Weighted Least-squares Solution Research Articles

Efficient Closed-Form Solutions for Visible Light Positioning in Low-Cost IoT Devices

Visible light positioning (VLP) has drawn great attention in the field of indoor positioning as light communication has been popularized in low-cost Internet-of-Things (IOT) devices. In this paper, we investigate the VLP problem using the received signal strength (RSS) and by only considering the line-of-slight (LOS) propagation. The RSS-based VLP problem is highly nonlinear, and its solutions may be trapped in local optima without a good initial guess. To circumvent this difficulty, we propose closed-form solutions of the VLP problem considering a known or unknown user orientation. By applying the weighted least squares (WLS) method, the closed-form solutions are divided into two stages. In the stage-one WLS solution, the nonlinear VLP problem is transformed into a pseudo-linear form by introducing some auxiliary variables, which are considered to be independent of each other. The estimates of the stage-one WLS solution are further refined in the stage-two WLS solution by exploiting the constrained relationships among these defined variables. The simulation results show that the stage-two WLS solution provides good estimates for the user position and orientation. The proposed stage-two WLS solution outperforms the existing methods especially at a high signal-to-noise ratio (SNR).

Gradient-descent iterative algorithm for solving exact and weighted least-squares solutions of rectangular linear systems

<abstract><p>Consider a linear system $ Ax = b $ where the coefficient matrix $ A $ is rectangular and of full-column rank. We propose an iterative algorithm for solving this linear system, based on gradient-descent optimization technique, aiming to produce a sequence of well-approximate least-squares solutions. Here, we consider least-squares solutions in a full generality, that is, we measure any related error through an arbitrary vector norm induced from weighted positive definite matrices $ W $. It turns out that when the system has a unique solution, the proposed algorithm produces approximated solutions converging to the unique solution. When the system is inconsistent, the sequence of residual norms converges to the weighted least-squares error. Our work includes the usual least-squares solution when $ W = I $. Numerical experiments are performed to validate the capability of the algorithm. Moreover, the performance of this algorithm is better than that of recent gradient-based iterative algorithms in both iteration numbers and computational time.</p></abstract>

Recursive Estimation of Multichannel Autoregressive Processes in Correlated Noise With Joint Instrumental Variable and Bias Compensation

This paper proposes a new recursive algorithm for estimating multichannel autoregressive (AR) processes in spatially correlated noise using joint extended instrumental variable (EIV) and bias compensation (BC). A weighted least squares solution (WLS) for solving the Yule-Walker (YW) equations from EIV and BC is proposed, while the noise covariance parameters is estimated via the least squares method. A splitting approach is proposed for solving this WLS problem based on matrix splitting, and it is further extended to a recursive algorithm, which can be solved via the variable forgetting factor (VFF)-EIV algorithm. A robust smoothing method for estimating the noise parameters and an adaptive model selection method for controlling the BC contribution are proposed for reducing the estimation variance. The validity of the proposed algorithm is evaluated using both synthetic and real data. Experimental results show that the proposed VFF-EBC algorithm gives higher accuracy than conventional recursive least squares (RLS) algorithm and the VFF-EIV method. It also provides faster tracking speed under sudden changes. The proposed method is applied to the BCI competition IV-2b EEG data set on motor imagery tasks for spectrum and effective connectivity estimation. The high-alpha and beta event-related desynchronization associated with the tasks can be clearly revealed. It was also applied to an auditory odd-ball data from the Open-fMRI database.

Noise resilient solution and its analysis for multistatic localization using differential arrival times

This paper addresses the multistatic localization problem using differential arrival times between the signals of direct and reflected paths. We consider two scenarios: one is that only the partial statistics of the signal propagation speed are known, and the other is that the propagation speed is completely not known. By transforming the measurement model, we begin with formulating two different weighted least squares (WLS) problems for the scenarios, which are recast as non-convex constrained optimization problems, with relationships among the optimization variables included in the constraints. To tackle the difficult non-convex problems, we relax them into convex semidefinite programs (SDPs) by applying the semidefinite relaxation technique and then tighten the relaxed SDP problems by adding second-order cone constraints to reach better solutions. Mean square error analysis validates that the WLS solutions are able to reach the hybrid Cramer-Rao lower bound accuracy under Gaussian noise when it is not quite large, implying that the proposed methods have potential to work well. We also derive the theoretical expressions of the solution biases for both scenarios, and then perform bias reduction by subtracting the estimated bias to achieve approximately unbiased solutions. Simulation results confirm that the proposed methods have superior performance over the existing methods.

Moving source localization using TOA and FOA measurements with imperfect synchronization

The clock bias and drift caused by imperfect time synchronization is a practical issue in localization problems. In this paper, we address the moving source localization problem using time-of-arrival and frequency-of-arrival measurements, where imperfect time synchronization is present. We transform the measurement models and formulate a nonlinear and non-convex weighted least squares (WLS) problem to jointly estimate the unknown source position, velocity, clock bias and drift. We then apply semidefinite relaxation (SDR) to the non-convex WLS problem to obtain a convex semidefinite program, which can be solved efficiently. We show through theoretical mean square error analysis that the WLS solution is able to attain the Cramer–Rao lower bound (CRLB) under small Gaussian noise condition. Simulation results show that the proposed SDR method has comparable performance with the maximum likelihood estimation method, and thus achieves the CRLB accuracy, when the noise is not too large.

An Efficient Estimator for Source Localization Using TD and AOA Measurements in MIMO Radar Systems

3-D target localization based on multi-input and multioutput radar systems is studied in this letter by utilizing time delay and angle of arrival measurements, which can be obtained by some local sensor signal processing methods. For such a positioning problem, the typical closed-form solution first establishes pseudolinear equation and then, applies the weighted least squares (WLS) technique to determine the final position. However, the WLS solution often suffers from performance degradation when the noise is large. To alleviate this problem, the cost function of the WLS is represented, and a quadratic constraint is imposed such that the expectation of the cost function can attain the minimum value at the true position. Moreover, simulation experiments show that the proposed method performs better than the state-of-art algorithms.

Methods to avoid being affected by non-zero closure phase in InSAR time series analysis in a multi-reference stack

Abstract This work discusses InSAR time series analysis using multi-reference stacks with exclusively short time intervals. A 5 months long stack of Sentinel-1 acquisitions over a high alpine territory that includes fast moving landslides is used. Use of the multi-look phases becomes problematic due to non-zero closure phase. The errors induced by non-zero phase closure can be avoided by the use of single-look phases for all scatterers, including distributed scatterers. A weighted least-squares solution of the deformation time-series is obtained using the multi-reference stack interferometric single-look phases. Quality control is done during the regression analysis, the conversion to the time series, and considering the temporal coherence. Spatial filtering of the single-look phase time series can be used to further reduce the phase noise present in areas with intermediate coherence levels.

Time-Delay-Based Target Localization in Wireless Sensor Network With Unknown Noise Covariance

This letter addresses the problem of target localization based on the time-delay (TD) measurements in a wireless sensor network (WSN) with a noncooperative illuminator of opportunity (IO), without assuming the prior knowledge of measurement noise covariance (MNC). The joint likelihood of target position and noise variances are built, based on which the target position and MNC can naturally be estimated by maximizing the joint likelihood. However, the resulting maximum likelihood (ML) problem is intractable due to its nonconvexity and nonlinearity. In order to overcome such difficulty, an alternately iterative optimization strategy is proposed in our approach. The approximate weighted least-squares solution and closed-form shrinkage estimator are introduced into the alternative estimation steps for the target location and MNC, respectively. In simulation experiments with 40 TD measurements, the performance of the proposed method, performing without knowledge of MNC, achieves about 8 dB average improvement over the traditional method without the MNC. Its performance is also close to that of the algorithms and Cramer–Rao lower bound (CRLB) working with known MNC, thus verifying the effectiveness of proposed approach.

Localisation of wireless nodes with partial connectivity in wireless sensor systems

Owing to their short communication range, wireless nodes in wireless sensor networks (WSNs) can exchange information with devices in their vicinity only. Thus, in sparse networks, the full connectivity of the network is rarely achieved. This renders a centralised approach towards localisation in WSNs useless. Moreover, the exploitation of a centralised algorithm for localisation compromises the scalability in dense networks. Thus, a decentralised, location-aware network with partial connectivity and hybrid (range and direction) measurements obtained between known sensors (reference sensors) and sensors at unknown locations (target sensors) is under focus. The decentralised location estimation is obtained using a linear least squares (LLS) approach and performance enhancements are achieved by introducing a weighing strategy to produce weighted least squares (WLS) estimates. This distributed estimation is made possible by designing a map stitching technique that forms the global map of the network from individual local maps of the wireless nodes without compromising the distributed nature of the network. In the analytical section of the study, theoretical mean squares error expression is derived for LLS estimation, and a Cramer–Rao lower bound is derived to bind the performance of the WLS solution. The algorithm's performance validation is conducted both theoretically and via simulations.

Distributed weighted least-squares estimation for networked systems with edge measurements

This paper studies the problem of distributed weighted least-squares (WLS) estimation for an interconnected linear measurement network with additive noise. Two types of measurements are considered: self measurements for individual nodes, and edge measurements for the connecting nodes. Each node in the network carries out distributed estimation by using its own measurement and information transmitted from its neighbours. We study two distributed estimation algorithms: a recently proposed distributed WLS algorithm and the so-called Gaussian Belief Propagation (BP) algorithm. We first establish the equivalence of the two algorithms. We then prove a key result which shows that the information matrix is always generalized diagonally dominant, under some very mild condition. Using these two results and some known convergence properties of the Gaussian BP algorithm, we show that the aforementioned distributed WLS algorithm computes the exact WLS solution asymptotically. A bound on its convergence rate is also presented.

Efficient source positioning method based on two stations using GROA and AOA measurements

By utilising the gain ratio of arrival (GROA) and angles of arrival (AOAs), three-dimensional source localisation with only two stations is studied. The weighted least squares (WLS) algorithm yields the typical closed-form solution, which is obtained by first establishing a set of linear equations and then determining the source position. However, the WLS solution exhibits poor performance at high noise levels. Hence, an efficient parameter augmented constraint method is proposed by extending the parameter space. The proposed method minimises the cost function by imposing a quadratic constraint such that the expectation of the cost function will attain its minimum value at the true solution. In addition, the proposed method yields an explicit algebraic solution and does not depend on a good initial guess; thus, there is no convergence problem. Theoretical analysis and simulation results verify the effectiveness of the proposed method.

A geometry conforming, isogeometric, weighted least squares (WLS) method for the neutron transport equation with discrete ordinate (S[formula omitted]) angular discretisation

This paper presents the application of isogeometric analysis (IGA) to the spatial discretisation of the multi-group, source iteration compatible, weighted least squares (WLS) form of the neutron transport equation with a discrete ordinate (SN) angular discretisation. The WLS equation is an elliptic, second-order form of the neutron transport equation that can be applied to neutron transport problems on computational domains where there are void regions present. However, the WLS equation only maintains conservation of neutrons in void regions in the fine mesh limit. The IGA spatial discretisation is based up non-uniform rational B-splines (NURBS) basis functions for both the test and trial functions. In addition a methodology for selecting the magnitude of the weighting function for void and near-void problems is presented. This methodology is based upon solving the first-order neutron transport equation over a coarse spatial mesh. The results of several nuclear reactor physics verification benchmark test cases are analysed. The results from these verification benchmarks demonstrate two key aspects. The first is that the magnitude of the error in the solution due to approximation of the geometry is greater than or equal to the magnitude of the error in the solution due to lack of conservation of neutrons. The second is the effect of the weighting factor on the solution which is investigated for a boiling water reactor (BWR) lattice that contains a burnable poison pincell. It is demonstrated that the smaller the area this weighting factor is active over the closer the WLS solution is to that produced by solving the self adjoint angular flux (SAAF) equation. Finally, the methodology for determining the magnitude of the weighting factor is shown to produce a suitable weighting factor for nuclear reactor physics problems containing void regions. The more refined the coarse solution of the first-order transport equation, the more suitable the weighting factor.

A Closed-Form Location Algorithm Without Auxiliary Variables for Moving Target in Noncoherent Multiple-Input and Multiple-Output Radar System

This paper focuses on the moving target localization and velocity measurement in incoherent centralized multiple-input and multiple-output (MIMO) radar systems with widely separated antennas in 3D space. In this paper, we assume that parameters such as time-of-arrival (TOA), frequency-of-arrival (FOA), azimuth angles and elevation angles have already been measured. With these measurements, a final closed-form solution of the target position and velocity can be obtained via the weighted least squares (WLS) method. When the target is located in the far field, due to poor system observability, a first-order Taylor expansion based on the WLS solution is necessary to obtain a more accurate and unbiased solution. Unlike the preceding papers which were based on the two-stage weighted least squares (2SWLS) method [1]-[3], in this paper, the angle information is introduced into the time delay equations and the Doppler frequency equations, so that the intermediate variables in the estimator can be eliminated [4], [5]. Meanwhile, the time delay equations and the Doppler equations are transformed into linear equations only related to the position and the speed of the target. This method, unlike 2SWLS-based methods [1]-[3], does not introduce auxiliary variables, so it does not require the decorrelation procedure. Simulation results show that the root mean-square error (RMSE) of position and velocity can reach Cramer-Rao lower bound (CRLB) when the noise is at a moderate level before the thresholding effect occurs.

Unification of optimal targeting methods in transcranial electrical stimulation

One of the major questions in high-density transcranial electrical stimulation (TES) is: given a region of interest (ROI) and electric current limits for safety, how much current should be delivered by each electrode for optimal targeting of the ROI? Several solutions, apparently unrelated, have been independently proposed depending on how “optimality” is defined and on how this optimization problem is stated mathematically. The least squares (LS), weighted LS (WLS), or reciprocity-based approaches are the simplest ones and have closed-form solutions. An extended optimization problem can be stated as follows: maximize the directional intensity at the ROI, limit the electric fields at the non-ROI, and constrain total injected current and current per electrode for safety. This problem requires iterative convex or linear optimization solvers. We theoretically prove in this work that the LS, WLS and reciprocity-based closed-form solutions are specific solutions to the extended directional maximization optimization problem. Moreover, the LS/WLS and reciprocity-based solutions are the two extreme cases of the intensity-focality trade-off, emerging under variation of a unique parameter of the extended directional maximization problem, the imposed constraint to the electric fields at the non-ROI. We validate and illustrate these findings with simulations on an atlas head model. The unified approach we present here allows a better understanding of the nature of the TES optimization problem and helps in the development of advanced and more effective targeting strategies.

Matrix-Based Algorithms for the Optimal Design of Variable Fractional Delay FIR Filters

This paper investigates the weighted least squares (WLS) and minimax design problems for variable fractional delay (VFD) FIR filters. For the WLS design, the general case of incorporating arbitrary nonnegative weighting functions is considered. The optimal solution is characterized by two matrix equations. An efficient algorithm using conjugate gradient (CG) techniques is proposed to solve the WLS solution. The proposed algorithm is guaranteed to converge to the optimal solution in a finite number of iterations. Moreover, an iterative reweighted least squares (IRLS) algorithm that uses the proposed CG algorithm as its iteration core is developed for the minimax design problem. In both of the algorithms, the filter coefficients are arranged as matrices, achieving a great saving in computations and memory space. The associated computational complexity is analyzed. Some design examples are provided and comparisons with existing methods show that the proposed ones are either computationally more efficient or can obtain the better filter performance, or both.

Efficient Weighted Least Squares Estimator for Moving Target Localization in Distributed MIMO Radar With Location Uncertainties

An asymptotically efficient estimator for determining the position and velocity of a moving target from time delay and Doppler shift measurements in the presence of location uncertainties in multiple-input multiple-output (MIMO) radar systems with widely separated antennas is presented. It is considered that the actual positions of transmitters and receivers are not available. By taking both the location and measurement errors into account, a novel closed-form two-stage weighted least-squares solution to the problem is developed and analyzed. In order to generate a substantial improvement in the performance of the method, a weighting matrix is employed at each stage of the process. The accuracy properties of the method as well as the Cramer–Rao lower bound (CRLB) for target localization accuracy are derived in the case of Gaussian observations. The proposed algorithm is shown analytically to achieve the CRLB under small noise conditions. Simulations are included to examine the algorithms performance and corroborate the theoretical developments.

Bias reduction for TDOA localization in the presence of receiver position errors and synchronization clock bias

Time difference of arrival (TDOA) localization does not require time stamping of the source signal and is playing an increasingly important role in passive location. In addition to measurement noise, receiver position errors and synchronization clock bias are two important factors affecting the performance of TDOA positioning. This paper proposes a bias-reduced solution for passive source localization using TDOA measurements in the presence of receiver position errors and synchronization clock bias. Like the original two-step weighted least-squares solution, the new technique has two stages. In the first stage, the proposed method expands the parameter space in the weighted least-squares (WLS) formulation and imposes a quadratic constraint to suppress the bias. In the second stage, an effective WLS estimator is given to reduce the bias generated by nonlinear operations. With the aid of second-order error analysis, theoretical biases for the original solution and proposed bias-reduced solution are derived, and it is proved that the proposed bias-reduced method can achieve the Cramér–Rao lower bound performance under moderate Gaussian noise, while having smaller bias than the original algorithm. Simulation results exhibit smaller estimation bias and better robustness for all estimates, including those of the source position, refined receiver positions, and clock bias vector, when the measurement noise or receiver position error increases.

Simulation of multilateration system based on Chan algorithm and conjugate gradient optimisation algorithm

In multilateration (MLAT) systems, the traditional Chan algorithm applies the theory of time-difference-of-arrival (TDOA) to solve the target position of the mathematical model. By introducing intermediate variables, the algorithm adopts a two-step weighted least-squares solution. The introduction of intermediate variables results in the target position equation producing a fuzzy solution, this reduces positioning accuracy. The conjugate gradient algorithm (CGA) is one of the most useful methods for solving large linear equations, it avoids solving the inverse of the matrix, whilst it 'speeds up' the solution of the target position. A four stations multi-point-positioning system mathematical model is established, and a new fusion algorithm Chan-CGA is applied to the MLAT system. Finally, the fusion algorithm is evaluated by simulation and compared with the Chan-Taylor algorithm.

Deriving time-series three-dimensional displacements of mining areas from a single-geometry InSAR dataset

This paper presents a method for deriving time-series three-dimensional (3-D) displacements of mining areas from a single-geometry interferometric synthetic aperture radar (InSAR) dataset (hereafter referred to as the SGI-based method). This is mainly aimed at overcoming the limitation of the traditional multi-temporal InSAR techniques that require SAR data from at least three significantly different imaging geometries to fully retrieve time-series 3-D displacements of mining areas. The SGI-based method first generates the multi-temporal observations of the mining-induced vertical subsidence from the single-geometry InSAR data, using a previously developed method for retrieving 3-D mining-related displacements from a single InSAR pair (SIP). The weighted least-squares solutions of the time series of vertical subsidence are estimated from these generated multi-temporal observations of vertical subsidence. Finally, the time series of horizontal motions in the east and north directions are estimated using the proportional relationship between the horizontal motion and the subsidence gradient of the mining area, on the basis of the SGI-derived time series of vertical subsidence. Seven ascending ALOS PALSAR images from the Datong mining area of China were used to test the proposed SGI-based method. The results suggest that the SGI-based method is effective. The SGI-based method not only extends the SIP-based method to time-series 3-D displacement retrieval from a single-geometry InSAR dataset, but also limits the uncertainty propagation from InSAR-derived deformation to the estimated 3-D displacements.

Rank/inertia approaches to weighted least-squares solutions of linear matrix equations

The well-known linear matrix equation AX=B is the simplest representative of all linear matrix equations. In this paper, we study quadratic properties of weighted least-squares solutions of this matrix equation. We first establish two groups of closed-form formulas for calculating the global maximum and minimum ranks and inertias of matrices in the two quadratical matrix-valued functions Q1−XP1X′ and Q2−X′P2X subject to the restriction trace[(AX−B)′W(AX−B)]=min, where both Pi and Qi are real symmetric matrices, i=1,2,W is a positive semi-definite matrix, and X′ is the transpose of X. We then use the rank and inertia formulas to characterize quadratic properties of weighted least-squares solutions of AX=B, including necessary and sufficient conditions for weighted least-squares solutions of AX=B to satisfy the quadratic symmetric matrix equalities XP1X′=Q1 an X′P2X=Q2, respectively, and necessary and sufficient conditions for the quadratic matrix inequalities XP1X′≻Q1 (≽Q1, ≺Q1, ≼Q1) and X′P2X≻Q2 (≽Q2, ≺Q2, ≼Q2) in the Löwner partial ordering to hold, respectively. In addition, we give closed-form solutions to four Löwner partial ordering optimization problems on Q1−XP1X′ and Q2−X′P2X subject to weighted least-squares solutions of AX=B.