主成分分析是在面對高維度類型的資料時，一種廣泛使用來達到降低資料維
度的方法。主成分分析在許多領域如訊號處理、氣象、機械工程等等領域皆有所
應用。然而，在這些領域中，有時我們感興趣的可能是資料分布尾端的變異而非
整體的變異。為了解決這個問題，不同於傳統使用的歐幾里德範數，Tran et al. (2014)
提出基於百分位及期望百分位所得之非對稱範數來尋找用來解釋資料尾端特徵之
主成分。為了探討及比較分布尾端部分及整體之主成分差異，我們主要有以下兩樣工
作。首先我們考慮在不同參數設定之函數型數據模擬，來探討期望百分位主成分
及傳統主成分之差異及關係。另外我們也計算並比較在冪跳躍模型下，當資料具
有不同相關性時，期望百分位主成分及傳統主成分所對應之特徵值之關係及表現。

Principal component analysis is an extensively used dimension reduction tool for many
kinds of high dimensional data. It is applied in many elds such as signal processing,
weather, mechanical engineering and other elds. In these elds practitioners may be
more interested in the tail variations of the data rather than the variations around the
mean. To handle this problem, an asymmetric norm based on quantiles or expectiles,
which are di erent from the classical L2-norm orthogonal projection, are used to nd
the principal components interpreting the tail character of the data. To investigate the
principal components in the tail part and central part of the distribution, we focus on two
works. First we consider the principal expectile component in functional data simulations
under many kinds of di erent settings to study the relationship between the PCA and
PEC. Second, we evaluate the eigenvalues of PEC and compare with the estimator of
conventional eigenvalues and the two corrected estimator of eigenvalues when the data
are correlated under the power spiked model.

論文審定書 i
誌謝 ii
摘要 iii
Abstract iv
1 Introduction 1
2 Principal expectile component 2
2.1 Expectile and asymmetric norms . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 The conventional PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 PCA in an asymmetric norm . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Algorithm for computing PEC . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 An example for PEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 The power spiked model 8
3.1 Noise-reduction methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Cross-data-matrix methodology . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Simulation study 10
4.1 Functional data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2 Simulation study of eigenvalue estimators . . . . . . . . . . . . . . . . . . . 15
5 Discussions and conclusions 16
References 17
A Appendix 18

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