标题： 货币的拓扑结构：用于外汇共移动性的持续同调分析：比较聚类研究 显示英文标题

标题： Topology of Currencies: Persistent Homology for FX Co-movements: A Comparative Clustering Study

摘要： 本研究探讨了拓扑数据分析（TDA）是否能在聚类货币行为方面提供传统统计方法之外的额外见解。 我们关注外汇（FX）市场，这是一个复杂系统，通常表现出非线性和高维动态，经典技术可能无法完全捕捉。 我们使用13种主要货币汇率（全部对欧元）的月度对数收益率，比较基于TDA派生特征与经典统计特征的聚类结果。 两种广泛使用的聚类算法，\(k\)-means和层次聚类，被应用于这两种特征类型，并通过轮廓系数和Calinski-Harabasz指数评估聚类质量。 我们的研究结果表明，基于TDA的特征聚类产生的聚类比基于传统统计特征的聚类更紧密且分离度更高，特别是在获得显著更高的Calinski-Harabasz分数方面。 然而，所有聚类方法都产生了适度的轮廓系数，这突显了对FX时间序列进行分组的内在难度。 在TDA与传统特征下的不同聚类组成表明，TDA捕捉到了货币共同变动中的结构模式，而传统方法可能会忽略这些模式。 这些结果强调了TDA作为分析金融时间序列的有价值补充工具，具有潜在的应用前景，在风险管理中理解结构共同变动至关重要。 显示英文摘要

摘要： This study investigates whether Topological Data Analysis (TDA) can provide additional insights beyond traditional statistical methods in clustering currency behaviours. We focus on the foreign exchange (FX) market, which is a complex system often exhibiting non-linear and high-dimensional dynamics that classical techniques may not fully capture. We compare clustering results based on TDA-derived features versus classical statistical features using monthly logarithmic returns of 13 major currency exchange rates (all against the euro). Two widely-used clustering algorithms, \(k\)-means and Hierarchical clustering, are applied on both types of features, and cluster quality is evaluated via the Silhouette score and the Calinski-Harabasz index. Our findings show that TDA-based feature clustering produces more compact and well-separated clusters than clustering on traditional statistical features, particularly achieving substantially higher Calinski-Harabasz scores. However, all clustering approaches yield modest Silhouette scores, underscoring the inherent difficulty of grouping FX time series. The differing cluster compositions under TDA vs. classical features suggest that TDA captures structural patterns in currency co-movements that conventional methods might overlook. These results highlight TDA as a valuable complementary tool for analysing financial time series, with potential applications in risk management where understanding structural co-movements is crucial.