Functional observability and subspace reconstruction in nonlinear systems

Abstract

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.

arXiv commentCodes are available in GitHub (https://github.com/montanariarthur/NonlinearObservability)

References in corpus (19)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• Recurrence Plots for the Analysis of Complex Systems
• Complex Network Approach for Recurrence Analysis of Time Series
• A mathematical framework for critical transitions: bifurcations, fast-slow systems and stochastic dynamics
• Controllability transition and nonlocality in network control
• Observability and Controllability of Nonlinear Networks: The Role of Symmetry
• Topological recognition of critical transitions in time series of cryptocurrencies
• State observation and sensor selection for nonlinear networks
• Functional observability and target state estimation in large-scale networks
• Crisis of the Chaotic Attractor of a Climate Model: A Transfer Operator Approach
• Resonances in a Chaotic Attractor Crisis of the Lorenz Flow
• Phase space reconstruction for non-uniformly sampled noisy time series
• A nonlinear graph-based theory for dynamical network observability
• Observability and Synchronization of Neuron Models
• Testing Dynamical System Variables for Reconstruction
• Limits on reconstruction of dynamical networks
• Recurrent Neural Networks for Partially Observed Dynamical Systems
• Particle filtering of dynamical networks: Highlighting observability issues
• Buffering variability in cell regulation motifs close to criticality

Cited by in corpus (6)

• Duality between controllability and observability for target control and estimation in networks
• Synchronizing Chaos using Reservoir Computing
• On the Popov-Belevitch-Hautus tests for functional observability and output controllability
• Nonlinear Functional Estimation: Functional Detectability and Full Information Estimation
• Efficient and faithful reconstruction of dynamical attractors using homogeneous differentiators
• Efficient gPC-based quantification of probabilistic robustness for systems in neuroscience