Koopman operator-based discussion on partial observation in stochastic systems
This work addresses the challenge of incomplete data in stochastic modeling, which is incremental as it extends existing deterministic frameworks to stochastic contexts.
The paper tackles the problem of partial observation in stochastic systems by applying Koopman operator theory, showing that delay embedding is beneficial and numerical experiments reveal a power-law relationship between accuracy and additive noise amplitude.
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 showed a power-law behavior of the accuracy for 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.