Estimating Correlated Angles Using the Hypertoroidal Grid Filter

Estimation for multiple correlated quantities generally requires considering a domain whose dimension is equal to the sum of the dimensions of the individual quantities. For multiple correlated angular quantities, considering a hyper-toroidal manifold may be required. Based on a Cartesian product of d equidistant one-dimensional grids for the unit circle, a grid for the d-dimensional hypertorus can be constructed. This grid is used for a novel filter. For n grid points, the update step is in O(n) for arbitrary likelihoods and the prediction step is in O(n2) for arbitrary transition densities. The run time of the latter can be reduced to O(n log n) for identity models with additive noise. In an evaluation scenario, the novel filter shows faster convergence than a particle filter for hypertoroidal domains and is on par with the recently proposed Fourier filters.