Weighted Least-squares Method Research Articles

Use of weighted least squares for geo-positioning using dual-satellite image pairs

ABSTRACTThis study investigates the use of weighted least squares (WLSs) estimation for geo-positioning using dual-satellite image pairs. Although many believe that the WLS method may be the optimal method for handling such pairs composed of different-resolution images, this study reveals that it has not been thoroughly validated using real satellite data and has obvious limitations for the space intersection of dual-satellite images, despite its potential. In addition, this article addresses the fact that the positioning accuracy may depend on the geometric conditions, as well as the resolution. This study visually and quantitatively checked the effect of the WLS method on the positioning accuracy using all the dual-satellite pairs available from two KOMPSAT-2, IKONOS, and QuickBird images. The results reveal that the WLS method is very effective for horizontal mapping. This observation was also valid in principle even for three rays, with a higher-resolution single image integrated into an existing stereo pair. In particular, it tends to cause a larger improvement when the pairs form a weaker geometry, which increases the spatial uncertainty near the intersection point. However, the WLS method did not improve the vertical mapping. It yielded a lower accuracy than the conventional approach, particularly in the case of three rays, owing to the weak convergence geometry created by integrating different-satellite images. This indicates that the WLS method must be conditionally accepted for mapping using dual-satellite images. This article demonstrates the potential and limitations of WLS estimation for finding the intersection of dual-satellite image pairs theoretically and experimentally, as well as its effects associated with the geometric conditions of imaging, visually and quantitatively.

Problems in Modelling Charge Output Accelerometers

Abstract The paper presents major issues associated with the problem of modelling change output accelerometers. The presented solutions are based on the weighted least squares (WLS) method using transformation of the complex frequency response of the sensors. The main assumptions of the WLS method and a mathematical model of charge output accelerometers are presented in first two sections of this paper. In the next sections applying the WLS method to estimation of the accelerometer model parameters is discussed and the associated uncertainties are determined. Finally, the results of modelling a PCB357B73 charge output accelerometer are analysed in the last section of this paper. All calculations were executed using the MathCad software program. The main stages of these calculations are presented in Appendices A−E.

Power System Static-State Estimation With Uncertain Network Parameters as Input Data

This paper introduces a novel Least Squares (LS) observer for the state estimation in power system under static conditions. The proposed state estimator takes into consideration not only the measurement uncertainty but also the uncertainty associated with all the network parameters used in the load flow model of the power grid. A comparison with the classical weighted LS method is reported.

Directivity control for regular polyhedron loudspeaker array based on weighted least-squares method using adjusted weight in spherical harmonics domain

We propose an adjusted weight least-squares method for directivity control of spherical loudspeaker arrays in the spherical harmonics domain. In previous work, we demonstrated that the sidelobes of the directivity pattern can be suppressed using the weighted least-squares method in the frequency domain. However, the gains of the calculated filter coefficients were very large at low frequencies. Consequently, to overcome this problem, we designed filter coefficients in the spherical harmonic domain. First, we set an ideal directivity pattern using a window function. Next, we obtain the filter based on the weighted least-squares method in the spherical harmonic domain with an initial weighting factor. Then, we update the weighting factor only on the side opposite that being faced based on the difference in sound pressure level between the ideal and the simulated radiation patterns. We then repeat this process until the difference level is satisfied by the condition of the update. The results of comparison of our proposed method with our previous method and the conventional least-squares method indicate that although the beam width in the look direction is wider than our previous method, our proposed method effectively prevents increasing filter coefficients by limiting order at low frequencies.

A Bias-Reduced Nonlinear WLS Method for TDOA/FDOA-Based Source Localization

We address the source localization problem by using both time-difference-of-arrival (TDOA) and frequency-difference-of-arrival (FDOA) measurements. We solve this problem in two steps, and in each step, we formulate a nonlinear weighted least squares (WLS) problem followed by a bias reduction scheme. In the first step, we formulate a nonlinear WLS problem using TDOA measurements only and derive the bias of the WLS solution, which is then used to develop an unbiased WLS solution by subtracting the bias from the WLS solution. In the second step, we formulate another nonlinear WLS problem by combining the results in the first step and the FDOA measurements. To avoid the potential risk of local convergence, this WLS problem is reduced to an approximate WLS problem, for which the globally optimal solution can be obtained. The bias of the WLS solution is also derived and then subtracted from the WLS solution to reduce the bias. Simulation results show that the bias of the proposed method is reduced and that the Cramér-Rao lower bound accuracy is also achieved.

Notes on testing equality and interval estimation in Poisson frequency data under a three-treatment three-period crossover trial

When the frequency of event occurrences follows a Poisson distribution, we develop procedures for testing equality of treatments and interval estimators for the ratio of mean frequencies between treatments under a three-treatment three-period crossover design. Using Monte Carlo simulations, we evaluate the performance of these test procedures and interval estimators in various situations. We note that all test procedures developed here can perform well with respect to Type I error even when the number of patients per group is moderate. We further note that the two weighted-least-squares (WLS) test procedures derived here are generally preferable to the other two commonly used test procedures in the contingency table analysis. We also demonstrate that both interval estimators based on the WLS method and interval estimators based on Mantel-Haenszel (MH) approach can perform well, and are essentially of equal precision with respect to the average length. We use a double-blind randomized three-treatment three-period crossover trial comparing salbutamol and salmeterol with a placebo with respect to the number of exacerbations of asthma to illustrate the use of these test procedures and estimators.

Reliable Urban Canyon Navigation Solution in GPS and GLONASS Integrated Receiver Using Improved Fuzzy Weighted Least-Square Method

Global Positioning System encounters many problems in urban canyons and hard environments because of obstacles that decrease the number of visible satellites in receiver view. So, integration with other satellite-based navigation systems such as Russian Global Navigation Satellite System is utilized in new receivers. Least square as the popular and usual method typically used for navigation solution associates all satellite information with the same weights. However, the satellite impacts in reliability and accuracy of the receiver outputs are different in real condition and can be weighted by intelligent factors. In this paper, an improved fuzzy-weighted least square method is proposed which weights the satellite based on the satellite effect on dilution of precision, elevation angle and a defined constellation factor. Experimental results show that the proposed method can calculate 2D position more reliable and accurate than other popular weighted least square methods. This improvement is more than 57.23 % in a defined figure of merit.

Effective impervious area for runoff in urban watersheds

AbstractEffective impervious area (EIA), or the portion of total impervious area (TIA) that is hydraulically connected to the storm sewer system, is an important parameter in determining actual urban runoff. EIA has implications in watershed hydrology, water quality, environment, and ecosystem services. The overall goal of this study is to evaluate the application of successive weighted least square (WLS) method to urban catchments with different sizes and various hydrologic conditions to determine EIA fraction. Other objectives are to develop insights on the data selection issues, EIA fraction, EIA/TIA ratio, and runoff source area patterns in urban catchments. The successive WLS method is applied to 50 urban catchments with different sizes from less than 1 ha to more than 2000 ha in Minnesota, Wisconsin, Texas, USA as well as Europe, Canada, and Australia. The average, median, and standard deviation of EIA fractions for the 42 catchments with residential land uses are found to be 0.222, 0.200, and 0.113, respectively. These values for the EIA/TIA ratio in the same 42 catchments are 0.50, 0.48, and 0.21, respectively. While the EIA/TIA results indicate the importance of EIA, 95% prediction interval of the mean EIA/TIA is found to be 0.07 to 0.93, which shows that using an average value for this ratio in each land use to determine EIA from TIA in ungauged urban watersheds can be misleading. The successive WLS method was robust and is recommended for determining EIA in gauged urban catchments. Copyright © 2016 John Wiley & Sons, Ltd.

Closed-Form FIR Filter Design with Accurately Controllable Cut-Off Frequency

To enhance the efficiency of designing finite impulse response (FIR) filters with a controllable cut-off frequency that possess excellent transfer characteristics, this paper proposes a closed-form filter design based on transfer characteristic compensation. First, a novel filter design based on a convolution window is presented, and the relationship between the spectrum of this window and the filter performance is elaborated. We then derive a three-stage filter design scheme that describes the design of an irregular filter, design of a compensation filter and filter summation. This scheme can be simplified into a closed-form design characterized by two analytic formulas by merging the intermediate steps. The configuration of a vital Kaiser window parameter is also derived. Numerical results show that the proposed closed-form design accurately controls the cut-off frequencies and exhibits a transfer performance comparable to the Remez design and the closed-form weighted least square (WLS) design. Moreover, our method is more efficiency than the closed-form WLS method for the design of high-order FIR filters.

Improved methods to estimate the effective impervious area in urban catchments using rainfall-runoff data

Summary Impervious surfaces are useful indicators of the urbanization impacts on water resources. Effective impervious area (EIA), which is the portion of total impervious area (TIA) that is hydraulically connected to the drainage system, is a better catchment parameter in the determination of actual urban runoff. Development of reliable methods for quantifying EIA rather than TIA is currently one of the knowledge gaps in the rainfall-runoff modeling context. The objective of this study is to improve the rainfall-runoff data analysis method for estimating EIA fraction in urban catchments by eliminating the subjective part of the existing method and by reducing the uncertainty of EIA estimates. First, the theoretical framework is generalized using a general linear least square model and using a general criterion for categorizing runoff events. Issues with the existing method that reduce the precision of the EIA fraction estimates are then identified and discussed. Two improved methods, based on ordinary least square (OLS) and weighted least square (WLS) estimates, are proposed to address these issues. The proposed weighted least squares method is then applied to eleven urban catchments in Europe, Canada, and Australia. The results are compared to map measured directly connected impervious area (DCIA) and are shown to be consistent with DCIA values. In addition, both of the improved methods are applied to nine urban catchments in Minnesota, USA. Both methods were successful in removing the subjective component inherent in the analysis of rainfall-runoff data of the current method. The WLS method is more robust than the OLS method and generates results that are different and more precise than the OLS method in the presence of heteroscedastic residuals in our rainfall-runoff data.

Quasi-distributed birefringence dispersion measurement for polarization maintain device with high accuracy based on white light interferometry.

A white light temporal interferometric technique for measurement of the quasi-distributed birefringence dispersion (BD) in a polarization maintain (PM) device with high accuracy based on weighted least square (WLS) method is presented. It is verified theoretically and experimentally that the accuracy of WLS method and the conventional ordinary least square (OLS) method both are proportional to the signal-to-noise ratio (SNR) of interferogram, whereas the WLS method holds a higher scaling factor because it is more suitable for heteroscedastic model that has unequal error variance. The experiment results show a repeatability of ~4.6 × 10(-5) ps/nm @ 1550 nm with WLS method for 100 sets of data, and ~4.3 × 10(-4) ps/nm with OLS method, for an interferogram with SNR of 30 dB. Besides, WLS method without iterative operation is carried out by using power spectrum of interferogram as weight value. The feasibility of this technique is demonstrated by distinguishing the quasi-distributed BD of every part for a packaged Y-waveguide with two 1m-long PM pigtails.

Marshall-Olkin Extended Exponential Distribution: Different Method of Estimations

This paper aims to estimation of the parameters and reliability characteristics for Marshall-Olkin extended exponential distribution (MOEED). We have used different estimation methods namely, method of maximum likelihood estimation (MLE), method of Maximum product spacing (MPS), method of least square (LSE) and weighted least square method (WLSE) to obtain the estimate of the parameters, reliability function and hazard function. Further, we have also provided 95% asymptotic conϐidence interval for the parameters. The performances of the estimators has been studied on the basis of Monte Carlo simulation and ϐinally, a real data set has been used for illustrative purpose of the study.

A new smart approach for state estimation of distribution grids considering renewable energy sources

The idea of smart grids has created new opportunities in the electrical networks for monitoring the system status. One of the valuable and significant techniques for monitoring the smart grids is state estimation. In this way, this paper proposes a sufficient state estimation algorithm for inclusive monitoring of the distribution systems in the presence of RESs (renewable energy sources). The proposed method is a hybrid technique using WLS (weighted least square) method and FA (firefly algorithm) to reach more reliable and accurate state estimation of the network. FA is equipped with new optimization operators that make it possible to solve the multi-modal problems using an automatic sub-division feature. In order to improve the overall search ability of the algorithm, a new two-phase modification method is proposed. The proposed hybrid method can estimate the voltage angle using the WLS method. The simulation results show more optimal cost function value with faster response with escaping from the several local optima of the problem.

Methods of linear multiple regression in a matrix form

The article contains a summary of three basic and two weighted linear multiple regression tech- niques in matrix form, together with the method of least squares of Gauss constitute a new tool re- gression analysis. The article contains a matrix formula that can be used to obtain equations of line- ar multiple regression and the basic weighted least-squares method to obtain regression equations without constant term and the method of obtaining the regression equations of general form. The article provides an example of use of matrix methods to obtain the coefficients of regression equa- tion of the general form, ie equation of equal performance.

Comparison of methods for analyzing binary repeated measures data: A simulation-based study (comparison of methods for binary repeated measures)

ABSTRACTIn this study, some methods suggested for binary repeated measures, namely, Weighted Least Squares (WLS), Generalized Estimating Equations (GEE), and Generalized Linear Mixed Models (GLMM) are compared with respect to power, type 1 error, and properties of estimates. The results indicate that with adequate sample size, no missing data, the only covariate being time effect, and a relatively limited number of time points, the WLS method performs well. The GEE approach performs well only for large sample sizes. The GLMM method is satisfactory with respect to type I error, but its estimates have poorer properties than the other methods.

Robust Visual Tracking Using Structurally Random Projection and Weighted Least Squares

Sparse representation-based visual tracking approaches have attracted increasing interests in the community in recent years. The main idea is to linearly represent each target candidate using a set of target and trivial templates, while imposing a sparsity constraint onto the representation coefficients. After we obtain the coefficients using $\ell _{1}$ -norm minimization methods, the candidate with the lowest error, when it is reconstructed using only the target templates and the associated coefficients, is considered as the tracking result. In spite of promising system performance widely reported, it is unclear if the performance of these trackers can be maximized. In addition, computational complexity caused by the dimensionality of the feature space limits these algorithms in real-time applications. In this paper, we propose a real-time visual tracking method based on structurally random projection (RP) and weighted least squares (WLS) techniques. In particular, to enhance the discriminative capability of the tracker, we introduce background templates to the linear representation framework. To handle appearance variations over time, we relax the sparsity constraint using a WLS method to obtain the representation coefficients. To further reduce the computational complexity, structurally RP is used to reduce the dimensionality of the feature space, while preserving the pairwise distances between the data points in the feature space. Experimental results show that the proposed approach outperforms several state-of-the-art tracking methods.

On the fundamental diagram for freeway traffic: A novel calibration approach for single-regime models

The speed-density or flow-density relationship has been considered as the foundation of traffic flow theory. Existing single-regime models calibrated by the least square method (LSM) could not fit the empirical data consistently well both in light-traffic/free-flow conditions and congested/jam conditions. In this paper, first, we point out that the inaccuracy of single-regime models is not caused solely by their functional forms, but also by the sample selection bias. Second, we apply a weighted least square method (WLSM) that addresses the sample selection bias problem. The calibration results for six well-known single-regime models using the WLSM fit the empirical data reasonably well both in light-traffic/free-flow conditions and congested/jam conditions. Third, we conduct a theoretical investigation that reveals the deficiency associated with the LSM is because the expected value of speed (or a function of it) is nonlinear with regard to the density (or a function of it).

MENGATASI HETEROSKEDASTISITAS PADA REGRESI DENGAN MENGGUNAKAN WEIGHTED LEAST SQUARE

In the regression analysis we need a method to estimate parameters to fulfill the BLUE characteristic. There are assumptions that must be fulfilled homoscedasticity one of which is a condition in which the assumption of error variance is constant (same), infraction from the assumptions homoskedasticity called heteroscedasticity. The Consequence of going heteroscedasticity can impact OLS estimators still fulfill the requirements of not biased, but the variant obtained becomes inefficient. So we need a method to solve these problems either by Weighted Least Square (WLS). The purpose of this study is to find out how to overcome heteroscedasticity in regression with WLS. Step of this research was do with the OLS analysis, then testing to see whether there heteroscedasticity problem with BPG method, the next step is to repair the beginning model by way of weighting the data an exact multiplier factor, then re-using the OLS procedure to the data that have been weighted, the last stage is back with a heteroscedasticity test BPG method, so we obtained the model fulfill the assumptions of homoskedasicity. Estimates indicate that the WLS method can resolve the heteroscedasticity, with exact weighting factors based on the distribution pattern of data seen.

Optimal Robust Linear Quadratic Regulator for Systems Subject to Uncertainties

In this technical note, a robust recursive regulator for linear discrete-time systems, which are subject to parametric uncertainties, is proposed. The main feature of the optimal regulator developed is the absence of tuning parameters in online applications. To achieve this purpose, a quadratic cost function based on the combination of penalty function and robust weighted least-squares methods is formulated. The convergence and stability proofs for the stationary system and a numerical comparative study among the standard linear quadratic regulator, guaranteed cost and H ∞ controllers are provided.

Inference of reactive transport model parameters using a Bayesian multivariate approach

AbstractParameter estimation of subsurface transport models from multispecies data requires the definition of an objective function that includes different types of measurements. Common approaches are weighted least squares (WLS), where weights are specified a priori for each measurement, and weighted least squares with weight estimation (WLS(we)) where weights are estimated from the data together with the parameters. In this study, we formulate the parameter estimation task as a multivariate Bayesian inference problem. The WLS and WLS(we) methods are special cases in this framework, corresponding to specific prior assumptions about the residual covariance matrix. The Bayesian perspective allows for generalizations to cases where residual correlation is important and for efficient inference by analytically integrating out the variances (weights) and selected covariances from the joint posterior. Specifically, the WLS and WLS(we) methods are compared to a multivariate (MV) approach that accounts for specific residual correlations without the need for explicit estimation of the error parameters. When applied to inference of reactive transport model parameters from column‐scale data on dissolved species concentrations, the following results were obtained: (1) accounting for residual correlation between species provides more accurate parameter estimation for high residual correlation levels whereas its influence for predictive uncertainty is negligible, (2) integrating out the (co)variances leads to an efficient estimation of the full joint posterior with a reduced computational effort compared to the WLS(we) method, and (3) in the presence of model structural errors, none of the methods is able to identify the correct parameter values.