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Quantifying uncertainty in state and parameter estimation.

Ulrich Parlitz1, Jan Schumann-Bischoff1, Stefan Luther1

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Summary
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This study quantifies the observability of dynamical system state variables and parameters from time series data. It introduces an uncertainty measure to identify estimable regions in state and parameter space.

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Area of Science:

  • Dynamical Systems Theory
  • Nonlinear Dynamics
  • Time Series Analysis
  • State Estimation

Background:

  • Observability of dynamical systems is crucial for understanding and predicting their behavior.
  • Estimating state variables and parameters from limited time series data presents significant challenges.
  • Existing methods often lack a quantitative measure of estimation uncertainty across the system's state and parameter space.

Purpose of the Study:

  • To develop and quantify a method for assessing the observability of state variables and parameters.
  • To introduce a measure of uncertainty for state and parameter estimation based on time series data.
  • To identify regions in state and parameter space where estimation is feasible or infeasible.

Main Methods:

  • Analysis of state variable and parameter observability using the Jacobian matrix of the delay coordinates map.
  • Introduction of a state- and parameter-dependent uncertainty measure for estimation.
  • Demonstration of the method on the Ikeda map and the Hindmarsh-Rose model.

Main Results:

  • A quantitative measure of observability uncertainty is established for dynamical systems.
  • The method successfully identifies regions of reliable state and parameter estimation from time series.
  • The Jacobian of the delay coordinates map proves effective for quantifying observability.

Conclusions:

  • The proposed uncertainty measure provides a robust framework for evaluating the observability of dynamical systems.
  • This approach enables targeted data acquisition and model refinement by highlighting estimable system components.
  • The demonstrated models confirm the practical applicability of the developed method in nonlinear dynamics.