Accurate uncertainty quantification is critical for robust and trustworthy predictions in many real-world applications. Bayesian Neural Networks (BNNs) provide a principled approach for modeling uncertainty but are often limited by the computational complexity of Bayesian inference. In this paper, we introduce a statistical linearization approach for multilayer feedforward BNNs. We demonstrate that this statistical linearization is equivalent to the Kalman Bayesian Neural Networks (KBNN) framework. This equivalence unifies these methodologies, providing a theoretical foundation for understanding the relationship between different BNN training approaches.

Efficiently solving the continuous-time signal and discrete-time observation filtering problem for chaotic dynamical systems presents unique challenges in that the advected distribution between observations may encounter a separatrix structure that results in the prior distribution being far from the observation or the distribution may become split into multiple disjoint components. In an attempt to sense and overcome these dynamical issues, as well as approximate a non-Gaussian distribution, a nudged particle filtering approach has been introduced. In the nudged particle filter method a control term is added, but has the potential drawback of degenerating the weights of the particles. To counter this issue, we introduce an intermediate resampling approach based on the modified Cramér-von Mises distance. The new method is applied to a challenging scenario of the non-chaotic, unforced nonlinear Duffing oscillator, which possesses a separatrix structure. Our results show that it consistently outperforms the standard particle filter with resampling and original nudged particle filter.

Densities separable in spherical coordinates have two advantages:$i$) the normalization constant is easy to compute, as the cumulative distribution can be decomposed into individual scalar integrals, and ii) an orthogonal inverse transform is directly available via a simple, scalar initial value problem and can be used to compute deterministic samples. We propagate uniform low-discrepancy sequences through that orthogonal inverse transform and obtain very homogeneous and even visually appealing deterministic samples. To demonstrate this technique, we exemplarily propose some spherical-coordinate-separable densities in$\mathbb{S}^{2}, \mathbb{R}^{2}$, and$\mathbb{R}^{3}$, including a non-isotropic modification of the von Mises-Fisher distribution. The proposed densities may be used, e.g., to represent uncertain radar measurements and for directional estimation. Furthermore, the framework presented herein allows quite simple design of various more densities tailored to a given scenario.

The solution to the state estimation problem is given by the Bayesian recursive relations (BRRs). Recently, ensemble Gaussian mixture filters have shown to be an accurate and consistent solution to the state estimation problem. In this type of filters, the BRRs are solved by approximating the state probability density function (PDF) via Gaussian mixtures (GMs) and point masses (PMs). Throughout the propagation and measurement update steps, the approximated state PDF is constantly switching between GMs and PMs. Therefore, a key step for this solution involves optimally sampling PMs from GMs. For onboard applications, verifiable and computationally inexpensive sampling techniques are crucial. In previous work, a deterministic sampling technique was developed by minimizing a distance metric known as the modified Cramér-von Mises distance (MCVMD), yielding a verifiable solution. However, the computationally feasibility of this solution for onboard use was not considered. This work introduces a new sampling strategy that is both deterministic and computationally inexpensive compared to MCVMD approach. By solving the approximate optimal transport problem via an iterative Sinkhorn-Knopp algorithm, this new technique is able to sub-optimally sample from a GM, providing a computationally inexpensive filter.

Particle filters are an important class of algorithms for Bayesian estimation. One of their drawbacks is the socalled particle degeneration where only very few particles with a meaningful weight remain after the filter step. This effect is typically remedied by regularly resampling the particles, yielding a set of equally weighted particles. This paper investigates an approach to deterministically sample particles from the proposal distribution in such a way to automatically have equally weighted particles at the end of the filter step. The proposed method is first motivated and presented for the one-dimensional case. Using the Radon transform and projected cumulative distributions, the one-dimensional algorithm is extended to multivariate problems. Some examples of the usefulness of the proposed algorithm are also shown.

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.

Bayesian machine learning models-especially Bayesian neural networks (BNNs)-offer powerful black-box approaches for prediction and uncertainty quantification. However, these models frequently exhibit inconsistent prediction quality across input regions, and conventional global metrics (e.g., the mean squared error (MSE)) are inadequate for capturing such local discrepancies. To overcome this limitation, we introduce a novel kernel-based framework for local calibration testing that assesses how well predicted distributions reflect both the function to be learned and inherent uncertainties. In our approach, spherical input-space kernels are used to define relevant subsets in the neighborhood of a point to be tested. This enables the online assessment of these localized regions using calibration metrics or statistical tests. By aggregating results across multiple kernel widths, our method yields both robust binary decisions and a continuous analysis over arbitrary inputs. Numerical experiments on single- and multi-dimensional regression tasks demonstrate the efficiency and scalability of our approach, underscoring its potential for real-time and large-scale applications.

In this paper, we introduce a novel Gaussian Assumed Density Filter (GADF) for high-quality state estimation in discrete-time stochastic nonlinear dynamic systems, with a primary focus on the measurement update. Rooted in optimal transport theory, the Wasserstein distance is employed as a powerful metric for comparing probability distributions. Building on this foundation, we utilize the unique, explicit Wasserstein barycentric interpolation between Gaussian distributions to parameterize an initial Gaussian Process (GP) in the joint measurement/prior state space. Deterministic samples drawn from the true joint measurement/state density are then used with likelihood-based parameter estimation techniques to optimize the parameters of this Gaussian Process. As a result, the derived Gaussian Process provides a local non-Gaussian approximation to the true joint density. This approach eliminates the need for a second Gaussian assumption on the joint density and avoids an explicit likelihood function, making it a higher-quality plug-in replacement for the commonly used Linear Regression Kalman Filter (LRKF).

In this paper, we present novel results and insights into tracking extended objects using the Stochastic Medial Axis Transform (SMAT). Unlike conventional methods that depend on explicit shape parameterization with basic priors, SMAT employs an implicit inside-out representation by constructing maximum inscribed circles within the object. This is achieved by simultaneously fitting two Bézier curves: one that defines the medial manifold, providing the centers of the maximum inscribed circles, and the other that characterizes the scalar thickness field, assigning positive radii to these centers. This dual-curve formulation leverages the concept of inverse skeletonization and offers a flexible, parametric shape model capable of tracking diverse shapes, whether convex or non-convex, symmetric or asymmetric. Furthermore, we obtain a closed-form likelihood function in 2D space that facilitates the application of advanced recursive Bayesian state estimators. Finally, we conduct two simulation studies to demonstrate and evaluate the effectiveness of the proposed approach.

In extended object tracking, basic parametric shapes such as ellipses and rectangles or non-parametric shape representations such as Fourier series or Gaussian processes can be utilized as shape priors. However, flexible non-parametric shape representations can be disproportionately detailed and computationally intensive for many applications. Therefore, we propose to adopt deformable superellipses for a low-dimensional and flexible representation of basic parametric shapes in this paper. We present a measurement model in 2D space that can cope with boundary and interior measurements simultaneously by recursively estimating an artificial noise variance for interior measurements. We investigate and compare the model in a simulated and real-world maritime scenario with the result that the combination of deformable superellipses and artificial measurement noise estimation performs better than state-of-the-art methods.

We propose a novel update step of a Gaussian mixture particle filter for nonlinear state estimation. The update procedure works as follows: First, unweighted samples are drawn in an optimal deterministic sense from a prior Gaussian mixture. These samples are then assigned weights from the likelihood function, and we compute higher-order moments from this samplebased posterior. These moment approximations converge with $L^{-1}$ instead of $L^{-1 / 2}$ as our samples are optimal deterministic. Finally, the continuous posterior approximation is determined as the Gaussian mixture that has minimal Fisher information under the constraint of having the aforementioned moments. To achieve this, we employ a closed-form solution of the Fisher information that involves Gaussian root mixture densities.

The construction of a continuous probability density function (pdf) that fits a set of samples is a frequently occurring task in statistics. This is an inherently underdetermined problem, that can only be solved by making some assumptions about the samples or the distribution to be estimated. This paper proposes a density estimation method based on the premise that each sample represents the same amount of probability mass of the underlying density. The estimated pdf is parameterized as the square of a polynomial spline, which makes further processing of the estimated density very efficient. This pdf is inherently nonnegative, ensuring a monotone cumulative distribution function, which makes it easy to generate samples from it through inverse transform sampling. Furthermore, it is cheap to evaluate and easy to integrate, making moment calculations fast. To find the coefficients of the polynomials that make up the spline, an optimization problem is derived. The Fisher information is used as a regularizer in this problem to select the solution that contains the least amount of information. The method is shown to work on samples from a variety of different one-dimensional probability distributions.

The earth’s magnetic field along railway tracks is strongly distorted by magnetic material in the vicinity, e.g., steel in rails and reinforced concrete. The resulting magnetic distortions are persistent in time and characteristic for a certain part of the track. Thus, these distortions can be seen as fingerprints that enable localization when a map of the magnetic field is available. This is particularity interesting for areas where global navigation satellite system (GNSS) signals are not available, such as tunnels. Unfortunately, creating the magnetic map in a GNSS-denied area requires a position reference system that is most likely not available. This paper addresses this problem with a graph-based simultaneous localization and mapping (SLAM) algorithm that uses only odometer and magnetometer measurements. The key idea of the proposed algorithm is to use the magnetic field to detect loop closures and to calculate the relative transformation between different nodes in the pose-graph. The algorithm is evaluated based on a data set recorded with the advanced TrainLab of the Deutsche Bahn traveling on a track in Berlin. Results show that the graph SLAM algorithm together with the magnetic loop closure detection reduces and bounds the position error of the odometry.

Bayesian Neural Networks (BNNs) offer a sophisticated framework for extending classical neural network point estimates to encompass predictive distributions. Despite the high potential of BNNs, established BNN training methods such as Variational Inference (VI) and Markov Chain Monte Carlo (MCMC) grapple with issues such as scalability and hyperparameter dependence. In addressing these issues, our research focuses on the fundamental elements of BNNs, in particular perceptrons and their predictive capabilities. We introduce a new perspective on the closed-form solution for backward-pass computation for the Bayesian perceptron and prove that the state-of-the-art solution is equivalent to statistical linearization. To assess the efficacy of Bayesian perceptrons and provide insights into their performance in distinct input space regions, a novel methodology utilizing k-d trees as a space partitioning method is introduced to evaluate prediction quality within specific input space regions.

In the past years, algorithms for 3D shape tracking using radial functions in spherical coordinates represented with different methods have been proposed. However, we have seen that mainly measurements from the lateral surface of the target can be expected in a lot of dynamic scenarios and only few measurements from the top and bottom parts leading to an error-prone shape estimate in the top and bottom regions when using a representation in spherical coordinates. We, therefore, propose to represent the shape of the target using a radial function in cylindrical coordinates, as these only represent regions of the lateral surface, and no information from the top or bottom parts is needed. In this paper, we use a Fourier-Chebyshev double series for 3D shape representation since a mixture of Fourier and Chebyshev series is a suitable basis for expanding a radial function in cylindrical coordinates. We investigate the method in a simulated and real-world maritime scenario with a CAD model of the target boat as a reference. We have found that shape representation in cylindrical coordinates has decisive advantages compared to a shape representation in spherical coordinates and should preferably be used if no prior knowledge of the measurement distribution on the surface of the target is available.

With the growing availability of high-resolution sensors, processing more than one detection per target becomes increasingly critical when tracking multiple extended objects. However, contemporary sensors often generate spurious detections that need to be considered. Naively employing standard multitarget trackers may result in poor tracking performance for multitarget–multidetection tracking in cluttered environments, and the relevant extensions are nontrivial. This paper introduces a version of the kernel symmetric measurement equation (SME) filter that considers both multidetections and clutter. For a simulated scenario, our novel filter achieved a higher accuracy than the global nearest neighbor (GNN) and a fast variant of the joint probabilistic data association filter (JPDAF).

Knowledge about human intention can be beneficial in many disciplines of robotics, such as collaborative manufacturing, prosthetics, or encountered-type haptics. Existing intention estimation approaches are either traditional and rely on handcrafted features and heuristics, or learning-based and tailored to very specific conditions. This paper attempts to combine the best of both worlds by making recurrent neural networks adaptable to different scenarios. To achieve this, the intention estimation problem is formulated as a probabilistic classification problem and two new data sets with real-world motion and eye-tracking data are presented. Based on this data, three real-time capable classifiers with different features regarding situational awareness and additional outputs are designed and evaluated against two competing approaches. The results show that two out of three classifiers lead to improved or equivalent performance compared to traditional approaches, while good generalization is maintained.

We present a new deterministic sampling method for arbitrary densities, unnormalized densities, and likelihoods. Our rejection-free and kernel-free method uses dense equal sphere packing of the volume under the density function (PoVuD). In order to obtain an ensemble that is better than independent random particles, we enforce some local homogeneity.

We propose a method for optimal Bayesian filtering with deterministic particles. In order to avoid particle degeneration, the filter step is not performed at once. Instead, the particles progressively flow from prior to posterior. This is achieved by splitting the filter step into a series of sub-steps. In each sub-step, optimal resampling is done by a map that replaces non-equally weighted particles with equally weighted ones. Inversions of the maps or monotonicity constraints are not required, greatly simplifying the procedure. The parameters of the mapping network are optimized w.r.t. a particle set distance. This distance is differentiable, and compares non-equally and equally weighted particles. Composition of the sequence of maps provides a final mapping from prior to posterior particles. Radial basis function neural networks are used as maps. It is important that no intermediate continuous density representation is required. The entire flow works directly with particle representations. This avoids costly density estimation.

We aim to approximate the distribution of the first-passage time of a particle moving according to a Gaussian process with increasing trend, i. e., the distribution of the first time a particle described, e.g., by a state-space model such as a constant-velocity or constant-acceleration model, arrives at a fixed location. Since the known approaches from the literature either consider processes from different families or lead to highly complex approximations, we seek a fast-to-compute method for the problem. Motivated by an engineering particle transport task for which we can assume that once a particle has arrived at this 10-cation it cannot move back, we derive an analytic approximation for the first-passage time probabilities and calculate its inverse cumulative distribution function analytically and the moments numerically. Furthermore, we propose a Gaussian approximation based on a linearization approach. The strengths and limitations of our methods are discussed and by comparison with Monte Carlo simulations, we show that in particular, the first one satisfies the requirements of engineering problems in terms of accuracy and computation time.

The control of brushless DC motors requires high-resolution angular position and accurate speed information. However, available sensor-based solutions only measure either the position or the speed directly, and then approximate the other numerically. In this work, a novel technique is presented to estimate both of these values simultaneously by sensing the stray magnetic field of the internal permanent magnets of the motor. However, achieving this requires the following two challenges to be addressed. First, the relationship between the magnetic field and the motor position is distorted by the rotational speed in a non-intuitive way, requiring careful modeling of these dependencies. Second, the derived model needs to consider that the angular position data is periodic by nature, but the magnetic field data and the angular speed data are linear (i.e., non-periodic). To achieve this, we introduce two different multidimensional regression models based on the Fourier series. Both models are first trained offline using reference data, and then used as a measurement function in a nonlinear estimator such as the EKF for online estimation. Evaluations show that both models outperform state-of-the-art techniques.

Nonlinear filtering is the most important aspect in state estimation with real-world systems. While the Kalman filter provides a simple though optimal estimate for linear systems, feasible filters for general systems are still subject of intensive research. The previously proposed Progressive Gaussian Filter PGF42 marked a new milestone, as it was able to efficiently compute an optimal Gaussian approximation of the posterior density in nonlinear systems [1]. However, for highly nonlinear systems where true posteriors are “banana-shaped” (e.g., cubic sensor problem) or multimodal (e.g., extended object tracking), even an optimal Gaussian approximation is an inadequate representation. Therefore, we generalize the established framework around the PGF42 from Gaussian to Gaussian mixture densities that are better able to approximate arbitrary density functions. Our filter simultaneously holds approximate Gaussian mixture and Dirac mixture representations of the same density, what we call coupled discrete and continuous densities (CoDiCo). For conversion between discrete and continuous representation, we employ deterministic sampling and the expectation-maximization (EM) algorithm, which we extend to deal with weighted particles.

We present a novel sample reduction scheme for random variables belonging to the SE(2) group by means of Dirac mixture approximation. For this, dual quaternions are employed to represent uncertain planar transformations. The Cramér-von Mises distance is modified as a smooth metric to measure the statistical distance between Dirac mixtures on the manifold of planar dual quaternions. Samples of reduced size are then obtained by minimizing the probability divergence via Riemannian optimization while interpreting the correlation between rotation and translation. We further deploy the proposed scheme for nonparametric modeling of estimates for nonlinear SE(2) estimation. Simulations show superior tracking performance of the sample reduction-based filter compared with Monte Carlo-based as well as parametric model-based planar dual quaternion filters.

Estimating orientations of objects in Euclidean space is an omnipresent challenge in robotics and autonomous systems. A useful representation of orientations involves unit quaternions. While the space of all unit quaternions forms a three-dimensional unit hypersphere, inverting the sign of a quaternion does not change the orientation described by it. Therefore, all possible orientations can be described by considering only a hemisphere of the unit hypersphere. In this paper, we propose a grid filter for arbitrary-dimensional unit hyperhemispheres and apply it to an orientation estimation task and another evaluation scenario. Our approach outperforms previous approaches that consider densities on the entire hypersphere.

Networks consisting of several spatially distributed sensor nodes are useful in many applications. While distributed processing of information can be more robust and flexible than centralized filtering, it requires careful consideration of dependencies between local state estimates. This paper proposes an algorithm to keep track of dependencies in decentralized systems where no dedicated fusion center is present. Specifically, it addresses double counting of measurement information due to intermediate fusion results as well as correlations due to common process noise and common prior information. To limit the necessary amount of data, this paper introduces a method to bound correlations partially, leading to a more conservative fusion result while reducing the necessary amount of data. Simulation studies compare the performance and convergence rate of the proposed algorithm to other state-of-the-art methods.

With the advent of cheap sensor technology, multisensor data fusion algorithms have been becoming a key enabler for efficient in-network processing of sensor data. The information filter, in particular, has proven useful due to its simple additive structure of the measurement update equations. In order to exploit this structure for an efficient in-network processing, each node in the network is supposed to locally process and combine data from its neighboring nodes. The aspired in-network processing, at first glance, prohibits efficient privacy-preserving communication protocols, and encryption schemes that allow for algebraic manipulations are often computationally too expensive. Partially homomorphic encryption schemes constitute far more practical solutions but are restricted to a single algebraic operation on the corresponding ciphertexts. In this paper, an additive-homomorphic encryption scheme is used to derive a privacy-preserving implementation of the information filter where additive operations are sufficient to distribute the workload among the sensor nodes. However, the encryption scheme requires the floating-point data to be quantized, which impairs the estimation quality. The proposed filter and the implications of the necessary quantization are analyzed in a simulated multisensor tracking scenario.

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.

In this paper, we present a novel nonlinear progressive filtering approach for estimatingSE(2) states represented by unit dual quaternions. Unlike previously published approaches, the measurement model no longer needs to be assumed as identity. Our solution utilizes deterministic sampling on a Bingham-like probability distribution, which has been adapted to simultaneously model orientation and translation. During the measurement update step, the estimate gets progressively updated. Our approach inherently incorporates the nonlinear structure ofSE(2) and enables a flexible measurement update step. We also give an evaluation for planar rigid body motion estimation with a case study that is close to real-world scenarios.

In this paper, we present a novel approach to perform simultaneous localization and mapping (SLAM) for planar motions based on stochastic filtering with dual quaternion particles using low-cost range and gyro sensor data. Here, SE(2) states are represented by unit dual quaternions and further get stochastically modeled by a distribution from directional statistics such that particles can be generated by random sampling. To build the full SLAM system, a novel dual quaternion particle filter based on Rao-Blackwellization is proposed for the tracking block, which is further integrated with an occupancy grid mapping block. Unlike previously proposed filtering approaches, our method can perform tracking in the presence of multi-modal noise in unknown environments while giving reasonable mapping results. The approach is further evaluated using a walking robot with on-board ultrasonic sensors and an IMU sensor navigating in an unknown environment in both simulated and real-world scenarios.

In a surveillance environment with high clutter, finding the correct measurement to track associations becomes extremely important for efficient target tracking. This study offers a novel algorithm to retrodict the data association probabilities at any past time instant, when the batch set of measurements is kept in memory. For the retrodiction procedure, the batch association cost is first written explicitly as a binary integer optimization problem with a quadratic cost function and it is shown that the relaxed form of the problem is convex. From the relaxed problem, a lower bound for the optimal association cost is derived, and this lower bound is used as the data association probabilities pertaining to that selected time instant in the past. Due to its consideration of the batch set of data in a retrospective manner, we will call this algorithm as Retrodictive Probabilistic Data Association, RPDA. For simplification of the mathematical analysis, a single point target with no missing measurements, i.e. PD= 1, is taken into account.

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.

In this paper, we present a new approach for solving the SLAM problem using the Ensemble Kalman Filter (EnKF). In contrast to other Kalman filter based approaches, the EnKF uses a small set of ensemble members to represent the state, thus circumventing the computation of the large covariance matrix traditionally used with Kalman filters, making this approach a viable application in high-dimensional state spaces. Our approach adapts techniques from the geoscientific community such as localization to the SLAM problem domain as well as using the Optimal Subpattern Assignment (OSPA) metric for data association. We then compare the results of our algorithm with an extended Kalman filter (EKF) and FastSLAM, showing that our approach yields a more robust, accurate, and computationally less demanding solution than the EKF and similar results to FastSLAM.

Source localization is of great importance for wireless sensor network applications. Locating emission sources using received signal strength (RSS) measurements is investigated in this paper. As RSS localization is a non-convex optimization problem, it is difficult to achieve global optima. Many optimization methods have been proposed to relax it to a convex optimization problem. Unlike these methods, we propose a convex combination scheme. By introducing a highly accurate linear approximation of a logarithmic function, the source location is represented by a convex combination of a set of virtual anchors. Then the original problem is relaxed to be a convex optimization problem of finding the optimal combination coefficients, which can be solved efficiently using constrained least squares. To obtain the virtual nodes, we construct parallel lines and use their intersections to form a convex polygon, which covers the source location with certain probability. The vertices of the polygon are taken as the virtual nodes. Numerical examples verify the performance of the proposed method in both localization accuracy and computational efficiency.

Estimation of periodic quantities such as angles or phase values is a common problem. However, standard approaches, for example the Kalman filter and extensions thereof, have difficulties when estimating periodic quantities. To address this problem, circular filtering algorithms have been proposed but they are limited to just a single angle. In order to deal with multiple, possibly correlated angles, toroidal filtering algorithms are necessary. We have previously proposed a bivariate filtering algorithm on the torus [1] that is limited to identity system and measurement models. In this paper, we show how the algorithm can be extended to handle nonlinear system and measurement models. The novel approach relies on the bivariate wrapped normal distribution for representing the uncertainty and it makes use of a deterministic sampling scheme for the torus. We provide a thorough evaluation of the proposed method using simulations.

A lot of performance evaluation metrics exist for nonlinear filters. At present, the most commonly used one is a single and incomprehensive metric of performance. This metric can continuously and quantitatively describe the performance of the nonlinear filters. But in many cases, we need to rank the performance of the filters. It is in general very hard to rank the filters just using a single metric. First, the rankings using a single metric at different times may be different. Then how to get a unique rank for all times? A typical existing solution is to average the single metric over all times. But it is easy to be dominated just by very large values at just some times. Second, a single metric is usually incomprehensive in measuring performance. To make the ranking more comprehensive, multiple metrics are usually needed. But how to get a comprehensive unique rank from the ranks, possibly conflicting with each other? In this paper, we propose a framework to rank multiple nonlinear filters using ranking vectors and voting fusion based on a single metric or multiple metrics. Illustrative examples show that this framework is very effective.

Decentralized data fusion is a challenging task. Either it is too difficult to maintain and track the information required to perform fusion optimally, or too much information is discarded to obtain informative fusion results. A well-known solution is Covariance Intersection, which may provide too conservative fusion results. A less conservative alternative is discussed in this paper, and generalizations are proposed in order to apply it to a wide class of fusion problems. The Inverse Covariance Intersection algorithm is about finding the maximum possible common information shared by the estimates to be fused. A bound on the possibly shared common information is derived and removed from the fusion result in order to guarantee consistency. It is shown that the conditions required for consistency can be significantly relaxed, and also other causes of correlations, such as common process noise, can be treated.

For distributed estimation, algorithms have to be specifically crafted to minimize communication between the sensor nodes. As an adjusted version of the regular Kalman filter, the distributed Kalman filter (DKF) allows for deriving optimal results while not requiring regular communication. To achieve this, the DKF requires that each node has full knowledge about the system model and measurement models of all nodes. However, the DKF is not sufficient if the characteristics of the errors in the system and measurement models are not purely stochastic. In this paper, we present a distributed version of a combined stochastic and set-membership Kalman filter. The proposed filter optimizes the approximations of the set-membership uncertainties and can even yield better results than the regular centralized filter.

With the ubiquity of information distributed in networks, performing recursive Bayesian estimation using distributed calculations is becoming more and more important. There are a wide variety of algorithms catering to different applications and requiring different degrees of knowledge about the other nodes involved. One recently developed algorithm is the distributed Kalman filter (DKF), which assumes that all knowledge about the measurements, except the measurements themselves, are known to all nodes. If this condition is met, the DKF allows deriving the optimal estimate if all information is combined in one node at an arbitrary time step. In this paper, we present an information form of the distributed Kalman filter (IDKF) that allows the use of explicit system inputs at the individual nodes while still yielding the same results as a centralized Kalman filter.

We present a particle filter for multi-object tracking that is based on the ideas of the Approximate Bayesian Computation (ABC) paradigm. The main idea is to avoid the explicit computation of the likelihood function by means of simulation. For this purpose, a large amount of particles in the state space is simulated from the prior, transformed into measurement space, and then compared to the real measurement by using an appropriate distance function, i.e., the OSPA distance. By selecting the closest simulated measurements and their corresponding particles in state space, the posterior distribution is approximated. The algorithm is evaluated in a multi-object scenario with and without clutter and is compared to a global nearest neighbour Kalman filter.

In this paper, we present a novel approach to parametric density estimation from given samples. The samples are treated as a parametric density function by means of a Dirac mixture, which allows for applying analytic optimization techniques. The method is based on minimizing a distance measure between the integral of the approximation function and the empirical cumulative distribution function (EDF) of the given samples, where the EDF is represented by the integral of the Dirac mixture. Since this minimization problem cannot be solved directly in general, a progression technique is applied. Increased performance of the approach in comparison to iterative maximum likelihood approaches is shown in simulations.

In nonlinear Bayesian estimation it is generally inevitable to incorporate approximate descriptions of the exact estimation algorithm. There are two possible ways to involve approximations: Approximating the nonlinear stochastic system model or approximating the prior probability density function. The key idea of the introduced novel estimator called Hybrid Density Filter relies on approximating the nonlinear system, thus approximating conditional densities. These densities nonlinearly relate the current system state to the future system state at predictions or to potential measurements at measurement updates. A hybrid density consisting of both Dirac delta functions and Gaussian densities is used for an optimal approximation. This paper addresses the optimization problem for treating the conditional density approximation. Furthermore, efficient estimation algorithms are derived based upon the special structure of the hybrid density, which yield a Gaussian mixture representation of the system state's density.

This paper addresses the problem of model-based reconstruction and parameter identification of distributed phenomena characterized by partial differential equations. The novelty of the proposed method is the systematic approach and the integrated treatment of uncertainties, which naturally occur in the physical system and arise from noisy measurements. The main challenge of accurate reconstruction is that model parameters, i.e., diffusion coefficients, of the physical model are not known in advance and usually need to be identified. Generally, the problem of parameter identification leads to a nonlinear estimation problem. Hence, a novel efficient recursive procedure is employed. Unlike other estimators, the so-called Hybrid Density Filter not only assures accurate estimation results for nonlinear systems, but also offers an efficient processing. By this means it is possible to reconstruct and identify distributed phenomena monitored by autonomous wireless sensor networks. The performance of the proposed estimation method is demonstrated by means of simulations.

Recursive calculation of the probability density function characterizing the state estimate of a nonlinear stochastic dynamic system in general cannot be performed exactly, since the type of the density changes with every processing step and the complexity increases. Hence, an approximation of the true density is required. Instead of using a single complicated approximating density, this paper is concerned with bounding the true density from below and from above by means of two simple densities. This provides a kind of guaranteed estimator with respect to the underlying true density, which requires a mechanism for ordering densities. Here, a partial ordering with respect to the cumulative distributions is employed. Based on this partial ordering, a modified Bayesian filter step is proposed, which recursively propagates lower and upper density bounds. A specific implementation for piecewise linear densities with finite support is used for demonstrating the performance of the new approach in simulations

In this paper we attempt to lay the foundation for a novel filtering technique for the fusion of two random vectors with imprecisely known stochastic dependency. This problem mainly occurs in decentralized estimation, e.g., of a distributed phenomenon, where the stochastic dependencies between the individual states are not stored. Thus, we derive parameterized joint densities with both Gaussian marginals and Gaussian mixture marginals. These parameterized joint densities contain all information about the stochastic dependencies between their marginal densities in terms of a parameter vector xi, which can be regarded as a generalized correlation parameter. Unlike the classical correlation coefficient, this parameter is a sufficient measure for the stochastic dependency even characterized by more complex density functions such as Gaussian mixtures. Once this structure and the bounds of these parameters are known, bounding densities containing all possible density functions could be found

Undirected cycles in Bayesian networks are often treated by using clustering methods. This results in networks with nodes characterized by joint probability densities instead of marginal densities. An efficient representation of these hybrid joint densities is essential especially in nonlinear hybrid net works containing continuous as well as discrete variables. In this article we present a unified representation of continuous, discrete, and hybrid joint densities. This representation is based on Gaussian and Dirac mixtures and allows for analytic evaluation of arbitrary hybrid networks without loosing structural in formation, even for networks containing clusters. Furthermore we derive update formulae for marginal and joint densities from a system theoretic point of view by treating a Bayesian network as a system of cascaded subsystems. Together with the presented mixture representation of densities this yields an exact analytic updating scheme