Comparison of continuous and discrete-time data-based modeling for hypoelliptic systems

TL;DR

This paper compares continuous-time stochastic differential equation models with discrete-time nonlinear autoregressive models for hypoelliptic systems, finding the discrete approach often yields better predictions with sparse data.

Contribution

It provides a systematic comparison of continuous and discrete data-based modeling approaches specifically for hypoelliptic systems, highlighting the advantages of discrete models in certain scenarios.

Findings

Discrete-time models outperform continuous models in predictive accuracy with sparse data.

Discrete approach is more robust to data sparsity in hypoelliptic systems.

The study offers insights into model selection for systems with partial observations.

Abstract

We compare two approaches to the predictive modeling of dynamical systems from partial observations at discrete times. The first is continuous in time, where one uses data to infer a model in the form of stochastic differential equations, which are then discretized for numerical solution. The second is discrete in time, where one directly infers a discrete-time model in the form of a nonlinear autoregression moving average model. The comparison is performed in a special case where the observations are known to have been obtained from a hypoelliptic stochastic differential equation. We show that the discrete-time approach has better predictive skills, especially when the data are relatively sparse in time. We discuss open questions as well as the broader significance of the results.