  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.
