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

Abstract The ecological literature often features phenomenological dynamic models lacking robust validation against observational data. Reverse engineering ecological models from data 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 especially when data resolution is low. This paper addresses the estimation of the (often) non‐linear deterministic and stochastic parts of Langevin models from sparsely sampled time series. We introduce a maximum likelihood estimation (MLE) inference method, termed Euler reconstruction, tailored for time series data with 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 splines to detect inherent nonlinearities in the unknown data‐generating system with high accuracy and acceptable computational burden. We applied both methods to a range of simulated, ecological, and climate datasets, with different data resolutions. We provide a practical measure (‘relaxation time’) to distinguish between different data resolutions. For simulated data, we show that the Euler reconstruction can accurately reveal the underlying system when data resolution is high, while Hermite reconstruction can recover the system even with low resolution. We recommend using Hermite reconstruction for real data even when the resolution is relatively high. Only when the resolution of real data is exceptionally high might Euler reconstruction suffice. We provide a MATLAB package and a tutorial to assist researchers in applying the method to their own data.