Performance analysis of the maximum a posteriori probability direction finding
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Abstract In this paper, the mean square error performance of the maximum a posteriori (MAP) probability direction finding by sensor array in terms of its Cramer‐Rao lower bound (CRLB) is analyzed. Based on the principle of Bayesian estimator, a log posteriori probability function is formed when the a priori knowledge of location is given. The Fisher information matrix (FIM) is found accordingly. It shows that the CRLB of the MAP estimator is much lower than that of maximum likelihood technique, especially when the element SNR is low and/or the number of snapshots is small. In addition, the CRLB remains at a relatively low level in terms of the variance of DOA of sources. It also shows that the location variance dominates the behavior of the MAP direction finder when locations of sources are Gaussian distributed.Keywords:
Cramér–Rao bound
Fisher information
This paper compares two performance indices for computing optimal observer paths for the bearings-only source localization and tracking problem, for constant velocity sources. Previous work on this problem is based on maximizing the determinant of the Fisher information matrix (FIM) of the estimation problem. This paper considers minimizing the trace of a weighted sum of the Cramer-Rao lower bound (CRLB) of current or future source position errors, and source velocity errors. Quasi-Newton optimization is used to compare optimal observer paths, given three distinct goals: minimizing current position error, velocity error, and future position error. Significant differences in optimal paths are observed, and the CRLB trace is found to yield smaller estimation ambiguity.< >
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Cramér–Rao bound
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Position (finance)
Non-line-of-sight propagation
Matrix (chemical analysis)
Parametric model
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The Cramer-Rao lower bound(CRLB) of nonlinear filtering for bearings-only tracking(BOT) is considered based on the measurements that contain stochastic missing observations.The measurement information comes from different detection probability signal channels in which the detection probability is less than 1.The model of CRLB for BOT in 3-D is derived by using the recursive Fisher information matrix(FIM) in multiple-input-multiple-output(MIMO) system.The theoretical formula involves the evaluation of the exponentially growing number of detection sequences.A detection reduction factor method in a sense of statistics is presented,and the result that the method here is always less than the theoretical CRLB is proved.In addition,an approximation of the theoretical bound for practical applications is proposed,which can reduce the computation load by analysis.
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High-accuracy position information is essential for emerging applications in cooperative networks. Compared with absolute coordinates, relative positions of nodes are more perti-nent for tasks like autonomous driving. In this paper, we establish a theoretical analysis framework for relative localization with unknown clock and orientation parameters. First, we specify three scenarios for relative localization in cooperative networks. The relative position estimation is proved to be a constrained optimization and the relative Cramér-Rao lower bound (CRLB) is derived as the constrained CRLB. Then we prove that the equivalent Fisher information matrix (EFIM) retains the information for relative localization of the concerned nodes. Moreover, we derive the Fisher information matrix for three scenarios and give the corresponding nullspace. Finally, the relative CRLB is proved to be the pseudo-inverse of the EFIM.
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In this paper we study the problem of traffic matrix estimation. The problem is ill-posed and thus some additional information has to be brought in to obtain an estimate. One common approach is to use the second moment statistics through a functional mean-variance relationship. We derive analytically the Fisher information matrix under this framework and obtain the Cramer-Rao lower bound (CRLB) for the variance of an estimator of the traffic matrix. Applications for the use of the CRLB are then demonstrated. From the bounds we can directly obtain confidence intervals for maximum likelihood estimates. Another use for the CRLB is the possibility to evaluate the efficiency of an estimator against the lower bound. A third possible application is to utilize the bounds in an approach to find the best placement for direct measurements of OD flows, so that it is optimal with regard to the traffic matrix estimation problem
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In estimation theory, it is known that prior knowledge of parameters can improve the Cramér-Rao lower bound (CRLB). In this paper, we study the influence of prior knowledge on the CRLB of the estimates of the parameters that describe the trajectory of a moving object (single molecule). Since the CRLB is obtained from the inverse of the Fisher information matrix, we present a general expression of the Fisher information matrix in terms of the image function, the object trajectory and the prior knowledge matrix. Applying this expression to an object moving linearly in a two-dimensional (2D) plane with two distinct cases of prior knowledge, explicit CRLB expressions are derived. From these expressions, we show that the improvement in the CRLB of the parameter estimates is dependent on which parameters are known.
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This paper compares two performance indices for computing optimal observer paths for the bearings-only source constant velocity sources. The problem is based on maximizing the determinant of the Fisher information matrix (FIM) of the estimation problem. It considers minimizing the trace of a weighted sum of the Cramer-Rao lower bound (CRLB) of current source position error. Quasi-Newton optimization is used to compare optimal observer paths, given the goal of minimizing current position error. Significant differences in optimal paths are observed, and the CRLB trace is found to yield smaller range error.< >
Cramér–Rao bound
Fisher information
TRACE (psycholinguistics)
Observer (physics)
Position (finance)
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This paper discusses an approach to using the Cramer Rao Lower Bound (CRLB) as a trajectory design tool for autonomous underwater vehicle (AUV) visual navigation. We begin with a discussion of Fisher Information as a measure of the lower bound of uncertainty in a simultaneous localization and mapping (SLAM) pose-graph. Treating the AUV trajectory as an non-random parameter, the Fisher information is calculated from the CRLB derivation, and depends only upon path geometry and sensor noise. The effect of the trajectory design parameters are evaluated by calculating the CRLB with different parameter sets. Next, optimal survey parameters are selected to improve the overall coverage rate while maintaining an acceptable level of localization precision for a fixed number of pose samples. The utility of the CRLB as a design tool in pre-planning an AUV survey is demonstrated using a synthetic data set for a boustrophedon survey. In this demonstration, we compare the CRLB of the improved survey plan with that of an actual previous hull-inspection survey plan of the USS Saratoga. Survey optimality is evaluated by measuring the overall coverage area and CRLB localization precision for a fixed number of nodes in the graph. We also examine how to exploit prior knowledge of environmental feature distribution in the survey plan.
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Abstract In this paper, the mean square error performance of the maximum a posteriori (MAP) probability direction finding by sensor array in terms of its Cramer‐Rao lower bound (CRLB) is analyzed. Based on the principle of Bayesian estimator, a log posteriori probability function is formed when the a priori knowledge of location is given. The Fisher information matrix (FIM) is found accordingly. It shows that the CRLB of the MAP estimator is much lower than that of maximum likelihood technique, especially when the element SNR is low and/or the number of snapshots is small. In addition, the CRLB remains at a relatively low level in terms of the variance of DOA of sources. It also shows that the location variance dominates the behavior of the MAP direction finder when locations of sources are Gaussian distributed.
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Fisher information
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The sensor-target placement plays a pivotal role in improving the localization accuracy; the commonly used evaluation function for this problem is built by Cramer-Rao lower bound (CRLB) or Fisher information matrix (FIM). Frame theory, which was developed recently, can provide a more easily tractable evaluation function, especially for some complex cases. In this letter, we derive an evaluation function by frame theory and prove its equivalence with that by CRLB and FIM. For angle-of-arrival based localization system, we present the optimal sensor augmentation problem with several fixed sensors and an arbitrary number of new sensors, and for arbitrary but fixed distances between new sensors and target. The conclusions are verified by several simulations.
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