Weighted Least Square Estimation Research Articles

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 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.

Linear-Array-MIMO SAR Tomography: An Autofocus Approach for Time-Variant and 3-D Space-Variant Motion Errors

Linear-array multiple-input–multiple-output (LA-MIMO) synthetic aperture radar (SAR) can obtain 3-D radar images by only one pass. However, it is sensitive to time-variant measurement errors of curved track and time-variant attitude angles, meaning that autofocus processing for the LA-MIMO SAR tomography is necessary. The existing autofocus methods cannot be used to estimate the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">time-variant</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3-D space-variant</i> motion errors (3-D SVME) of the LA-MIMO SAR. To solve this problem, a new autofocus approach based on multiple local autofocusing and the LA-MIMO SAR time-variant motion error estimation is proposed. First, the local motion error estimation based on the fast local spectral analysis (SPECAN) 3-D imaging and the maximum contrast optimization 2-D local autofocusing is performed to estimate the local time-variant motion errors. Then, based on the linear-array motion error model, the time-variant 3-D trajectory deviations of the array center and attitude angles are estimated by the weighted least square estimation (WLSE) to solve the 3-D SVMEs. Last, the 3-D fast factorized backprojection (FFBP) is performed to obtain the well-focused 3-D image of the whole beam. The proposed approach has been applied for the tomography of a new crawler-type unmanned-ground-vehicle (UGV) LA-MIMO SAR. Both the simulation and real data experiments verify the effectiveness of the proposed approach.

An Assessment of Incorporating Log-Logistic Testing Effort Into Imperfect Debugging Delayed S-Shaped Software Reliability Growth Model

Software reliability growth models (SRGM) are employed to aid us in predicting and estimating reliability in the software development process. Many SRGM proposed in the past claim to be effective over previous models. While some earlier research had raised concern regarding use of delayed S-shaped SRGM, researchers later indicated that the model performs well when appropriate testing-effort function (TEF) is used. This paper proposes and evaluates an approach to incorporate the log-logistic (LL) testing-effort function into delayed S-shaped SRGMs with imperfect debugging based on non-homogeneous Poisson process (NHPP). The model parameters are estimated by weighted least square estimation (WLSE) and maximum likelihood estimation (MLE) methods. The experimental results obtained after applying the model on real data sets and statistical methods for analysis are presented. The results obtained suggest that performance of the proposed model is better than the other existing models. The authors can conclude that the log-logistic TEF is appropriate for incorporating into delayed S-shaped software reliability growth models.

Stress-Strength Reliability for Exponentiated Inverted Weibull Distribution with Application on Breaking of Jute Fiber and Carbon Fibers.

For the first time and by using an entire sample, we discussed the estimation of the unknown parameters θ1, θ2, and β and the system of stress-strength reliability R=P(Y < X) for exponentiated inverted Weibull (EIW) distributions with an equivalent scale parameter supported eight methods. We will use maximum likelihood method, maximum product of spacing estimation (MPSE), minimum spacing absolute-log distance estimation (MSALDE), least square estimation (LSE), weighted least square estimation (WLSE), method of Cramér-von Mises estimation (CME), and Anderson-Darling estimation (ADE) when X and Y are two independent a scaled exponentiated inverted Weibull (EIW) distribution. Percentile bootstrap and bias-corrected percentile bootstrap confidence intervals are introduced. To pick the better method of estimation, we used the Monte Carlo simulation study for comparing the efficiency of the various estimators suggested using mean square error and interval length criterion. From cases of samples, we discovered that the results of the maximum product of spacing method are more competitive than those of the other methods. A two real‐life data sets are represented demonstrating how the applicability of the methodologies proposed in real phenomena.

Data-Quality Aware Distribution State Estimation Using Maximum Normal Measurement Rate

State estimation application in the distribution system is being revisited with the availability of additional sensor data enabled by the distribution automation and massive sensors installation in the electric power grid. Some of these sensor data have measurement error, data quality problems, and different reporting rate. The traditional weighted least square estimation (WLSE) method is used to find best fit between the system model and the measured data. WLSE will typically give low performance and poor accuracy when some measurements have excessively large error compared to the rest of the measurements. This work proposes a maximum normal measurement rate (MNMR) based state estimation with simultaneous bad data detection method and considering three-phase unbalanced systems. The simulation results for an IEEE test system and an industrial feeder demonstrate the MNMR-based state estimation provides better performance even in the presence of multiple bad measurement data.

On Generalized Linear Exponential Distribution: Different Methods of Estimation

This paper concerns with various techniques for estimations from the generalized linear exponential distribution (GLED) that can be used for modeling bathtub, increasing and decreasing hazard rate (HR) behavior and was first proposed by [3]. This distribution is important since it contains as special sub-models some widely well-known distributions such as the exponential distribution (ED), the Rayleigh distribution (RD), the linear exponential distribution (LED), and the Weibull distribution (WD). The various techniques for estimations can be considered as maximum likelihood estimation (MLE), least-square estimation (LSE), weighted least square estimation (WLSE), Cramer Von-Mises estimation (CVME), and Anderson Darling estimation (ADE). These methods of estimations are used to estimate the unknown parameters of the well-known GLED. Two applications are used to show that the GLED is a viable distribution in modeling lifetime data and to compare the varying methods of estimations based on the Kolmogorov-Simnorov test with the corresponding P-value to show the optimal method. Finally, a simulation study is presented to compare the varying methods of estimation based on the mean square error (MSE) and the average absolute bias (AAB).

A closed-form distributed source localisation method using TDOA and GROA

ABSTRACTThis paper proposes a closed-form distributed passive source localisation method using time difference of arrival (TDOA) and gain ratios of arrival (GROA) measurements. After removing the unknown parameters in TDOA equations by using GROA measurements, the proposed method applies two-step weighted least-square (LS) minimisations. First, the TDOA and GROA equations are transformed into a set of pseudo-linear equations by introducing additional parameters, and a weighted LS estimation is used to obtain a rough estimate; secondl, this method exploits additional parameters to refine the estimate through another weighted LS estimation. The application of weighting matrix leads to an approximate maximum likelihood estimator and produces a substantial improvement in source localisation accuracy. Both the theoretical analysis and simulation results indicate that the proposed method can achieve the Cramer–Rao Lower Bound at a moderate noise level and outperform the existing methods in terms of localisation accuracy.

Data fusion-based distributed Prony analysis

This paper presents a distributed Prony analysis algorithm using data fusion approach. This classic approach can be found in Kalman filter's measurement update. Distributed optimization algorithms, e.g., alternating direction method of multipliers (ADMM), suitable for constrained optimization problems, have been proposed in the previous literature to develop distributed architecture. In this article, we show that Prony analysis, a non-constrained least square estimation (LSE) problem, can be solved using the classic data fusion approach. Compared to the iterative distributed optimization algorithms (e.g., ADMM and subgradient methods), data fusion takes only one step. There is no need for iteration and there is no issue related to convergence. This approach leads to a distributed Prony analysis architecture which requires a much less demanding communication system (the bandwidth can be less than 0.1Hz) compared to the conventional centralized Prony analysis for multiple channels which requires a bandwidth of 30Hz. The application discussed in this paper is to identify oscillation modes from real-world phasor measurement unit (PMU) data and further reconstruct signals. A key technical challenge to implement Prony analysis for signals from multiple channels is the difficulty to identify the noise characteristics of each channel. In this paper, a method is proposed to identify the noise covariances, which leads to the construction of a weighted least square estimation (WLSE) problem. This problem is solved through a distributed architecture. The effectiveness of the proposed distributed Prony analysis is demonstrated through case study results. The accuracy of the estimation is improved in one order compared with the centralized Prony analysis.

A novel adaptive image zooming scheme via weighted least-squares estimation

A critical issue in image interpolation is preserving edge detail and texture information in images when zooming. In this paper, we propose a novel adaptive image zooming algorithm using weighted least-square estimation that can achieve arbitrary integer-ratio zoom (WLS-AIZ) For a given zooming ratio n, every pixel in a low-resolution (LR) image is associated with an n × n block of high-resolution (HR) pixels in the HR image. In WLS-AIZ, the LR image is interpolated using the bilinear method in advance. Model parameters of every n×n block are worked out through weighted least-square estimation. Subsequently, each pixel in the n × n block is substituted by a combination of its eight neighboring HR pixels using estimated parameters. Finally, a refinement strategy is adopted to obtain the ultimate HR pixel values. The proposed algorithm has significant adaptability to local image structure. Extensive experiments comparing WLS-AIZ with other state of the art image zooming methods demonstrate the superiority of WLS-AIZ. In terms of peak signal to noise ratio (PSNR), structural similarity index (SSIM) and feature similarity index (FSIM), WLS-AIZ produces better results than all other image integer-ratio zoom algorithms.

얼굴의 대칭성을 이용하여 조명 변화에 강인한 2차원 얼굴 인식 시스템 설계

In this paper, we propose Two-Dimensional Robust Face Recognition System Realized with the Aid of Facial Symmetry with Illumination Variation. Preprocessing process is carried out to obtain mirror image which means new image rearranged by using difference between light and shade of right and left face based on a vertical axis of original face image. After image preprocessing, high dimensional image data is transformed to low-dimensional feature data through 2-directional and 2-dimensional Principal Component Analysis (2D)2PCA, which is one of dimensional reduction techniques. Polynomial-based Radial Basis Function Neural Network pattern classifier is used for face recognition. While FCM clustering is applied in the hidden layer, connection weights are defined as a linear polynomial function. In addition, the coefficients of linear function are learned through Weighted Least Square Estimation(WLSE). The Structural as well as parametric factors of the proposed classifier are optimized by using Particle Swarm Optimization(PSO). In the experiment, Yale B data is employed in order to confirm the advantage of the proposed methodology designed in the diverse illumination variation

Design of K-means clustering-based polynomial radial basis function neural networks (pRBF NNs) realized with the aid of particle swarm optimization and differential evolution

In this paper, we introduce an advanced architecture of K-means clustering-based polynomial Radial Basis Function Neural Networks (p-RBF NNs) designed with the aid of Particle Swarm Optimization (PSO) and Differential Evolution (DE) and develop a comprehensive design methodology supporting their construction. The architecture of the p-RBF NNs comes as a result of a synergistic usage of the evolutionary optimization-driven hybrid tools. The connections (weights) of the proposed p-RBF NNs being of a certain functional character and are realized by considering four types of polynomials. In order to design the optimized p-RBF NNs, a prototype (center value) of each receptive field is determined by running the K-means clustering algorithm and then a prototype and a spread of the corresponding receptive field are further optimized through running Particle Swarm Optimization (PSO) and Differential Evolution (DE). The Weighted Least Square Estimation (WLSE) is used to estimate the coefficients of the polynomials (which serve as functional connections of the network). The performance of the proposed model and the comparative analysis involving models designed with the aid of PSO and DE are presented in case of a nonlinear function and two Machine Learning (ML) datasets

공간 탐색 최적화 알고리즘을 이용한 K-Means 클러스터링 기반 다항식 방사형 기저 함수 신경회로망: 설계 및 비교 해석

In this paper, we introduce an advanced architecture of K-Means clustering-based polynomial Radial Basis Function Neural Networks (p-RBFNNs) designed with the aid of SSOA (Space Search Optimization Algorithm) and develop a comprehensive design methodology supporting their construction. In order to design the optimized p-RBFNNs, a center value of each receptive field is determined by running the K-Means clustering algorithm and then the center value and the width of the corresponding receptive field are optimized through SSOA. The connections (weights) of the proposed p-RBFNNs are of functional character and are realized by considering three types of polynomials. In addition, a WLSE (Weighted Least Square Estimation) is used to estimate the coefficients of polynomials (serving as functional connections of the network) of each node from output node. Therefore, a local learning capability and an interpretability of the proposed model are improved. The proposed model is illustrated with the use of nonlinear function, NOx called Machine Learning dataset. A comparative analysis reveals that the proposed model exhibits higher accuracy and superb predictive capability in comparison to some previous models available in the literature.

A study of testing‐effort dependent inflection S‐shaped software reliability growth models with imperfect debugging

PurposeThe purpose of this paper is to investigate how to incorporate the exponentiated Weibull (EW) testing‐effort function (TEF) into inflection S‐shaped software reliability growth models (SRGMs) based on non‐homogeneous Poisson process (NHPP). The aim is also to present a more flexible SRGM with imperfect debugging.Design/methodology/approachThis paper reviews the EW TEFs and discusses inflection S‐shaped SRGM with EW testing‐effort to get a better description of the software fault detection phenomenon. The SRGM parameters are estimated by weighted least square estimation (WLSE) and maximum‐likelihood estimation (MLE) methods. Furthermore, the proposed models are also discussed under imperfect debugging environment.FindingsExperimental results from three actual data applications are analyzed and compared with the other existing models. The findings reveal that the proposed SRGM has better performance and prediction capability. Results also confirm that the EW TEF is suitable for incorporating into inflection S‐shaped NHPP growth models.Research limitations/implicationsThis paper presents the WLSE results with equal weight. Future research may be carried out for unequal weights.Practical implicationsSoftware reliability modeling and estimation are a major concern in the software development process, particularly during the software testing phase, as unreliable software can cause a failure in the computer system that can be hazardous. The results obtained in this paper may facilitate the software engineers, scientists, and managers in improving the software testing process.Originality/valueThe proposed SRGM has a flexible structure and may capture features of both exponential and S‐shaped NHPP growth models for failure phenomenon.

Power system state estimation using singular value decomposition

This paper presents a new fundamental static state estimation algorithm using weighted least square (WLS) estimation, which is based on singular value decomposition (SVD) rather than the normal equations. In addition, this paper also uses linear WLS estimation, which is quicker than non-linear. The SVD approach does not require the whole network system to be observable prior to estimation. It can provide a solution even if the system under consideration is partially observable. The simulation study was performed on the IEEE 14-bus test system. The simulation results, both linear and non-linear WLS, have shown that the SVD approach can provide a solution even when ill-conditioned occurred while the normal equation approach failed to give satisfactory results. In addition, the SVD approach can identify which parts of the network are unobservable islands.

Decoupled State-Estimation in Energy Control Centres

Abstract STATE-ESTIMATION (SE) is the process of assigning a value to an unknown system state-variable based on measurements from that system according to some criteria. Usually, the process involves imperfect measurements inherent in the process and therefore estimating the system state is based on a statistical criteria that estimates the true value of the state-variable so as to minimize or maximize the selected criteria. A commonly used and familiar criteria is that of minimizing the sum of squares of the differences between the estimated and ‘true’ (i.e. measured) values. Power System Static State-Estimation (PSSE) is a proven technique for providing modern energy control centres with reliable real-time data bases for security monitoring and on-line control. The state-estimation processes real-time raw measurements to obtain optimal estimates of voltage magnitudes and angles at all buses, which determines the static state of the Power System (PS). The estimation problem of PSSE is usually formulated as a nonlinear least-square problem. Among state estimators based on the normal equations approach, the weighted least square (WLS) estimator provides reliable estimates but is slow, mainly due to its requirement for updating and factorization of the full-matrix in every iteration. The decoupled state-estimator requires less computation time per iteration and less storage requirement than WLS. The concept of p- δ, Q-V decoupling technique as applied in load flow studies is extended to state estimation technique also which reduces the storage requirement and also the time for convergence. The most common approach to the SE problem is to formulate it as a nonlinear problem usually solved by an iterative procedure based on successive linearization; in each iteration one has to solve a linear least-squares problem. Associated with WLS estimation is a set of redundant equations are to be solved to yield the solution. These equations are ‘Normal Equations’ and are solved using Cholesky factorization. The normal equations are very satisfactory with respect to preserving sparsity, but they may give rise to numerical ill-conditioning. Methods using orthogonal transformations, such as those of Householder have far better stability properties. The Method of Data Equations which uses the Householder orthogonal transformation is more efficient than the conventional method of normal equations. Also, it is less prone to computer round-off errors and is more numerically stable. The standard approach to the solution of the WLS state estimation in power system is the iterative normal equations method. Occassional ill-conditioning has been experienced with this method. Alternative approaches to state-estimation are based on ‘orthogonal transformation’. These methods have improved numerical behaviour, but generally their sparsity suffers when the measurement redundancy is high. In this paper the decoupled estimator based on transformation method is presented using different algorithms and the results agree with the method of normal equations. Measurements are transformed into new ‘measurements’ that are functions of the states and of the original measurements. In PSSE redundancy rate is an important aspect. Estimator obtained from various redundancy rates, by considering active and reactive redundancy seperately obtained by uniformly distributing the measurements over the system is presented. Line flows and bus injection measurements required for stateestimation studies, have been obtained from load-flow computations. Error (noise) terms (σ = 0.01 and 0.001) are added to these measurements. The error vector being simulated using a random number generator. Convergence tolerance of 0.001 PU and 0.001 are used for V and δ, respectively. Results obtained for various combinations of measurements; and with different order of processing these measurements have been presented. Measurements are classified as. Type 1, 2 : Power injections. Type 3, 4 : Power flows. Type 5 : Voltage magnitudes. Different cases analysed are. i) Case 1 : 1, 2, 3, 4 & 5 Type measurements. ii) Case 2 : 1, 3, 4 and 5 Type measurements. iii) Case 3 : 1, 2, 3 and 4 Type measurements. iv) Case 4 : 3 and 5 Type measurements. It is important that at least few voltage measurements (NBUS/ 10 to NBUS/5) must be included while estimating the states. When line-flows only are used as measurements, solution takes large number of iterations to converge. Comparative results obtained from ‘method of Normal equations’ and ‘Data equations’ have been presented. Illustrative examples considered are IEEE-14 bus system and 5-bus sytem.