從多變量方法延伸至函數型數據的線上轉折點偵測

Abstract

函數性數據，例如醫學領域中的心電圖 (ECG) 信號，或氣象領域中隨時間記錄的天氣變量，通常以連續且無限維度的曲線形式呈現，已成為當今最常見的數據形式之一。然而，儘管其重要性日益增加，針對這類數據的線上轉折點偵測 (OCPD) 方法在現有文獻中的討論相對有限。相比之下，多變量數據則擁有許多已發展成熟的 OCPD 方法。因此，我們在仔細審視現有文獻後，決定將四種無分配假設的多變量 OCPD 方法延伸應用至函數性數據，希望能有效地檢測出連續數據流中的異常。模擬實驗中，我們使用兩個常用的基準——平均運行長度和平均檢測延遲——來評估這些方法的表現，並通過這兩個指標來驗證其可靠性和效率。最後，我們通過比較這些改進方法之間的效能差異，並進一步探討導致這些差異的可能因素。這項研究不僅為函數性數據的線上監控系統的發展作出了貢獻，也為實時分析函數性數據提供了有價值的見解和一些潛在可行的方法。

Functional data, such as Electrocardiogram (ECG) signals in the medical field, or weather-related variables recorded over time in the meteorological field, are often presented as continuous and infinite-dimensional curves, and have become a common form of data. However, despite its relevance, online change point detection (OCPD) for this datatype has received limited attention in the existing literature. On the other hand, multivariate data have plenty of its own OCPD methods; therefore, after a thorough survey, we decided to extend four nonparametric multivariate OCPD methods to accommodate functional data characteristics, aiming to successfully detect those anomalies in continuous data streams. Eventually, we evaluate the performance of these adapted methods against two benchmarks commonly employed in online settings: average run length and average detection delay. These metrics provide insights into the reliability and efficiency of these methods. Our work compares the performance between the extended methods and native functional data OCPD techniques via simulations, and further discusses the differences. This research contributes to the ongoing development of robust online monitoring tools for functional data, and offers valuable perspectives for potential method candidates and practical implementations in real-time analysis for functional data.