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This study introduces a Bayesian framework for identifying dynamical systems from time-series data. This approach offers robust model selection and uncertainty quantification, outperforming traditional sparse regression methods.

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Area of Science:

  • Dynamical Systems Theory
  • Statistical Inference
  • Machine Learning

Background:

  • Time-series data analysis is crucial for understanding complex systems.
  • Traditional methods for dynamical system identification often lack robust uncertainty quantification.
  • Sparse regression techniques offer model interpretability but may struggle with complex noise models.

Purpose of the Study:

  • To present a Bayesian maximum a posteriori (MAP) framework for dynamical system identification.
  • To provide a theoretical justification for regularization terms in system identification.
  • To compare Bayesian algorithms with existing sparse regression methods.

Main Methods:

  • Developed a Bayesian MAP framework for dynamical system identification.
  • Equated the framework to generalized Tikhonov regularization.
  • Employed joint MAP and variational Bayesian approximation algorithms.
  • Compared performance against LASSO, ridge regression, and SINDy algorithms.

Main Results:

  • The Bayesian framework provides a rational basis for residual and regularization terms.
  • Bayesian inference allows for model ranking, uncertainty quantification, and hyperparameter estimation.
  • The posterior Gaussian norm serves as a robust metric for quantitative model selection.
  • Bayesian methods demonstrated superior performance in identifying dynamical systems with various noise types.

Conclusions:

  • The proposed Bayesian MAP framework offers a principled approach to dynamical system identification.
  • It provides enhanced capabilities for model selection and uncertainty quantification compared to existing methods.
  • The framework is particularly effective for systems with Gaussian or Laplace noise.