A moment-based Kalman filtering approach for estimation in ensemble systems

A persistent challenge in tasks involving large-scale dynamical systems, such as state estimation and error reduction, revolves around processing the collected measurements. Frequently, these data suffer from the curse of dimensionality, leading to increased computational demands in data processing methodologies. Recent scholarly investigations have underscored the utility of delineating collective states and dynamics via moment-based representations. These representations serve as a form of sufficient statistics for encapsulating collective characteristics, while simultaneously permitting the retrieval of individual data points. In this paper, we reshape the Kalman filter methodology, aiming its application in the moment domain of an ensemble system and developing the basis for moment ensemble noise filtering. The moment system is defined with respect to the normalized Legendre polynomials, and it is shown that its orthogonal basis structure introduces unique benefits for the application of Kalman filter for both i.i.d. and universal Gaussian disturbances. The proposed method thrives from the reduction in problem dimension, which is unbounded within the state-space representation, and can achieve significantly smaller values when converted to the truncated moment-space. Furthermore, the robustness of moment data toward outliers and localized inaccuracies is an additional positive aspect of this approach. The methodology is applied for an ensemble of harmonic oscillators and units following aircraft dynamics, with results showcasing a reduction in both cumulative absolute error and covariance with reduced calculation cost due to the realization of operations within the moment framework conceived.