Weighted Least-squares Algorithm Research Articles

Constrained Weighted Least-Squares Algorithms for 3-D AOA-Based Hybrid Localization

Constrained Weighted Least-Squares Algorithms for 3-D AOA-Based Hybrid Localization

Defect Isolation from Whole to Local Field Separation in Complex Interferometry Fringe Patterns through Development of Weighted Least-Squares Algorithm

In this paper, based on Gaussian 1σ-criterion and histogram segmentation, a weighted least-squares algorithm is applied and validated on digital holographic speckle pattern interferometric data to perform phase separation on the complex interference fields. The direct structural diagnosis tool is used to investigate defects and their impact on a complex antique wall painting of Giotto. The interferometry data is acquired with a portable off-axis interferometer set-up with a phase-shifted reference beam coupled with the object beam in front of the digital photosensitive medium. A digital holographic speckle pattern interferometry (DHSPI) system is used to register digital recordings of interferogram sequences over time. The surface is monitored for as long as it deforms prior to returning to its initial reference equilibrium state prior to excitation. The attempt to separate the whole vs. local defect complex amplitudes from the interferometric data is presented. The main aim is to achieve isolation and visualization of each defect’s impact amplitude in order to obtain detailed documentation of each defect and its structural impact on the surface for structural diagnosis purposes.

Quantification analysis of mural defects in digital holography with fundamental voice excitation based on local phase field separation

Previous collaborative studies have shown that fringe patterns by optical interference way are used as a direct diagnostic tool for qualitative analysis of defects in Culture Heritage. Nevertheless, the complexity of the fringe patterns prevents the accurate quantification of defects. In this study, a phase separation method based on the weighted least-squares algorithm is proposed to conveniently isolate the local phase distribution from the whole, complex fringe pattern. This method is examined to accurately quantify the deformation and strain distribution of visible and invisible defects. In experimental work, the Non-destructive Acoustic excitation and Digital Holography (NDA-DH) detection is employed for the superficial and subsurface defects in the Ruting murals of the Forbidden City, Beijing. Under the impact of fundamental human voice, the frequency range of the acoustic excitation was considered from 100–800 Hz. All results show that the suggested phase separation method allows direct observation of defects as well as quantified surface deformation providing a precious documentation to the conservators.

Automatic elimination of phase aberrations in digital holography based on Gaussian 1σ- criterion and histogram segmentation.

We propose a numerical and automatic quadratic phase aberration elimination method in digital holography for phase-contrast imaging. A histogram segmentation method based on Gaussian 1σ-criterion is used to obtain the accurate coefficients of quadratic aberrations using the weighted least-squares algorithm. This method needs no manual intervention for specimen-free zone or prior parameters of optical components. We also propose a maximum-minimum-average-standard deviation (MMASD) metric to quantitatively evaluate the effectiveness of quadratic aberration elimination. Simulation and experimental results are demonstrated to verify the efficacy of our proposed method over the traditional least-squares algorithm.

Development of electrical power transmission system linear hybrid state estimator based on circuit analysis techniques

Electrical power systems are in rapid evolution, necessitating advanced monitoring, control, and protection aspects connected to an accurate knowledge of the state of the grid. State estimation is one-way such knowledge could be achieved by estimating voltage magnitudes and phase angles at all buses in electrical networks. Electric utilities are deploying advanced Phasor Measurement Units (PMUs) in grids with conventional Remote Terminal Units (RTUs). Integrating measurements from the two types of devices for state estimation is inevitable. In the past, hybrid state estimators have been based on nonlinear models associated with various challenges, such as using nonlinear state estimation algorithms with high computational time, slow convergence, and initialization problems. For this reason, there is a need to develop hybrid linear mathematical models that eliminate issues posed by nonlinearity models. This paper presents a novel mathematical formulation that gives a linear measurement model for hybrid state estimation based on the standard steady-state branch model. The developed model is then tested in MATLAB using three standard transmission test cases (IEEE 14-BUS, IEEE 30-BUS, and IEEE 57-BUS) with available measurements subjected to different errors. Notably, the performance of the developed model based on two different state estimation algorithms, Weighted Least Square (WLS) and Weighted Least Absolute Value (WLAV), are compared. The simulation results obtained for voltage magnitudes and phase angles for all buses in the three scenarios compare well with reference values. Using Normalized Cumulative Error (NCE) as a performance index, the developed linear measurement model gives less cumulative error than the conventional nonlinear model. With measurements subjected to bad data (such as negative values for measured variables), the developed model performs better using WLAV than the WLS estimation algorithm. The computational time for all three networks is notably less when using the WLAV algorithm than the WLS algorithm.

A weighted GPS positioning algorithm for urban canyons using dual-polarised antennae

ABSTRACT The positioning performance of the Global Positioning System (GPS) is severely degraded in urban canyons due to buildings blocking or attenuating signals. This paper proposes a weighted GPS positioning method using a dual-polarisation antenna in order to deal with this issue. The GPS signals are first classified and labelled into NLOS signals and LOS/MP signals by considering the right-hand circular polarised (RHCP) and the left-hand circular polarised (LHCP) signal strengths. Then, a Bayesian optimisation-based Gaussian process is used to fit the pseudorange error related to the input of the carrier-to-noise ratio of the RHCP QUOTE() and the elevation angle of the LOS/MP signals. The pseudorange errors of the LOS/MP signals are therefore predicted based on the rules extracted from the fitting results. The positioning solutions are then obtained from the weighted least-squares algorithm, with the weighting strategies being based on the predicted pseudorange errors. Static positioning results in deep urban areas showed that the horizontal root-mean-square error (RMSE) positioning has been improved by 64.6% and 57.3%, respectively, while 3D RMSEwas improved by 74.3% and 48.1%, respectively, compared with traditional single-point positioning results.

Asymptotic convergence of a distributed weighted least squares algorithm for networked systems with vector node variables

This paper studies the convergence properties of a recently proposed distributed algorithm for weighted least-squares (WLS) estimation in networked systems. This algorithm is suitable for large-scale networks with a vector parameter (variable) in each node of the network. By establishing the connection between this algorithm and the celebrated Gaussian Belief Propagation (BP) algorithm for statistical learning with scalar variables, asymptotic convergence of the algorithm is established under the assumption of generalised block diagonal dominance. This result generalises the known asymptotic convergence result of the Gaussian BP algorithm for networks with scalar variables. By extending the notion of diagonal dominance to block matrices, we are able to generalise the so-called walk-sum approach for convergence analysis of the Gaussian BP algorithm to this distributed WLS algorithm and show a similar asymptotic convergence property for networks with vector parameters. The significance of our work is that it gives theoretical guarantee for the distributed WLS algorithm for a new class of large-scale networked systems with vector parameters.

WLS algorithm for UAV navigation in satellite‐less environments

This work addresses one of the major challenges in the field of Unmanned Aerial Vehicles (UAVs), which is related to UAV navigation in 3-dimensional indoor environments. This topic is highly relevant due to limited-to-no availability of satellite signals that are usually the main sources used to govern the movement of UAVs in outdoor environments. Instead, the algorithm proposed in this work resorts to terrestrial radio measurements between the UAV and a set of stationary reference points to accomplish accurate navigation. The radio measurements are first exploited to give origin to range information that is used to calculate intersection points between a suitable subset of spheres, which serve as a coarse estimate of the UAV's position. From the intersection points, both azimuth and elevation angle estimates are obtained at each reference point. Finally, by taking advantage of estimated bearing information and applying simple geometry, the localisation problem is formulated according to the Weighted Least Squares (WLS) criterion and solved in closed form. The authors’ simulation results show good performance of the proposed approach, matching or even outperforming the state-of-the-art methods in terms of localisation accuracy with significantly lower computational complexity.

IpDFT-Tuned Estimation Algorithms for PMUs: Overview and Performance Comparison

The Interpolated Discrete Fourier Transform (IpDFT) is one of the most popular algorithms for Phasor Measurement Units (PMUs), due to its quite low computational complexity and its good accuracy in various operating conditions. However, the basic IpDFT algorithm can be used also as a preliminary estimator of the amplitude, phase, frequency and rate of change of frequency of voltage or current AC waveforms at times synchronized to the Universal Coordinated Time (UTC). Indeed, another cascaded algorithm can be used to refine the waveform parameters estimation. In this context, the main novelty of this work is a fair and extensive performance comparison of three different state-of-the-art IpDFT-tuned estimation algorithms for PMUs. The three algorithms are: (i) the so-called corrected IpDFT (IpDFTc), which is conceived to compensate for the effect of both the image of the fundamental tone and second-order harmonic; (ii) a frequency-tuned version of the Taylor Weighted Least-Squares (TWLS) algorithm, and (iii) the frequency Down-Conversion and low-pass Filtering (DCF) technique described also in the IEEE/IEC Standard 60255-118-1:2018. The simulation results obtained in the P Class and M Class testing conditions specified in the same Standard show that the IpDFTc algorithm is generally preferable under the effect of steady-state disturbances. On the contrary, the tuned TWLS estimator is usually the best solution when dynamic changes of amplitude, phase or frequency occur. In transient conditions (i.e., under the effect of amplitude or phase steps), the IpDFTc and the tuned TWLS algorithms do not clearly outperform one another. The DCF approach generally returns the worst results. However, its actual performances heavily depend on the adopted low-pass filter.

Modified MM Algorithm and Bayesian Expectation Maximization-Based Robust Localization Under NLOS Contaminated Environments

Robust localization methods that employ distance measurements to predict the position of an emitter are proposed in this paper. The occurrence of outliers due to the non-line-of sight (NLOS) propagation of signals can drastically degrade the localization performance in crowded urban areas and indoor situations. Hence, robust positioning methods are considered to mitigate the effects of outliers. Specifically, localization methods based on robust statistics are considered. Modified multi-stage ML-type method (MM) based weighted least squares (WLS), maximum a posteriori (MAP) expectation maximization (EM) WLS and variational Bayes (VB) EM WLS algorithms are developed under various outlier-contaminated environments. Simulation results show that the position estimation accuracy of the proposed modified MM WLS method, which uses the novel weight, is higher than that of the other methods under most outlier-contaminated conditions. Furthermore, the MAP-EM WLS and VB-EM WLS methods are the most accurate among algorithms that do not require statistical testing. Additionally, the mean square error (MSE) and asymptotic unbiasedness of the proposed algorithms are analyzed.

Convergence of Self-Tuning Regulators under Conditional Heteroscedastic Noises with Unknown High-Frequency Gain

In the classical theory of self-tuning regulators, it always requires that the conditional variances of the systems noises are bounded. However, such a requirement may not be satisfied when modeling many practical systems, and one significant example is the well-known ARCH (autoregressive conditional heteroscedasticity) model in econometrics. The aim of this paper is to consider self-tuning regulators of linear stochastic systems with both unknown parameters and conditional heteroscedastic noises, where the adaptive controller will be designed based on both the weighted least-squares algorithm and the certainty equivalence principle. The authors will show that under some natural conditions on the system structure and the noises with unbounded conditional variances, the closed-loop adaptive control system will be globally stable and the tracking error will be asymptotically optimal. Thus, this paper provides a significant extension of the classical theory on self-tuning regulators with expanded applicability.

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.

Doppler-aided positioning in GNSS receivers - A performance analysis

The main objective of Global Navigation Satellite Systems (GNSS) is to precisely locate a receiver based on the reception of radio-frequency waveforms broadcasted by a set of satellites. Given delayed and Doppler shifted replicas of the known transmitted signals, the most widespread approach consists in a two-step algorithm. First, the delays and Doppler shifts from each satellite are estimated independently, and subsequently the user position and velocity are computed as the solution to a Weighted Least Squares (WLS) problem. This second step conventionally uses only delay measurements to determine the user position, although Doppler is also informative. The goal of this paper is to provide simple and meaningful expressions of the positioning precision. These expressions are analysed with respect to the standard WLS algorithms, exploiting the Doppler information or not. We can then evaluate the performance improvement brought by a joint frequency and delay positioning procedure. Numerical simulations assess that using Doppler information is indeed effective when considering long observation times, and particularly useful in challenging scenarios such as urban canyons (constrained satellite visibility) or near indoor situations (weak signal conditions which need long integration times), thus providing new insights for the design of robust and high-sensitivity receivers.

Time Difference of Arrival (TDoA) Localization Combining Weighted Least Squares and Firefly Algorithm

Time difference of arrival (TDoA) based on a group of sensor nodes with known locations has been widely used to locate targets. Two-step weighted least squares (TSWLS), constrained weighted least squares (CWLS), and Newton–Raphson (NR) iteration are commonly used passive location methods, among which the initial position is needed and the complexity is high. This paper proposes a hybrid firefly algorithm (hybrid-FA) method, combining the weighted least squares (WLS) algorithm and FA, which can reduce computation as well as achieve high accuracy. The WLS algorithm is performed first, the result of which is used to restrict the search region for the FA method. Simulations showed that the hybrid-FA method required far fewer iterations than the FA method alone to achieve the same accuracy. Additionally, two experiments were conducted to compare the results of hybrid-FA with other methods. The findings indicated that the root-mean-square error (RMSE) and mean distance error of the hybrid-FA method were lower than that of the NR, TSWLS, and genetic algorithm (GA). On the whole, the hybrid-FA outperformed the NR, TSWLS, and GA for TDoA measurement.

Least squares orbit estimation including atmospheric density uncertainty consideration

Density uncertainty is the major driver of unrealistic covariances for objects in low Earth orbits. The analytic propagation of uncertainties in the neutral atmospheric density to resulting uncertainties in the orbital position and velocity has only received little attention in the literature so far.The main contribution of the paper at hand is the analytic development of an orbital state-vector error variance-covariance matrix that models the propagation of uncertainties in atmospheric density to the orbital state-vector error in the Geocentric Celestial Reference Frame (GCRF). Also extensions of the classical batch weighted least squares (WLS) and the sequential batch weighted least squares algorithms, which allow to incorporate this covariance matrix as process-noise, are presented.Numerical simulations with three different semi-empirical models are provided to validate the derivations. It is shown that the extension of the WLS-algorithm in combination with the density uncertainty GCRF covariance matrix is able to consistently perform orbit and covariance estimation, which is not the case without density uncertainty consideration in a classical WLS algorithm.

Robust LMedS-Based WLS and Tukey-Based EKF Algorithms Under LOS/NLOS Mixture Conditions

In this paper, we present robust localization algorithms that use range measurements. The least median of squares (LMedS)-weighted least squares (WLS), LMedS-spherical simplex unscented transform (SSUT) based WLS and Tukey-based extended Kalman filter (EKF) algorithms are proposed for line-of-sight (LOS)/non-line-of-sight (NLOS) mixture environments. First, the LMedS solution is obtained, and then sensors are predicted to be LOS or LOS/NLOS mixture sensors. The range observation predicted as an outlier is replaced with the estimated distance obtained using the LMedS algorithm. Subsequently, the two-step WLS method is executed using these new distance measurements. In the Tukey-based EKF method, Tukey's risk function and the 3-σ edit rule are employed in the innovation step. Furthermore, the mean square error (MSE) analysis of the proposed algorithms is performed. We demonstrate that the positioning accuracy of the proposed methods is higher than that of conventional methods through extensive simulation.

Measurement Data Fusion Based on Optimized Weighted Least-Squares Algorithm for Multi-Target Tracking

The outliers remove, the classification of effective measurements, and the weighted optimization method of the corresponding measurement are the main factors that affect the positioning accuracy based on range-based multi-target tracking in wireless sensor networks. In this paper, we develop an improved weighted least-square algorithm based on an enhanced non-naive Bayesian classifier (ENNBC) method. According to the ENNBC method, the outliers in the measurement data are removed effectively, dataset density peaks are found quickly, and remaining effective measurements are accurately classified. The ENNBC method improves the traditional direct classification method and took the dependence among continuous density attributes into account. Four common indexes of classifiers are used to evaluate the performance of the nine methods, i.e., the normal naive Bayesian, flexible naive Bayesian (FNB), the homologous model of FNB (FNB ROT ), support vector machine, k-means, fuzzy c-means (FCM), possibilistic c-means, possibilistic FCM, and our proposed ENNBC. The evaluation results show that ENNBC has the best performance based on the four indexes. Meanwhile, the multi-target tracking experimental results show that the proposed algorithm can reduce the root-mean-squared error of the position compared with the extended Kalman filter. In addition, the proposed algorithm has better robustness against large localization and tracking errors.

WLS Localization Using Skipped Filter, Hampel Filter, Bootstrapping and Gaussian Mixture EM in LOS/NLOS Conditions

We present robust range-based localization algorithms for which range measurements are used to estimate the location parameter. Non-line-of-sight (NLOS) propagation of signal can deteriorate the estimation performance severely in the indoor and crowded urban areas. A study for localization has been intensively performed in the line-of-sight (LOS) conditions, but the work for the positioning in the mixed LOS/NLOS environments is comparatively rare. Thus, we aim at the robust localization in the LOS/NLOS mixture environments. The Hampel and skipped filters-based weighted least squares (WLS) methods are proposed for situations where the variance for inliers is unknown in LOS/NLOS mixture environments. For the unsupervised clustering algorithm, Gaussian mixture expectation maximization-based WLS algorithm is utilized. It is demonstrated that the positioning accuracy of the proposed methods is higher than that of conventional methods through extensive simulation.

Effects of nuisance variables selection on target localisation accuracy in multistatic passive radar

The target localisation problem is considered using the bistatic range measurement in multistatic passive radar. The well-known two-step weighted least-squares algorithm is widely used to locate a single target, in which the nuisance variables are normally introduced in the first step. Whereas the nuisance variables can be categorised as either target–transmitter ranges based or target–receiver ranges based. The influence of nuisance variables selection on target positioning accuracy is investigated in this Letter. Some basic criteria on choosing nuisance variables are presented and validated through simulations.

Inverse synthetic aperture radar imaging of maneuvering target with distributed high resolution radars

The high-resolution inverse synthetic aperture radar (ISAR) imaging of the maneuvering target with small-angle measurements is expected, which can be achieved by using the radar operating at a higher central frequency. However, because the defocus induced by the spatially variant phase error becomes significant with the resolution improvement and the central frequency increasing, the global phase correction is invalid. Moreover, due to the scattering characteristics of aspect-dependent scatterers and small-angle measurements, we can rarely obtain the rough outline of the maneuvering target. To deal with these difficulties, we propose an ISAR imaging method for the maneuvering target with distributed high resolution radars. To mitigate the spatially variant phase error, we correct phase errors in patches based on the weighted least-squares algorithm, which makes the image well-focused. Then, the ISAR imaging method with distributed radars is presented to obtain the fusion image, with the object to capture enough scatterers to represent the maneuvering target. The fusion image could provide more details. Real data have verified the effectiveness of the proposed method.