本文考慮函數型線性模型 Y(t) = β0(t) + Xβ(t) + ϵ(t)，其中 ϵ(t) 為具有異質性和相關性的誤差。我們進一步假設模型中的 β(t) 為某些未知的本質編碼 Φ(t) 之線性組合，亦即 β(t) = ΓΦ(t)，其中 Γ 稱為本質效應參數。因不加任何條件時，Φ(t) 會有無窮多組解，故本論文主要探討如何在某些關於 Φ(t) 的直交和長度限制式下，合理地定義 Φ(t) 和 Γ，以便能對其做估計。本文將本質編碼視為投影方向，先將函數型線性模型投影到本質編碼上轉成單變量線性模型，再藉由比較這些單變量線性模型，提出兩個定義 Φ(t) 的合理準則，其中一個準則與費雪線性判別分析有一致性，另一準則則能極大化 X 解釋函數型資料之變異量。本文在給定估計出的本質編碼下，估計本質效應參數，並檢定其是否顯著。最後，我們將這兩個準則應用到晶圓厚度模擬資料上，以比較不同準則和不同共變異函數
下，所估得的本質編碼之差異。

In this work, we consider the functional linear model Y (t) = β 0 (t) + Xβ(t) + ϵ(t), where ϵ(t) is an error curve assumed to be heterogeneous and correlated over t. We assume that the vector of coefficient functions β(t) is a linear combination of some unknown essence codings Φ(t), i.e., there exists a matrix of parameters Γ such that β(t) = ΓΦ(t). The parameters in Γ are called essence effects. Because both Γ and Φ(t) are unknown, the equation β(t) = ΓΦ(t) has infinitely many solutions of Γ and Φ(t). In this thesis, we impose some reasonable criteria on this equation to uniquely define Γ and Φ(t) under some orthogonality and norm constraints. We also develop the sample versions of these criteria for the purpose of estimating Φ(t). They are achieved by regarding every essence coding as a projection direction, and sequentially projecting the functional linear model onto every essence coding to transform it into a univariate linear model. We compare the performance of the univariate linear models generated from different projections in interpreting data to propose two criteria for choosing the best Φ(t). The first criterion has a close connection with Fisher’s linear discriminant analysis, while the second one is related to functional principal component analysis, i.e., it can identify the projection direction that maximizes the variation of Y (t) explained by X. For the analysis concerning Γ, including estimation and testing, we suggest a linear-model approach conditioned on the estimator of Φ(t). We also illustrate
our methods using simulated functional data of wafer thickness, and compare the estimated essence codings obtained under different criteria and different covariance structures of ϵ(t).

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