Posterior Cramer-Rao Lower Bound Research Articles

Node Topology Effect on Target Tracking Based on UWSNs Using Quantized Measurements.

On one hand, due to the energy and bandwidth constraint of underwater wireless sensor networks (UWSNs), local data quantization/compression is not only a necessity, but also an integral part of the design of UWSNs; on the other hand, since underwater nodes provide measurements for target tracking based on UWSNs, node topology, which is made up of the underwater nodes, may affect the performance of target tracking. This paper studies the effect of node topology on the target tracking in UWSNs using quantized measurements. Firstly, by using the knowledge of geometry, the effects of four typical topologies on target tracking using quantized measurements are analyzed qualitatively. The four typical topologies include two nodes are close to each other, three nodes are close to each other, three nodes are co-linear, and three nodes form a regular triangle. Secondly, under the condition of quantized measurements, the relationship between the posterior Cramer-Rao lower bound (PCRLB) and node's position is derived to evaluate the arbitrary topology. Thirdly, our target tracking scheme consisting of the optimal topology selection scheme by minimizing PCRLB, the optimal fusion center selection scheme by minimizing energy consumption, and the multisensor particle filter with quantized measurements is designed. Last, simulation results show the effectiveness of the proposed scheme.

Consensus-based distributed dynamic sensor selection in decentralised sensor networks using the posterior Cramér–Rao lower bound

The paper considers the problem of dynamic sensor scheduling for non-linear tracking problems in distributed sensor/agent networks (AN/SN), where channel limitations restrict how many sensors can simultaneously participate in the estimation mechanism. Commonly referred to as sensor selection, the basic objective is to select a subset of sensors from available sensors that minimises the estimation error. The posterior Cramer–Rao lower bound (PCRLB) has recently been proposed as an effective sensor selection criteria. Nevertheless, existing PCRLB-based selectors are limited to centralised and hierarchical networks, and when extended to distributed architectures use approximate expressions. First, the non-conditional distributed PCRLB (dPCRLB) that considers observations and state variables to be random is used to derive the non-conditional sensor-selector. The dPCRLB expressions we derive are optimal. Unlike the non-conditional PCRLB, its conditional counterpart is a function of the past history of observations and is a more accurate representation of the system׳s performance. Second, the paper generalises the non-conditional dPCRLB sensor-selector to its conditional dPCRLB version. Both dPCRLB sensor-selectors use raw observations adding significant communication overhead. Third, the paper extends the conditional dPCRLB framework to quantised observations and develops the quantised version of the conditional dPCRLB sensor-selector. Numerical simulations verify the efficiency of our distributed dynamic sensor-selectors.

Full- and reduced-order distributed Bayesian estimation analytical performance bounds

Motivated by the resource management problem in nonlinear multisensor tracking networks, the paper derives online, distributed estimation algorithms for computing the posterior Cramer-Rao lower bound (PCRLB) for full-order and reduced-order distributed Bayesian estimators without requiring a fusion center and with nodal communications limited to local neighborhoods. For both cases, Riccati-type recursions are derived that sequentially determine the global Fisher information matrix (FIM) from localized FIMs of the distributed estimators. We use particle filter realizations for these bounds and quantify their performance for data fusion problems through Monte Carlo simulations.

A Particle Filter Approach to Approximate Posterior Cramer-Rao Lower Bound: The Case of Hidden States

A Particle Filter Approach to Approximate Posterior Cramer-Rao Lower Bound: The Case of Hidden States

Energy-Efficient Target Tracking in Wireless Sensor Networks: A Quantized Measurement Fusion Framework

Optimizing the design of tracking system under energy and bandwidth constraints in wireless sensor networks (WSN) is of paramount importance. In this paper, the problem of collaborative target tracking in WSNs is considered in a framework of quantized measurement fusion. First, the measurement in each local sensor is quantized by probabilistic quantization scheme and transmitted to a fusion center (FC). Then, the quantized messages are fused and sequential importance resampling (SIR) particle filtering is employed to estimate the target state. In the FC, quantized measurement fusion via both augmented approach and weighted approach is investigated. For both approaches, the closed-form solution to the optimization problem of bandwidth scheduling is given, where the total energy consumption is minimized subject to a constraint on the fusion performance. Finally, posterior Cramer-Rao lower bounds (CRLBs) on the tracking accuracy using quantized measurement fusion are derived. Simulation results reveal that both approaches perform very closely to the posterior CRLB while obtaining average communication energy saving up to 72.8% and 45.1%, respectively.

Exploring Limitations of Return-Based Exposure Estimation for Hedge Funds

Return-based methods are frequently used by risk managers to estimate the exposures of hedge funds due to limited transparency. This article proposes a method to assess the limitations of such methods using the posterior Cramer-Rao bound (PCRB) concept. This PCRB methodology allows one to quantify the maximum achievable accuracy of exposure estimates without having to compare computed estimates to actual exposures. This is particularly useful in situations where exposure data are unavailable. The main determinants of the PCRB are shown to be the variance of the hedge fund exposures and the ratio of market variance to idiosyncratic error variance. To illustrate the application of the methodology, the authors provide five examples in which they compare the PCRB to the computed estimation error using real hedge fund data.

Posterior Cramér–Rao Bound for Anchorless Tracking

We derive the posterior Cramer-Rao bound (PCRB) for tracking in anchorless dynamic networks consisting of static nodes at unknown locations and mobile nodes. We show that this is a type of PCRB under parametric constraints where the parameter space cannot be modeled by equation constraints. We propose a novel framework for modeling the constrained parameter manifold, as a factor manifold under the actions of a group. The approach we propose is generic, and can be used to derive lower error bounds for other parameter estimation problems where the parametric constraints cannot be optimally described by equation constraints.

A Particle Filter Approach to Approximate Posterior Cramer-Rao Lower Bound: The Case of Hidden States

The posterior Cramer-Rao lower bound (PCRLB) derived in [1] provides a bound on the mean square error (MSE) obtained with any nonlinear state filter. Computing the PCRLB involves solving complex, multi-dimensional expectations, which do not lend themselves to an easy analytical solution. Furthermore, any attempt to approximate it using numerical or simulation-based approaches require a priori access to the true states, which may not be available, except in simulations or in carefully designed experiments. To allow recursive approximation of the PCRLB when the states are hidden or unmeasured, a new approach based on sequential Monte-Carlo (SMC) or particle filters (PFs) is proposed. The approach uses SMC methods to estimate the hidden states using a sequence of the available sensor measurements. The developed method is general and can be used to approximate the PCRLB in nonlinear systems with non-Gaussian state and sensor noise. The efficacy of the developed method is illustrated on two simulation examples, including a practical problem of ballistic target tracking at reentry phase.

Channel-Aware Tracking in Multi-Hop Wireless Sensor Networks with Quantized Measurements

A channel-aware target tracking approach is proposed for multi-hop sensor networks based on M-bit quantized measurements. For two cases where the fusion center has the knowledge of the instantaneous fading channel gains and where only the knowledge of fading channel statistics is available, particle filtering (PF)-based channel-aware tracking algorithms are developed. Furthermore, the posterior Cramer-Rao lower bounds (PCRLBs) for the tracking filters are derived. The improved tracking accuracy and robustness of the proposed approach are demonstrated through simulations.

A Multiple-Detection Joint Probabilistic Data Association Filter

Most conventional target tracking algorithms assume that a target can generate at most one measurement per scan. However, there are tracking problems where this assumption is not valid. For example, multiple detections from a target in a scan can arise due to multipath propagation effects as in the over-the-horizon radar (OTHR). A conventional multitarget tracking algorithm will fail in these scenarios, since it cannot handle multiple target-originated measurements per scan. The Joint Probabilistic Data Association Filter (JPDAF) uses multiple measurements from a single target per scan through a weighted measurement-to-track association. However, its fundamental assumption is still one-to-one. In order to rectify this shortcoming, this paper proposes a new algorithm, called the Multiple-Detection Joint Probabilistic Data Association Filter (MD-JPDAF) for multitarget tracking, which is capable of handling multiple detections from targets per scan in the presence of clutter and missed detection. The multiple-detection pattern, which can account for many-to-one measurement set-to-track association rather than one-to-one measurement-to-track association, is used to generate multiple detection association events. The proposed algorithm exploits all the available information from measurements by combinatorial association of events that are formed to handle the possibility of multiple measurements per scan originating from a target. The MD-JPDAF is applied to a multitarget tracking scenario with an OTHR, where multiple detections occur due to different propagation paths as a result of scattering from different ionospheric layers. Experimental results show that multiple-detection pattern based probabilistic data association improves the state estimation accuracy. Furthermore, the tracking performance of the proposed filter is compared against the Posterior Cramer-Rao Lower Bound (PCRLB), which is explicitly derived for the multiple-detection scenario with a single target.

Distributed Particle Filter Implementation With Intermittent/Irregular Consensus Convergence

Motivated by non-linear, non-Gaussian, distributed multi-sensor/agent navigation and tracking applications, we propose a multi-rate consensus/fusion based framework for distributed implementation of the particle filter (CF/DPF). The CF/DPF framework is based on running localized particle filters to estimate the overall state vector at each observation node. Separate fusion filters are designed to consistently assimilate the local filtering distributions into the global posterior by compensating for the common past information between neighboring nodes. The CF/DPF offers two distinct advantages over its counterparts. First, the CF/DPF framework is suitable for scenarios where network connectivity is intermittent and consensus can not be reached between two consecutive observations. Second, the CF/DPF is not limited to the Gaussian approximation for the global posterior density. A third contribution of the paper is the derivation of the exact expression for computing the posterior Cramer–Rao lower bound (PCRLB) for the distributed architecture based on a recursive procedure involving the local Fisher information matrices (FIMs) of the distributed estimators. The performance of the CF/DPF algorithm closely follows the centralized particle filter approaching the PCRLB at the signal to noise ratios that we tested.

Localization in Wireless Sensor Networks: Byzantines and Mitigation Techniques

Wireless Sensor Networks (WSNs) are vulnerable to Byzantine attacks in which malicious sensors send falsified information to the Fusion Center (FC) with the goal of degrading inference performance. In this paper, we consider Byzantine attacks for the location estimation task in WSNs using binary quantized data. Posterior Cramer-Rao Lower Bound (PCRLB) is used to characterize the performance of the network. Two kinds of attack strategies are considered: Independent and Collaborative attacks. We determine the fraction of Byzantine attackers in the network that make the FC incapable of utilizing sensor information to estimate the target location. Optimal attacking strategies for given attacking resources are also derived. Furthermore, we propose two schemes for the mitigation of Byzantine attacks. The first scheme is based on a Byzantine Identification method under the assumption of identical local quantizers. We show with simulations that the proposed scheme identifies most of the Byzantines. In order to improve the performance, we propose a second scheme in conjunction with our identification scheme where dynamic non-identical threshold quantizers are used at the sensors. We show that it not only reduces the location estimation error but also makes the Byzantines `ineffective' in their attack strategy.

Decentralized Conditional Posterior Cramér–Rao Lower Bound for Nonlinear Distributed Estimation

Motivated by the decentralized adaptive resource management problems, the letter derives recursive expressions for online computation of the conditional decentralized posterior Cramer-Rao lower bound (PCRLB). Compared to the non-conditional PCRLB, the conditional PCRLB is a function of the past history of observations made and, therefore, a more accurate representation of the estimator's performance and, consequently, a better criteria for sensor selection. Previous algorithms to compute the conditional PCRLB are limited to centralized architectures. The letter addresses this gap. Our simulations verify the optimality of the conditional dPCRLB by comparing it with the centralized conditional PCRLB in bearing-only tracking applications.

A Comparison of Error Bounds for a Nonlinear Tracking System with Detection Probability Pd < 1

Error bounds for nonlinear filtering are very important for performance evaluation and sensor management. This paper presents a comparative study of three error bounds for tracking filtering, when the detection probability is less than unity. One of these bounds is the random finite set (RFS) bound, which is deduced within the framework of finite set statistics. The others, which are the information reduction factor (IRF) posterior Cramer-Rao lower bound (PCRLB) and enumeration method (ENUM) PCRLB are introduced within the framework of finite vector statistics. In this paper, we deduce two propositions and prove that the RFS bound is equal to the ENUM PCRLB, while it is tighter than the IRF PCRLB, when the target exists from the beginning to the end. Considering the disappearance of existing targets and the appearance of new targets, the RFS bound is tighter than both IRF PCRLB and ENUM PCRLB with time, by introducing the uncertainty of target existence. The theory is illustrated by two nonlinear tracking applications: ballistic object tracking and bearings-only tracking. The simulation studies confirm the theory and reveal the relationship among the three bounds.

Posterior Cramer-Rao Bound for Inertial Sensors Enhanced Mobile Positioning Under The Random Walk Motion Model

This letter presents a new posterior Cramer-Rao bound (PCRB) for inertial sensors enhanced mobile positioning, which performs hybrid data fusion of parameters including position estimates, pedestrian step size, pedestrian heading, and the knowledge of random walk motion model. Moreover, a non-matrix closed form of the PCRB is derived without position estimates. Finally, our numerical results show that when the accuracy of step size and heading measurements is high enough, the knowledge of random walk model becomes redundant.

Cognitive Radar: Step Toward Bridging the Gap Between Neuroscience and Engineering

In this paper, we describe a cognitive radar (CR) that mimics the visual brain. Although the visual brain and radar are different in that the visual brain does not transmit a probing signal to the environment while the active radar greatly relies on the probing signal it transmits to the environment, both of them are observers of the surrounding environment. As such, there is much that we can learn from the visual brain in building a new generation of CRs that outperform traditional radars. In this paper, we confine the discussion, in both analytic and experimental terms, to CR aimed at target tracking. From a theoretical perspective, using the posterior Cramer-Rao lower bound (PCRLB), it is shown that a cognitive tracking radar has the potential to improve tracking performance significantly. In particular, computer experiments are presented, which demonstrate that CR can indeed go beyond the theoretical limits of traditional active radars (TARs) as well as fore-active radars (FARs); the latter are radars equipped with feedback from the receiver to the transmitter. Moreover, computer experiments are presented to demonstrate another practical benefit resulting from the combined use of memory and executive attention in CR for a target-tracking application. Specifically, it is shown that with the provision of these two cognitive processes, the transition in switching from one transmit waveform to another goes forward in a smooth manner. Such a capability is beyond that of TAR or FAR.

Performance bounds of extended target tracking in cluttered environments

In this paper, we consider the problem of the performance bound of a nonlinear filtering problem corresponding to tracking an extended target in cluttered environments (i.e., with false alarms and missed detections). The high resolution sensor obtains the measurements of the position and the extent of the extended target whose shape is modeled by an ellipse. The posterior Cramer-Rao lower bound (PCRLB) provides a useful tool to evaluate the best achievable performance of the nonlinear filtering problem. The bounds of the traditional kinematic state estimation are calculated using the point and the extended target model. It is shown in this paper that the bound of extended target tracking is smaller than that of point model because more information is utilized. The bounds are calculated to examine the influence of the measuring accuracy, the geometry between the sensor and the target, the prior knowledge of the target, and the environmental circumstance. In a cluttered environment, the PCRLB is calculated by IRF (information reduction factor), MSC (measurement sequence conditioning), and MESC (measurement existence sequence conditioning) approaches. The simulation results also illustrate the relationship of the three methods.

New Conditional Posterior Cramér-Rao Lower Bounds for Nonlinear Sequential Bayesian Estimation

The recursive procedure to compute the posterior Cramer-Rao lower bound (PCRLB) for sequential Bayesian estimators, derived by Tichavsky , provides an off-line performance bound for a general nonlinear filtering problem. Since the corresponding Fisher information matrix (FIM) is obtained by taking the expectation with respect to all the random variables, this PCRLB is not well suited for online adaptive resource management for dynamic systems. For online estimation performance evaluation in a nonlinear system, the concept of conditional PCRLB was proposed by Zuo in 2011. In this paper, two other online conditional PCRLBs are proposed which are alternatives to the one proposed by Zuo Numerical examples are provided to show that the three online bounds, namely the conditional PCRLB proposed by Zuo and the two conditional PCRLBs proposed in this paper, are very close to one another.

Multiple Unresolved Target Localization and Tracking using Colocated MIMO Radars

In this paper localization and tracking of multiple unresolved targets using a colocated multiple input multiple output (MIMO) radar is addressed. The commonly-used model for colocated MIMO radars is modified in order to guarantee the observability in received measurements. Then, a maximum likelihood estimator is derived for localizing multiple targets falling within a certain resolution cell. The Cramer-Rao lower bound (CRLB) for localization with the new model is also derived. For the tracking part, a multiple-hypothesis-based approach is used to deal with the uncertainty in target state estimation. In addition, an unscented Kalman filter (UFK) based estimator is used to tackle the nonlinearity in the measurement model. Finally, the posterior CRLB (PCRLB) is derived to evaluate the consistency of tracking results. Simulation results confirm the superiority of the proposed approach in resolving multiple targets over using the standard localization results for tracking.

Impact of Flight Disturbances on Airborne Radar Tracking

This correspondence is devoted to the study of the impact of flight disturbances due to the atmospheric turbulence on airborne radar tracking. The performance of a turbulence-ignorant tracking filter is assessed in a turbulent simulated scenario. The turbulence is modelled according to the Dryden model, a correlated, zero-mean, Gaussian random process. A quasi-ellipsoidal racetrack course is chosen for the air platform that carries the radar. The performance of the turbulence-ignorant tracking algorithm is analyzed in terms of mean value and standard deviation of the estimation errors for each component of the position and velocity vectors and compared with the posterior Cramer-Rao lower bound (PCRLB), evaluated for the ideal case of absence of platform vibrations.