結合鄰近資訊與模糊分群之函數型轉折點偵測方法

Abstract

在函數型資料的轉折點分析中，分群是一種常見用以揭示資料結構變化的位置的方法。本方法提出結合模糊分群與移動窗口的變點偵測方法，旨在有效辨識資料中的結構改變。我們設計了兩種方法：第一種結合分群結果與隸屬值標準差的變異，用以偵測潛在轉折點，第二種則利用單邊窗口分群所獲得的隸屬值分佈，透過Kullback–Leibler 散度衡量差異程度，進一步找出變化位置。上述兩種方法皆透過分群結果與結構變異指標的交集，篩選出真實轉折點。此外，我們也提出投票機制來穩定分群數的選擇。透過五種模擬情境的驗證，本方法在面對變異幅度不一致與資料異質性時，仍展現良好的穩健性。最後，本研究亦應用於高維度空氣污染資料，顯示該方法在非函數型資料上仍具實用性。未來可進一步擴展至即時資料的應用，並探討其他分群方法，以提升於不同資料類型下的適應能力。

Clustering is a common approach in change point analysis for functional data, often used to identify structural variations within datasets. This paper proposes a novel method that integrates fuzzy clustering with a sliding window framework to detect potential change points. We develop two approaches: the first combines changes in cluster assignments with variations in the standard deviation of membership values to identify candidate change points. The second utilizes membership distributions derived from onesided window clustering and applies the Kullback–Leibler divergence to quantify differences, thereby locating mean shifts. Both approaches filter true change points by intersecting clustering results with indicators of structural change. Additionally, we introduce a voting-based mechanism to stabilize the selection of the number of clusters. Through extensive simulations under five scenarios, the proposed method demonstrates strong robustness, effectively handling uneven shift magnitudes and heterogeneous data structures. Finally, an empirical study using high dimensional air pollution data further illustrates the method＇s practical applicability. Future research may extend this framework to online settings and explore alternative clustering techniques to enhance adaptability across various data types.