本篇論文探討函數型資料的單因子變異數分析(one-way ANOVA for functional data)問題，並提出新的檢定統計量，包括函數局部與全域的檢定統計量，其中使用局部多項式迴歸(local polynomial regression)方法於函數的估計上，建立單一函數、群平均函數與總平均函數之估計。透過單一函數、整群函數與全體函數之誤差平方和(sum of squared errors)，來定義函數組內與組間的變異量，提出的檢定統計量形式仿效傳統的單因子變異數分析，將組間變異量除以組內變異量，且證明新的檢定統計量會近似F分配。模擬中除了觀察提出之統計量在不同情況下，型一錯誤(type I error)和檢定力(power)等的變化外，還與文獻上的其它方法比較，且針對函數型資料有相關性時，加入generalized least squares以提升檢定力。最後則將方法應用在高速公路的交通資料上，並進行有關2016年高雄美濃地震的案例研究。

We propose and study a new test for the one-way ANOVA problem for functional data based on local polynomial regression with hierarchical ANOVA structures. Exact local and global ANOVA expressions are obtained for estimating individual curves, a mean curve within a group of functions, and an overall mean curve for all groups of functions. The proposed test statistics, mimicking the classical one-way ANOVA, have a form of comparing between-group SSE to within-group SSE, and are shown to have asymptotic F-distributions. Simulation studies are presented to compare the proposed global ANOVA test with some tests in the literature. A real data example on traffic flows illustrate the proposed tests.

摘要
目錄
1 緒論
2 背景介紹
2.1 函數型資料. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 函數型資料的單因子變異數分析. . . . . . . . . . . . . . . . . . . 2
2.3 局部多項式迴歸. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.4 局部多項式迴歸的變異數分析. . . . . . . . . . . . . . . . . . . . . 6
2.5 其它文獻回顧. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5.1 Wishart 分配. . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5.2 Wishart 過程. . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5.3 Two-Cumulant Matched F-Approximation . . . . . . . . . . 8
3 階層變異數分析與檢定方法
3.1 階層變異數分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 檢定統計量與漸近分配. . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2.1 局部單因子變異數分析. . . . . . . . . . . . . . . . . . . . 16
3.2.2 全域單因子變異數分析. . . . . . . . . . . . . . . . . . . . 20
4 模擬
4.1 模擬結果. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 文獻方法比較. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 提升布朗誤差之檢定力. . . . . . . . . . . . . . . . . . . . . . . . . 31
5 實際資料分析
6 結論與後續研究
7 附錄
7.1 證明. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7.1.1 定理3.2. 證明. . . . . . . . . . . . . . . . . . . . . . . . . 39
7.1.2 定理3.3. 證明. . . . . . . . . . . . . . . . . . . . . . . . . 41
7.2 模擬結果. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.2.1 模型(I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.2.2 模型(II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7.2.3 模型(III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.2.4 模型(IV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.2.5 文獻方法比較. . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.2.6 真實共變異矩陣. . . . . . . . . . . . . . . . . . . . . . . . 52
7.2.7 圖. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
參考文獻

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