Comparison of the sparse-grid quadrature rule and the cubature rule in nonlinear filtering

In this paper, the recently developed sparse-grid quadrature filter is compared with the cubature Kalman filter. The relation between the sparse-grid quadrature rule and the cubature rule is revealed. It can be shown that arbitrary degree cubature rules can be obtained by the projection of the sparse-grid quadrature rule. Since both rules can achieve an arbitrary high degree of accuracy, they are more accurate than the conventional third-degree cubature rule and the unscented transformation. In addition, they are computationally more efficient than the Gauss-Hermite quadrature rule and the Monte-Carlo method when they are used to calculate the Gaussian type integrals in the nonlinear filtering. The comparison of these rules is demonstrated by a benchmark numerical integration example.