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

Abstract

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.

Data-driven discovery of stochastic dynamical equations of collective motion

“…small-group sized) descriptions of a simple spatiallyexplicit local-alignment-based model of collective motion. To do so, we adopted a novel data-driven equation discovery approach [65,66]. For the class of spatial models we considered, a focal individual interacts with k randomly chosen neighbours within a radius R. Our results reveal broad consistency between the mean-field theory and the spatially explicit models.…”

“…An algorithm called STLSQ (sequentially thresholded least squares) [66] is used for sparse regression: the procedure involves iteratively fitting a regression model and eliminating terms that have fitted coefficients below a certain threshold. For more details on the regression procedure, see [65,66].…”

“…This state-of-theart method enables the construction of dynamical system models from the high-resolution time series of a system variable (e.g. order parameter of collective motion) [65][66][67]. Here, the goal is to directly discover the governing SDEs, starting from observed (or simulated) time series, while incorporating physical constraints imposed by the symmetries of the system.…”