Decentralized Estimation in an Inhomogeneous Sensing Environment

We consider the distributed estimation of an unknown vector signal in a bandwidth constrained sensor network with a fusion center (FC). Due to power and bandwidth limitations, each sensor compresses its data in order to minimize the amount of information that needs to be communicated to the FC. In this context, we design a linear decentralized estimation scheme (DES), where each sensor linearly encodes its observations before the transmission to the FC, which performs a minimum mean squared error (MMSE) estimation for the unknown vector signal based on the received messages. When the channels between sensors and the FC are orthogonal, it has been shown previously that the complexity of designing the optimal encoding matrices is NP-hard in general. In this paper, we study the optimal design of linear DES for the case of non- orthogonal multiple access channel (MAC) under both bandwidth and power constraints. We show that when the MAC between sensors and the FC is noiseless, the resulting problem has a closed-form solution, while in the noisy MAC case, the problem can be efficiently solved by semi-definite programming (SDP).

In classical estimation usually unbiased estimators are used. This is mainly because the bias term in classical biased estimators in general depends on the parameter to be estimated. However, recently a considerable amount of research has been spent on improving unbiased estimators by introducing a bias, e.g. based on a minimax optimization strategy. In this work we follow this idea of introducing a bias, but we describe a different strategy for optimizing the estimators' performance. Although we stick to classical estimation, we show that the Bayesian linear minimum mean square error estimator can be brought into the same algebraic form as the resulting biased estimator improving the best linear unbiased estimator. This not only emphasizes the fact that this approach leads to betters estimators than the minimax approach on average over all parameters, but also can be seen as another way of reducing the gap between classical and Bayesian estimation.

We consider distributed estimation algorithms in a large Wireless Sensor Network (WSN) with a star topology. The local estimates computed by each sensor must be collected in a fixed amount of time by the network's Fusion Center (FC) and processed to produce a global estimate. When the amount of time available is not sufficient to collect a local estimate from every node, a strategy that enables the most reliable estimates to be collected first must be developed. In this paper, we develop such an algorithm for a WSN whose FC uses a Best Linear Unbiased Estimator (BLUE) and collects the local estimates on a collision channel. The proposed schemes use population splitting that is based on the reliabilities of the local estimates. The time available for data collection is divided into frames and each frame is subdivided into slots. The slots in the first frame are used to collect the most reliable local estimates, the next frame of slots is used to collect the next most reliable set of local estimates, etc. As the performance of the schemes depends on the reliability thresholds, the number of bits representing the estimates, and the number of time slots in a frame, we formulate time-constrained optimization problems and derive methods to obtain the optimal values of these parameters. An interesting result shows that the optimal reliability thresholds do not maximize the channel's throughput.

A distributed state estimation scheme based on quantized observations in wireless sensor network (WSN) is proposed. Unlike the SOI-KF approach, we address the state estimation problem in two steps: firstly, local sensors and the fusion center apply the decentralized estimation scheme (DES) to design the quantized message function and the fusion function; secondly, the fusion center constructs a linear filter to obtain the state estimation by minimizing an upper bound of the error power norm. The optimal filter is computed by the linear matrix inequality (LMI) solver. An example is presented to demonstrate the efficiency of our method.

In this paper, we study the energy-efficient distributed estimation problem for a wireless sensor network where a physical phenomena that produces correlated data is sensed by a set of spatially distributed sensor nodes and the resulting noisy observations are transmitted to a fusion center via noise-corrupted channels. We assume a Gaussian network model where (i) the data being sensed at different sensors are correlated and the correlation structure (in the form of a correlation matrix) is known at the fusion center, (ii) the links between the local sensors and the fusion center are subject to multipath fading plus AWGN, and the fading gains are available to the receiver node, and (iii) the central node uses the squared error distortion metric. We first determine the optimum power-distortion regions assuming (i) a multiple-letter, and (ii) a single-letter square distortion characterization. Next, for the two distortion characterization, we investigate the performance of an uncoded transmission approach where the noisy observations are only amplified-and-forwarded to the fusion center. At the fusion center, two different estimators are considered: (i) minimum mean-square error estimator (MMSE) that exploits the correlation, and (ii) best linear unbiased estimator (BLUE) that does not require or exploit the knowledge of the correlation matrix. For both estimators, we solve for the optimal power allocation that results in a minimum total transmission power while satisfying some distortion level for the estimate (for both multiple-letter and single-letter distortion metrics). The numerical comparisons between the two schemes indicate that the MMSE requires less power to attain the same distortion provided by the BLUE. Furthermore, comparisons between power-distortion region achieved by the theoretically optimum system and the uncoded system indicates that the performance gap between the two system becomes small for low level of correlation between the sensor observations. If observations at all sensor nodes are uncorrelated, the uncoded system attains the theoretically optimum system performance.

Wu et al. [Computational comparison for weighted moments estimators and BLUE of the scale parameter of a Pareto distribution with known shape parameter under type II multiply censored sample, Appl. Math. Comput. 181 (2006), pp. 1462–1470] proposed the weighted moments estimators (WMEs) of the scale parameter of a Pareto distribution with known shape parameter on a multiply type II-censored sample. They claimed that some WMEs are better than the best linear unbiased estimator (BLUE) based on the exact mean-squared error (MSE). In this paper, the general WME (GWME) is proposed and the computational comparison of the proposed estimator with the WMEs and BLUE is done on the basis of the exact MSE for given sample sizes and different censoring schemes. As a result, the GWME is performing better than the best estimator among 12 WMEs and BLUE for all cases. Therefore, GWME is recommended for use. At last, one example is given to demonstrate the proposed GWME.

MSE superiority of Bayes and empirical Bayes estimators in two generalized seemingly unrelated regressions

This paper investigates the problem of adaptive power allocation for distributed best linear unbiased estimation (BLUE) of a random parameter at the fusion center (FC) of a wireless sensor network (WSN). An optimal power-allocation scheme is proposed that minimizes the $L^2$-norm of the vector of local transmit powers, given a maximum variance for the BLUE estimator. This scheme results in the increased lifetime of the WSN compared to similar approaches that are based on the minimization of the sum of the local transmit powers. The limitation of the proposed optimal power-allocation scheme is that it requires the feedback of the instantaneous channel state information (CSI) from the FC to local sensors, which is not practical in most applications of large-scale WSNs. In this paper, a limited-feedback strategy is proposed that eliminates this requirement by designing an optimal codebook for the FC using the generalized Lloyd algorithm with modified distortion metrics. Each sensor amplifies its analog noisy observation using a quantized version of its optimal amplification gain, which is received by the FC and used to estimate the unknown parameter.

We consider a distributed reception scenario where the transmitter transmits a signal and multiple geographically separated receive nodes forward the processed version (using only a small number of bits) of their received signal to the fusion center. The fusion center tries to decode the transmitted signal only based on the forwarded bits from the distributed nodes. We show that there is a strong connection between distributed reception and channel coding and develop the unified framework of coded diversity techniques for distributed reception. To enable efficient encoding at the distributed nodes, we focus on linear block codes that achieve the Griesmer bound with equality to have the maximum diversity gain.

A bound is established for the Euclidean norm of the difference between the best linear unbiased estimator and any linear unbiased estimator in the general linear model. The bound involves the spectral norm of the difference between the dispersion matrices of the two estimators, and the residual sum of squares, all evaluated at the assumed model, but is independent of the provenance of the observation vector at hand. The bound, a straightforward consequence of first principles in Gauss–Markov theory, generalizes previous results on the difference between the best linear unbiased estimator and the ordinary least-squares estimator. In a numerical example from repeated precise levelling, the bound is used to analyse the sensitivity of estimates of vertical motion to the choice of estimator.

Large-scale channel matrices in massive multiple-input multiple-output (MIMO) systems increase the difficulty of channel estimation. Conventional best linear unbiased estimator (BLUE) algorithms include multiple inverse operations for large-scale matrices; however this typically reduces algorithm efficiency. Efficiency can be improved by converting inverse operations into a system of linear equations, which are more easily solved for large matrices. This approach is used in the conjugate gradient (CG) method, which exhibits higher efficiency and precision when solving for approximate optimal solutions. As such, we propose a hybrid conjugate gradient (CG BLUE) algorithm, an improved version of the conventional BLUE technique. This algorithm reduces the dimensions of the channel model through a vectorization transform and uses the CG method to improve algorithm process efficiency during channel estimation for massive MIMO systems. Simulated results of NMSE and algorithm processing time show the proposed CG-BLUE algorithm resulted in nearly one order of magnitude higher calculation efficiency while maintaining the same level of accuracy as conventional BLUE.

We investigate the limits of communication over the discrete-time additive white Gaussian noise (AWGN) channel, when the channel output is quantized using a small number of bits. We first provide a proof of our recent conjecture on the optimality of a discrete input distribution in this scenario. Specifically, we show that for any given output quantizer choice with K quantization bins (i.e., a precision of log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> K bits), the input distribution, under an average power constraint, need not have any more than K + 1 mass points to achieve the channel capacity. The cutting-plane algorithm is employed to compute this capacity and to generate optimum input distributions. Numerical optimization over the choice of the quantizer is then performed (for 2-bit and 3-bit symmetric quantization), and the results we obtain show that the loss due to low-precision output quantization, which is small at low signal-to-noise ratio (SNR) as expected, can be quite acceptable even for moderate to high SNR values. For example, at SNRs up to 20 dB, 2-3 bit quantization achieves 80-90% of the capacity achievable using infinite-precision quantization.

This paper tabulates: coefficients and relative s-efficiency of the best linear unbiased estimator (BLUE) of the scale parameter of the Rayleigh distribution for type II censored samples of size N=20(5)40, r=0(1)4 (number of observations censored from the left) and s=0(1)4 (number of observations censored from the right); and ranks, coefficients, variances, and relative s-efficiencies of the BLUE of a based on a selected few order statistics (k) for sample size N=20(1)40 and k=2(1)4. These estimators have the minimum variance among the BLUE of a based on the same number of order statistics. As compared to maximum likelihood estimators (MLE) and approximate MLE, the s-efficiency of BLUE of /spl rho/ is very high. When estimating the parameter using only a few observations, the k-optimum BLUE of /spl rho/ is the only choice, as the MLE of /spl rho/ is not available. Therefore, these tables for coefficients of BLUE of /spl rho/ based on censored samples and few observations for moderately large samples, have many applications. >

This paper presents a quantitative performance evaluation method for the maximum a posteriori (MAP) state estimate fusion algorithm. Under ideal conditions where data association is assumed to be perfect, it has been shown that the MAP or best linear unbiased estimate (BLUE) fusion formula provides the best linear minimum mean squared estimate (LMMSE) given local estimates under the linear Gaussian assumption for a static system. However, for a dynamic system where fusion is recursively performed by the fusion center on local estimates generated from local measurements, it is not obvious how the MAP algorithm will perform. In the past, several performance evaluation methods have been proposed for various fusion algorithms, including simple convex combination, cross-covariance combination, information matrix, and MAP fusion. However, not much has been done to quantify the steady state behavior of these fusion methods for a dynamic system. The goal of this work is to present analytical fusion performance results for MAP state estimate fusion without extensive Monte Carlo simulations, using an approach developed for steady state performance evaluation for track fusion. Two different communication strategies are considered: fusion with and without feedback to the sensors. Analytic curves for the steady state performance of the fusion algorithm for various communication patterns are presented under different operating conditions.

In this paper, an analysis of a target location estimation\nsystem using the best linear unbiased estimator (BLUE) for high\nperformance radar systems is presented. In synthetic environments,\nwe are here concerned with three key elements of radar system\nmodeling, which makes radar systems operates accurately in strategic\nsituation in virtual ground. Radar Cross Section (RCS) modeling\nis used to determine the actual amount of electromagnetic waves\nthat are reflected from a tactical object. Pattern Propagation Factor\n(PPF) is an attenuation coefficient of the radar equation that contains\nthe reflection from the surface of the earth, the diffraction, the\nrefraction and scattering by the atmospheric environment. Clutter is\nthe unwanted echoes of electronic systems. For the data fusion of\noutput results from radar detection in synthetic environment, BLUE\nis used and compared with the mean values of each simulation results.\nSimulation results demonstrate the performance of the radar system.