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The Jacobian matrix of a dynamical system describes its response toperturbations. Conversely one can estimate the Jacobian matrix by carefullymonitoring how the system responds to environmental noise. Here we present aclosed form analytical solution for the calculation of a system's Jacobian froma timeseries. Being able to access a system's Jacobian enables us to perform abroad range of mathematical analyses by which deeper insights into the systemcan be gained. Here we consider in particular the computation of the leadingJacobian eigenvalue as an early warning signal for critical transition. Toillustrate this approach we apply it to ecological meta-foodweb models, whichare strongly nonlinear dynamical multi-layer networks. Our analysis shows thataccurate results can be obtained, although the data demand of the method isstill high.

A closed form for Jacobian reconstruction from timeseries and its application as an early warning signal in network dynamics