A parameter estimation method using linear response statistics: Numerical scheme.

Area of Science:

• Stochastic Differential Equations
• Numerical Analysis
• Computational Statistics

Background:

• Parameter estimation for Itô diffusion models is crucial for understanding complex systems.
• Existing methods often rely on equilibrium statistics, which can be computationally intensive or inaccurate.
• A novel parameter estimation framework based on response statistics was recently developed by the authors.

Purpose of the Study:

• To present a numerical method for implementing the authors' recently formulated parameter estimation technique using response statistics.
• To address the computational challenges in parameter estimation for Itô drift diffusions.
• To establish the theoretical foundation for the convergence of the proposed numerical method.

Main Methods:

• Formulating the parameter estimation problem as a nonlinear least-squares problem.
• Employing a polynomial surrogate model for response statistics to avoid repeated model solving in iterative schemes.
• Establishing the existence of minimizers for approximate polynomial least-squares problems.

Main Results:

• The proposed numerical method is implemented on Langevin dynamics and stochastically forced gradient flow models.
• The method demonstrates superior performance compared to conventional approaches using equilibrium statistics.
• Practical considerations like response operator selection and parameter space reduction are discussed.

Conclusions:

• The developed numerical method offers an efficient and accurate approach for parameter estimation in Itô diffusion models.
• The use of polynomial surrogate models significantly improves computational efficiency.
• The findings suggest a promising alternative to traditional parameter estimation techniques in various scientific applications.