Generalized unscented transformation for forecasting non-Gaussian processes

The observations of linear and nonlinear physical processes are subject to random errors, which can be represented by a wide variety of probability distributions. In contrast, most estimation and inference techniques rely on a Gaussian assumption, which may limit our ability to make model-based predictions. There is a need for data assimilation methods that can capture and leverage the higher moments of these physical processes for state estimation and forecasting. In this paper, we develop the generalized unscented transform (GenUT), which uses a minimal number of sample points to accurately capture elements of the higher moments of most probability distributions. Constraints can be analytically enforced on the sample points while guaranteeing at least second-order accuracy. The GenUT is widely applicable to non-Gaussian distributions, which can substantially improve the assimilation of observations of nonlinear physics, such as the modeling of infectious diseases.