Sydney Dynamics Group Seminar: Caroline Wormell, University of Sydney -- A statistical test for infinite-data eigenvalues of data-driven operators

Seminar : Sydney Dynamics group Seminar 
Day/time : Thursday, March 12 at 4:00 PM
Location : SMRI seminar room (A12 Macleay Room 301) 
Speaker : Dr. Caroline Wormell, University of Sydney 

Title : A statistical test for infinite-data eigenvalues of data-driven operators 

Abstract : Emergent patterns can be extracted from dynamical systems through the
eigenvalues and eigenvectors of the system's Koopman operator.  The Koopman operator is
a linear composition operator, and so it is very easy to estimate from a data sample,
e.g.  some pairs (x_m,f(x_m)), and many have tried this with some success using methods
such as Dynamic Mode Decomposition.  A drawback of doing this is that it can be hard to
tell whether an eigenvalue will persist as the amount of data is taken to infinity.  

In this talk I will present a very general approach for studying spectral errors of
data-driven least-squares approximations of matrices.  Knowledge of the infinite-data
limit allows us to generate a pseudo-spectrum-like function on the complex plane that
allows us to predict where true eigenvalues are likely to go under the finite data
error.  In the other direction, by reprocessing our finite data sample, we can test
statistically for the location of the true eigenvalues.  This gives us a rigorous way to
assess whether the patterns we extract from finite data are likely to be signal or
noise.