Itô-vector projection filter for exponential families

In this paper, we study the application of Itô-vector projection [1] to the optimal filtering problem. The algorithm projects one SDE to another, possibly lower dimensional, SDE by minimizing an Itô–Taylor expansion of the local projection error's L² norm. We explicitly derive the projection filter equation for a general class of parametric densities, and then specifically apply it to exponential families. We demonstrate that for the case where the measurement drift function is in the span of the natural statistics, the Itô-vector projection filter (IVPF) coincides with the Stratonovich-projection filter (SPF) [2]. We then compare the performance of the IVPF against the SPF (with both being implemented using the Gaussian bijection proposed in [3] and the sparse Gauss–Patterson numerical integration) for two-dimensional optimal filtering problem to show the effectiveness of the proposed algorithm. We vary the measurement drift function to four different functions that are not in the span of natural statistics. Based on one hundred Monte Carlo simulations for each measurement drift, we found that their performances are comparable, with the IVPF potentially offering a slightly more robust performance. However, in our current numerical implementation, the SPF consistently outperforms the IVPF in terms of speed.