給定 Φˆ (t) 下，對 Γ 做估計及檢定，以辨識出重要的本質效應。本論文亦考慮將截距項 β0(t) 拆成本質編碼的線性組合之模型，以及考慮模型 Y (t) = β0(t) + Xγ0 + Xβ(t) + ε(t)，其中 γ0 不為 t 的函數，並比較它們之
間分析結果的差異。最後，我們將本論文所發展的方法，應用於一筆晶圓厚度的實驗數據上，對此數據估計出本質編碼，並辨識其所對應之重要本質效應。

In this work, we consider the functional linear model Y (t) = β0(t) +Xβ(t) + ε(t), where the response Y (t) is a random function of t, the model matrix X is scalar, and ε(t) is an error function of t assumed to be homogeneous and independent over t. We assume that the vector of coefficient functions β(t) is a linear combination of some unknown essence codings Φ(t). That is, there exists a matrix of parameters Γ such that β(t) = ΓΦ(t). The parameters in Γ are called essence effects. In this thesis, we are interested in identifying meaningful and important essence codings
and effects. Because the equation β(t) = ΓΦ(t) has infinitely many solutions of Γ and Φ(t), we propose a criterion for this equation to uniquely define and estimate Γ and Φ(t). This criterion is developed by treating essence codings as projection directions and projecting the functional linear model onto every essence codings to transform it into a univariate linear model. This criterion sequentially maximizes the amount of variation in the response of this univariate linear model explained by the model matrix X to identify and estimate important projection directions.
For the analysis about the essence effects in Γ, including estimation and testing, we suggest to perform them by conditioning on the essence coding estimators Φˆ (t). In this thesis, we further consider the model in which the intercept function β0(t) is also expressed as a linear combination of essence codings, and the model Y (t) = β0(t) + Xγ0 + Xβ(t) + ε(t), where γ0 is a vector of unknown constants over t. The analysis procedures of essence codings and effects under the above two methods are developed and then compared to the first one. Finally, the methods developed in this thesis are applied on some functional data of wafer thickness to estimate essence codings and identify important essence effects.

1 緒論 1
2 文獻回顧 4
2.1 函數型線性模型 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 多變量線性模型 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 研究方法 6
3.1 模型介紹 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 本質編碼之定義準則 . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 本質效應參數之估計與檢定 . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4 反映常數項函數之本質編碼 . . . . . . . . . . . . . . . . . . . . . . . . 13
3.5 與 t 無關之效應 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 晶圓研磨實驗之數據分析 16
4.1 原始晶圓厚度數據之分析結果 . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 反映常數項函數之本質編碼的分析結果 . . . . . . . . . . . . . . . . . . . . 24
4.3 與 t 無關之效應的分析結果 . . . . . . . . . . . . . . . . . . . . . . . . 28
5 結論 32
參考文獻 33

[1] Johnson, R. A. and Wichern, D. W. (2007). Applied Multivariate Statistical Analysis, 6th edition. Pearson Prentice Hall, New Jersey.
[2] Peng, P.-R. (2018). “Identifying essence codings and effects in functional linear models with heterogeneous and correlated errors,” Master thesis, National Tsing Hua University, Hsinchu, Taiwan.
[3] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd edition. Springer, New York.