We present analytical expressions for predicting the mean square error (MSE) performance of the Kalman filter (KF) and Kalman smoother (KS) when applied to fixed, non-random state trajectories. Unlike the standard Bayesian setting, where the state is modeled as a random process, some practical applications-such as benchmark evaluations in target tracking-rely on deterministic state trajectories. In such cases, the KF and KS become biased and inconsistent (in a non-random/frequentist sense), and their standard covariance estimates no longer reflect actual estimation errors. To address this issue, we derive batch and horizon-recursive MSE expressions for KF and KS to predict their performance without relying on Monte Carlo simulations. Our approach also accounts for measurement model mismatch, allowing performance evaluation under model misspecification. Simulation results validate the accuracy of the proposed analytical expressions, demonstrating their agreement with empirical MSE values.

Bayesian filtering problem is considered in linear Gaussian systems with unknown inverse Wishart distributed process noise covariance. A Bayesian filter is formulated to approximate the joint posterior for the state and the process noise covariance. This involves utilizing moment matching and a scale Gaussian mixture approximation of the$\boldsymbol{t}$-distribution, The proposed filter distinguishes itself by being non-iterative, setting it apart from existing Bayesian solutions given in the literature. The algorithm's performance is demonstrated through its application to a scenario where a target is tracked in two dimensions. Simulation results indicate that the proposed filter achieves similar or better performance compared to state-of-the-art solutions while demanding a reduced computational load.

This paper presents a new state-space model and tracking filter for Riemannian angle tracking on the unit sphere. We consider a scenario where noisy angle-only measurements (azimuth and elevation) of a target, moving with nearly constant velocity, are obtained by a stationary sensor. The angular state, composed of the angular position vector constrained on the unit sphere and the angular velocity vector defined over the tangent plane at the angular position, is tracked. Since the target's nearly constant velocity motion in 3D space induces correlated acceleratory motion on the unit sphere, we develop a state-space model that accounts for this acceleration. A tracking filter based on the Riemannian generalization of the unscented Kalman filter (UKF) given in the literature is proposed and implemented using this model. Simulation results demonstrate that the developed angle tracker outperforms two benchmark Riemannian filters, one based on a (Riemannian generalization of) nearly constant velocity model and the other on a (Riemannian generalization of) nearly constant acceleration model, particularly for close-range and/or fast-moving targets.

Providing efficient yet accurate statistical models is a challenging problem in many applications. When elementary models are not sufficiently descriptive, mixtures of densities can be used. A complexity management issue arises when mixture models are employed: the number of components should be a trade-off between the complexity and the accuracy of the model. However, in general, it is not obvious how to determine the right number of mixture components for a specific application. In a previous work, theoretical foundations to address such a topic have been laid, grounded on the use of the Composite Transportation Dissimilarity between mixtures, and a preliminary criterion to manage the complexity of a mixture model has been proposed. In this paper, additional theoretical insights are provided that allow to formulate a novel adaptive mixture reduction algorithm. Numerical tests show that in most cases the new algorithm constitutes a significant improvement over the previous one.

It has been shown lately that any "standard" Bayesian lower bound (BLB) on the mean squared error (MSE) of the Weiss-Weinstein family (WWF) admits a "tighter" form which upper bounds the "standard" form. Applied to the Bayesian Cramér-Rao bound (BCRB), this result suggests to redefine the concept of efficient estimator relatively to the tighter form of the BCRB, an update supported by a noteworthy example. This paper lays the foundation to revisit some Bayesian estimation problems where the BCRB is not tight in the asymptotic region.

In a sensor network compensation of the correlated information caused by previous communication is of utmost interest for distributed estimation. In this paper, we investigate different information decorrelation approaches that can be applied when using an interacting multiple model filter in a local sensor node. The related decorrelation and the corresponding fusion operations are discussed. The different approaches are compared on a simple distributed single maneuvering target tracking example.

In this paper, recursive Bobrovsky-Zakai bounds for filtering, prediction and smoothing of nonlinear dynamic systems are presented. The similarities and differences to an existing Bobrovsky-Zakai bound in the literature for the filtering case are highlighted. The tightness of the derived bounds are illustrated on a simple example where a linear system with non-Gaussian measurement likelihood is considered. The proposed bounds are also compared with the performance of some well-known filters/predictors/smoothers and other Bayesian bounds.

An approximate extended target tracking (ETT) measurement update is derived for random matrix extent representation with measurement noise. The derived update uses Taylor series approximations. The performance of the proposed update methodology is illustrated on a simple ETT scenario and compared to alternative updates in the literature.

In this paper, marginal versions of the Bayesian Bhattacharyya lower bound (BBLB), which is a tighter alternative to the classical Bayesian Cramér- Rao bound, for discrete-time filtering are proposed. Expressions for the second and third-order marginal BBLBs are obtained and it is shown how these can be approximately calculated using particle filtering. A simulation example shows that the proposed bounds predict the achievable performance of the filtering algorithms better.

Parametric Cramér-Rao lower bounds (CRLBs) are given for discrete-time systems with non-zero process noise. Recursive expressions for the conditional bias and mean-square-error (MSE) (given a specific state sequence) are obtained for Kalman filter estimating the states of a linear Gaussian system. It is discussed that Kalman filter is conditionally biased with a non-zero process noise realization in the given state sequence. Recursive parametric CRLBs are obtained for biased estimators for linear state estimators of linear Gaussian systems. Simulation studies are conducted where it is shown that Kalman filter is not an efficient estimator in a conditional sense.

A marginal version of the Weiss-Weinstein bound (WWB) is proposed for discrete-time nonlinear filtering. The proposed bound is calculated analytically for linear Gaussian systems and approximately for nonlinear systems using a particle filtering scheme. Via simulation studies, it is shown that the marginal bounds are tighter than their joint counterparts.

Cramér-Rao lower bounds (CRLBs) are proposed for deterministic parameter estimation under model mismatch conditions where the assumed data model used in the design of the estimators differs from the true data model. The proposed CRLBs are defined for the family of estimators that may have a specified bias (gradient) with respect to the assumed model. The resulting CRLBs are calculated for a linear Gaussian measurement model and compared to the performance of the maximum likelihood estimator for the corresponding estimation problem.

In this letter, we propose a general framework for greedy reduction of mixture densities of exponential family. The performances of the generalized algorithms are illustrated both on an artificial example where randomly generated mixture densities are reduced and on a target tracking scenario where the reduction is carried out in the recursion of a Gaussian inverse Wishart probability hypothesis density (PHD) filter.

We present an adaptive smoother for linear state-space models with unknown process and measurement noise covariances. The proposed method utilizes the variational Bayes technique to perform approximate inference. The resulting smoother is computationally efficient, easy to implement, and can be applied to high dimensional linear systems. The performance of the algorithm is illustrated on a target tracking example.

A marginal version of the enumeration Bayesian Cramér-Rao Bound (EBCRB) for jump Markov systems is proposed. It is shown that the proposed bound is at least as tight as EBCRB and the improvement stems from better handling of the nonlinearities. The new bound is illustrated to yield tighter results than BCRB and EBCRB on a benchmark example.

Monitoring and tracking human activities around restricted areas is an important issue in security and surveillance applications. The movement of different parts of the human body generates unique micro-Doppler features which can be extracted effectively using joint time-frequency analysis. In this paper, we describe the simultaneous tracking of both location and micro-Doppler features of a human using particle filters (PF). The results obtained using the data from a 77 GHz radar prove the successful usage of particle filters in tracking micro-Doppler features of the human gait.