PyDaddy: A Python package for discovering stochastic dynamical equations from timeseries data

Most real-world ecological dynamics, ranging from ecosystem dynamics to collective animal movement, are inherently stochastic in nature. Stochastic differential equations (SDEs) are a popular modelling framework to model dynamics with intrinsic randomness. Here, we focus on the inverse question: If one has empirically measured time-series data from some system of interest, is it possible to discover the SDE model that best describes the data. Here, we present PyDaddy (PYthon library for DAta Driven DYnamics), a toolbox to construct and analyze interpretable SDE models based on time-series data. We combine traditional approaches for data-driven SDE reconstruction with an equation learning approach, to derive symbolic equations governing the stochastic dynamics. The toolkit is presented as an open-source Python library, and consists of tools to construct and analyze SDEs. Functionality is included for visual examination of the stochastic structure of the data, guided extraction of the functional form of the SDE, and diagnosis and debugging of the underlying assumptions and the extracted model. Using simulated time-series datasets, exhibiting a wide range of dynamics, we show that PyDaddy is able to correctly identify underlying SDE models. We demonstrate the applicability of the toolkit to real-world data using a previously published movement data of a fish school. Starting from the time-series of the observed polarization of the school, pyDaddy readily discovers the SDE model governing the dynamics of group polarization. The model recovered by PyDaddy is consistent with the previous study. In summary, stochastic and noise-induced effects are central to the dynamics of many biological systems. In this context, we present an easy-to-use package to reconstruct SDEs from timeseries data.