Splitting Gaussian Densities to Minimize Variance Along a Direction of Nonlinearity
Amit Kumar · 2022
Nonlinear functions of random vectors are frequently used in signal processing, and especially in state space tracking algorithms. Many of these algorithms require a way of estimating the probability density of the state vector at the output of the nonlinear function. Algorithms derived from Kalman Filter, such as Extended Kalman Filter and Unscented Kalman Filter, are popular choices for this, but they only estimate mean and covariance which may be insufficient to describe the non-Gaussian densities. On the other hand, Monte Carlo methods such as particle filters can be more capable but require much more computation. Gaussian mixture filters aim to strike a balance between these two approaches. They offer more flexibility than the filters in the Kalman Filter family by being able to approximate any smooth density arbitrarily well, and at the same time typically require far less computation than the Monte Carlo methods. The number of components in a Gaussian mixture is often chosen to balance the trade-off between computation and accuracy. When necessary, new components are typically created by splitting one of the components in a single direction. This work proposes a new method for determining the direction of split that minimizes the variance of the new components along the direction of nonlinearity. This results in more localized linear function approximation that generally improves accuracy. The proposed direction of split is close to optimal, and performs better than popular alternatives in several examples detailed in this thesis.