Weighted Least-squares Method Research Articles

State Estimation of Radial Distribution Systems Based on Multiple Legendre Neural Networks

The conventional weighted least square (WLS) method is the most effective technique used in the state estimation of high voltage transmission system. Unfortunately, the application of WLS in radial distribution network encounter difficulties due to the inherent characteristics of these systems, such as the low measurement redundancy and high r/x ratio of the distribution systems. Given the structure of bulky systems that require a bulky number of measurements, the use of artificial neural networks is considered an effective alternative to estimate these values using a lesser number of measurements than conventional techniques. Due to state estimation based on ANN technique, the time-consuming gain matrix manipulation and pseudo measurements required in the conventional WLS method are no longer necessary. The efficiency of deep learning neural networks such as multi-layer perceptron (MLP) and Legendre neural network (LeNN) depends on the position of the measurements and the number of neural networks applied. Determining the applicable number of neural networks to ensure high estimation accuracy plays an important role in the estimation process. This aspect is addressed in this study, where multiple neural networks are used to improve performance compared to a single neural network. The results obtained indicate that determining the applicable number of neural networks depends on several factors such as the position of the measurements and the diversity of the data. The application of LeNN on state estimation of a 69-bus radial distribution network is used as an illustrative example to explain the distinctive feature of the proposed technique.

An Optimal Strategy for Estimating Weibull distribution Parameters by Using Maximum Likelihood Method

Several methods have been used to estimate the Weibull parameters such as least square method (LSM), weighted least square method (WLSM), method of moments (MOM), and maximum likelihood (MLE). The maximum likelihood method is the most popular method (MLE). Newton-Raphson method has been applied to solve the normal equations of MLE’s in order to estimate the Weibull parameters. The method was used to find the optimal values of the Weibull distribution parameters for which the log-likelihood function is maximized. We tried to find the approximation solution to the normal equations of the MLE’s because there is no close form for get analytical solution. In this work, we tried to carry out a study that show the difference between two strategies to solve the MLE equations using Newton-Raphson algorithm. Both two strategies are provided an optimal solution to estimate the Weibull distribution parameters but which one more easer and which one converges faster. Therefore, we applied both strategies to estimate the Weibull’s shape and scale parameters using two different types of data (Real and simulation). We compared between the results that we got by applying the two strategies. Two studies have been done for comparing and selecting the optimal strategy to estimate Weibull distribution parameters using maximum likelihood method. We used some measurements to compare between the results such as number of steps for convergence (convergence condition), the estimated values for AIC, BIC and the RMSE value. The results show the numerical solution that we got by applying first strategy convergence faster than the solution that we got by applying second strategy. Moreover, the MRSE estimated by applying the first strategy is lower than the MRSE estimated by applying second strategy for the simulation study with different noise levels and different samples size.

Comparison of Weights in Weighted Least Square Method For Handling Heteroscedasticity on Multiple Regression Model

Regression analysis is the most popular and commonly used to determine causality between two or more variables. In regression analysis there are several assumptions that must be held, so that the property of the best linear unbiased estimator (BLUE) is still guaranteed. In fact, we often found violations of the assumptions. One of them was violations of the homoscedasticity or occurs heteroscedasticity. The impact of heteroscedasticity in the regression model is that the ordinary least square (OLS) estimator no longer has a minimum variance although still linear and unbiased. To handle this, weighted least square (WLS) regression is used instead, which giving weights on the observations. But the problem often encountered is choosing which the best weight in WLS method. This paper aimed to compare and determine the best weight among 1/X, 1/ , 1/Y and 1/s in multiple regression model. Human development index factors data, which were obtained from the Indonesian Central Bureau of Statistics, were used. The results showed that the best weight on human development index data was 1/s. The coefficient of determination was 98.7% indicating that the model was very good.

Three-Dimensional Signal Source Localization with Angle-Only Measurements in Passive Sensor Networks

Some passive sensors can provide only relative angles of a signal source. To obtain the signal source location, multiple passive sensors can be constructed into a passive sensor network through communication links. This paper investigates the source localization problem with angle-only measurements in three-dimensional space. First, we present an intersection localization method, which estimates the target position by minimizing the sum of distances between lines formed by angle-only measurements. It has the same target position estimate as the widely used least-squares (LS) method, but with a lower computational cost. Furthermore, considering the differences in measurement accuracy of sensors, the weighted least-squares (WLS) algorithm can achieve better localization performance than the LS method. Unfortunately, since the coefficient matrix and the noise vector are correlated, the WLS method is biased. The bias-compensation WLS (BCWLS) method is also presented in this paper to reduce the bias by estimating the correlation between the coefficient matrix and the pseudolinear noise vector. To evaluate the performance of the presented algorithms, numerical simulations are conducted, indicating that the superiority of the intersection localization method in computational cost and the superiority of the BCWLS method in localization accuracy.

A Weighted Least-Squares Method for Non-Asymptotic Identification of Markov Parameters from Multiple Trajectories

Markov parameters play a key role in system identification. There exists many algorithms where these parameters are estimated using least-squares in a first, pre-processing, step, including subspace identification and multi-step least-squares algorithms, such as Weighted Null-Space Fitting. Recently, there has been an increasing interest in non-asymptotic analysis of estimation algorithms. In this contribution we identify the Markov parameters using weighted least-squares and present non-asymptotic analysis for such estimator. To cover both stable and unstable systems, multiple trajectories are collected. We show that with the optimal weighting matrix, weighted least-squares gives a tighter error bound than ordinary least-squares for the case of non-uniformly distributed measurement errors. Moreover, as the optimal weighting matrix depends on the system’s true parameters, we introduce two methods to consistently estimate the optimal weighting matrix, where the convergence rate of these estimates is also provided. Numerical experiments demonstrate improvements of weighted least-squares over ordinary least-squares in finite sample settings.

Robust localisation methods based on modified skipped filter weighted least squares algorithm

AbstractRobust localisation techniques that utilise distance observations to determine the location are focused upon. In urban environments with limited visibility and high population density, the presence of non‐line‐of‐sight signals can introduce a positive measurement bias, negatively affecting the accuracy of estimation. To resolve this problem caused by multipath effects, robust localisation techniques have been explored, specifically the skipped filter weighted least squares (WLS) method for localisation. However, the squared estimation bias of the transformed distance estimate of the existing skipped filter WLS method is high in the low signal‐to‐noise ratio condition owing to the second‐order noise terms. Therefore, the modified skipped filter WLS methods are proposed to reduce the squared estimation bias of transformed distance estimate. First, the closed‐form modified skipped filter WLS method uses the maximum likelihood estimate (MLE) to reduce the squared estimation bias of the transformed distance estimate. In addition, the modified skipped filter WLS method using the online ML and online expectation maximisation (EM) algorithms are introduced whose advantage is that they do not require the number of Gaussian components unlike the existing Gaussian mixture model EM algorithm. The mean square error analysis of proposed closed‐form skipped filter WLS and existing skipped filter WLS methods is performed. Furthermore, the localisation accuracy of the proposed techniques is found to outperform that of competing algorithms via simulation results.

A novel state estimation method for distribution networks with low observability based on linear AC optimal power flow model

The state estimation (SE) problem is one of the key functions in real-time control and management of modern electrical distribution networks. However, due to the complex network configuration and the limited number of measuring devices in these networks, the SE problem faces many challenges. In this manuscript, a new state estimation methodology is proposed based on the AC optimal power flow (ACOPF) model for distribution networks with a small number of real-time measurements. The proposed method is formulated in the form of a mixed integer linear programming problem (MILP), where the estimated states are regarded as the decision variables in the optimization problem. The load consumption values are also assumed as the decision variables where the estimation limits of these variables are determined by pseudo-measurements taken from the historical data. The main objective function is to maximize the state estimation accuracy, while the power flow equations are defined as the problem constraints. The efficiency of the proposed methodology is evaluated by implementing on the test networks and comparing the estimation results accuracy with the conventional weighted least squares (WLS) method. The obtained numerical results show the superior performance of OPF-based SE model to WLS method. Moreover, linearizing the formulation demonstrates an acceptable precision while ensuring convergence and optimum solution in low observability conditions.

Research on evaluation index method of cloud-network convergence capability

There is no measurable and evaluable index system for cloud-network convergence that provides guidance and reference for the subsequent construction and development of cloud-network convergence. It is a big project to select and evaluate the indexes of cloud-network convergence, which requires suitable index selection and index evaluation schemes. Based on analytic hierarchy process (AHP) and entropy weight method, this paper proposes an improved AHP (i-AHP) index selection scheme and index evaluation scheme leveraging the years of experts’ experience, the geometric mean and the least square method. The improved weighted least square method (WLSM) is finally proved to be more stable for index evaluation scheme by adding abnormal data. In addition, the index weight obtained by the index evaluation scheme with WLSM are provided as a reference for the future development of cloud-network convergence. The simulation results show that the proposed scheme is superior to the existing index evaluation scheme and can avoid the weight deviation caused by the disturbance and fluctuation of abnormal data.

The Weighted Least-Squares Collocation Method for Elastic Wave Obstacle Scattering Problems

Scattering problems have wide applications in the medical and military fields. In this paper, the weighted least-squares (WLS) collocation method based on radial basis functions (RBFs) is developed to solve elastic wave scattering problems, which are governed by the Navier equation and the Helmholtz equations with coupled boundary conditions. The perfectly matched layer (PML) technique is used to truncate the unbounded domain into a bounded domain. The WLS method is constructed by setting the collocation points denser than the trial centers and imposing different weights on different types of boundary conditions. The WLS method can overcome the matrix singularity problem encountered in the Kansa method, and the convergence rate of WLS is [Formula: see text] for Sobolev kernel with kernel smoothness [Formula: see text]. Furthermore, compared with the finite element method (FEM) and the Kansa method, WLS can provide higher accuracy and more stable solutions for relatively large angular frequencies. The numerical example with a circular obstacle is used to verify the effectiveness and convergence behavior of the WLS. Besides, the proposed scheme can easily handle irregular obstacles and obtain stable results with high accuracy, which is validated through experiments with ellipse and kite-shaped obstacles.

Automated component analysis in DOSY NMR using information criteria

This study introduces a model selection technique based on Bayesian information criteria for estimating the number of components in a mixture during Diffusion-Ordered Spectroscopy (DOSY) Nuclear Magnetic Resonance (NMR) data analysis. As the accuracy of this technique is dependent on the efficiency of parameter estimators, we further investigate the performance of the Weighted Least Squares (WLS) and Maximum a Posteriori (MAP) estimators. The WLS method, enhanced with meticulously tuned L2-regularization, effectively detects components when the difference in self-diffusion coefficients is more than two-fold, especially when the component with the smaller coefficient has a larger weight ratio. The MAP method, strengthened by a substantial database of prior information, exhibits outstanding precision, decreasing this threshold to 1.5 times. Both estimators provide weight ratio estimates with standard deviations of approximately around 1 percentage point, although the MAP method tends to overestimate the component with a larger self-diffusion coefficient. Deviations from the expected values can exceed 10 percentage points, often due to inaccuracies in component detection. The error estimates are determined using data resampling techniques derived from a large-scale 1000-point experiment and an additional five measurements from a single-component mixture. This approach allowed us to thoroughly examine data distribution characteristics, thereby laying a robust groundwork for future refinement efforts.

Elliptic Localization of a Moving Object by Transmitter at Unknown Position and Velocity: A Semidefinite Relaxation Approach

This paper investigates the elliptic localization for moving object problem from time delay (TD) and Doppler frequency shift (DFS) measurements, where the transmitter position and velocity are unknown. The transmitter is not perfectly time syncronized such that unknown offsets exist in the TD and DFS measurements. We propose to jointly estimate the object and transmitter positions and velocities and the offsets. Using the TD and DFS measurements from both the indirect and direct paths between the transmitter and the receivers, we formulate a non-convex weighted least squares (WLS) problem. Local convergence may occur when solving the non-convex WLS problem, implying that good estimate is not guaranteed. Thus, we relax the non-convex WLS problem into a convex semidefinite program by applying semidefinite relaxation (SDR). Moreover, we theoretically show that the performance can be improved by using multiple transmitters as compared to that using single transmitter, although more unknown parameters are introduced. We then extend the proposed SDR method to handle the multiple transmitters case. Finally, the mean square error analysis is provided to show that the proposed WLS method reaches the Cramer-Rao lower bound accuracy under small Gaussian noise condition. Simulation results validate the theoretical analysis and show the superior performance over the existing methods.

Closed-Form Rectilinear Emitters Localization With Multiple Sensor Arrays: Phase Alignment and Optimal Weighted Least-Square Method

Position determination algorithms of rectilinear emitters (also called strictly second-order noncircular emitters) with multiple sensor arrays (SAs) allow for higher localization accuracy and more achievable degrees of freedom compared to the conventional version with circular emitters. However, it also brings additional computational complexity and noise. In this article, we develop a position determination algorithm of rectilinear emitters in closed-form on the basis of phase alignment and optimal weighted least square (OWLS), which is termed the PAOWLS method. First, we utilize the polynomial root-finding method to obtain the angle of arrival (AOA) and estimate rectilinear phases to align AOA. Then, we construct a least-square (LS) constraint to estimate the positions of emitters directly (termed the PALS method). Finally, we construct a weighted LS constraint and deduce the optimal weight to compensate for the position estimation bias. Extensive numerical results verify the outstanding performance of the proposed algorithm in terms of localization accuracy, resolution capability, and computational complexity.

Distribution System State Estimation and False Data Injection Attack Detection with a Multi-Output Deep Neural Network

Distribution system state estimation (DSSE) has been introduced to monitor distribution grids; however, due to the incorporation of distributed generations (DGs), traditional DSSE methods are not able to reveal the operational conditions of active distribution networks (ADNs). DSSE calculation depends heavily on real measurements from measurement devices in distribution networks. However, the accuracy of real measurements and DSSE results can be significantly affected by false data injection attacks (FDIAs). Conventional FDIA detection techniques are often unable to identify FDIAs into measurement data. In this study, a novel deep neural network approach is proposed to simultaneously perform DSSE calculation (i.e., regression) and FDIA detection (i.e., binary classification) using real measurements. In the proposed work, the classification nodes in the DNN allow us to identify which measurements on which phasor measurement unit (PMU), if any, were affected. In the proposed approach, we aim to show that the proposed method can perform DSSE calculation and identify FDIAs from the available measurements simultaneously with high accuracy. We compare our proposed method to the traditional approach of detecting FDIAs and performing SE calculations separately; moreover, DSSE results are compared with the weighted least square (WLS) algorithm, which is a common model-based method. The proposed method achieves better DSSE performance than the WLS method and the separate DSSE/FDIA method in presence of erroneous measurements; our method also executes faster than the other methods. The effectiveness of the proposed method is validated using two FDIA schemes in two case studies: one using a modified IEEE 33-bus distribution system without DGs, and the other using a modified IEEE 69-bus system with DGs. The results illustrated that the accuracy and F1-score of the proposed method are better than when performing binary classification only. The proposed method successfully detected the FDIAs on each PMU measurement. Moreover, the results of DSSE calculation from the proposed method has a better performance compared to the regression-only method, and the WLS methods in the presence of bad data.

TDOA –FDOA method for tracking of a moving target in a distributed sensor scenario

Tracking of an underwater target in ocean environment is very important in both military and civilian applications. This paper focuses on the localization and tracking of a moving target by a passive multi-static sonar system using TOA, Time Difference and Frequency Difference of Arrival measurements. An algorithm is developed for an acoustic sensor distributed network of five sensors and an acoustic moving source. TDOAFDOA method is used to track a moving source usingTime Difference of Arrival (TDOA) and Frequency Difference of Arrival (FDOA) measurements, as a closed form solution. The highly nonlinear measurement equations of TDOA and FDOA are first converted into pseudo linear equations. This is achieved by introducing extra parameters. These equations are solved using a weighted least squares (WLS) method. Five distributed stationary receivers are used to track a moving acoustic target. The moving target parameters estimated from the above WLS method are improved by an Unscented Kalman Filter. The mean square errors of these estimates are compared and the results are analyzed.

Received Signal Strength‐Based Localization for Vehicle Distance Estimation in Vehicular Ad Hoc Networks (VANETs)

Vehicular ad hoc networks (VANETs) are an eminent area of intelligent transportation systems (ITS) which includes vehicle tracking, positioning, and emergency warnings. For protective applications, vehicle localization in the urban area is a significant problem. Global Positioning Systems (GPS) are one of the many solutions that have been offered, although they do not offer accuracy. To locate a target vehicle accurately, a unique approach called received signal strength‐ (RSS‐) based localization scheme has been proposed. By detecting signals within its range, it establishes communication through roadside units (RSUs) and determines the typical RSS. Following the discovery of the RSS, the RSS‐based localization algorithm helped in determining the precise position of the vehicle. The high signal‐to‐noise ratio of the proposed algorithm, which is derived from neighbouring RSUs, is its key component. The vehicle’s Cramer Rao lower bound (CRLB) is examined after its position is discovered. Various experimentations considering dynamic vehicles ranging from 2 to 100, localization error, ranging error, and throughput have been conducted which demonstrates that the proposed algorithm (RSS) has better results (87%, 80%, and 120) than other well‐established least squares (LS) and weighted least squares methods (WLS).

Analisis Pendapatan Menggunakan Metode Weighted Least Square (Wls) dengan Fungsi Pembobot Huber

The Weighted Least Square (WLS) method is a development of the OLS method which will provide a more accurate solution than the OLS method when outliers are identified in the data. It is possible that the model generated by the OLS method still contains outliers, while the WLS method minimizes outliers in the data. The income result determined using the OLS method is less than the income result using the WLS method, which means that if a fixed value is given (according to the standard) for each result of the factors that affect the income result in the OLS method, the average fisherman income is IDR 2,225 .220. While the average income of fishermen using the WLS method is Rp. 1,015,840. There were two outliers identified using the OLS method and after using the WLS method there were no outliers.

Time-difference-of-arrival positioning based on visible light communication for harbor-border inspection

In view of the complexity of port ship supervision and the influence of external factors such as electromagnetic interference in harbor-border inspection, an efficient system combining an unmanned aerial vehicle (UAV) and visible light positioning (VLP) is proposed for locating maritime targets. In this system, a rotatable receiver with five photodetectors (PDs) installed obliquely on UAV is designed for expanding the positioning range and allowing a lower flight altitude. On this basis, we propose the Chan-Taylor (CT) method based on time difference of arrival (TDOA) for target positioning. First, the localization problem is reformulated as a weighted least squares (WLS) problem and provides a good initial estimate via the two-step WLS (TWLS) method. Then, based on Taylor expansion of TDOA equations, estimated error is calculated using the initial estimate, which can correct the estimated position of the target iteratively. To offset the error, weighted centroid CT (WCCT) is proposed by endowing different weights based on error difference to estimated results. For further improving accuracy, a restricted-region fingerprinting positioning based on CT (CT-RFP) is proposed. In restricted area determined by CT, a certain number of fingerprints is generated based on received signal strength (RSS) for matching. Simulation results show that CT is significantly improved over the previous methods. Compared with TWLS, the accuracy of CT is improved by 49.71%. For WCCT, the maximum error is reduced from 8.65 to 6.91 cm, which effectively reduces the influence of error. Moreover, CT-RFP can achieve an accuracy within millimeter level via the appropriate number of fingerprints and ensemble runs of CT, even at high noise levels.

Exponential Transformed Inverse Rayleigh Distribution: Statistical Properties and Different Methods of Estimation

In this article a generalization of the inverse Rayleigh distribution has been addressed by using DUS transformation, named as Exponential Transformed Inverse Rayleigh (ETIR) distribution. Some of the statistical properties of this newly proposed distribution like mode, quantiles, moment, moment generating function, survival and hazard rate function have been studied comprehensively. To estimate the parameter of this distribution, four different estimation procedures, such as maximum likelihood estimation (MLE), maximum product spacing method (MPS), least square method (LSE) and weighted least square method (WLSE) are briefly discussed. Performance of these estimates are compared using extensive simulations. As an application point of view the model superiority is verified through two real datasets.

Bistatic Radar Observations Correlation of LEO Satellites Considering J2 Perturbation

Space debris near Earth severely interferes with the development of space, and cataloging space objects is increasingly important. Since optical telescopes and radars used to detect space debris only provide short-arc observations, mathematical algorithms are needed to solve problems in the correlation of observations. In this work, an efficient mathematical algorithm based on J2 analytic solutions is put forward. Initial orbit determination (IOD) serves as the starter and orbit determination (OD) with the weighted least-squares method (WLSM) is used to improve the accuracy of the estimated orbit. Meanwhile, the effect of the weight of different observation types is analyzed. The correlation criteria for bistatic radar observations are accordingly developed. Lastly, the variation in and evolution of the error of bistatic radar ranging are discussed.

A multi-objective programming approach to Weibull parameter estimation

Weibull distribution is widely used in various areas such as life tables, failure rates, and definition of wind speed distribution. Therefore, parameter estimation for the Weibull distribution is an important problem in many real data applications. The least square (LS), the weighted least square (WLS) and the maximum likelihood (ML) are the most popular methods for the parameter estimation in the Weibull distribution. In this study, based on the LS, WLS and ML estimation methods, a multi-objective programming approach is proposed for the parameter estimation of two-parameter Weibull distribution. This new approach evaluates together LS, WLS and ML methods in the estimation process. NSGA-II method, which is a multi-objective heuristic optimization method, is used to solve the proposed multi-objective estimation model. To evaluate the applicability and performance of the proposed approach, a detailed Monte Carlo simulation study based on deficiency criteria and a real data application are designed. The results illustrated that the proposed multi-objective programming approach provides quite accurate parameter estimates for the two parameter Weibull distribution with respect to deficiency criteria.