Efficient Identification of Nonlinear Stochastic Dynamical Systems using Meta-State-Space Learning

Alapfogalmak

A novel meta-state-space representation and identification method that can efficiently model and identify general nonlinear stochastic systems with accuracy close to the theoretical limit.

Kivonat

The paper introduces a novel meta-state-space (MSS) representation for a wide class of nonlinear stochastic systems. The MSS representation describes the evolution of the complete probability distribution of the system state as a deterministic function of the meta-state, which can be seen as a parameterization of the state probability distribution.

The key highlights are:

1. Derivation of the MSS representation, which shows that the stochastic system dynamics can be captured by a deterministic meta-state transition function.
2. Formulation of an efficient data-driven identification method for MSS models using maximum a posteriori (MAP) estimation.
3. Parameterization of the MSS model components using artificial neural networks, which enables learning highly nonlinear meta-state dynamics and output distributions.
4. Demonstration on a challenging nonlinear stochastic system identification problem, where the proposed method achieves performance close to the theoretical limit.

The MSS representation and identification approach provide a general framework for modeling and identification of complex stochastic dynamical systems without restrictive structural assumptions. This opens up new possibilities for efficient data-driven modeling of a wide range of stochastic processes.

Statisztikák

The system is described by the following nonlinear stochastic state-space model:
xt+1 = α(xt, et)xt + ut
yt = xt
where α(xt, et) = 0.3 + 0.7e^(-(xt+et)^2) and the process noise et is i.i.d. with a uniform distribution U(-0.5, 0.5).

Idézetek

"The meta-state-space representation is also able to describe the PDF of the output trajectories."
"The meta-state transition function is deterministic, which allows capturing the stochastic process representation via a deterministic model."
"The proposed identification method can obtain models with a log-likelihood close to the theoretical limit even for highly nonlinear, highly stochastic systems."

Főbb Kivonatok

Meta-State-Space Learning: An Identification Approach for Stochastic Dynamical Systems

by Gerb... : arxiv.org 05-02-2024

https://arxiv.org/pdf/2307.06675.pdf

Mélyebb kérdések

How can the meta-state-space representation be extended to capture the joint distribution of the output sequence, e.g., p(y1, y2, ..., yn|u1, u2, ..., un)

To extend the meta-state-space representation to capture the joint distribution of the output sequence, we can introduce a subspace encoder that can encode the entire output sequence into a single meta-state vector. This subspace encoder can be trained to map the entire output sequence to a meta-state vector that encapsulates the joint distribution of the outputs. By incorporating this subspace encoder into the meta-state-space framework, we can model and identify the joint distribution of the output sequence, denoted as p(y1, y2, ..., yn|u1, u2, ..., un).

What are the theoretical limits on the accuracy of meta-state-space models in terms of the minimum required meta-state dimension for a given stochastic system

The theoretical limits on the accuracy of meta-state-space models are related to the minimum required meta-state dimension for a given stochastic system. The accuracy of the model is influenced by the complexity of the system dynamics and the level of detail required to capture the system's behavior accurately. In general, the minimum required meta-state dimension for a stochastic system depends on the level of uncertainty and variability in the system's dynamics. As the complexity and nonlinearity of the system increase, a higher-dimensional meta-state representation may be needed to accurately capture the system's behavior. However, the exact theoretical limits on the minimum required meta-state dimension for a given system are challenging to determine and may vary depending on the specific characteristics of the system.

Can the meta-state-space framework be applied to model and identify stochastic systems with time-varying or non-stationary statistics

The meta-state-space framework can be applied to model and identify stochastic systems with time-varying or non-stationary statistics by incorporating adaptive mechanisms into the meta-state transition function and output distribution. By allowing the meta-state dynamics to adapt to changes in the system's statistics over time, the meta-state-space model can effectively capture the evolving nature of the stochastic system. This adaptability can be achieved by updating the meta-state transition function and output distribution parameters based on the observed data, enabling the model to track and represent the changing statistics of the system accurately. Additionally, techniques such as online learning and recursive estimation can be employed to continuously update the meta-state-space model in real-time as the system dynamics evolve.