適應式貪婪分段法於函數變異偵測中的應用

Abstract

在統計分析中，變異點偵測是一項關鍵議題，尤其在面對函數型資料時，其挑戰性更為明顯。過去的研究多半仰賴固定基底投影與強假設條件，可能導致對結構性變異的偵測能力不足。為解決此問題，本研究提出一套適應式貪婪分段法，結合差異強化協方差算子與迭代式基底選擇策略，能有效對應不同方向的變異並提升變異點定位的精確度。 本方法的主要優勢包括：(1) 能動態調整基底方向，使其更貼近資料中的結構性變化；(2) 無須預先設定變異點個數或限制每段僅含一個變異點；(3) 在模擬與實證資料中皆展現出優於傳統方法的表現，且在高噪音或時間相依的情境下依然能維持良好表現。 透過一系列模擬實驗與實際資料分析，本研究驗證了所提方法在變異點數目估計、位置偵測準確率與過／低偵測情境下的穩定表現。未來可望應用於氣候分析、生物訊號監控與智慧生活等多元領域。

Change-point detection is a critical topic in statistical analysis, especially for functional data where traditional projection-based methods may fail to capture structural shifts accurately. To address this challenge, we propose an Adapted Greedy Segmentation (AGS) framework that integrates discrepancy-enhanced covariance operators with an iterative basis refinement procedure. This approach allows the basis functions to adaptively align with the directions of structural changes, thereby improving detection sensitivity and localization accuracy. The proposed method offers several advantages:(1) it automatically selects basis directions that reflect segment-wise changes;(2) it does not rely on restrictive assumptions such as the "at most one change-point" condition;(3) it demonstrates improved empirical performance over existing methods, and maintains reliable detection accuracy under both high-noise and temporally dependent settings. Extensive simulation studies and real data applications confirm the robustness and effectiveness of the AGS approach. The results suggest that our method achieves superior detection accuracy compared to conventional projection-based and fully functional techniques, and holds promise for broader applications in climate research, biomedical monitoring, and smart environments.