Point-mass filters solve Bayesian recursive relations by approximating probability density functions of a system state over grids of discrete points. The approach suffers from the curse of dimensionality. The exponential increase of the number of the grid points can be mitigated by application of low-rank approximations of multidimensional arrays. Tensor train decompositions represent individual values by the product of matrices. This paper focuses on selected issues that are substantial in state estimation. Namely, the contamination of the density approximations by negative values is discussed first. Functional decompositions of quadratic functions are compared with decompositions of discretised Gaussian densities next. In particular, the connection of correlation with tensor train ranks is explored. Last, the consequences of interpolating the density values from one grid to a new grid are analysed.

This paper focuses on the multi-target tracking using the Stone Soup framework. In particular, we aim at evaluation of two multi-target tracking scenarios based on the simulated class-B dataset and ADS-B class-A dataset provided by OpenSky Network. The scenarios are evaluated w.r.t. selection of a local state estimator using a range of the Stone Soup metrics. Source code with scenario definitions and Stone Soup set-up are provided along with the paper.

This paper uses multi-object tracking methods known from the radar tracking community to address the problem of pedestrian tracking using 2D bounding box detections. The standard point-object (SPO) model is adopted, and the posterior density is computed using the Poisson multi-Bernoulli mixture (PMBM) filter. The selection of the model parameters rooted in continuous time is discussed, including the birth and survival probabilities. Some parameters are selected from the first principles, while others are identified from the data, which is, in this case, the publicly available MOT-17 dataset. Although the resulting PMBM algorithm yields promising results, a mismatch between the SPO model and the data is revealed. The model-based approach assumes that modifying the problematic components causing the SPO model-data mismatch will lead to better modelbased algorithms in future developments.

State-space estimation and tracking rely on accurate dynamical models to perform well. However, obtaining an accurate dynamical model for complex scenarios or adapting to changes in the system poses challenges to the estimation process. Recently, augmented physics-based models (APBMs) appear as an appealing strategy to cope with these challenges where the composition of a small and adaptive neural network with known physics-based models (PBM) is learned on the fly following an augmented state-space estimation approach. A major issue when introducing data-driven components in such a scenario is the danger of compromising the meaning (or interpretability) of estimated states. In this work, we propose a novel constrained estimation strategy that constrains the APBM dynamics close to the PBM. The novel state-space constrained approach leads to more flexible ways to impose constraints than the traditional APBM approach. Our experiments with a radar-tracking scenario demonstrate different aspects of the proposed approach and the trade-offs inherent in the imposed constraints.

Gaussian mixture filters for nonlinear systems usually rely on severe approximations when calculating mixtures in the prediction and filtering step. Thus, offline approximations of noise densities by Gaussian mixture densities to reduce the approximation error have been proposed. This results in exponential growth in the number of components, requiring ongoing component reduction, which is computationally complex. In this paper, the key idea is to approximate the true transition density by an axis-aligned Gaussian mixture, where two different approaches are derived. These approximations automatically ensure a constant number of components in the posterior densities without the need for explicit reduction. In addition, they allow a trade-off between estimation quality and computational complexity.

Non-linear filters consider probability density functions in various non-parametric representations. They often suffer from the curse of dimensionality. Computation of weights over a grid of points becomes infeasible even for low dimensions. Filters processing data produced in different sensor nodes provide their own probability densities. Combination of such densities is desired. A favourite paradigm is to construct a fused density as a weighted arithmetic or geometric mean of the individual densities. This paper prospects the fusion for tensor train representation of densities produced by point-mass filters. In this representation, the weights are neither evaluated for a whole grid nor fully stored in the memory of the filters. Aspects of tensor-train-based fusion are discussed, such as computation of auxiliary characteristics and experience with numerical examples.

This paper deals with state estimation of nonlinear stochastic dynamic models. In particular, the stochastic integration rule, which provides asymptotically unbiased estimates of the moments of nonlinearly transformed Gaussian random variables, is reviewed together with the recently introduced stochastic integration filter (SIF). Using SIF, the respective multi-step prediction and smoothing algorithms are developed in full and efficient square-root form. The stochastic-integration-rule-based algorithms are implemented in Python (within the Stone Soup framework) and in MATLAB® and are numerically evaluated and compared with the well-known unscented and extended Kalman filters using the Stone Soup defined tracking scenario.

This paper deals with the initial integrity evaluation of the navigation information provided by the multilayer GNSS and LEO-PNT constellation. Although, the global satellite navigation systems (GNSS) play indispensable role in almost all aspects of today’s society, their signals are prone to intentional or accidental interference. Therefore, low Earth orbit (LEO) constellations aiming at position, navigation, and timing (PNT) solution have recently been introduced as their extension. The LEO-PNT constellations are planned to contain hundreds of SVs with better interference resilience and geometric diversity. As a consequence, the multi-layer GNSS and LEO-PNT constellation was shown to offer more accurate PNT solution. In this paper, we focus on another important aspect of the multi-layer navigation information, which is its integrity assessment. In particular, we analyse possible dependencies between GNSS and LEO-PNT constellations and their impact on the integrity evaluated using the solution separation. The analysis is supported by the numerical simulations using GPS and LEO-PNT constellations with 32 and 441 satellites, respectively.

A first-principle single-object model is proposed for pedestrian tracking. It is assumed that the extent of the moving object can be described via known statistics in 3D, such as pedestrian height. The proposed model thus need not constrain the object motion in 3D to a common ground plane, which is usual in 3D visual tracking applications. A nonlinear filter for this model is implemented using the unscented Kalman filter (UKF) and tested using the publicly available MOT-17 dataset. The proposed solution yields promising results in 3D while maintaining excellent results when projected into the 2D image. Moreover, the estimation error covariance matches the true one. Unlike conventional methods, the introduced model parameters have convenient meaning and can readily be adjusted for a problem.

The paper deals with measures of nonlinearity. In state estimation, they are utilized i) to select a suitable state estimation algorithm by assessing the nonlinearity of a system model, ii) to adapt the estimation algorithm structure or parameters, or iii) to indicate the possible effect of strong nonlinearity that leads to estimate credibility loss. This paper summarizes the state of the art of nonlinearity measures, focusing on the mean-square-error-based measure of nonlinearity. Its weak point is illustrated, and based on this, requirements for the new measure of nonlinearity are formulated. A new nonlinearity measure that is both unitless and normalized is designed. Its properties are demonstrated using numerical tracking experiments.

Subjective Bayesian estimation perceives probability density functions as expert opinions. Among various rules for combining the opinions, the product and the weighted geometric mean of densities are prominent. Nevertheless, closed-form representations are scarce and non-parametric approaches often suffer from the curse of dimensionality. This paper prospects the fusion of densities represented by non-parametric marginal densities and a parametric Gaussian copula. The explicit reconstruction of the joint densities followed by an optimisation step is avoided. A cheap approximate combination is proposed instead. The combination of marginal densities is tuned by a Gaussian term, while the proposed copula parameter uses moments of the marginal densities. The presented examples illustrate the approximative nature of the approach for non-Gaussian densities and highlight some numerical issues.

This paper deals with the resilient time provision based on an ensemble of clocks. In particular, the emphasis is laid on the combination of clock outputs and detecting possible faults. Two classes of fault detection methods, namely model-based and AI/ML-based, are discussed and analysed. In addition, a novel fault detection technique based on the solution separation principle is proposed and tailored for the area of the time provision. Selected fault detection methods are numerically evaluated using a model of an atomic clock ensemble.

Common visual tracking algorithms make use of measurement models whose parameters need to be specified. These are, namely, measurement noise covariance related to spatial error of detections provided by a visual detection algorithm, probability of detection, and expected number of clutter detections. The measurement model parameters are often hand selected, using no data-based knowledge. This paper proposes a technique to estimate the parameters by reliably associating detections to annotations in each video frame. The technique is verified on the publicly available MOT-17 dataset.

Common visual tracking algorithms make use of bounding box (BB) motion models. These models are parameterized by quantities such as noise covariance parameters corresponding to the evolution of the position, velocity, aspect ratio, width, or height of the bounding box, or their respective velocities. The noise covariance parameters are often hand-selected, using no principled knowledge regarding various aspects such as camera pose or frame rate. This paper aims to analyze how these aspects influence parameter estimates obtained from annotated datasets that are well-known to the visual tracking community. To obtain the estimates, the recently developed measurement difference method (MDM) is modified and used.

Active fault diagnosis (AFD) of stochastic large-scale systems in multiple model framework involves two stages: offline and online. In the offline stage, an excitation input generator is designed based on a Bellman function. In the online stage, the generator is utilized together with an estimator of the model indices. A similar estimator is used in the offline stage for the Bellman function calculation using the value iteration technique. However, due to the high dimensions of information states of the associated perfect state information problem, the estimator in the offline stage must involve approximations. The paper provides the relations for the estimate calculation using the Bayesian recursive relations, proposes four algorithms, and studies effects of such approximations on the AFD decisions. In particular, the quality of the model index estimates is analyzed using a power network model.

Linear fusion of estimates is a basic tool for combining probabilistic data. If the correlation of estimation errors is unknown, the fusion performance is evaluated with respect to the worst case. Inverse Covariance Intersection fusion is a rule for combining two estimates with partially known crosscorrelation matrix. This paper generalises the rule to fusing multiple estimates. First, the generalised assumption and the essential theory are presented. A suboptimal solution with a simple parametrisation is derived next and it is shown to be better than the solution for unknown correlation. Finally, a recursive fusion of multiple estimates is designed.

This paper is devoted to the Bayesian state estimation of the nonlinear stochastic dynamic systems. The stress is laid on the numerical solution to the Bayesian recursive relations by the point-mass filter for a class of state-space models with linear dynamics and nonlinear measurement. In particular, a novel reliable technique for convolution computation is proposed. The technique combines the standard point-mass-based convolution with a density-weighted integration to provide accurate results even for systems with small state noise. Several implementations of the technique are developed, theoretically analysed, and evaluated in a numerical study.

The paper deals with the state estimation of nonlinear stochastic systems with additive Gaussian noises by means of the Gaussian filters leveraging numerical integration rules. The filters were derived under the assumption of the joint state and measurement predictive density being Gaussian, which is violated by the system nonlinearity. Such violation can hardly be monitored by the standard Gaussian filters, which re-generate a new set of points for each involved numerical integration to accommodate their variance increase due to the additive noises. The paper proposes a stochastic integration filter algorithm that modifies the points instead of their resampling and thus admits reusing the points in the next time steps. The distribution of the points can thus bear more information than just the first two moments in case of the standard Gaussian filters. The acquired information is then utilized for the Gaussian assumption monitoring purposes. In the event of the assumption violation, the filter may change its behavior. As a by-product of reusing the points, the computational costs of the proposed filter are significantly reduced compared to the standard stochastic integration filter.

The paper deals with the active fault diagnosis of large scale stochastic systems with faults modeled as mutually dependent Markov chains. The system is described by multiple models representing fault-free and faulty behavior of the system. The aim of the active fault detector in addition to detecting the faults is to excite the system to improve the detection quality. The algorithm consists of two stages: the off-line design of the Bellman function providing the optimal excitation and the on-line estimation, which generates the decisions and selects the optimal excitation according to the Bellman function. In particular, the paper focuses on the online estimation and proposes an algorithm in the hierarchical architecture. The local nodes estimate the continuous state of the subsystems, select the optimal excitations and send local likelihoods to the central node. The central node generates the decisions and submits the respective model probabilities to the local nodes. The performance of the proposed algorithm is validated using a simple numerical example.

A recently proposed algorithm of fusion under partially known correlations of estimation errors has been proved to outperform the classic Covariance Intersection algorithm, which was proposed for the case of no knowledge of correlation. This paper shows that the assumptions of the recently proposed algorithm are rather strict with respect to the classic one. Namely, the mean square error (MSE) matrices of the two state estimates cannot be upper-bounded arbitrarily. A relaxation of the assumption that the matrices have to be known exactly is discussed, as well as an iterative fusion of multiple estimates, and several examples dealing with dependent errors are provided.

The paper focuses on state estimation of discrete-time nonlinear stochastic dynamic systems with a special focus on the stochastic integration filter. The filter is an representative of the Gaussian filter and computes the state and measurement predictive moments by making use of a stochastic integration rule. As a result, the calculated values of the moments are random variables and exhibit favorable asymptotic properties. The paper analyzes theoretical consequences of using stochastic integration rules and proposes several modifications that improve the performance of the stochastic integration filter. As the filter requires multiple iterations of the stochastic rule, its computational costs are higher in comparison with other Gaussian filters. To reduce the costs, several modifications are proposed in the paper, which are also concerned with numerical stability issues. The proposed modifications are illustrated using both static and dynamic numerical examples used in target tracking.

Optimal fusion of estimates that are computed in a distributed fashion is a challenging task. In general, the sensor nodes cannot keep track of the cross-correlations required to fuse estimates optimally. In this paper, a novel technique is presented that provides the means to reconstruct the required correlation structure. For this purpose, each node computes a set of deterministic samples that provides all the information required to reassemble the cross-covariance matrix for each pair of estimates. As the number of samples is increasing over time, a method to reduce the size of the sample set is presented and studied. In doing so, communication expenses can be reduced significantly, but approximation errors are possibly introduced by neglecting past correlation terms. In order to keep approximation errors at a minimum, an appropriate set size can be determined and a trade-off between communication expenses and estimation quality can be found.

The paper deals with state estimation of nonlinear stochastic dynamic discrete-time systems with a special focus on the stochastic integration filter. The filter is an instance of Gaussian filters, which for strongly nonlinear systems may provide inconsistent estimates. Primarily, optimistic inconsistent estimates, which overrate quality of the point estimate, are inappropriate in many applications where estimate integrity is crucial. In this paper, a technique for an estimate consistency monitoring for detection of optimistic estimates is proposed based on entropy. For the purpose of the entropy computation, a probabilistic analysis of the stochastic integration filter behavior is carried out. The proposed consistency monitoring is illustrated in a numerical example.

In decentralised estimation, locally measured data are processed locally and the local filters are unaware of the other ones. Due to the lack of the global knowledge, the fusion of the local estimates cannot utilise the correlations of the local estimate errors in the computation of the fused mean square error matrix. For this reason, algorithms of fusion under unknown correlations have been designed to provide upper bounds of these matrices. This paper reveals a fundamental relation between the upper bounds and Minkowski sums of ellipsoids. The obtained insight improves the comprehension of the fusion algorithms. Application of the ellipsoidal calculus is illustrated using elemental examples.

The paper introduces a novel approach to an estimator design, the cooperative filter design, for state estimation of nonlinear systems. The approach is based on the idea of combining estimates of several different approximate (and thus sub-optimal) nonlinear filters, which are configured to perform the same task. Within the concept, two strategies are proposed, namely the cooperative estimation and cooperative monitoring. The strategies have the potential of an improvement of the estimation performance in terms of accuracy and consistency, which was confirmed by a numerical illustration.

The paper deals with the state estimation of nonlinear stochastic dynamic systems. The stress is laid on the assessment of the estimate error, which is caused by the violation of the estimator design assumptions. The assessment is based on measures comparing estimators actual working conditions and the assumptions under which the estimators have been proposed. In particular, the measures of nonlinearity and non-Gaussianity are discussed. The measures are briefly introduced and selected typical representatives are detailed with respect to their implementation. Performance of the measures is evaluated in the framework of the Gaussian filters in a numerical study.

The aim of this article is to design a moment transformation for Student-t distributed random variables, which is able to account for the error in the numerically computed mean. We employ Student-t process quadrature, an instance of Bayesian quadrature, which allows us to treat the integral itself as a random variable whose variance provides information about the incurred integration error. Advantage of the Student-t process quadrature over the traditional Gaussian process quadrature, is that the integral variance depends also on the function values, allowing for a more robust modelling of the integration error. The moment transform is applied in nonlinear sigma-point filtering and evaluated on two numerical examples, where it is shown to outperform the state-of-the-art moment transforms.

The paper deals with the nonlinear state estimation of stochastic dynamic systems with a special focus on coping with outliers appearing in the system. A new stochastic integration Student's-t filter is developed based on the generic Student's-t filter and assuming the density of random variables present in the model and the conditional density of the state be Student's-t distributed. For evaluation of the integrals with Student's-t weights present in the filter relations, the stochastic integration rule is used. In contrast to other integration rules, it provides asymptotically exact values of the integrals. Performance of the proposed stochastic integration Student's-t filter is illustrated using a numerical simulation involving the coordinated-turn motion model.