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

TL;DR

This paper demonstrates that Topological Data Analysis (TDA) enhances clustering of currency exchange rates by capturing structural patterns in FX co-movements, outperforming traditional statistical features in cluster compactness and separation.

Contribution

It introduces TDA-based features for FX data clustering and shows they produce more meaningful clusters than classical statistical features, offering new insights into currency co-movement structures.

Findings

TDA-based clustering yields higher Calinski-Harabasz scores.

TDA captures structural patterns overlooked by traditional methods.

All methods show modest Silhouette scores, indicating clustering difficulty.

Abstract

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.