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In analytical chemistry, we often perform repetitive measurements to detect and minimize inaccuracies caused by both determinate and indeterminate errors. Despite the cares we take, the presence of random errors means that repeated measurements almost never have exactly the same magnitude. The collective difference between these measurements - observed values - and the estimated or expected value is called uncertainty. Uncertainty is conventionally written after the estimated or expected value.
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Related Experiment Video

Updated: Jun 15, 2025

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
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Dynamical system identification, model selection, and model uncertainty quantification by Bayesian inference.

Robert K Niven1, Laurent Cordier2, Ali Mohammad-Djafari3

  • 1School of Engineering and Technology, The University of New South Wales, Canberra, ACT 2600, Australia.

Chaos (Woodbury, N.Y.)
|August 27, 2024
PubMed
Summary
This summary is machine-generated.

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.