  We proposed a data-driven approach to dissect multivariate time series in
order to discover multiple phases underlying dynamics of complex systems. This
computing approach is developed as a multiple-dimension version of Hierarchical
Factor Segmentation(HFS) technique. This expanded approach proposes a
systematic protocol of choosing various extreme events in multi-dimensional
space. Upon each chosen event, an empirical distribution of event-recurrence,
or waiting time between the excursions, is fitted by a geometric distribution
with time-varying parameters. Iterative fittings are performed across all
chosen events. We then collect and summarize the local recurrent patterns into
a global dynamic mechanism. Clustering is applied for partitioning the whole
time period into alternating segments, in which variables are identically
distributed. Feature weighting techniques are also considered to compensate for
some drawbacks of clustering. Our simulation results show that this expanded
approach can even detect systematic differences when the joint distribution
varies. In real data experiments, we analyze the relationship from returns,
trading volume, and transaction number of a single, as well as of multiple
stocks in S&P500. We can successfully not only map out volatile periods but
also provide potential associative links between stocks.
