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Stochastic differential equations as a tool to regularize the parameter estimation problem for continuous time

Jacob Leander1, Torbjörn Lundh2, Mats Jirstrand3

  • 1Fraunhofer-Chalmers Centre, Chalmers Science Park, SE-412 88 Gothenburg, Sweden; Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden.

Mathematical Biosciences
|March 18, 2014
PubMed
Summary

Estimating parameters in ordinary differential equations is improved by using stochastic differential equations. This approach reduces local minima in objective functions, enhancing parameter estimation accuracy and convergence.

Keywords:
Extended Kalman filterFitzHugh–NagumoLotka–VolterraOrdinary differential equationsParameter estimationStochastic differential equations

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Area of Science:

  • Computational Biology
  • Mathematical Modeling
  • Systems Biology

Background:

  • Parameter estimation in ordinary differential equations (ODEs) from discrete data is challenging due to local minima in objective functions.
  • Traditional ODE models may struggle to accurately represent systems with inherent noise or uncertainty.
  • Local minima can hinder the convergence of optimization algorithms, leading to suboptimal parameter estimates.

Purpose of the Study:

  • To investigate the use of stochastic differential equations (SDEs) to regularize objective functions in parameter estimation.
  • To demonstrate how incorporating noise into the model can overcome local minima problems in ODE parameter estimation.
  • To improve the convergence and accuracy of parameter estimation for dynamical systems.

Main Methods:

  • Transitioning from ordinary differential equations (ODEs) to stochastic differential equations (SDEs) for parameter estimation.
  • Utilizing the extended Kalman filter as a state estimator for SDE models.
  • Employing sensitivity equations for accurate gradient calculation in the objective function.
  • Testing the method on the FitzHugh-Nagumo and Lotka-Volterra models using in silico data.

Main Results:

  • Stochastic differential equations regularize objective functions by reducing the number of local minima.
  • Improved convergence rates were observed in parameter estimation procedures.
  • The method demonstrated robustness, maintaining predictions close to data even with incorrect parameters.
  • Successful application to both the FitzHugh-Nagumo and Lotka-Volterra systems.

Conclusions:

  • Stochastic differential equations offer a powerful tool for regularizing objective functions in parameter estimation problems.
  • This approach effectively mitigates the issue of local minima, facilitating the use of efficient gradient-based optimization methods.
  • The proposed method enhances the reliability and accuracy of parameter estimation for complex dynamical systems.