Robust Linear Estimation Research Articles

Does Capital Structure Affect Profitability of Listed Ghanaian Firms?

ABSTRACT: Capital structure is one of the crucial decisions managers must make that can affect a firm's short-term and long-term value maximization prospects. Though extant studies have shown capital structure significantly affects profits, researchers have struggled to find an optimal mix of capital structure ideal for value maximization due to the influence of macro and micro factors. There is therefore the need for this to be properly studied contextually. Building on the Pecking Order Theory, this study examines the capital structure effects on profitability from a short-term, long-term, and total debt financing perspective on listed Ghanaian firms between 2003 and 2013. An unbalanced panel data of 23 listed manufacturing and service firms that are listed on the Ghanaian Stock Exchange are subjected to robust maximum likelihood heteroskedastic linear estimation models. Three main regressions that are run explain the short-term debt, long-term debt and total debt effect on firm profitability. The results show that short-term debt exhibits a positive effect on return on equity (ROE), whilst the relationship reverses for long-term debt, indicating that short-term financing was beneficial to firms rather than long-term financing. Also, total debt exhibited a negative and non-significant effect on return on equity. The author also found that firm liquidity exhibited a more pronounced positive effect on profitability for firms with short-term debt than long-term debt. Financial risk negatively affected the returns of only firms with short-term debt. Market share had a pronounced positive effect on ROE across firms whilst firm age exhibited a less pronounced positive effect on firm profits. Firm size and growth had increasing profit effects but was not significant in the dataset. Financial risk also negatively affected profits, but this was only significant in firms with short-term debt structures. This study finds that managers need to adopt more short-term debt financing rather than long-term financing as the former leads to more profits. There is a need for policymakers to develop long-term debt financing structures to compete in the Ghanaian debt market. It is also recommended that managers exercise due care when supplementing equity capital with debt.

Robust Total Least Squares Estimation Method for Uncertain Linear Regression Model

In data analysis and modeling, least squares and total least squares are both mathematical optimization techniques. It is noteworthy that both the least squares method and the total least squares method are used to deal with precise and random data. However, when the given data are not random, or when the data are imprecise, and only the range of the data is available, the traditional linear regression method cannot be used. This paper presents an uncertain total least squares estimation method and an uncertain robust total least squares linear regression method based on uncertainty theory and total least squares method. The uncertain total least squares estimation can fully consider errors in the given data and the uncertain robust total least squares linear regression method can effectively eliminate outliers in the data. It is possible to obtain a more reasonable fitting effect with both of these methods, as well as to solve the predicted value and the confidence interval with these two methods. In terms of robust total least squares linear regression estimation, both uncertain total least squares regression estimation and uncertain robust total least squares regression estimation are feasible based on numerical examples. There are more accurate fitting equations and more reliable results with uncertain robust least squares linear regression estimation.

Graph neural networks on factor graphs for robust, fast, and scalable linear state estimation with PMUs

As phasor measurement units (PMUs) become more widely used in transmission power systems, a fast state estimation (SE) algorithm that can take advantage of their high sample rates is needed. To accomplish this, we present a method that uses graph neural networks (GNNs) to learn complex bus voltage estimates from PMU voltage and current measurements. We propose an original implementation of GNNs over the power system’s factor graph to simplify the integration of various types and quantities of measurements on power system buses and branches. Furthermore, we augment the factor graph to improve the robustness of GNN predictions. This model is highly efficient and scalable, as its computational complexity is linear with respect to the number of nodes in the power system. Training and test examples were generated by randomly sampling sets of power system measurements and annotated with the exact solutions of linear SE with PMUs. The numerical results demonstrate that the GNN model provides an accurate approximation of the SE solutions. Furthermore, errors caused by PMU malfunctions or communication failures that would normally make the SE problem unobservable have a local effect and do not deteriorate the results in the rest of the power system.

How do global financial markets affect the green bond markets? Evidence from different estimation techniques

The green bond market has significantly improved in recent years thanks to the development of financial instruments and the rising climate change concerns. Given this backdrop, this paper investigates the effects of returns in different financial markets, i.e. the United States Treasury Bonds, the Standard & Poor’s stock market, the United States Dollar, Gold, Crude Oil, and Bitcoin on the Green Bond returns (the Standard & Poor’s Green Bond Index) from September 17, 2014, to September 1, 2022. The results from the robust linear and machine learning estimators indicate that the returns of the United States Treasury Bonds and the United States Dollar are negatively related to the Green Bond returns. Meanwhile, Gold returns positively affect Green Bond returns. The quantile regression estimations of Machado–Santos Silva also show that these findings are valid in different quantiles. The paper also discusses policy implications related to climate change and the development of financial instruments to promote green investments.

A robust spline approach in partially linear additive models

Partially linear additive models generalize linear regression models by assuming that the relationship between the response and a set of explanatory variables is linear on some of the covariates, while the other ones enter into the model through unknown univariate smooth functions. The harmful effect of outliers either in the residuals or in the covariates involved in the linear component has been described in the situation of partially linear models, that is, when only one nonparametric component is involved. When dealing with additive components, the problem of providing reliable estimators when atypical data arise is of practical importance motivating the need of robust procedures. Based on this fact, a family of robust estimators for partially linear additive models that combines B-splines with robust linear MM-regression estimators is proposed. Under mild assumptions, consistency results and rates of convergence for the proposed estimators are derived. Furthermore, the asymptotic normality for the linear regression estimators is obtained. A Monte Carlo study is carried out to compare, under different models and contamination schemes, the performance of the robust MM-proposal based on B-splines with its classical counterpart and also with a quantile approach. The obtained results show the benefits of using the robust MM-approach. The analysis of a real data set illustrates the usefulness of the proposed method.

Temporal and Spatial Taylor's Law: Application to Japanese Subnational Mortality Rates

AbstractTaylor's law is a widely observed empirical pattern that relates the variances to the means of population densities. We present four extensions of the classical Taylor's law (TL): (1) a cubic extension of the linear TL describes the mean–variance relationship of human mortality at subnational levels well; (2) in a time series, long-run variance measures not only variance but also autocovariance, and it is a more suitable measure than variance alone to capture temporal/spatial correlation; (3) an extension of the classical equally weighted spatial variance takes account of synchrony and proximity; (4) robust linear regression estimators of TL parameters reduce vulnerability to outliers. Applying the proposed methods to age-specific Japanese subnational death rates from 1975 to 2018, we study temporal and spatial variations, compare different coefficient estimators, and interpret the implications. We apply a clustering algorithm to the estimated TL coefficients and find that cluster memberships are strongly related to prefectural gross domestic product. The time series of spatial TL coefficients has a decreasing trend that confirms the narrowing gap between rural and urban mortality in Japan.

Decentralized robust state estimation for hybrid AC/DC distribution systems with smart meters

Hybrid AC/DC distribution systems are becoming a popular means to accommodate the increasing penetration of distributed energy resources and flexible loads. This paper proposes a decentralized robust state estimation (DRSE) method for hybrid AC/DC distribution systems using multiple sources of data. In the proposed decentralized implementation framework, a unified robust linear state estimation model is derived for each AC and DC regions, where the regions are connected via AC/DC converters and only limited information exchange is needed. In this context, estimation accuracy may be suffering due to linearization. To enhance the estimation accuracy, a deep neural network (DNN) based on the smart meter data is used to extract hidden system statistical information and allow deriving nodal power injections that keep up with the real-time measurement update rate. This provides the way of integrating smart meter data, SCADA measurements and zero injections together for state estimation. Simulations on two hybrid AC/DC distribution systems show that the proposed DRSE has only slight accuracy loss by the linearization formulation but offers robustness of suppressing bad data automatically, as well as benefits of improving computational efficiency.

Probability-Constrained robust estimation for a random parameter with stochastic model uncertainties

This paper investigates the problem of estimating a random parameter in a linear observation model when the transformation matrix suffers stochastic uncertainties. In practice, the uncertainties can be either the bounded uncertainties or the Gaussian uncertainties. The existing minimax robust estimation method is only applicable to the bounded uncertainties, and minimizes the worst-case mean squared error (MSE) over the region of uncertainties, irrespective of the probability of such worst-case scenario, so it tends to be overly conservative. A probability-constrained (PC) robust linear estimation method is proposed in this paper to minimize the MSE with a guaranteed probability, which can handle bounded uncertainties as well as Gaussian uncertainties. However, the formulated problem is not analytically tractable due to the unspecified distribution of the uncertainties, so two approximate PC robust estimators are derived by tackling two upper bounds of the original optimization problem and converting them to semidefinite programming problems with the safe tractable approximation techniques. The comparative analysis reveals that there is a tradeoff between the MSE performance and computation complexity in the proposed two PC estimators. The Monte Carlo simulations are used to corroborate the theoretical results, which demonstrate that the proposed PC estimators are robust to the uncertainties compared to the nominal linear minimum mean squared error (LMMSE) estimator and the nominal best linear unbiased estimator (BLUE), and are less conservative compared to the traditional minimax robust estimator.

Robust estimation and inference for general varying coefficient models with missing observations

This paper considers estimation and inference for a class of varying coefficient models in which some of the responses and some of the covariates are missing at random and outliers are present. The paper proposes two general estimators—and a computationally attractive and asymptotically equivalent one-step version of them—that combine inverse probability weighting and robust local linear estimation. The paper also considers inference for the unknown infinite-dimensional parameter and proposes two Wald statistics that are shown to have power under a sequence of local Pitman drifts and are consistent as the drifts diverge. The results of the paper are illustrated with three examples: robust local generalized estimating equations, robust local quasi-likelihood and robust local nonlinear least squares estimation. A simulation study shows that the proposed estimators and test statistics have competitive finite sample properties, whereas two empirical examples illustrate the applicability of the proposed estimation and testing methods.

Estimation fusion for distributed multi-sensor systems with uncertain cross-correlations

ABSTRACTThis paper addresses the estimation fusion problem in distributed multi-sensor systems with uncertain cross-covariance among local estimation errors. A robust linear estimation fusion method is proposed in the sense of minimising the worst mean square error of the fused estimator over the uncertain normalised cross-covariances (NCC). The weighted coefficient matrices of the fused estimator can be obtained by solving a semi-definite programming problem. This estimation fusion method is suitable for the situations with completely unknown NCC or partly known NCC. Two fusion estimators for the uncertain NCC with partly known prior information are presented. Some numerical simulations are provided to show the good performance of the proposed estimators.

A Fast and Robust Linear State Estimator for Very Large Scale Interconnected Power Grids

This paper presents a computationally efficient and statistically robust linear state estimator for very large scale interconnected power grids that are made observable by synchronized phasor measurements. The developed estimator has some desirable properties such as robustness against gross errors in phasor measurements (bad data), computational performance that remains minimally affected by growing system size or bad data as long as the size of the largest control area remains bounded, and the fact that independent system operators associated with control areas do not have to share majority of their real-time data or measurements while the estimator solves for the entire system. The estimator is implemented and tested on a small (30-bus) and a large (2917-bus) utility test systems and results are provided.

A Hybrid State Estimator Based on SCADA and PMU Measurements for Medium Voltage Distribution System

With the increasing importance of renewable energy and flexible loads, the operation of the distribution system is becoming more stochastic and complex, and it is necessary to monitor the power system in real-time. Considering the gradual applications of intelligent electronic devices in the distribution systems, a hybrid state estimator based on supervisory control and data acquisition (SCADA) and phasor measurement unit (PMU) measurements is proposed in this paper, which consists of the improved robust estimation and linear state estimation. At the time of SCADA data acquisition, the improved robust estimation combining the SCADA measurements with PMU measurements is performed. To eliminate the effect of bad data, the internal student residual method is introduced, and the robust thresholds are adjusted adaptively. Then the linear state estimation is performed at the time of PMU data acquisition based on the results of the previous estimation time and the PMU measurements, which can quickly correct the robust estimation results and track the changes of the distribution system. Finally, the effectiveness and performance of the proposed method are verified in a modified IEEE 33-bus distribution system and a real distribution system in China.

Real‐time harmonic estimation using a novel hybrid technique for embedded system implementation

Summary In this paper, an analytical technique for efficient estimation of harmonics generated by power systems is presented. A robust linear estimator known as H∞ filter is applied to extract the amplitudes of harmonics. As well, phases are estimated using a nonlinear algorithm, namely, cubature Kalman filter. Both methods are based upon space modeling of the studied signals leading to improved numerical stability. The proposed H∞-cubature Kalman filter algorithm has the capability of estimating power system time varying waveforms even in highly noisy measurements. Synthetic and real signals are used for testing the proposed algorithm, and accuracy of the estimation is reported by statistical indices. Simulations in MATLAB are compared with unscented Kalman filter to represent the quality of estimation. Real-time implementation of the proposed algorithm is conducted using C++ programming code in hardware-in-the-loop laboratory. The setup consists of an embedded computer set and dedicated hardware. Results show satisfactory performance of the proposed method for practical application in power quality and protection cases.

Competitive Robust Estimation for Uncertain Linear Dynamic Models

In this paper, two types of robust linear estimation problems for dynamic channel model uncertainty are considered with the aim of characterizing (in computationally effective ways) competitive robust estimators, i.e., robust estimators that improve on the pointwise performance of minimax MSE (mean-squared error) estimators over the uncertain model set, at the expense of a moderate increase in the worst case MSE. The first one corresponds to the minimization of the worst case value of an approximate-regret function defined by a quadratic approximation of the “lower MSE envelope” on the uncertain model set. The second one corresponds to minimizing the nominal MSE error while ensuring that the worst case estimation error does not exceed a prescribed value. For uncertain classes defined by $H_{2}$ -norm balls, it is shown that these two types of estimation problems can be recast as “semidefinite programming problems (SDPs, for short).” Numerical examples are presented for both the case of linear, finite-dimensional model classes (FIRs of a given length) and the case of nonparametric uncertain sets of causal, real-rational frequency-responses, suggesting that these two types of estimators can be attractive alternatives to the min–max MSE estimator. For the case of spectral-norm (in the finite-dimensional case) or $H_{\infty }$ -norm (in the nonparametric case), the worst case MSE and approximate-regret for each candidate estimator are replaced by upper bounds obtained by Lagrangian relaxation and (somewhat conservative) versions of the estimation problems previously mentioned are posed. It is shown that these problems can also be recast as SDPs.

Adaptive rendering based on a weighted mixed-order estimator

In this paper, we propose a novel adaptive rendering method to robustly handle noise artifacts and outliers of Monte Carlo ray tracing by combining the Nadaraya---Watson and robust local linear estimators while efficiently preserving fine details. Our method first constructs a sparse robust local linear estimator in feature space (normal,texture,etc.), while also removing spike noise. Then, we utilize the Nadaraya---Watson estimator to filter the outlier-free image. We generate the final image by interpolating the values of each estimator at each pixel with weights that are inversely proportional to the estimated mean squared errors. Lastly, we distribute additional samples to the regions with higher estimated mean squared errors if sampling budget remains. We have demonstrated that our estimator outperforms previous methods visually and numerically.

Low-Distortion MMSE Speech Enhancement Estimator Based on Laplacian Prior

The most well-known conventional speech enhancement algorithms introduce unwanted artifact noise and speech distortion to the enhanced signal. Reducing the effects of such issues require more robust linear and non-linear estimators. This paper proposes new optimum linear and non-linear Laplacian distribution-based estimators. The proposed estimators are derived based on a minimum mean squared error (MMSE) sense to minimize the distortion in different conditions of the underlying speech. Thus, artifact noise is reduced without compromising the noise reduction process. The analytical solutions of the Laplacian distribution-based estimators, linear bilateral Laplacian gain estimator (LBLG), and non-linear bilateral Laplacian gain estimator (NBLG), are presented. The proposed estimators are implemented in three steps. First, the observation signal is decorrelated through a real transform domain to obtain its transform coefficients. Second, the proposed estimators are applied to estimate the clean speech signal from the noisy signal in the decorrelated domain. Finally, the inverse of the real transform is applied to obtain the original speech signal in the time domain. Two conditions in these estimators account for interference events between the speech signal and noise coefficients in the decorrelated domain. Moreover, a mathematical aspect of mean square error of LBLG is evaluated, which presents a significant improvement over other methods. Furthermore, a comprehensive description of the whole variations of the LBLG and NBLG gains characteristics is presented. A comparative evaluation is performed with effective quality metrics, segmental signal-to-noise ratio and perceptual evaluation of speech quality, to demonstrate the advantage and effectiveness of the proposed estimators. The performance of the proposed estimators outperformed other methods, which are the traditional MMSE approach, perceptually motivated Bayesian estimator, dual gain Wiener estimator, and dual MMSE estimator in terms of different objective measurements.

Robust Visual Tracking via Least Soft-Threshold Squares

In this paper, we propose an online tracking algorithm based on a novel robust linear regression estimator. In contrast to existing methods, the proposed least soft-threshold squares (LSS) algorithm models the error term with the Gaussian-Laplacian distribution, which can be efficiently solved. For visual tracking, the Gaussian-Laplacian noise assumption enables our LSS model to handle the normal appearance change and outlier simultaneously. Based on the maximum joint likelihood of parameters, we derive an LSS distance metric to measure the difference between an observation sample and a dictionary of positive templates. Compared with the distance derived from ordinary least squares methods, the proposed metric is more effective in dealing with the outliers. In addition, we provide insights on the relationships among the LSS problem, Huber loss function, and trivial templates, which facilitate better understandings of the existing tracking methods. Finally, we develop a robust tracking algorithm based on the LSS distance metric with an update scheme and negative templates, and speed it up with a particle selection mechanism. Experimental results on numerous challenging image sequences demonstrate that the proposed tracking algorithm performs favorably than the state-of-the-art methods.

Robust best linear estimator for Cox regression with instrumental variables in whole cohort and surrogates with additive measurement error in calibration sample

Biomedical researchers are often interested in estimating the effect of an environmental exposure in relation to a chronic disease endpoint. However, the exposure variable of interest may be measured with errors. In a subset of the whole cohort, a surrogate variable is available for the true unobserved exposure variable. The surrogate variable satisfies an additive measurement error model, but it may not have repeated measurements. The subset in which the surrogate variables are available is called a calibration sample. In addition to the surrogate variables that are available among the subjects in the calibration sample, we consider the situation when there is an instrumental variable available for all study subjects. An instrumental variable is correlated with the unobserved true exposure variable, and hence can be useful in the estimation of the regression coefficients. In this paper, we propose a nonparametric method for Cox regression using the observed data from the whole cohort. The nonparametric estimator is the best linear combination of a nonparametric correction estimator from the calibration sample and the difference of the naive estimators from the calibration sample and the whole cohort. The asymptotic distribution is derived, and the finite sample performance of the proposed estimator is examined via intensive simulation studies. The methods are applied to the Nutritional Biomarkers Study of the Women's Health Initiative.

Reduced-rank estimation for ill-conditioned stochastic linear model with high signal-to-noise ratio

Reduced-rank approach has been used for decades in robust linear estimation of both deterministic and random vector of parameters in linear model y=Hx+ϵn. In practical settings, estimation is frequently performed under incomplete or inexact model knowledge, which in the stochastic case significantly increases mean-square-error (MSE) of an estimate obtained by the linear minimum mean-square-error (MMSE) estimator, which is MSE-optimal among linear estimators in the theoretical case of perfect model knowledge. However, the improved performance of reduced-rank estimators over MMSE estimator in estimation under incomplete or inexact model knowledge has been established to date only by means of numerical simulations and arguments indicating that the reduced-rank approach may provide improved performance over MMSE estimator in certain settings. In this paper we focus on the high signal-to-noise ratio (SNR) case, which has not been previously considered as a natural area of application of reduced-rank estimators. We first show explicit sufficient conditions under which familiar reduced-rank MMSE and truncated SVD estimators achieve lower MSE than MMSE estimator if singular values of array response matrix H are perturbed. We then extend these results to the case of a generic perturbation of array response matrix H, and demonstrate why MMSE estimator frequently attains higher MSE than reduced-rank MMSE and truncated SVD estimators if H is ill-conditioned. The main results of this paper are verified in numerical simulations.

Robust Linear Regression Analysis— A Greedy Approach

The task of robust linear estimation in the presence of outliers is of particular importance in signal processing, statistics and machine learning. Although the problem has been stated a few decades ago and solved using classical (considered nowadays) methods, recently, it has attracted more attention in the context of sparse modeling, where several notable contributions have been made. In the present manuscript, a new approach is considered in the framework of greedy algorithms. The noise is split into two components: a) the inlier bounded noise and b) the outliers, which are explicitly modeled by employing sparsity arguments. Based on this scheme, a novel efficient algorithm (Greedy Algorithm for Robust Denoising—GARD), is derived. GARD alternates between a least square optimization criterion and an Orthogonal Matching Pursuit (OMP) selection step that identifies the outliers. The case where only outliers are present has been studied separately, where bounds on the Restricted Isometry Property guarantee that the recovery of the signal via GARD is exact. Moreover, theoretical results concerning convergence as well as the the recovery of the support of the sparse outlier vector and derivation of error bounds in the case of additional bounded noise are discussed. Finally, we provide extensive simulations, which demonstrate the comparative advantages of the new technique.