Functional observability and subspace reconstruction in nonlinear systems

Time-series analysis is fundamental for modeling and predicting dynamical behaviors from time-
ordered data, with applications in many disciplines such as physics, biology, finance, and engineering.
Measured time-series data, however, are often low dimensional or even univariate, thus requiring
embedding methods to reconstruct the original system’s state space. The observability of a system
establishes fundamental conditions under which such reconstruction is possible. However, complete
observability is too restrictive in applications where reconstructing the entire state space is not
necessary and only a specific subspace is relevant. Here, we establish the theoretic condition to
reconstruct a nonlinear functional of state variables from measurement processes, generalizing the
concept of functional observability to nonlinear systems. When the functional observability condition
holds, we show how to construct a map from the embedding space to the desired functional of
state variables, characterizing the quality of such reconstruction. The theoretical results are then
illustrated numerically using chaotic systems with contrasting observability properties. By exploring
the presence of functionally unobservable regions in embedded attractors, we also apply our theory
for the early warning of seizure-like events in simulated and empirical data. The studies demonstrate
that the proposed functional observability condition can be assessed a priori to guide time-series
analysis and experimental design for the dynamical characterization of complex systems.