Gaussian-Based Parametric Bijections For Automatic Projection Filters

The automatic projection filter is a recently developed numerical method for projection filtering that makes use of sparse-grid integration and automatic differentiation. However, its accuracy is highly dependent on the accuracy of the cumulant-generating function calculation done via sparse-grid integration, which in turn depends on the selection of the bijection from the canonical hypercube to the state space. In this paper, we propose two new adaptive parametric bijections for the automatic projection filter. The first bijection relies on the minimization of Kullback--Leibler divergence, whereas the second method employs the sparse-grid Gauss--Hermite quadrature formula. These bijections let the sparse-grid nodes move adaptively with the high-density region of the state space, resulting in a substantially improved approximation while using only a small number of quadrature nodes. The practical applicability of the methodology is illustrated in three simulated nonlinear filtering problems.