The suboptimal method via probabilists' Hermite polynomials to solve nonlinear filtering problems

In this paper, we shall investigate a novel suboptimal nonlinear filtering with augmented states via probabilists' Hermite polynomials (HP). The estimation of the original state can be extracted from the augmented one. Our method is motivated by the so-called Carleman approach (Germani et al., 2007). The novelty of our paper is to augment the original state with its probabilists' HPs, instead of its powers as in Carleman approach. Then we form a bilinear system of the first ν generalized Hermite polynomials (gHP) to yield the degree-ν approximation. We demonstrate that the neglect of the probabilists' gHPs with high degree is more reasonable by showing that the expectation of the HPs with degree n tends to zero, as n goes to infinity, if the density function belongs to certain function class. Moreover, we discuss the choice of the scaling and translating factors to yield better resolution. The benchmark example, 1d cubic sensor problem with zero initial condition, has been numerically solved by various methods, including the most widely used extended Kalman filter and particle filter. Our method with adaptive scaling factor outperforms the other methods in accuracy.