Copula-based convolution for fast point-mass prediction

Abstract

This paper deals with the state estimation of the nonlinear stochastic dynamic discrete-in-time models by a numerical solution to the Bayesian recursive relations represented by the point-mass filter (PMF). In particular, emphasis is placed on the development of the fast convolution, which reduces computational complexity of the PMF prediction step by the orders of magnitude for models with a diagonal form of the dynamic equation. The copula-based convolution decomposes the joint conditional density into the marginal densities (allowing efficient prediction) and an easy-to-calculate copula density function. As a consequence, it has the linear growth of its computational complexity with the state dimension, which is in a contrast with the exponential growth of the standard convolution complexity in PMF methods. The proposed fast convolution is analysed and illustrated in a numerical study for a static example and a dynamic terrain-aided navigation scenario. An exemplary implementation of the proposed convolution is provided along with the paper.