Article,
An information-geometric approach to feature extraction and moment reconstruction in dynamical systems
(2020)cite arxiv:2004.02172.
Abstract
We propose a dimension reduction framework for feature extraction and moment
reconstruction in dynamical systems that operates on spaces of probability
measures induced by observables of the system rather than directly in the
original data space of the observables themselves as in more conventional
methods. Our approach is based on the fact that orbits of a dynamical system
induce probability measures over the measurable space defined by (partial)
observations of the system. We equip the space of these probability measures
with a divergence, i.e., a distance between probability distributions, and use
this divergence to define a kernel integral operator. The eigenfunctions of
this operator create an orthonormal basis of functions that capture different
timescales of the dynamical system. One of our main results shows that the
evolution of the moments of the dynamics-dependent probability measures can be
related to a time-averaging operator on the original dynamical system. Using
this result, we show that the moments can be expanded in the eigenfunction
basis, thus opening up the avenue for nonparametric forecasting of the moments.
If the collection of probability measures is itself a manifold, we can in
addition equip the statistical manifold with the Riemannian metric and use
techniques from information geometry. We present applications to ergodic
dynamical systems on the 2-torus and the Lorenz 63 system, and show on a
real-world example that a small number of eigenvectors is sufficient to
reconstruct the moments (here the first four moments) of an atmospheric time
series, i.e., the realtime multivariate Madden-Julian oscillation index.
