應用於異質定義域之多變量函數型資料的線性判別分析

Abstract

在本論文中，我們聚焦於多變量函數型資料的分類問題，此類資料定義於可能異質的定義域上。雖然已有眾多方法被提出以進行函數型資料的分類，但大多數僅針對單變量函數，或是定義於相同定義域上的多個函數。然而，當函數型變數定義於異質的定義域時，直接套用此類方法可能並不恰當。為此，我們提出一個整合 multivariate functional principal component analysis (MFPCA, Happ and Greven, 2018) 與 sensible functional linear discriminant analysis (SFLDA, Chen and Jiang, 2018) 的統一方法，稱為 multivariate SFLDA (M-SFLDA)，可用於分類收集於異質定義域上的多變量函數型資料。具體而言，首先利用 MFPCA 整合多個函數變數，並同時建構類內共變異數；接著使用 SFLDA 辨識判別方向；最後在投影結果上套用 linear discriminant analysis (LDA) 進行分類。模擬研究涵蓋不同雜訊程度與變數間相關性結構，結果顯示本方法在僅使用兩個成分的情況下，仍能穩定且有效地達成準確分類。與傳統方法，即先以 MFPCA 降維再以 LDA 對主成分分數進行分類，簡稱 MFPCA+LDA 相比，M-SFLDA 在變數定義於異質定義域的情境下表現更為優異。此外，我們亦將本方法應用於一筆實際資料，以建立能區分酗酒者與對照組的分類規則。結果顯示，M-SFLDA 在包含可能無資訊變數的情況下，仍能維持穩定且良好的分類表現。附錄提供了模擬研究的補充資料與結果。

In this thesis, we focus on the classification problem for multivariate functional data defined on possibly heterogeneous domains. While numerous methods have been proposed for functional data classification, most are for univariate functions or for multiple functions on the same domain. However, when the functional variables are on heterogeneous domains, directly applying such methods may not be proper. Thus, we propose a method that integrates multivariate functional principal component analysis (MFPCA, Happ and Greven, 2018) and sensible functional linear discriminant analysis (SFLDA, Chen and Jiang, 2018) into a unified framework, named multivariate SFLDA (M-SFLDA), which is capable of classifying multivariate functional data collected on heterogeneous domains. First, MFPCA is applied to integrate multiple functional variables, and a byproduct, the within-class covariance, is constructed. SFLDA is then employed to identify the discriminant directions. Finally, linear discriminant analysis (LDA) is applied to the resulting projections. Simulation studies with various noise levels and correlation structures demonstrate that the proposed method is both stable and effective in achieving accurate classification with two components. Compared to the conventional approach, referred to as MFPCA+LDA, which performs MFPCA for dimension reduction and applies LDA to the principal component scores for classification, M-SFLDA consistently outperforms, particularly in the scenario where the functional variables are on heterogeneous domains. The proposed method is applied to a real dataset to construct a classification rule capable of discriminating between alcoholic and control subjects based on their functional recordings. The results demonstrate that M-SFLDA achieves decent classification performance and remains stable even when possibly uninformative variables are included. Appendices include additional materials and results of the simulation studies.