Variational Gaussian filter (VGF) approximates the intractable posterior of the state of a non-linear non-Gaussian system using a single Gaussian density normally found through Kullback-Leibler divergence minimization. This paper focuses on the VGFs whose measurement update is realized by employing the natural gradient descent (NGD). Under the assumption that the state predictive distribution is also Gaussian, we re-examine the iterative NGD-based measurement update under two different parameterizations of the Gaussian posterior. The first one consists of the mean and covariance, while the other comprises the mean and precision matrix (i.e., the inverse of the covariance). Their NGD-based update rules are derived in an alternative but unified way using matrix calculus. They are compared against each other and with the one developed using the natural parameterization of the Gaussian density. Important new insights are obtained. Modifications to the established update rules, which guarantee the positive definiteness of the covariance/precision matrix of the Gaussian posterior, are re-visited as well. Simulations are used to corroborate the theoretical results and evaluate the performance of the developed algorithms in range-bearing tracking.

This paper introduces a new variational Gaussian filtering approach for estimating the state of a nonlinear dynamic system. We first assume that the predictive distribution of the state is Gaussian and derive an iterative method for updating the state posterior in the natural parameter space through KullbackLeibler divergence minimization. The obtained update rule is the same as that of the conjugate-computation variational inference technique in Bayesian learning. The derivation here is simpler and more insightful. We then impose a Wishart prior on the inverse of the state prediction covariance to take into account the impact of approximating the state predictive distribution using a Gaussian density on the state posterior estimation. The prediction covariance is identified jointly with the state using variational inference and the established state posterior update rule to achieve the desired Gaussian filtering. Simulation study examines the performance of the proposed filtering framework in target tracking based on bearing and range measurements.

The quotient of two multivariate Gaussian densities can be written as an unnormalized Gaussian density, which has been applied in some recently developed multiple-model fixed-interval smoothing algorithms. However, this expression is invalid if instead of being positive definite, the covariance of the unnormalized Gaussian density is indefinite (i.e., it has both positive and negative eigenvalues) or undefined (i.e., computing it requires inverting a singular matrix). This paper considers approximating the quotient of two Gaussian densities in this case using two different approaches to mitigate the caused numerical problems. The first approach directly replaces the indefinite covariance of the unnormalized Gaussian density with a positive definite matrix nearest to it. The second approach computes the approximation through solving, using the natural gradient, an optimization problem with a Kullback-Leibler divergence-based cost function. This paper illustrates the application of the theoretical results by incorporating them into an existing smoothing method for jump Markov systems and utilizing the obtained smoothers to track a maneuvering target.

The Schmidt-Kalman filter (SKF) achieves filtering consistency in the presence of biases in system dynamic and measurement models through accounting for their impacts when updating the state estimate and covariance. However, the performance of the SKF may break down when the measurements are subject to non-Gaussian and heavy-tail noise. To address this, we impose the Wishart prior distribution on the precision matrix of measurement noise, such that the measurement likelihood now has heavier tails than the Gaussian distribution to deal with the potential occurrence of outliers. Variational inference is invoked to establish analytically tractable methods for computing the posterior of the system state, system biases, and the measurement noise precision matrix. The principle of the SKF considers the effect of system biases but does not actively estimate them when two variants of outlier-robust SKFs are incorporated. We evaluate their performance in terms of estimation accuracy and filtering consistency using simulations and real-world data. Promising results are obtained.

The Gaussian mixture model (GMM) has been extensively investigated in nonlinear/non-Gaussian filtering problems. This paper presents two enhancements for GMM-based nonlinear filtering techniques, namely, the adaptive ordering of the measurement update and normalized innovation square (NIS)-based mixture component management. The first technique selects the order of measurement update by maximizing the marginal measurement likelihood to improve performance. The second approach takes the filtering history of a mixture component into account and prunes those components with NIS larger than a threshold to eliminate their impact on the filtering posterior. The advantage of the proposed enhancements is illustrated via simulations that consider source tracking using the time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements received at two unmanned aerial vehicles (UAVs). A GMM-cubature quadrature Kalman filter (CQKF) is implemented and its performances with different measurement update and mixture component management strategies are compared. The superior performance obtained via the use of the two proposed techniques is demonstrated.