Computational Topology and Time-series Analysis

Monday, February 10, 2014 - 3:15pm - 4:05pm
Keller 3-180
Elizabeth Bradley (University of Colorado)
Most of the traditional time-series analysis techniques that are used
to study trajectories from nonlinear dynamical systems involve
state-space reconstructions and clever approximations of asymptotic
quantities, all in the context of finite and often noisy data. Few of
these techniques work well in the face of nonstationarity. Embedding
a time series that samples different dynamical systems at different
times, for instance---and then calculating a long-term Lyapunov
exponent---does not make sense.

Computational topology offers some important advantages in situations
like this. Topological descriptions of structure are inherently more
qualitative, and thus more robust, than more-rigid geometrical
characterizations. Even when the data contain finite amounts of noise, computational
topology can produce provable results [Day et al. 2008, Mischaikow et
al. 1999], and it is naturally immune to changes of scale and
orientation that skew the data. Faced with a time
series that samples a number of different dynamical systems, one can detect regime changes by looking for shifts in the
topological structure of the reconstructed dynamics: e.g., nearby
points whose immediate future paths are significantly different. This
segmentation strategy works in some situations where others do
not---if the different components of the signal overlap in state
space, for instance. Once a signal has been segmented into
components, one can compute topological signatures of those
components---e.g., with witness complexes and Conley index theory.

The notion of persistence can be leveraged to make appropriate choices
for the different free parameters in these algorithms. It may also be
possible to take advantage of persistence in order to handle the
additional challenges that arise when the data arrive in a stream and
must be analyzed 'on the fly.' In this situation, one does not have
the luxury of post facto analysis of the full data set. Rather, one
must detect shifts immediately--and then immediately start building up
a new model from the incoming stream.

This is joint work with Jim Meiss, Vanessa Robins, and Zach Alexander.
MSC Code: