Filtering in Triplet Markov Chain Model in the Presence of Non-Gaussian Noise with Application to Target Tracking

Abstract

In hidden Markov chain (HMC) models, widely used for target tracking, the process noise and measurement noise are in general assumed to be independent and Gaussian for mathematical simplicity. However, the independence and Gaussian assumptions do not always hold in practice. For instance, in a typical radar tracking application, the measurement noise is correlated over time as the sampling frequency of a radar is generally much higher than the bandwidth of the measurement noise. In addition, target maneuvers and measurement outliers imply that the process noise and measurement noise are non-Gaussian. To solve this problem, we resort to triplet Markov chain (TMC) models to describe stochastic systems with correlated noise and derive a new filter under the maximum correntropy criterion to deal with non-Gaussian noise. By stacking the state vector, measurement vector, and auxiliary vector into a triplet state vector, the TMC model can capture the complete dynamics of stochastic systems, which may be subjected to potential parameter uncertainty, non-stationarity, or error sources. Correntropy is used to measure the similarity of two random variables; unlike the commonly used minimum mean square error criterion, which uses only second-order statistics, correntropy uses second-order and higher-order information, and is more suitable for systems in the presence of non-Gaussian noise, particularly some heavy-tailed noise disturbances. Furthermore, to reduce the influence of round-off errors, a square-root implementation of the new filter is provided using QR decomposition. Instead of the full covariance matrices, corresponding Cholesky factors are recursively calculated in the square-root filtering algorithm. This is more numerically stable for ill-conditioned problems compared to the conventional filter. Finally, the effectiveness of the proposed algorithms is illustrated via three numerical examples.