本文考慮函數型線性模型 $Y(t)$ = $\beta_{0}(t)$ + $\mathbf{X}\bm{\beta}(t)$ + $\epsilon(t)$，其中$Y(t)$為函數型反應變數，$\mathbf{X}$為純量型解釋變數，$\bm{\beta}(t)$為係數函數，$\epsilon(t)$ 為具同質性和獨立性之誤差。我們進一步假設模型中的$\bm{\beta}(t)$ 為某些未知的本質編碼$\Phi(t)$ 之線性組合，亦即 $\bm{\beta}(t)$ = $\Gamma\Phi(t)$，其中$\Gamma$ 稱為本質效應參數。本文主要探討的是在反應變數中有遺失值時，如何估計本質編碼與本質效應。透過改寫Liao (2018)中估計本質編碼的估計式，我們得知若能先估計出係數函數$\bm{\beta}(t)$，便可由係數函數估得本質編碼與本質效應。因為本文探討的數據中有遺失值，無法藉由逐點最小平方法估計係數函數$\bm{\beta}(t)$，故本文採用文獻中處理不規則型函數型資料的方法，發展出三種做法來估得$\widehat{\bm{\beta}}(t)$，再用其估計出本質編碼與本質效應。最後，我們將這些分析方法，應用到我們生成的兩筆有遺失值的模擬數據上，並與無遺失值時的分析結果做比對。

In this work, we consider the functional linear model $Y(t) = \beta_{0} + \mathbf{X}\bm{\beta}(t) + \epsilon(t)$, where the response $Y(t)$ is a random function of $t$, the model matrix $\mathbf{X}$ is scalar, $\bm{\beta}(t)$ is the vector of coefficient functions, and $\epsilon(t)$ is an error function of $t$ assumed to be homogeneous and independent over $t$. We assume that $\bm{\beta}(t)$ is a linear combination of some unknown essence codings $\Phi(t)$. That is, there exists a matrix of parameters $\Gamma$ such that $\bm{\beta}(t) = \Gamma\Phi(t)$. The parameters in $\Gamma$ are called essence effects. In this thesis, we discuss the estimation of essence codings and the analyses of essence effects when some response values are missing.
By expressing the estimation criterion of essence codings in Liao (2018) as a function of ${\bm{\beta}}(t),$ we conclude that when there are missing values in the response, the estimation problem of essence codings and essence effects can be resolved if ${\bm{\beta}}(t)$ can be estimated. Because the pointwise least square estimator of ${\bm{\beta}}(t)$ is infeasible under the missing circumstance, we adopt some methods in the literature for irregular functional data to develop three different methods for estimating $\bm{\beta}(t)$, and use the estimate $\hat{\bm{\beta}}(t)$ to further estimate essence codings and essence effects. The methods developed in this thesis are demonstrated using two simulated datasets with missing values and the analysis results are compared with that of no missing values.

1 緒論1
2 文獻回顧5
2.1 本質編碼與效應之估計. . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 相關性法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 加法混合模型法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 局部線性迴歸法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 研究方法15
3.1 本質編碼與效應之估計. . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 相關性法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 加法混合模型法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 局部線性迴歸法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 數據分析20
4.1 晶圓厚度模擬資料. . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 晶圓模擬資料分析結果. . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2.1 數據一. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2.2 數據二. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5 結論與建議32
參考文獻33

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in semivarying coefficient models for longitudinal/ clustered data.
The Annals of Statistics, 44(5), 1988-2017.
[2] Liao, Y.-F. (2018). Identifying essence codings and effects in functional linear
models with homogeneous and independent errors, Master thesis, National
Tsing Hua University, Hsinchu, Taiwan.
[3] Scheipl, F., Staicu, A. M., and Greven, S. (2015). Functional additive mixed
models. Journal of Computational and Graphical Statistics, 24(2), 477-501.
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