Koopman operator-based discussion on partial observation in stochastic systems

Abstract

It is sometimes difficult to achieve a complete observation for a full set of observables, and partial observations are necessary. For deterministic systems, the Mori-Zwanzig formalism provides a theoretical framework for handling partial observations. Recently, data-driven algorithms based on the Koopman operator theory have made significant progress, and there is a discussion to connect the Mori-Zwanzig formalism with the Koopman operator theory. In this work, we discuss the effects of partial observation in stochastic systems using the Koopman operator theory. The discussion clarifies the importance of distinguishing the state space and the function space in stochastic systems. Even in stochastic systems, the delay-embedding technique is beneficial for partial observation, and several numerical experiments show a power-law behavior of error with respect to the amplitude of the additive noise. We also discuss the relation between the exponent of the power-law behavior and the effects of partial observation.

26 pages, 5 figures

References in corpus (15)

• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• Applied Koopmanism
• Chaos as an Intermittently Forced Linear System
• Ergodic theory, Dynamic Mode Decomposition and Computation of Spectral Properties of the Koopman operator
• A discrete approach to stochastic parametrization and dimensional reduction in nonlinear dynamics
• Tensor-based dynamic mode decomposition
• Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism
• Multidimensional approximation of nonlinear dynamical systems
• A Priori Estimation Of Memory Effects In Coarse-Grained Nonlinear Systems Using The Mori-Zwanzig Formalism
• Understanding causation via correlations and linear response theory
• Extended duality relations between birth-death processes and partial differential equations
• Data Driven Learning of Mori-Zwanzig Operators for Isotropic Turbulence
• Duality in stochastic processes from the viewpoint of basis expansions
• Permutation of Tensor-Train Cores for Computing Moments on Stochastic Differential Equations
• Redundant basis interpretation of Doi-Peliti method and an application