We propose a Bayesian framework for the received-signal-strength-based cooperative localization problem with unknown path loss exponent. Our purpose is to infer the marginal posterior of each unknown parameter: the position or the path loss exponent. This probabilistic inference problem is solved using message passing algorithms that update messages and beliefs iteratively. For numerical tractability, we combine the variable discretization and Monte-Carlo-based numerical approximation schemes. To further improve computational efficiency, we develop an auxiliary importance sampler that updates the beliefs with the help of an auxiliary variable. An important ingredient of the proposed auxiliary importance sampler is the ability to sample from a normalized likelihood function. To this end, we develop a stochastic sampling strategy that mathematically interprets and corrects an existing heuristic strategy. The proposed message passing algorithms are analyzed systematically in terms of computational complexity, demonstrating the computational efficiency of the proposed auxiliary importance sampler. Various simulations are conducted to validate the overall good performance of the proposed algorithms.

Posterior Cramér-Rao bounds (CRBs) are derived for the estimation performance of three Gaussian process-based state-space models. The parametric CRB is derived for the case with a parametric state transition and a Gaussian process-based measurement model. We illustrate the theory with a target tracking example and derive both parametric and posterior filtering CRBs for this specific application. Finally, the theory is illustrated with a positioning problem, with experimental data from an office environment where the obtained estimation performance is compared to the derived CRBs.

The ensemble Kalman filter (EnKF) is a Monte Carlo-based implementation of the Kalman filter (KF) for extremely high-dimensional, possibly nonlinear, and non-Gaussian state estimation problems. Its ability to handle state dimensions in the order of millions has made the EnKF a popular algorithm in different geoscientific disciplines. Despite a similarly vital need for scalable algorithms in signal processing, e.g., to make sense of the ever increasing amount of sensor data, the EnKF is hardly discussed in our field. This self-contained review is aimed at signal processing researchers and provides all the knowledge to get started with the EnKF. The algorithm is derived in a KF framework, without the often encountered geoscientific terminology. Algorithmic challenges and required extensions of the EnKF are provided, as well as relations to sigma point KF and particle filters. The relevant EnKF literature is summarized in an extensive survey and unique simulation examples, including popular benchmark problems, complement the theory with practical insights. The signal processing perspective highlights new directions of research and facilitates the exchange of potentially beneficial ideas, both for the EnKF and high-dimensional nonlinear and non-Gaussian filtering in general.

We study cooperative sensor network localization in a realistic scenario where 1) the underlying measurement errors more probably follow a non-Gaussian distribution; 2) the measurement error distribution is unknown without conducting massive offline calibrations; and 3) non-line-of-sight identification is not performed due to the complexity constraint and/or storage limitation. The underlying measurement error distribution is approximated parametrically by a Gaussian mixture with finite number of components, and the expectation-conditional maximization (ECM) criterion is adopted to approximate the maximum-likelihood estimator of the unknown sensor positions and an extra set of Gaussian mixture model parameters. The resulting centralized ECM algorithms lead to easier inference tasks and meanwhile retain several convergence properties with a proof of the "space filling" condition. To meet the scalability requirement, we further develop two distributed ECM algorithms where an average consensus algorithm plays an important role for updating the Gaussian mixture model parameters locally. The proposed algorithms are analyzed systematically in terms of computational complexity and communication overhead. Various computer based tests are also conducted with both simulation and experimental data. The results pin down that the proposed distributed algorithms can provide overall good performance for the assumed scenario even under model mismatch, while the existing competing algorithms either cannot work without the prior knowledge of the measurement error statistics or merely provide degraded localization performance when the measurement error is clearly non-Gaussian.

A marginal version of the enumeration Bayesian Cramer-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.

The dependence of radio signal propagation on the environment is  well known, and both statistical and deterministic methods have been presented in the literature. Such methods are either based on randomised or actual reflectors of radio signals. In this work, we instead aim at estimating the location of the reflectors based on geo-localised radio channel impulse reponse measurements and using methods from synthetic aperture radar (SAR). Radio channel data measurements from 3GPP E-UTRAN have been used to verify the usefulness of the proposed approach. The obtained images show that  the estimated reflectors are well correlated with the aerial map of the environment. Also, which part of the trajectory contributed to different reflectors have been estimated with promising results.

We consider time-of-arrival based robust geolocation in harsh line-of-sight/non-line-of-sight environments. Herein, we assume the probability density function (PDF) of the measurement error to be completely unknown and develop an iterative algorithm for robust position estimation. The iterative algorithm alternates between a PDF estimation step, which approximates the exact measurement error PDF (albeit unknown) under the current parameter estimate via adaptive kernel density estimation, and a parameter estimation step, which resolves a position estimate from the approximate log-likelihood function via a quasi-Newton method. Unless the convergence condition is satisfied, the resolved position estimate is then used to refine the PDF estimation in the next iteration. We also present the best achievable geolocation accuracy in terms of the Cramér-Rao lower bound. Various simulations have been conducted in both real-world and simulated scenarios. When the number of received range measurements is large, the new proposed position estimator attains the performance of the maximum likelihood estimator (MLE). When the number of range measurements is small, it deviates from the MLE, but still outperforms several salient robust estimators in terms of geolocation accuracy, which comes at the cost of higher computational complexity.

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 Cramer-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 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.

In this paper, marginal versions of the Bayesian Bhattacharyya lower bound (BBLB), which is a tighter alternative to the classical Bayesian Cramer-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.

In this paper, the derivation of the Bayesian Bhattacharyya bound for discrete-time filtering as proposed ina paper by Reece and Nicholson is revisited. It turns out that the results presented in the aforementioned contribution are incorrect, as some expectations appearing in the information matrix recursions are missing. This paper gives a generalized derivation of the N-th order Bayesian Bhattacharyya bound and presents corrected expressions for the case N = 2. A nonlinear toy example is used to illustrate the results

Classification of motion mode (walking, running, standing still) and device mode (hand-held, in pocket, in back-pack) is an enabler in personal navigation systems for the purpose of saving energy and design parameter settings and also for its own sake. Our main contribution is to publish one of the most extensive datasets for this problem, including inertial data from eight users, each one performing three pre-defined trajectories carrying four smartphones and seventeen inertial measurement units on the body. All kind of metadata is available such as the ground truth of all modes and position. A second contribution is the first study on a joint classifier of motion and device mode, respectively, where preliminary but promising results are presented.

This paper deals with state inference and parameter identification in Jump Markov Non-Linear System. The state inference problem is solved efficiently using a recently proposed Rao-Blackwellized Particle Filter, where the discrete state is integrated out analytically. Within the RBPF framework, Recursive Maximum Likelihood parameter identification is performed using gradient ascent algorithms. The proposed learning method has the advantage over (online) Expectation Maximization methods, that it can be easily applied to cases where the probability density functions defining the Jump Markov Non-Linear System are not members of the exponential family. Two benchmark problems illustrate the parameter identification performance.

Assessing the fundamental performance limitationsin Bayesian filtering can be carried out using the parametricCram´er-Rao bound (CRB). The parametric CRB puts a lowerbound on mean square error (MSE) matrix conditioned on aspecific state trajectory realization. In this work, we derive theparametric CRB for state-space models, where the measurementequation is modeled by a Gaussian process regression.These models appear, for instance in proximity report-basedpositioning, where proximity reports are obtained by hardthresholding of received signal strength (RSS) measurements, thatare modeled through Gaussian process regression. The proposedparametric CRB is evaluated on selected state trajectories andfurther compared with the positioning performance obtained bythe particle filter. The results corroborate that the positioningaccuracy achieved in this framework is close to the parametricCRB.

In this paper, recent results on the evaluation of the Bayesian Cramer-Rao bound for jump Markov systems are presented. In particular, previous work is extended to jump Markov systems where the discrete mode variable enters into both the process and measurement equation, as well as where it enters exclusively into the measurement equation. Recursive approximations are derived with finite memory requirements as well as algorithms for checking the validity of these approximations are established. The tightness of the bound and the validity of its approximation is investigated on a couple of examples.

We consider personal navigation systems in devices equipped with inertial sensors and GPS, where we propose an improved Pedestrian Dead Reckoning (PDR) algorithm that learns gait parameters in time intervals when position estimates are available, for instance from GPS or an indoor positioning system (IPS). A novel filtering approach is proposed that is able to learn internal gait parameters in the PDR algorithm, such as the step length and the step detection threshold. Our approach is based on a multi-rate Kalman filter bank that estimates the gait parameters when position measurements are available, which improves PDR in time intervals when the position is not available, for instance when passing from outdoor to indoor environments where IPS is not available. The effectiveness of the new approach is illustrated on several real world experiments. 

Parametric Cramer-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.

The well-known cooperative localization algorithm, ‘sum-product algorithm over a wireless network’ (SPAWN) hastwo major shortcomings, a relatively high computationalcomplexity and a large communication load. Using the Gaus-sian mixture model with a model selection criterion and thesigma-point (SP) methods, we propose the SPAWN-SP toovercome these problems. The SPAWN-SP easily accommo-dates different localization scenarios due to its high flexibilityin message representation. Furthermore, harsh LOS/NLOSenvironments are considered for the evaluation of coopera-tive localization algorithms. Our simulation results indicatethat the proposed SPAWN-SP demonstrates high localizationaccuracy in different localization scenarios, thanks to its highflexibility in message representation.

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.

The Ensemble Kalman filter (EnKF) is a standard algorithm in oceanography and meteorology, where it has got thousands of citations. It is in these communities appreciated since it scales much better with state dimension n than the standard Kalman filter (KF). In short, the EnKF propagates ensembles with N state realizations instead of mean values and covariance matrices and thereby avoids the computational and storage burden of working on n×n matrices. Perhaps surprising, very little attention has been devoted to the EnKF in the signal processing community. In an attempt to change this, we present the EnKF in a Kalman filtering context. Furthermore, its application to nonlinear problems is compared to sigma point Kalman filters and the particle filter, so as to reveal new insights and improvements for high-dimensional filtering algorithms in general. A simulation example shows the EnKF performance in a space debris tracking application.

In this paper, we aim to relate different Bayesian Cramér-Rao bounds which appear in the discrete-time nonlinear filtering literature in a single framework. A comparative theoretical analysis of the bounds is provided in order to relate their tightness. The results can be used to provide a lower bound on the mean square error in nonlinear filtering. The findings are illustrated and verified by numerical experiments where the tightness of the bounds are compared.

We study received signal strength-based cooperative localization in wireless sensor networks. We assume that the measurement noise fits a contaminated Gaussian model so as to take into account some outlier conditions. In addition, some environment-dependent parameters are assumed to be unknown. We propose an expectation-maximization based algorithm for robust centralized network localization without offline training. As benchmark for comparison, we express the best achievable localization accuracy in terms of the Cramér-Rao bound. Experimental results demonstrate the advantages of the proposed algorithm as compared to some representative algorithms.

The Bayesian Cramer Rao Bound (BCRB) is de­rived for nonlinear state space models with dependent process and measurement noise processes. It generalizes the previously BCRB for the case of dependent noise. Two different dependence structures appearing in literature are considered, leading to two different recursions for BCRB. The special cases of Gaussian noise, and linear models are presented separately. Simulations demonstrate that correct treatment of dependencies is important for both filtering algorithms and the BCRB.

The Expectation-Maximization (EM) algorithm in combination with particle filters is a powerful tool that can solve very complex problems, such as parameter estimation in general nonlinear non-Gaussian state space models. We here apply the recently proposed online EM algorithm to parameter estimation in jump Markov models, that contain both continuous and discrete states. In particular, we focus on estimating process and measurement noise distributions being modeled as mixtures of members from the exponential family.

This report contains supplementary material for the paper [1], and gives detailed proofs of all lemmas and theorems that could not be included into the paper due to space limitations. The notation is adapted from the paper.

[1] C. Fritsche, U. Orguner, and F. Gustafsson, “Bobrovsky-Zakai bound for filtering, prediction and smoothing ofnonlinear dynamic systems,” in International Conference on Information Fusion (FUSION), Cambridge, UK, Jul.2018, pp. 1–8.

In this report, the derivation of the Bayesian Bhattacharyya bound for discrete-time filtering as proposed by Reece and Nicholson [1] is revisited. It turns out that the general results presented in [1] are incorrect, as some expectations appearing in the information matrix recursions are missing. This report presents the corrected results and it is argued that the missing expectations are only zero in a number of special cases. A nonlinear toy example is used to illustrate when this is not the case.

This report contains supplementary material for the paper, and gives detailed proofs of all lemmas and propositions that could not be included into the paper due to space limitations. The notation is adaptedfrom the paper.

In this paper we are concerned with nonlinear systems subject to a conditionally linear, Gaussian sub-structure. This structure is often exploited in high-dimensional state estimation problems using the marginalized (aka Rao-Blackwellized) particle filter. The main contribution in the present work is to show how an efficient filter can be derived by exploiting this structure within the auxiliary particle filter. Based on a multisensor aircraft tracking example, the superior performance of the proposed filter over conventional particle filtering approaches is demonstrated.