利用拓撲資料分析進行具抗離群值能力之時間序列變化點偵測

Abstract

本研究提出一套結合時間延遲嵌入、Mapper 演算法與 DBSCAN 密度式分群的線上變點偵測框架。有別於傳統以均值或變異數等低階統計量為基礎的偵測方法，本方法將參考資料的分布結構以高維嵌入空間中的原型球集合加以表示，使偵測器得以對均值偏移、變異數變化與分布形態改變等多種類型的結構轉變保持敏感。此外，本方法在建構原型結構前，先透過 Mapper 圖的雜訊標記機制對參考資料進行自動清洗，有效去除孤立離群值對參考模型的干擾，使偵測器在參考資料受污染的情境下仍能維持穩定的偵測能力。 透過模擬實驗，本研究於六種情境（均值變化、變異數變化與分布形態變化，各分為參考資料乾淨與受污染兩種條件）下與三種基準方法（滑動視窗、最大均值差異法與判別式漂移偵測法）進行比較。實驗結果顯示，本方法為唯一在所有情境與參考資料條件下均能維持高偵測成功率的方法，並在變異數變化與分布形態變化情境下具有最短的條件平均偵測延遲，優勢在參考資料受污染時尤為顯著。最後，本方法亦應用於 2020 年新冠疫情期間的標準普爾 500 指數資料，驗證其在真實金融時間序列中識別市場狀態轉換的實用性。

This thesis proposes an online change-point detection framework that integrates time-delay embedding, the Mapper algorithm, and DBSCAN density-based clustering. Unlike conventional detectors that monitor low-order statistics such as the mean or variance, the proposed method represents the reference distribution as a collection of prototype balls in a high-dimensional embedding space, enabling sensitivity to a broad class of structural changes---including mean shifts, variance changes, and distributional structure changes---without requiring parametric assumptions about the form of the change. A key component of the framework is an integrated reference cleaning step, which uses the noise labels produced by DBSCAN within each Mapper interval to identify and remove isolated outliers from the reference data before the prototype structure is constructed. This mechanism preserves detection performance when the reference data are contaminated, a setting in which competing methods that lack such a step degrade substantially. The proposed method is evaluated against three baseline detectors---Sliding Window, Maximum Mean Discrepancy (MMD), and Discriminative Drift Detection (D3)---across six benchmark scenarios spanning mean, variance, and distributional structure changes under both clean and contaminated reference conditions. Experimental results show that the proposed method is the only detector to achieve consistently high success rates across all scenarios and reference conditions. Its advantage is most pronounced in variance and distributional structure change scenarios, where moment-based detectors have limited power, and under reference contamination, where MMD's performance degrades substantially due to distortion of its kernel bandwidth estimate. The method is further validated on S\&P~500 index data surrounding the 2020 COVID-19 market crash, demonstrating its applicability to real financial time series where the type of distributional change is unknown in advance and reference data quality cannot be guaranteed.