Influence of prior knowledge on the accuracy limit of parameter estimation in single-molecule fluorescence microscopy
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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.Keywords:
Cramér–Rao bound
Fisher information
Matrix (chemical analysis)
The Cramer-Rao lower bound (CRLB) provides a useful tool for evaluating the performance of parameter estimation techniques. Several techniques for the computation of the CRLB for ARMA and AR-plus-noise models are presented. It is shown that the CRLB can be expressed as an explicit function of the model parameters.
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Localization is an important research problem in WSNs (wireless sensor networks) and many WSN localization algorithms have been proposed in the literature. For a single sensor node whose anchors' positions are known, the location estimation error lower bound can be computed by using the CRLB (Cramer-Rao Lower Bound). However, it is still unclear to the research community what the localization error lower bound is from a network point of view. In this paper, we derive a lower bound of the expected localization error for a network whose sensor nodes and anchors are randomly distributed according to a Poisson point process. We show that the lower bound of the expected localization error for a network is a function of the anchor density and the variance of anchor-to-sensor distance measurements.
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Poisson point process
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In view of the existence of fixed measurement error and target's maneuver, the corresponding theoretic lower bound (denoted by Cramer-Rao lower bound (CRLB)) for bearings-only target tracking is investigated. We consider the effect of both target's maneuver and fixed measurement error on the calculation of CRLB, and a recursive formulation of the CRLB for the biased bearings-only maneuvering target tracking is obtained. Finally, Monte Carlo simulations are conducted and some useful conclusions are drawn.
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Tracking (education)
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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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Fisher information
Position (finance)
Matrix (chemical analysis)
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Target localization is one of the basic functions of wireless sensor networks. In the target localization algorithms, the problem of how many sensors are needed and how they are deployed is what we are concerned about in this paper. The authors of (K.C. Ho et al., 2007) present the mathematical expression of Cramer-Rao lower bound for the localization error, which is the smallest lower bound of any unbiased estimation. However it is difficult to be used to find out the relation between sensor number and Cramer-Rao lower bound directly. In this paper the relation between sensor number and Cramer- Rao lower bound is revealed by simulation. And how the topology affects the Cramer-Rao lower bound is also presented.
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A Cramer-Rao type lower bound for a class of systems with faulty measurements is presented. Lower bounds for both the state and the Markovian interruption variables of the system are derived, based on the recently presented sequential version of the Cramer-Rao lower bound (CRLB) for general nonlinear systems. To facilitate the calculation of the lower bound for this class of systems, the discrete distribution of the fault indicators is approximated by a continuous one and the lower bound is obtained via a limiting process applied to the approximating system. The results presented in this paper facilitate a relatively simple calculation of a nontrivial lower bound for the state vector of systems with faulty measurements. The CRLB-type lower bound for the interruption process variables is trivially zero, however, a non-trivial, non-CRLB-type bound for these variables has been recently presented elsewhere by the authors.
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This paper surveys some recent results regarding the Cramer-Rao Lower Bound (CRLB) — the requirements for its existence and its extension to the situation where the parameter's likelihood function (LF) support depends on the parameter to be estimated. In the latter case the conventional CRLB does not hold in general. First we revisit the derivation of the CRLB to elucidate the necessary and sufficient conditions needed for the estimation bound. Following this, the situation of parameter-dependent LF support is considered and the Cramer-Rao-Leibniz Lower Bound (CRLLB) is shown to have an additional term that follows from the Leibniz integral rule, hence the name of this recently obtained bound. This result is the key to provide the estimation bound for the case of uniformly distributed measurement noise over a finite interval. It is also shown that in this case the CRLLB is unattainable. Other examples are also discussed, like the truncated Gaussian measurement noise, in which case the CRLLB is shown to be attained. We also present two examples of apparent “super-efficient estimators” where the estimators' variances are smaller than the conventional CRLB and show that these examples have correct CRLLBs, thus setting straight the myth of “super-efficiency”.
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Gaussian Noise
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The theoretical Cramer-Rao lower bound (CRLB) for bearings-only maneuvering target tracking is derived in the case where the observation measurements are lost in a random fashion. Two binary variables are introduced to model two events respectively, one which the target maneuvers or not and another that the target is detected or missed. The corresponding recursive formula for theoretical CRLB is then derived based on the sequential version of the CRLB for general nonlinear systems. The theoretical formula suffers from heavy calculation load of the Fisher information matrix (FIM) while the constant probability of detection is less than unity. An approximation of the theoretical bound is proposed. In addition, a detection reduction factor bound is presented and proved to be less than the theoretical CRLB. The results are illustrated with a numerical example.
Cramér–Rao bound
Fisher information
Constant (computer programming)
Tracking (education)
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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.
Cramér–Rao bound
Fisher information
Matrix (chemical analysis)
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In this paper we have provided some of evidence work of the authors R.N.Hasan, O.A Tantawy in 2016 [1] thy given proof of the concept between two bound soft sets & subsets of soft elements real numbers also was concluded an upper bound and lower bound by using two sequences of soft element real numbers which is  In this paper, supposed to extend R.N.Hasan, O.A Tantawy work but here we are given new notion and proof for upper limit superior and lower limit inferior with two sequences and subsequences for the conclude new proof after recalled that, which is upper limit superior and lower limit inferior By this we have proved above new two theorems and one proposition & strengthen the example.AMS Subject Classification: 06D72, 40A05, 54A40 Â
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