# 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.

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:

37M10

Keywords: