On clustering financial time series: a need for distances between dependent random variables

TL;DR

This paper explores advanced clustering methods for financial time series, emphasizing the importance of defining appropriate distances between dependent random variables to improve risk and portfolio analysis.

Contribution

It introduces a new approach for clustering financial time series using distances that capture dependencies, extending beyond covariance-based methods.

Findings

Hierarchical clustering is statistically consistent for financial data.

Proposes a broader application of clustering to incorporate all information in dependent processes.

Highlights the need for new distance measures between dependent random variables.

Abstract

The following working document summarizes our work on the clustering of financial time series. It was written for a workshop on information geometry and its application for image and signal processing. This workshop brought several experts in pure and applied mathematics together with applied researchers from medical imaging, radar signal processing and finance. The authors belong to the latter group. This document was written as a long introduction to further development of geometric tools in financial applications such as risk or portfolio analysis. Indeed, risk and portfolio analysis essentially rely on covariance matrices. Besides that the Gaussian assumption is known to be inaccurate, covariance matrices are difficult to estimate from empirical data. To filter noise from the empirical estimate, Mantegna proposed using hierarchical clustering. In this work, we first show that this procedure is statistically consistent. Then, we propose to use clustering with a much broader application than the filtering of empirical covariance matrices from the estimate correlation coefficients. To be able to do that, we need to obtain distances between the financial time series that incorporate all the available information in these cross-dependent random processes.