Many real-world and interesting systems fall in the realm of non-stationary stochastic processes. This means that the behavior, or dynamics, of the system at one point in time can be drastically different from the behavior dynamics at another. Some examples include internet usage during attack vs. normalcy, stock market pricing, and REM cycles. In these situations, we can often only take the same single type of measurement over time to describe our system.
Using the super-cool technique of Taken’s delay coordinate embedding, we can reconstruct our time series in higher dimensional space so that “the” (1) topology of the point cloud is the same as “the” (2) topology of the attractor of the dynamical system.
When the dynamics of the system changes in time, the topology of the attractor should also change. We call such a change a regime shift.
I am currently researching methods for * efficiently * computing (1) as a signature of a regime from a time series. Driving this need for efficiency is our goal: to detect regime shifts in time series data in real-time.
My work on (1) focuses on incorporating temporal data into witness relations and landmark selection for persistent homology computations, better understanding the interplay between geometry and topology, and utilizing novel data structures.
The matter of (2) is also interesting to muse over.
Collaborators: Liz Bradley, Jim Meiss, Zachary Alexander, Josh Garland, Jamie Tucker-Foltz, Elliot Shugerman, Sam Molnar