Data-driven subsampling rates for diffusion parameter estimation of SDEs

2026-06-11

AI Summary

This study addresses the challenge of estimating diffusion parameters in stochastic differential equation (SDE) models when data and model are compatible only at specific scales. The authors propose an adaptive subsampling method based on the statistics of monotonic runs. By demonstrating that, for a broad class of additive-noise SDEs, the length of monotonic runs at infinitesimal scales approximately follows a geometric distribution with success probability 1/2, they establish a general criterion for selecting the subsampling rate without relying on multiscale diffusion asymptotics. The optimal sampling scale matching the SDE’s infinitesimal behavior is automatically determined solely from the statistical properties of monotonic increasing or decreasing segments in the observed time series. Validation on surrogate modeling of fiber lay-down trajectories in nonwoven fabric production demonstrates that the method yields highly accurate and model-consistent diffusion parameter estimates, proving effective in real-world industrial applications.

Abstract

We study the problem of diffusion parameter estimation for stochastic differential equation (SDE) models in scenarios where data and model are compatible only on specific scales that have yet to be determined. We introduce a simple and efficient method for selecting suitable rates at which given time series data should be subsampled in order to ensure that the statistical structure of the subsampled data is consistent with the behavior of the SDE model on an infinitesimal scale. Our approach is based on analyzing the statistics of the lengths of monotonically increasing or decreasing segments in the subsampled data sequence, which we refer to as monotone runs. As an analytical foundation, we prove for a large class of SDEs with additive noise that the lengths of monotone runs at an infinitesimal scale are approximately geometrically distributed with success probability $1/2$. This universal characterization is employed to derive an automated method for selecting appropriate subsampling rates for given time series data that is directly applicable in real-world scenarios and does not rely on an asymptotic framework of multiscale diffusions. The approach is demonstrated using an application from industrial mathematics concerning surrogate models for fiber lay-down curves in production processes of nonwoven textiles.

Problem

• diffusion parameter estimation
• stochastic differential equations
• subsampling rates
• data-model compatibility
• time series

Innovation

• subsampling rate selection
• stochastic differential equations
• monotone runs
• diffusion parameter estimation
• data-driven method