In the natural sciences, differential equations are widely used to describe dynamical systems. The discovery and verification of such models from data has become a fundamental challenge of science today. From a statistical point of view, we distinguish two problems: parameter estimation and structure search. In parameter estimation, we start from a given differential equation and estimate the parameters from noisy data that are observed at discrete time points. The estimate depends nonlinearly on the parameters. This poses both statistical and computational challenges and makes the task of structure search even more ambitious. Existing methods use either standard model selection techniques or various types of sparsity enforcing regularization, hence focusing on predictive performance. In this work, we develop novel methodology for structure search in ordinary differential equation models. Exploiting ideas from causal inference, we propose to rank models not only by their predictive performance, but also by taking into account stability, i.e., their ability to predict well in different experimental settings. Based on this model ranking we also construct a ranking of individual variables reflecting causal importance. It provides researchers with a list of promising candidate variables that may be investigated further in interventional experiments. Our ranking methodology (both for models and variables) comes with theoretical asymptotic guarantees and is shown to outperform current state-of-the art methods based on extensive experimental evaluation on simulated data. Practical applicability of the procedure is illustrated on a not yet published biological data set. Our methodology is fully implemented. Code will be provided online and will also be made available as an R package.
View on arXiv