Microstate initialisation of dynamical systems using localised ensemble Kalman filters

Often in socio-economic systems, we possess sensible micro-level interaction rules governing their dynamics; however, observations of the system are sparse and only available in aggregation. In this paper, we show how to initialize a dynamical system represented by a model (eg.  agent-based model, initial-value problem, etc) of individual agents embedded in some topology (eg. a network, geographical space, etc) where only macro observations are available. To do so, we apply an ensemble Kalman Filter (EnKF) adapted with model-space covariance localization to assimilate the states of the agents sequentially.  We also formulate the initialization problem as a  Bayesian estimation problem and, under several approximations, we derive the EnKF equations. We remark that this approach could be useful for initialising agent-based models because it needs a  small number of model simulations and treats the dynamical system as a black box.   We validate the EnKF on a high-dimensional approximation of the chaotic Mackey-Glass system and in the Hegselmann-Krause nonlinear agent-based model for opinion dynamics, where we find accurate microstate initialisations for both cases. In the later model,  we show that the localisation technique greatly improves the accuracy of the EnKF.