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.

翻译：本研究探讨了拓扑数据分析（TDA）能否在聚类货币行为方面提供超越传统统计方法的额外洞见。我们聚焦于外汇市场，这是一个复杂的系统，常表现出非线性、高维度的动态特征，而经典技术可能无法完全捕捉这些特征。我们使用13种主要货币汇率（均相对于欧元）的月度对数收益率，比较了基于TDA提取特征与基于经典统计特征的聚类结果。两种广泛使用的聚类算法——\(k\)-均值和层次聚类，被应用于两类特征上，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特征的聚类比基于传统统计特征的聚类产生了更紧凑、分离度更好的簇，尤其获得了显著更高的Calinski-Harabasz分数。然而，所有聚类方法得到的轮廓系数均不高，这凸显了对外汇时间序列进行分组的固有难度。TDA特征与经典特征下不同的簇构成表明，TDA捕捉到了货币联动性中传统方法可能忽略的结构性模式。这些结果凸显了TDA作为分析金融时间序列的一种有价值的补充工具，在理解结构性联动至关重要的风险管理等领域具有潜在应用价值。