Reconstructing Langevin systems from high and low-resolution time series using Euler and Hermite reconstructions

The ecological literature often features phenomenological dynamic models lacking robust validation against observational data. Reverse engineering is an alternative approach, where time series data are utilized to infer or fit a stochastic differential equation. This process, known as system reconstruction, presents significant challenges. This paper addresses the estimation of the (often) non-linear deterministic and stochastic parts of Langevin models, one of the simplest yet commonly used stochastic differential equations. We introduce a Maximum Likelihood Estimation (MLE) inference method, termed Euler reconstruction, tailored for time series data with medium to high resolution. However, the Euler approach is not reliable for low-resolution data. To fill the gap for sparsely sampled data, we present an MLE inference method pioneered by Aït-Sahalia, that we term Hermite reconstruction. We employ a powerful modeling framework utilizing splines to detect inherent nonlinearities in the unknown data-generating system to achieve high accuracy with minimal computational burden. Applying Euler and Hermite reconstructions to a range of simulated, ecological, and climate datasets, we demonstrate their efficacy and versatility. We provide a user-friendly tutorial and a MATLAB package called the ‘MATLAB reconstruction package’.