Posterior Cramer-Rao Lower Bound Research Articles

SENSOR SELECTION FOR TARGET TRACKING IN SENSOR NETWORKS

This paper addresses the sensor selection problem which is a very important issue where many sensors are available to track a target. In this problem, we need to select an appropriate group of sensors at each time to perform tracking in a wireless sensor network (WSN). As the theoretical tracking performance is bounded by posterior Cramer-Rao lower bound (PCRLB), it is used as a criterion to select sensors. Based on the PCRLB, sensor selection algorithms with and without sensing range constraint are developed. Without sensing range limit, exhaustive enumeration is flrst adopted to search all possible combinations for sensor selection. To reduce complexity of enumeration, second, we restrict the selected sensors to be within a flxed area in the WSN. With sensing range constraint, a circle will be drawn with the help of communication range for sensor selection. In a similar manner, two approaches, namely, selecting all sensors inside the circle or using enumeration to select sensors within the circle are presented. The efiectiveness of the proposed methods is validated by computer simulation results in target tracking for WSNs.

Posterior Cramer-Rao lower bounds for multitarget bearings-only tracking

Usually, only the Cramer-Rao lower bound (CRLB) of single target is taken into consideration in the state estimate of passive tracking systems. As for the case of multitarget, there are few works done due to its complexity. The recursion formula of the posterior Cramer-Rao lower bound (PCRLB) in multitarget bearings-only tracking with the three kinds of data association is presented. Meanwhile, computer simulation is carried out for data association. The final result shows that the accuracy probability of data association has an important impact on the PCRLB.

Target Tracking by Particle Filtering in Binary Sensor Networks

We present particle filtering algorithms for tracking a single target using data from binary sensors. The sensors transmit signals that identify them to a central unit if the target is in their neighborhood; otherwise they do not transmit anything. The central unit uses a model for the target movement in the sensor field and estimates the target's trajectory, velocity, and power using the received data. We propose and implement the tracking by employing auxiliary particle filtering and cost-reference particle filtering. Unlike auxiliary particle filtering, cost-reference particle filtering does not rely on any probabilistic assumptions about the dynamic system. In the paper, we also extend the method to include estimation of constant parameters, and we derive the posterior Cramer-Rao bounds (PCRBs) for the states. We show the performances of the proposed methods by extensive computer simulations and compare them to the derived bounds.

Geoacoustic Environment Tracking Using Kalman and Particle Filters

This paper addresses the problem of tracking the geoacoustic environmental parameters such as the sound speeds in water, sediment and the bottom, sediment attenuation, density and thickness. The tracking is based on using acoustic measurements within an extended Kalman (EKF), unscented Kalman (UKF), and particle filter (PF) framework. The acoustic field is computed using a normal mode code which introduced a varying degree of nonlinearity depending on the environmental parameter of interest. Posterior Cramer‐Rao lower bounds (PCRLB) are used to compute the tracking performances of the filters, including the filter efficiencies, divergence statistics, and computational complexities. The results showed that some of the parameters such as the water column parameters can be tracked by Kalman filters, however, the tracking performance of the Kalman filters was limited by the highly nonlinear relation between the sediment/bottom parameters and the acoustic field and non‐Gaussian densities of these parameters. Particle filters proved to be very promising in tracking sediment layer parameters, even in the abruptly changing environments.

Performance measure for Markovian switching systems using best-fitting Gaussian distributions

This paper considers the theoretical posterior Cramer-Rao lower bound (PCRLB) for the case of tracking a manoeuvring target with Markovian switching dynamics. In a recent article [2] it was proposed to calculate the PCRLB conditional on the manoeuvre sequence and then determine the bound as a weighted average, giving an unconditional PCRLB. However, we demonstrate that this approach can produce an overly optimistic lower bound, because the sequence of manoeuvres is implicitly assumed known. Motivated by this, we develop a general approach and derive a closed-form estimate of the PCRLB in the case of Markovian switching systems. The basis of the approach is to, at each time step, replace the multi-modal prior target probability density function (pdf) with a best-fitting Gaussian (BFG) approximation. We present a recursive formula for calculating the mean and covariance of this Gaussian distribution, and demonstrate how the covariance increases as a result of the potential manoeuvres. We are then able to calculate the PCRLB for this BFG model using an existing Riccati-like recursion. Because of the BFG approximation, we are no longer guaranteed a bound and so we refer to our estimate as an "error performance measure" rather than a bound. The presented approach is applied both to filtering and smoothing cases. The simulation results indicate a very close agreement between the proposed performance measure and the error performance of an interacting multiple model estimator.

Posterior Cramer-Rao lower bounds for bearing-only tracking

In the state estimation of passive tracking systems, the traditional approximate expression for the Cramer-Rao lower bound (CRLB) does not take two factors into consideration, that is, measurement origin uncertainty and state noise. Such treatment is only valid in ideal situation but it is not feasible in actual situation. In this article, considering the two factors, the posterior Cramer-Rao lower bound (PCRLB) recursion expression for the error of bearing-only tracking is derived. Then, further analysis is carried out on the PCRLB. According to the final result, there are four main parameters that play a role in the performance of the PCRLB, that is, measurement noise, detection probability, state noise and clutter density, amongst which the first two have greater impact on the performance of the PCRLB than the others.

Posterior Cramér-Rao bounds analysis for passive target tracking

For the problem of deterministic parameter estimate, the theoretical lower bound of estimate error is the Cramer-Rao bound; while for random parameter, the lower bound of estimate error is generally termed by Posterior Cramer-Rao Bound (PCRB). Under the background of passive tracking where the target’s state can be seen as a time-varying random parameter, PCRB of the state estimate error is analyzed in this paper, and the relation between PCRB and varied condition is also fully investigated using different simulation examples. The presented analytical method provides a theoretical base for performance assessment of all kinds of suboptimal estimate algorithms used in practice.

Mobility Tracking in Cellular Networks Using Particle Filtering

Mobility tracking based on data from wireless cellular networks is a key challenge that has been recently investigated both from a theoretical and practical point of view. This paper proposes Monte Carlo techniques for mobility tracking in wireless communication networks by means of received signal strength indications. These techniques allow for accurate estimation of mobile station's (MS) position and speed. The command process of the MS is represented by a first-order Markov model which can take values from a finite set of acceleration levels. The wide range of acceleration changes is covered by a set of preliminary determined acceleration values. A particle filter and a Rao-Blackwellised particle filter are proposed and their performance is evaluated both over synthetic and real data. A comparison with an extended Kalman filter (EKF) is performed with respect to accuracy and computational complexity. With a small number of particles the RBPF gives more accurate results than the PF and the EKF. A posterior Cramer Rao lower bound (PCRLB) is calculated and it is compared with the filters' root- mean-square error performance.

Expectation Maximization Based GPS/INS Integration for Land-Vehicle Navigation

Integration of global positioning system (GPS) and inertial navigation system (INS) provides continuous positioning information of high accuracy due to the synergistic effect of both systems. While a Kalman filter is usually employed to fuse the GPS and INS measurements, this approach requires a priori knowledge on the stochastic and deterministic parameters of both systems. In practice, these unknown parameters are often determined by trial and error. We propose an expectation-maximization (EM) method here to estimate these unknowns in a maximum likelihood (ML) framework. In particular, we employ a delta operator model to approximate the continuous-time system instead of the conventional shift operator model. The proposed method achieves simultaneous positioning and unknown parameter estimation. To assess the performance of the proposed method, we derive the posterior Cramer-Rao bound (PCRB) of our model and compare the performance with adaptive Kalman filtering technique. Both real and simulated data arc used to validate the effectiveness of the proposed EM-based method.

PCRLB-based multisensor array management for multitarget tracking

In this paper we consider the general problem of managing an array of sensors in order to track multiple targets in the presence of clutter. There are three complicating factors. The first is that because of physical limitations (e.g., communication bandwidth) only a small subset of the available sensors can be utilized at any one time. The second complication is that the associations of measurements to targets/clutter are unknown. The third complication is that the total number of targets in the surveillance region is unknown and possibly time varying. It are these second and third factors that extend previous work [ Tharmarasa, R., Kirubarajan, T., and Hernandez, M. L. Large-scale optimal sensor array management for multitarget tracking. IEEE Transactions on Systems, Man, and Cybernetics, to be published.]. Hence sensors must be utilized in an efficient manner to alleviate association ambiguities and to allow accurate estimation of the states of a varying number of targets. We pose the problem as a bi-criterion optimization with the two objectives of (1) controlling the posterior Cramer-Rao lower bound ((PCRLB) which provides a measure of the optimal achievable accuracy of target state estimation), and (2) maximizing the probability of detecting new targets. Only recently have expressions for multitarget PCRLBs been determined [Hue, C, Le Cadre, J.-P., and Perez, P]. Performance analysis of two sequential Monte Carlo methods and posterior Cramer-Rao bounds for multitarget tracking. In Proceedings of the 5th International Conference on Information Fusion, vol. 1, Annapolis, MD, July 2002, 464-473.], and the necessary simulation techniques are computationally expensive. However, in this paper we show the existence of a multitarget information reduction matrix (IRM) which can be calculated off-line in most cases. Additionally, we propose some approximations that further reduce the computational load. We present solution methodologies that, in simulations, are shown to determine efficient utilization strategies for the available sensor resources, with some sensors selected to track existing targets and others given the primary task of surveillance in order to identify new threats.

Closed-form posterior Cramer-Rao bounds for bearings-only tracking

We address the classical bearings-only tracking problem (BOT) for a single object, which belongs to the general class of nonlinear filtering problems. Recently, algorithms based on sequential Monte-Carlo methods (particle filtering) have been proposed. As far as performance analysis is concerned, the posterior Cramer-Rao bound (PCRB) provides a lower bound on the mean square error. Classically, under a technical assumption named "asymptotic unbiasedness assumption", the PCRB is given by the inverse Fisher information matrix (FIM). The latter is computed using Tichavsky's recursive formula via Monte-Carlo methods. Two major problems are studied here. First, we show that the asymptotic unbiasedness assumption can be replaced by an assumption which is more meaningful. Second, an exact algorithm to compute the PCRB is derived via Tichavsky's recursive formula without using Monte-Carlo methods. This result is based on a new coordinate system named logarithmic polar coordinate (LPC) system. Simulation results illustrate that PCRB can now be computed accurately and quickly, making it suitable for sensor management applications

Multisensor deployment using PCRLBS, incorporating sensor deployment and motion uncertainties

Recently, a general framework for sensor resource deployment (Hernandez, et. al. 2004) has been shown to allow efficient and effective utilization of a multisensor system. The basis of this technique is to use the posterior Cramer-Rao lower bound (PCRLB) to quantify and control the optimal achievable accuracy of target state estimation. In the original formulation (Hernandez, et. al. 2004) it was assumed that the sensor locations were known without error. In the current paper, the authors extend this framework by addressing the issues of imperfect sensor placement and uncertain sensor movement (e.g., sensor drift). The crucial consideration is then how these two forms of uncertainty are factored into the sensor management strategy. If unaccounted for, these uncertainties will render the output of the resource manager inaccurate and overoptimistic. The authors adjust the PCRLB to account for sensor location uncertainty, and we also allow for measurement origin uncertainty due to missed detections and false alarms. The work is motivated by the problem of tracking a submarine by adaptively deploying sonobuoys from a helicopter. Simulation results are presented to show the advantages of accounting for sensor location uncertainty within this focal domain of antisubmarine warfare. The authors note that the generic nature of the technique allows it to be utilized within other problem domains, including tracking ground-based targets using unattended ground sensors (UGSs) or unmanned aerial vehicles (UAVs)

PCRLB for tracking in cluttered environments: measurement sequence conditioning approach

We consider the problem of calculating the posterior Cramer-Rao lower bound (PCRLB) for tracking in cluttered domains in which there can be both missed detections and false alarms. We present a novel formulation of the PCRLB in which we initially determine a bound conditional on the sequence of measurements available. We then create an unconditional bound as a weighted average of these conditional PCRLBs. This new bound is proven to be less optimistic than the standard formulation of the PCRLB for cluttered environments recently developed (Zhang and Willett, 2001 and Hernandez et al., 2002) and will therefore better predict optimal estimator performance. At each stage, the conditional PCRLB must evaluate the effect of the uncertain measurements, and we extend previous work (Hernandez et al., 2002) to show that the measurement origin uncertainty manifests itself as a single information reduction factor (IRF) that is dependent on the number of measurements available. We also present some useful approximations when the false alarm rate is low. Simulations then consider the problems of 1) determining the CRLB for the point of impact of a ballistic missile, and 2) determining the PCRLB for tracking a nearly constant-velocity (NCV) target in a high clutter environment. In each case, we compare the new bound with the standard approach, and as expected the new CRLB/PCRLB can be seen to be less optimistic. Moreover, in case 1) we compare the new CRLB with a heuristic bound specially constructed for this problem, and a maximum likelihood estimator (MLC). The new bound both compares favorably with the heuristic bound, and shows close agreement with the performance of the MLE. The new bound is therefore an accurate predictor of filter performance in this case. In example 2) we demonstrate some interesting features of the new theory. Of particular interest we determine both precisely when the new bound will be significantly greater than the standard bound and when the two bounds will be virtually identical. This is useful in determining when the new approach, with its greater computational burden, should be preferred to the established approach. We conclude that the novel PCRLB formulation introduced herein represents an exciting development in the determination of RMSE performance bounds in cluttered environments.

Posterior Cramer-Rao bounds for multi-target tracking

This study is concerned with multi-target tracking (MTT). The Cramer-Rao lower bound (CRB) is the basic tool for investigating estimation performance. Though basically defined for estimation of deterministic parameters, it has been extended to stochastic ones in a Bayesian setting. In the target tracking area, we have thus to deal with the estimation of the whole trajectory, itself described by a Markovian model. This leads up to the recursive formulation of the posterior CRB (PCRB). The aim of the work presented here is to extend this calculation of the PCRB to MTT under various assumptions.

Improved sequential monte carlo filtering for ballistic target tracking

We present in this correspondence an improved sequential Monte Carlo (SMC) filter for ballistic target tracking with random, time-varying ballistic coefficient. The proposed tracker is a sampling/importance resampling (SIR) filter that uses an optimized importance function to combat particle degeneracy, and also incorporates an additional measurement-driven Markov chain Monte Carlo (MCMC) move step to prevent particle impoverishment. Simulation results show that, using significantly fewer particles than previously reported in the literature for similar tracking problems, the root mean-square error (RMSE) curves for the proposed optimized SIR filter approach the square root of the ideal posterior Cramer-Rao lower bound (PCRLB).

An Expectation–Maximization-Based Interacting Multiple Model Approach for Cooperative Driving Systems

In this paper, we present a novel combined sensor registration and fusion approach for cooperative driving in intelligent transportation systems (ITSs). A realistic augmented registration and fusion-state space model in three dimensions is first developed for dissimilar sensors. In order to have unbiased sensor registration parameter estimates, the expectation-maximization (EM) algorithm is incorporated with the extended Kalman filter (EKF) to give simultaneous state and parameter estimates. Furthermore, the interacting multiple model (IMM) filter is introduced here for collaborative driving in order to deal with the jumping model problem occurred in different vehicles driving status. To evaluate the registration and fusion performance, a new recursive relationship is derived theoretically for computing the posterior Cramer-Rao bound (PCRB). It is shown by simulation that the proposed EM-IMM-EKF method has a more robust estimation performance than the conventional approach. The performance is furthermore verified by comparing the mean square error with the PCRB.

Space-time registration of radar and ESM using unscented Kalman filter

Space and time alignments are the prerequisites for the successful fusion of multiple sensors. A space-time registration model is proposed to estimate the system biases and to perform time synchronization together for mobile radar and electronic support measure (ESM) systems. A space-time registration model for radar and ESM is first developed, and an unscented Kalman filter (UKF) is proposed to estimate the space-time biases and target states simultaneously. The posterior Cramer-Rao bounds (PCRBs) are derived for the proposed UKF registration algorithm for ESM detection probability less than or equal to one. Theoretical analyses are performed to evaluate the accuracy and robustness of the proposed method. Computer simulations show that the UKF registration algorithm is indeed effective and robust for different radar and ESM tracking scenarios.

Multisensor resource deployment using posterior Cramer-Rao bounds

The development of a general framework for the systematic management of multiple sensors in target tracking in the presence of clutter is described. The basis of the technique is to quantify, and subsequently control, the accuracy of target state estimation. The posterior Cramer-Rao lower bound (PCRLB) provides the means of achieving this aim by enabling us to determine a bound on the performance of all unbiased estimators of the unknown target state. The general approach is then to use optimization techniques to control the measurement process in order to achieve accurate target state estimation. We are concerned primarily with the deployment and utilization of limited sensor resources. We also allow for measurement origin uncertainty, with sensor measurements either target-generated or false alarms. An example in which the aim is to track a submarine by deploying a series of constant false-alarm rate passive sonobuoys is presented. We show that by making some standard assumptions, the effect of the measurement origin uncertainty can be expressed as a state-dependent information reduction factor which can be calculated off-line. This enables the Fisher information matrix (FIM) to be calculated quickly, allowing Cramer-Rao bounds to be utilized for real-time, dynamic sensor management. The sensor management framework is shown to determine deployment strategies that enable the target to be accurately localized, and at the same time efficiently utilize the limited sensor resources.

Optimal estimation and Cramer-Rao bounds for partial non-Gaussian state space models

Partial non-Gaussian state-space models include many models of interest while keeping a convenient analytical structure. In this paper, two problems related to partial non-Gaussian models are addressed. First, we present an efficient sequential Monte Carlo method to perform Bayesian inference. Second, we derive simple recursions to compute posterior Cramer-Rao bounds (PCRB). An application to jump Markov linear systems (JMLS) is given.

Fast computation of the exact FIM for deterministic signals in colored noise

A fast algorithm for computing the exact finite-sample Fisher information matrix (FIM) for the parameters of a deterministic signal observed in Gaussian AR noise is derived. In the case of a harmonic signal with random phases, closed-form expressions for the finite-sample posterior Cramer-Rao bound (PCRB) are established. It is shown that the fast algorithm is also useful for computing the conditional CRB when the additive noise is a non-Gaussian AR process. It is seen that the asymptotic CRB may deviate significantly from the exact CRB even when the data length is moderate, whereas the PCRB, which is easy to compute, provides a better approximation. Theoretical results are illustrated via numerical evaluation of the different lower bounds.