以迭代學習模式偵測函數型變化之有效投影空間

Abstract

功能型資料之變點偵測在現代統計分析中具有重要性，並廣泛應用於訊號處理與時間序列分析等領域。隨著資料維度與結構日益複雜，如何在高維或連續型資料中準確辨識平均函數的變化，已成為一項具挑戰性的問題。傳統方法多仰賴將資料投影至低維子空間，例如功能型主成分分析（FPCA），但其投影方向主要反映變異結構，未必能捕捉與平均變化最相關的資訊。

為此，本文採用中央分布偏離子空間（Central Distribution Deviation Subspace, CDDS），以刻劃與平均變化相關之方向，並據此建構具變化資訊的低維表示。我們進一步提出一套迭代學習架構，透過交替進行變點偵測與子空間更新，使估計逐步貼近實際的變化結構。

最後，透過模擬實驗與既有方法進行比較，本文所提出的方法在變點偵測能力、位置估計準確度及檢定力方面均展現較佳表現。此外，本文將所提出的方法應用於英格蘭中部氣溫資料，以說明其在實際資料分析中的應用性。

Functional change point detection has become an important problem in modern statistical analysis, with applications in areas such as signal processing, time series analysis, and functional data analysis. Traditional approaches often rely on projecting functional observations onto low-dimensional subspaces using variance-based methods such as Functional Principal Component Analysis (FPCA). However, such projections are not necessarily aligned with structural changes in the data and may fail to capture directions that are most informative for detecting change points.

To address this limitation, we adopt the concept of a Central Distribution Deviation Subspace (CDDS), which characterizes a low-dimensional subspace that captures mean changes associated with change points. Building on this notion, we propose an iterative learning framework that adaptively estimates change-informative projection directions. The proposed method alternates between estimating change points and updating projection directions, thereby progressively refining a representation that better captures mean changes in the data.

Numerical experiments demonstrate that the proposed approach achieves higher detection power and more accurate change-point localization than existing projection-based methods. In addition, an application to the Hadley Centre Central England Temperature dataset illustrates the practical applicability of the proposed approach.