Optimal Transport vs. Fisher-Rao distance between Copulas for Clustering Multivariate Time Series (1604.08634v2)

Abstract: We present a methodology for clustering N objects which are described by multivariate time series, i.e. several sequences of real-valued random variables. This clustering methodology leverages copulas which are distributions encoding the dependence structure between several random variables. To take fully into account the dependence information while clustering, we need a distance between copulas. In this work, we compare renowned distances between distributions: the Fisher-Rao geodesic distance, related divergences and optimal transport, and discuss their advantages and disadvantages. Applications of such methodology can be found in the clustering of financial assets. A tutorial, experiments and implementation for reproducible research can be found at www.datagrapple.com/Tech.

• The paper demonstrates that Wasserstein (Optimal Transport) distances yield more stable clustering results compared to the sensitivity of Fisher-Rao in high-dependence scenarios.
• The paper establishes a copula-based framework leveraging Sklar’s Theorem to encode dependencies for clustering multivariate time series.
• The paper reveals through experiments with Gaussian copulas that Optimal Transport maintains robustness under noise and parameter uncertainty relative to Fisher-Rao.

Analysis of Distances Between Copulas for Clustering Multivariate Time Series

This paper presents a comprehensive paper on the comparative efficacy of using Optimal Transport and Fisher-Rao distances in the clustering of multivariate time series through the lens of copula theory. Specifically, it examines how these distances perform when applied to copulas, which model the dependence structure between random variables.

The research is positioned within a well-defined methodological framework targeting the challenge of clustering objects represented by multivariate time series. A copula-based approach is advocated, leveraging these statistical functions to encapsulate dependency between series. The crux of this investigation is the determination of an appropriate distance measure to compare copulas, with the Wasserstein (Optimal Transport) distance and Fisher-Rao distance being the focal points of comparison.

Key Methodological Insights

1. Copula Utilization: The paper highlights the utility of copulas in encoding the dependence between variables, a crucial aspect considering the multivariate nature of the data. Sklar's Theorem is emphasized as the foundational theory linking univariate margins to form multivariate distributions.
2. Distance Metrics: The investigation into the performance of distance metrics focuses heavily on their respective geometrical properties and suitability in capturing dependence nuances. The Fisher-Rao distance, stemming from Information Geometry, and the Wasserstein distance, rooted in Optimal Transport theory, are scrutinized for their computational tractability and interpretative ability.
3. Practical Constraints: While the methodology demonstrates robustness through non-parametric, noise-resistant traits, the Fisher-Rao metric exhibits sensitivity to high dependence scenarios. This sensitivity can lead to computational inefficiencies and suboptimal clustering results due to the high curvature in the parameter space involved when strong correlations are present.

Experimental Evaluation

The experiments conducted showcase the practicality of these theoretical insights using Gaussian copulas as test cases, selected for their analytical tractability. Fisher-Rao and Wasserstein distances were compared for sensitivity relative to correlation strengths. The results revealed that the Fisher-Rao distance and its associated divergences provide less intuitive clustering for strong dependencies, as opposed to the Wasserstein metric which maintains stability across varying correlation strengths.

These observations underline the optimal nature of Wasserstein distances for practical applications where estimating copula parameters involves significant uncertainty. In contrast, Fisher-Rao’s sensitivity to parameter variations can produce counter-intuitive clusters, especially in finance where high correlation prevalance is common.

Implications and Future Directions

The practical implications of this paper are substantial, especially for applications where dependence structures within time series are pivotal, such as financial asset clustering. For theoretical advancements, the findings suggest a re-evaluation of the applicability of Fisher-Rao distances in environments characterized by strong dependencies.

For future research, the paper suggests further examination of non-Gaussian copulas and potential extensions into integrating kernel methods in the copula comparison framework. This approach could yield better embedding into Hilbert spaces, offering nuanced insights into dependence metrics.

In summary, this paper provides critical insights into the selection of distance measures for clustering dependent observations and highlights the nuanced balance required in selecting the right metric based on domain-specific data characteristics.