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

2025-10-22

AI Summary

Conventional statistical methods struggle to capture the nonlinear, high-dimensional dynamic structures underlying currency co-movements in foreign exchange markets. Method: This study investigates the applicability of Topological Data Analysis (TDA) to currency behavior clustering. Using monthly log-returns of 13 major currencies, we construct a persistent homology feature space and systematically compare k-means and hierarchical clustering under TDA-derived metrics against traditional ones—namely Pearson correlation and Euclidean distance. Clustering quality is evaluated via silhouette coefficient and Calinski-Harabasz index. Contribution/Results: TDA-based features significantly enhance inter-cluster separation and intra-cluster compactness, yielding an average 32% improvement in Calinski-Harabasz scores and revealing structural dependencies overlooked by conventional approaches. To our knowledge, this is the first systematic application of TDA to multi-currency linkage analysis, establishing a novel topological modeling paradigm for financial time series.

Technology Category

• Machine Learning: Clustering
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Natural Language Processing: Machine Translation, Multilinguality, Cross-Lingual NLP

Application Category

• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• Comparing TDA with traditional methods for clustering currency behaviors
• Capturing structural patterns in FX co-movements that conventional methods miss
• Evaluating clustering quality using Silhouette and Calinski-Harabasz scores

Innovation

• Uses persistent homology for currency co-movements analysis
• Compares TDA features with classical statistical features
• Applies k-means and hierarchical clustering algorithms