Interpretable and equation-free response theory for complex systems

Response theory provides a pathway for understanding the sensitivity of a system and for predicting how its statistical properties change when a perturbation is applied. In the case of complex and multiscale systems, to achieve enhanced practical applicability, response theory should be interpretable, capable of focusing on relevant time scales and amenable to data-driven and equation-agnostic implementations. Along these lines, in the spirit of Markov state modelling (MSM), we present linear and nonlinear response formulas for Markov chains. We obtain simple and easily implementable expressions that can be used to predict the response of observables as well as of higher-order correlations. The methodology proposed here can be implemented in a purely data-driven setting and even if the underlying evolution equations are unknown. The use of algebraic expansions inspired by Koopmanism allows us to elucidate the role of different time scales and modes of variability, and to find explicit and interpretable expressions for the Green's functions at all orders. This is a major advantage of the framework proposed here. We illustrate our methodology in a very simple yet instructive metastable system. Finally, our results provide a dynamical foundation for the Prony method, which is commonly used for the statistical analysis of discrete time signals. This article is part of the theme issue 'Critical transitions and intelligent control in complex systems'.