Fast Bayesian reconstruction of chaotic dynamical systems via extended Kalman filtering

Area of Science:

• Dynamical Systems and Chaos Theory
• Computational Statistics
• Bayesian Inference

Background:

• Estimating parameters in chaotic dynamical systems is challenging due to slow convergence in traditional Bayesian methods like the Gibbs sampler.
• High posterior correlations in chaotic maps hinder the efficiency of existing Markov chain Monte Carlo (MCMC) algorithms.

Purpose of the Study:

• To introduce an improved Markov chain Monte Carlo (MCMC) algorithm for efficient posterior computation in chaotic dynamical systems.
• To enhance parameter estimation for chaotic maps by addressing the slow convergence issues of the Gibbs sampler.

Main Methods:

• Developed a novel MCMC algorithm incorporating the extended Kalman filter.
• The extended Kalman filter is utilized to compute the likelihood function by integrating out unknown system states.
• Applied the new algorithm and the Gibbs sampler to the logistic, tent, and Moran-Ricker maps for comparative analysis.

Main Results:

• The proposed MCMC algorithm demonstrates significantly improved efficiency compared to the Gibbs sampler.
• Performance was evaluated using computational time (CPU) and integrated autocorrelation time, with the new method showing superior results.
• The integration of the extended Kalman filter effectively mitigates issues of slow convergence caused by high posterior correlations.

Conclusions:

• The enhanced MCMC algorithm offers a more efficient approach for posterior computation in chaotic dynamical systems.
• This method provides a valuable tool for researchers working with Bayesian inference in the field of chaos theory.
• The improved efficiency has practical implications for analyzing complex chaotic models more rapidly.