From macro to micro and back: Microstates initialization from chaotic aggregate time series

When modeling real-world systems, suitable interaction rules governing their dynamics are often available. However, data is usually just accessible in aggregation, so valuable information about these dynamics is lost. In this work we tackle the problem of inferring the microstates of systems of which we know their model, but where there is limited information about them. More specifically, we take short, noisy, univariate time series aggregated non-linearly from deterministic chaotic dynamical systems, and we develop a method to infer initial conditions that reproduce the observed data. This method, the Microstate Initialization Procedure (MIP), consists of minimizing the mean-square error between the data and the model simulations by approaching a solution through a system’s strange attractor and then refining this solution using gradient descent techniques. We validate the MIP on the Lorenz and Mackey-Glass systems by making out-of-sample predictions that outperform their Lyapunov characteristic times. Finally, we analyze the predicting power of the MIP concerning the length of the observed time series, where we find a critical transition for the Mackey-Glass system.