The Jacobian matrix is central to the linear stability of a dynamical system. Its eigenvectors determine preferential directions in the dynamics. The real part of the eigenvalues gives the rate of relaxation, while the imaginary part tells whether oscillations are present and their frequency. The Jacobian entries provide insight into the underlying processes, interactions, and even causal relations between the components of the system.
However, the Jacobian is typically constructed from a model and is not directly observable from data. Here we investigate whether it can be inferred from the time evolution of the system in the presence of noise and under which assumptions this is possible. Using covariance matrices together with the linear noise approximation, also known as the Van Kampen expansion, we show that the Jacobian can be inferred with good accuracy. We also examine whether the stationary covariance matrix at zero time lag is sufficient, or if temporal correlations are required. In this context, we address when oscillations can be distinguished from noise in the data, and what limits this separation.
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