Weighted Least-squares Estimation Research Articles

Optimal mascon geometry in estimating mass anomalies within Greenland from GRACE

The mascon approach is a well-known technique to estimate mass anomalies in Greenland using GRACE satellite gravity data. It partitions the area of interest into laterally-homogeneous patches (mascons). An important aspect of the mascon approach is the chosen geometry of mascons. So far, its impact has not been fully understood. In this study, we use a full-scale numerical study and real data analysis to identify the optimal strategy (primarily, the size of the mascons) for the extrac- tion of mass anomalies over the Greenland drainage systems from GRACE monthly solutions. We use ordinary and weighted least-squares techniques to estimate the mascon parameters. The weighted least-squares estimator uses the full noise covari- ance matrices of monthly GRACE models, i.e. is designed to suppress random noise in the estimates. In addition, the zero-order Tikhonov regularization is applied. Four types of quantities of interest, which are associated with di erent temporal scales, are investigated in this study: monthly mass anomalies, mean mass anomalies per calendar month, inter-annual mass variations, and long-term linear trends. We show that the dominant error sources are random errors and parameterization (model) errors, as well as the bias introduced by the regularization. Errors in long-term lin- ear trend estimates are dominated by parameterization errors, whereas the role of random errors increases with the decreasing temporal scale. The best solutions are Downloaded from https://academic.oup.com/gji/advance-article-abstract/doi/10.1093/gji/ggy242/5040767 by guest on 05 July 2018 2 J. Ran, P. Ditmar and R. Klees obtained when the territory of Greenland is split into at least 23 mascons (the area of each one being 90; 000 km2). The usage of smaller mascons does not worsen the solutions in most cases, which is explained by the application of the regularization. Usage of larger mascons leads in most cases to inferior results due to the impact of parameterization errors. The application of the weighted least-squares estimator noticeably improves the quality of the solutions, with the exception of long-term linear trends estimated at the drainage system scale. In addition, we considered the long-term linear trend estimates integrated over entire Greenland. It is shown that the best results are obtained in that case when no regularization is applied. The results of real GRACE data processing are consistent with those obtained in the numerical study.

Weighted least-squares estimation for the subcritical Heston process

Abstract We simultaneously estimate the four parameters of a subcritical Heston process. We do not restrict ourselves to the case where the stochastic volatility process never reaches zero. In order to avoid the use of unmanageable stopping times and a natural but intractable estimator, we use a weighted least-squares estimator. We establish strong consistency and asymptotic normality for this estimator. Numerical simulations are also provided, illustrating the favorable performance of our estimation procedure.

Incorporating the covariance effect in modelling batch flotation kinetics

In this paper, recovery covariances were used to estimate the kinetic parameters from batch flotation tests to account for the lack of statistical independence and homoscedasticity in the cumulative recoveries. Non-linear parameter estimations were compared by using unweighted least squares estimation (ULSE), weighted least squares estimation (WLSE) and non-linear generalized least squares estimation (NLGLSE). Three autocorrelated time-recovery curves were used as base case to simulate theoretical kinetic response. Single Rate Constant, Rectangular and Gamma models were employed to describe the kinetic response.The NLGLSE allowed for significant precision improvements in the parameter estimation with respect to ULSE and WLSE, under known and constant covariance estimates. The variability of kSRC (SRC), kmax (Rectangular) and kmean (Gamma) decreased by approximately 40% with respect to ULSE, and 28% in comparison to WLSE. For R∞, the dispersion decreased 33% in comparison to ULSE, and 17% regarding WLSE. The limitations of ULSE and WLSE were caused by a lack of validity of the assumptions of independence and homoscedasticity in the time-recovery curves. The advantages of NLGLSE were only observed with accurate estimators of the covariance matrix, which were obtained in the simulations and in a laboratory flotation test that involved 11 replicates.Incorporating the covariance matrix in the parameter estimation allowed for improvements in the kinetic characterization. Thus, uncertainties related to the ore potential and circuit sizing (obtained from R∞ and the rate constant estimates) might be decreased using accurate covariance estimators in the objective functions.

How to deal with the high condition number of the noise covariance matrix of gravity field functionals synthesised from a satellite-only global gravity field model?

The posed question arises for instance in regional gravity field modelling using weighted least-squares techniques if the gravity field functionals are synthesised from the spherical harmonic coefficients of a satellite-only global gravity model (GGM), and are used as one of the noisy datasets. The associated noise covariance matrix, appeared to be extremely ill-conditioned with a singular value spectrum that decayed gradually to zero without any noticeable gap. We analysed three methods to deal with the ill-conditioned noise covariance matrix: Tihonov regularisation of the noise covariance matrix in combination with the standard formula for the weighted least-squares estimator, a formula of the weighted least-squares estimator, which does not involve the inverse noise covariance matrix, and an estimator based on Rao’s unified theory of least-squares. Our analysis was based on a numerical experiment involving a set of height anomalies synthesised from the GGM GOCO05s, which is provided with a full noise covariance matrix. We showed that the three estimators perform similar, provided that the two regularisation parameters each method knows were chosen properly. As standard regularisation parameter choice rules do not apply here, we suggested a new parameter choice rule, and demonstrated its performance. Using this rule, we found that the differences between the three least-squares estimates were within noise. For the standard formulation of the weighted least-squares estimator with regularised noise covariance matrix, this required an exceptionally strong regularisation, much larger than one expected from the condition number of the noise covariance matrix. The preferred method is the inversion-free formulation of the weighted least-squares estimator, because of its simplicity with respect to the choice of the two regularisation parameters.

A novel bearing-assisted TDOA-GROA approach for passive source localization

PurposeThe passive source localization (PSL) problem using angles of arrival (AOA), time differences of arrival (TDOA) or gain ratios of arrival (GROA) is generally nonlinear and nontrival. In this research, the purpose of this paper is to design an accurate hybrid source localization approach to solve the PSL problem. The inspiration is drawn from the fact that the bearings, TDOAs and GROAs are complementary in terms of their geometry properties.Design/methodology/approachThe maximum-likelihood (ML) method is reexamined by using hybrid measurements. Being assisted by the bearings, a new hybrid weighted least-squares (WLS) method is then proposed by jointly utilizing the bearing, TDOA and GROA measurements.FindingsTheoretical performance analysis illustrates that the mean-square error of the ML or WLS method can attain the Cramér-Rao lower bound for Gaussian noise over small error region. However, the WLS method has much lower computational complexity than the ML algorithm. Compared with the AOA-only, TDOA-only, AOA-TDOA, TDOA-GROA methods, the localization accuracy can be greatly improved by combining the AOAs, TDOAs and GROAs, especially for some specific geometries.Originality/valueA novel bearing-assisted TDOA-GROA method is proposed for source localization, and a new hybrid WLS estimator is presented inspired from the fact that the bearings, TDOAs and GROAs are complementary in terms of their geometry properties.

Super-resolution Doppler beam sharpening method using fast iterative adaptive approach-based spectral estimation

Doppler beam sharpening (DBS) is a critical technology for airborne radar ground mapping in forward-squint region. In conventional DBS technology, the narrow-band Doppler filter groups formed by fast Fourier transform (FFT) method suffer from low spectral resolution and high side lobe levels. The iterative adaptive approach (IAA), based on the weighted least squares (WLS), is applied to the DBS imaging applications, forming narrower Doppler filter groups than the FFT with lower side lobe levels. Regrettably, the IAA is iterative, and requires matrix multiplication and inverse operation when forming the covariance matrix, its inverse and traversing the WLS estimate for each sampling point, resulting in a notably high computational complexity for cubic time. We propose a fast IAA (FIAA)-based super-resolution DBS imaging method, taking advantage of the rich matrix structures of the classical narrow-band filtering. First, we formulate the covariance matrix via the FFT instead of the conventional matrix multiplication operation, based on the typical Fourier structure of the steering matrix. Then, by exploiting the Gohberg–Semencul representation, the inverse of the Toeplitz covariance matrix is computed by the celebrated Levinson–Durbin (LD) and Toeplitz-vector algorithm. Finally, the FFT and fast Toeplitz-vector algorithm are further used to traverse the WLS estimates based on the data-dependent trigonometric polynomials. The method uses the Hermitian feature of the echo autocorrelation matrix R to achieve its fast solution and uses the Toeplitz structure of R to realize its fast inversion. The proposed method enjoys a lower computational complexity without performance loss compared with the conventional IAA-based super-resolution DBS imaging method. The results based on simulations and measured data verify the imaging performance and the operational efficiency.

Inherently Stable Weighted Least-Squares Estimation of Common Acoustical Poles With the Application in Feedback Path Modeling Utilizing a Kautz Filter

In adaptive feedback cancellation, the feedback path must be modeled precisely using as few adaptive parameters as possible to reduce computational complexity and enable rapid convergence. To reduce the number of adaptive parameters, the feedback path is usually modeled with a transfer function composed of an invariant component and a variable component using all-pole, all-zero, and pole-zero filters. These filters may be inefficient, requiring a large number of parameters for their specification, particularly in reverberant environments. In this letter, we present a weighted least-squares algorithm to precisely estimate the common poles of feedback paths, and we then model the invariant basis functions employing a Kautz filter. The Kautz filter is defined by a set of the fixed poles and a corresponding set of tap output weights. The fixed poles are associated with prominent peaks that are common to the measured feedback path frequency responses. The algorithm guarantees unconditionally the stability of the estimated poles of the inferred model. The experimental results using measured acoustic feedback paths from a two-microphone, behind-the-ear hearing aid show that the proposed method provides an accurate model of the feedback paths for a variety of acoustic environments that were not employed in estimating the common poles.

Robust Bayesian linear regression with application to an analysis of the CODATA values for the Planck constant

Weighted least-squares estimation is commonly applied in metrology to fit models to measurements that are accompanied with quoted uncertainties. The weights are chosen in dependence on the quoted uncertainties. However, when data and model are inconsistent in view of the quoted uncertainties, this procedure does not yield adequate results.When it can be assumed that all uncertainties ought to be rescaled by a common factor, weighted least-squares estimation may still be used, provided that a simple correction of the uncertainty obtained for the estimated model is applied. We show that these uncertainties and credible intervals are robust, as they do not rely on the assumption of a Gaussian distribution of the data. Hence, common software for weighted least-squares estimation may still safely be employed in such a case, followed by a simple modification of the uncertainties obtained by that software. We also provide means of checking the assumptions of such an approach.The Bayesian regression procedure is applied to analyze the CODATA values for the Planck constant published over the past decades in terms of three different models: a constant model, a straight line model and a spline model. Our results indicate that the CODATA values may not have yet stabilized.

Helmert transformation with mixed geodetic and Cartesian coordinates

It is common in geodetic practice that station geodetic coordinates in an old frame are measured/estimated by adjusting conventional leveling and vertical networks (together with a vertical datum), whereas their Cartesian coordinates in a new frame are measured/estimated by adjusting modern 3D Global Navigation Satellite System (GNSS) networks. To estimate Helmert transformation parameters between the two frames, one can simply convert the geodetic coordinates to their Cartesian counterparts in the same frame, and then perform a least-squares transformation with Cartesian coordinates in both frames. This stepwise approach is not optimal because of nonlinearity in the first step. In this work, a direct transformation is conducted. A functional model with mixed geodetic and Cartesian measurements is followed, and a realistic stochastic model considering measurement errors in both frames is adopted. Correlations between common and non-common stations are also taken into account. Weighted least-squares estimates of the transformation parameters and transformed/adjusted coordinates in the new frame are derived in detail, the latter representing the enlarged network of the new frame. Simulations are conducted, and the results validate the superiority of the proposed method compared with the stepwise method.

An underwater measurement and control network centralized data fusion localization algorithm based on Chan-algorithm method

In order to exploit the development potential of current measurement and control equipment and to build an omnibus underwater measurement and control network with higher precision, according to the working characteristics of the system itself, an underwater measurement and control network centralized data fusion localization algorithm based on Chan-algorithm method is proposed. It is an algorithm that coarsely calculates the target position according to the time of arrival by using the weighted least squares estimation(WLS) method at first; second, constructs new error vectors on the basis of the relationship between the position and the time delay information; last, resolves the target position from the vectors above by using WLS method again. The result of research demonstrates that the algorithm realizes the data fusion method of multiple sets of underwater measurement and control equipment, which could improve the precision of localization globally. The precision of the proposed method is much better than the data fusion localization algorithm purely based on the time of arrival.

Heuristic monitoring method for sparsely measured distribution grids

There is an increasing interest in operating the power system close to its limits in order to avoid grid reinforcements. Distribution management requires the knowledge of grid state parameters, but outfitting grids with a large amount of measurements is costly. Therefore, we developed a new heuristic monitoring method (HMM) for balanced grids that relies only on few mandatory measurements and enables a fast way to monitor the grid for off-limit conditions. Due to a new formulation of the power flow equations, it has a low computational complexity for radial grids. The method analyzes the network topology; network buses are categorized and sorted into branches. Depending on the location of available voltage measurements, the bus powers of the corresponding branches are adjusted iteratively to better fit the measured voltage. To test the performance of our new algorithm, we design an evaluation process to compare our approach with the standard weighted least squares (WLS) state estimation (SE) method. Simulation results on artificial and real unmeshed distribution grids on the medium voltage (MV) level show very promising results, outperforming the WLS estimator even with a high amount of distributed generation (DG).

Inference for Large Financial Systems

We treat the parameter estimation problem for mean-field models of large interacting financial systems such as the banking system and a pool of assets held by an institution or backing a security. We develop an asymptotic inference approach that addresses the scale and complexity of such systems. Harnessing the weak convergence results developed for mean-field financial systems in the literature, we construct an approximate likelihood for large systems. The approximate likelihood has a conditionally Gaussian structure, enabling us to design an efficient numerical method for its evaluation. We provide a representation of the corresponding approximate estimator in terms of a weighted least-squares estimator, and use it to analyze the large-system and large-sample behavior of the estimator. Numerical results for a mean-field model of systemic financial risk highlight the efficiency and accuracy of our estimator.

State estimator for electrical distribution systems based on an optimization model

State estimation (SE) has become an important aspect of real-time management of electrical distribution systems (EDS). However, it is a challenging problem, mainly due to the network model complexity and the lack of a reliable measurement infrastructure capable to cope with a variety of network configurations. This work presents an Extended Optimal Power Flow (E-OPF) for SE in EDS that considers different network configurations. The active and reactive power loads are handled as extended variables in the optimization model that are constrained by upper and lower bounds obtained from historical data. The objective function combines the state estimation error (SEE) from the known Weighted Least Squares (WLS) approach with additional indexes related to the state variables, which improve the SE process. The combination of different indexes and the handling of power loads as optimization variables without needing precise historical data are novelty for SE and consist of the main contributions for an EDS that, in general, has few available data. The computational efficiency when different topologies are considered and in situations where the WLS estimator might fail is also a suitable feature of the proposed approach. The known 16-bus and 33-bus test systems are used to assess the proposed methodology. An illustrative application of the proposed approach in a Phasor Measurement Unit (PMU) allocation problem is also presented.

Estimate-Merge-Technique-based algorithms to track an underwater moving target using towed array bearing-only measurements

Bearing-only passive target tracking is a well-known underwater defence issue dealt in the recent past with the conventional nonlinear estimators like extended Kalman filter (EKF) and unscented Kalman filter (UKF). It is being treated now-a-days with the derivatives of EKF, UKF and a highly sophisticated particle filter (PF). In this paper, two novel methods based on the Estimate Merge Technique are proposed. The Estimate Merge Technique involves a process of getting a final estimate by the fusion of a posteriori estimates given by different nonlinear estimates, which are in turn driven by the towed array bearing-only measurements. The fusion of the estimates is done with the weighted least squares estimator (WLSE). The two novel methods, one named as Pre-Merge UKF and the other Post-Merge UKF, differ in the way the feedback to the individual UKFs is applied. These novel methods have an advantage of less root mean square estimation error in position and velocity compared with the EKF and UKF and at the same time require much lesser number of computations than that of the PF, showing that these filters can serve as an optimal estimator. A testimony of the afore-mentioned advantages of the proposed novel methods is shown by carrying out Monte Carlo simulation in MATLAB R2009a for a typical war time scenario.

Swedish lessons: How important are ICT and R&D to economic growth?

We investigate to what extent ICT and R&D capital are associated with value added growth in the Swedish business sector. By estimating output elasticities based on data for 47 different industries for the period 1993–2012 we show that ICT and R&D capital are significantly associated with value added for most specifications. We also divide ICT capital into hardware and software capital. To our knowledge, this distinction has not been made in any previous study at the industry level. In this case only the estimated elasticity of software is significantly different from zero. Finally, we use the growth accounting framework to compare the contribution from ICT and R&D to value added growth when output elasticities are based on either income shares or weighted least squares (WLS) estimates. The contribution of ICT to value added growth increases from 0.9 to 1.5 percentage points when WLS estimates are used instead of income shares.

Minimax robust designs for wavelet estimation of nonparametric regression models with autocorrelated errors

We discuss the construction of designs for estimation of nonparametric regression models with autocorrelated errors when the mean response is to be approximated by a finite order linear combination of dilated and translated versions of the Daubechies wavelet bases with four vanishing moments. We assume that the parameters of the resulting model will be estimated by weighted least squares (WLS) or by generalized least squares (GLS). The bias induced by the unused components of the wavelet bases, in the linear approximation, then inflates the natural variation of the WLS and GLS estimates. We therefore construct our designs by minimizing the maximum value of the average mean squared error (AMSE). Such designs are said to be robust in the minimax sense. Our illustrative examples are constructed by using the simulated annealing algorithm to select an optimal [Formula: see text]-point design, which are integers, from a grid of possible values of the explanatory or design variable [Formula: see text]. We found that the integer-valued designs we constructed based on GLS estimation, have smaller minimum loss when compared to the designs for WLS or ordinary least squares (OLS) estimation, except when the correlation parameter [Formula: see text] approaches 1.

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.

Minimum Variance Distortionless Response Estimators for Linear Discrete State-Space Models

For linear discrete state-space models, under certain conditions, the linear least-mean-squares filter estimate has a convenient recursive predictor/corrector format, aka the Kalman filter. The purpose of this paper is to show that the linear minimum variance distortionless response (MVDR) filter shares exactly the same recursion, except for the initialization which is based on a weighted least-squares estimator. If the MVDR filter is suboptimal in mean-squared error sense, it is an infinite impulse response distortionless filter (a deconvolver) which does not depend on the prior knowledge (first- and second-order statistics) on the initial state. In other words, the MVDR filter can be pre-computed and its behaviour can be assessed in advance independently of the prior knowledge on the initial state.

Online Local Input Selection Through Evolving Heterogeneous Fuzzy Inference System

Recently, online input selection has gained an increasing attention in evolving fuzzy models. In this paper, we proposed a new evolving fuzzy system referred to as evolving heterogeneous fuzzy inference system (eHFIS), which can simultaneously perform local input selection and system identification in an evolving and integrative manner. The introduced eHFIS is structured by some fuzzy rules with different effective input variables. This was achieved through inclusion of some parameters (local input selectors) in the structure of the Takagi–Sugeno system. An online learning algorithm is proposed to identify the eHFIS, where 1) the premise parameter learning and rule evolution take place with the usage of an incremental and evolving clustering for partitioning the data space; 2) a local input selection strategy based on switching to a neighboring model is adopted, and then, all fuzzy rules with the same input structure form a new category; and 3) for each category, the parameters of linear models, in consequent parts, are updated by weighted recursive fuzzily weighted least-squares estimator. The performance of the proposed eHFIS is evaluated and compared through several simulations on hand made as well as real-life datasets.

Resurrecting weighted least squares

This paper shows how asymptotically valid inference in regression models based on the weighted least squares (WLS) estimator can be obtained even when the model for reweighting the data is misspecified. Like the ordinary least squares estimator, the WLS estimator can be accompanied by heteroskedasticity-consistent (HC) standard errors without knowledge of the functional form of conditional heteroskedasticity. First, we provide rigorous proofs under reasonable assumptions; second, we provide numerical support in favor of this approach. Indeed, a Monte Carlo study demonstrates attractive finite-sample properties compared to the status quo, in terms of both estimation and inference.