Fitting Effective Diffusion Models to Data Associated with a "Glassy Potential": Estimation, Classical Inference Procedures and Some Heuristics

TL;DR

This paper explores methods for estimating parameters of diffusion models from atomistic simulation data, focusing on maximum likelihood estimation, approximation of transition densities, and heuristics for nonlinear models, with applications to glassy potentials.

Contribution

It introduces a heuristic technique for approximating nonlinear diffusion models and evaluates the accuracy of Ait-Sahalia's transition density expansions in glassy potential scenarios.

Findings

Ait-Sahalia's expansions effectively capture transition density curvature in simulated glassy potentials.

Approximate transition densities enable goodness-of-fit testing and confidence interval estimation.

Heuristic local modeling provides a practical approach for nonlinear diffusion estimation.

Abstract

A variety of researchers have successfully obtained the parameters of low dimensional diffusion models using the data that comes out of atomistic simulations. This naturally raises a variety of questions about efficient estimation, goodness-of-fit tests, and confidence interval estimation. The first part of this article uses maximum likelihood estimation to obtain the parameters of a diffusion model from a scalar time series. I address numerical issues associated with attempting to realize asymptotic statistics results with moderate sample sizes in the presence of exact and approximated transition densities. Approximate transition densities are used because the analytic solution of a transition density associated with a parametric diffusion model is often unknown.I am primarily interested in how well the deterministic transition density expansions of Ait-Sahalia capture the curvature of the transition density in (idealized) situations that occur when one carries out simulations in the presence of a "glassy" interaction potential. Accurate approximation of the curvature of the transition density is desirable because it can be used to quantify the goodness-of-fit of the model and to calculate asymptotic confidence intervals of the estimated parameters. The second part of this paper contributes a heuristic estimation technique for approximating a nonlinear diffusion model. A "global" nonlinear model is obtained by taking a batch of time series and applying simple local models to portions of the data. I demonstrate the technique on a diffusion model with a known transition density and on data generated by the Stochastic Simulation Algorithm.