Approximate Bayes learning of stochastic differential equations.

Area of Science:

• Statistics
• Dynamical Systems
• Machine Learning

Background:

• Stochastic differential equations (SDEs) model systems with inherent randomness.
• Estimating drift and diffusion functions is crucial for understanding SDE dynamics.
• Existing methods often struggle with unobserved states or require dense data.

Purpose of the Study:

• To develop a flexible nonparametric method for estimating drift and diffusion functions in SDEs.
• To address challenges posed by sparse observations and unobserved latent dynamics.
• To provide a robust framework for analyzing complex systems described by SDEs.

Main Methods:

• Utilized Gaussian processes as flexible models for drift and diffusion functions.
• Employed Gaussian process regression for direct estimation from dense data.
• Developed an approximate expectation maximization algorithm for sparse data.
• Approximated posterior states using a piecewise linearized Ornstein-Uhlenbeck process.
• Applied sparse Gaussian process approximation for maximum a posteriori drift estimation.

Main Results:

• Successfully estimated drift and diffusion functions from observed state vectors.
• The method effectively handles both dense and sparse observational data.
• The approximate expectation maximization algorithm accurately models latent dynamics.
• Sparse Gaussian process approximation improved drift estimation efficiency.

Conclusions:

• The proposed nonparametric approach offers a powerful tool for SDE analysis.
• Gaussian processes provide a flexible and effective modeling strategy.
• The developed algorithm enhances the ability to infer system dynamics from limited data.