Least-squares Estimation Research Articles

BER Reduction Using Partial-Elements Selection in IRS-UAV Communications With Imperfect Phase Compensation

This article considers minimizing the communications bit error rate (BER) of unmanned aerial vehicles when assisted by intelligent reflecting surfaces. By noting that increasing the number of IRS elements in the presence of phase errors does not necessarily improve the system’s BER, it is crucial to use only the elements that contribute to reducing such a parameter. To this end, we propose an efficient algorithm to select the elements that can improve BER. The proposed algorithm has lower complexity and comparable BER to the optimum selection process, which is an NP-hard problem. The accuracy of the estimated phase is evaluated by deriving the probability distribution function (PDF) of the least-square channel estimator, and showing that the PDF can be closely approximated by the von Mises distribution at high signal-to-noise ratios. The obtained analytical and simulation results show that using all the available reflectors can significantly deteriorate the BER, and thus, partial element selection is necessary. It is shown that, in some scenarios, using about 26% of the reflectors provides more than tenfold BER reduction. The number of selected reflectors may drop to only 10% of the total elements. As such, the unassigned 90% of the elements can be allocated to serve other users, and the overhead associated with phase information is significantly reduced.

On the Least Squares Estimation of Multiple-Threshold-Variable Autoregressive Models

Most threshold models to-date contain a single threshold variable. However, in many empirical applications, models with multiple threshold variables may be needed and are the focus of this article. For the sake of readability, we start with the Two-Threshold-Variable Autoregressive (2-TAR) model and study its Least Squares Estimation (LSE). Among others, we show that the respective estimated thresholds are asymptotically independent. We propose a new method, namely the weighted Nadaraya-Watson method, to construct confidence intervals for the threshold parameters, that turns out to be, as far as we know, the only method to-date that enjoys good probability coverage, regardless of whether the threshold variables are endogenous or exogenous. Finally, we describe in some detail how our results can be extended to the K-Threshold-Variable Autoregressive (K-TAR) model, K > 2. We assess the finite-sample performance of the LSE by simulation and present two real examples to illustrate the efficacy of our modeling.

Parotid gland evaluation of menopausal women with xerostomia using the iterative decomposition of water and fat with echo asymmetry and least-squares estimation (IDEAL-IQ) method of MRI: a pilot study.

This study aimed to analyze the quantitative fat fraction (FF) of the parotid gland in menopausal females with xerostomia using the iterative decomposition of water and fat with echo asymmetry and least-squares estimation (IDEAL-IQ) method. A total 138 parotid glands of 69 menopausal females were enrolled in our study and participants were divided into normal group and xerostomia group. The xerostomia group was divided into those with or without Sjögren's syndrome. Participants underwent IDEAL-IQ sequences of MRI and the stimulated salivary flow test (s-SFR). The unpaired t-test was used to compare the FFs between the normal and xerostomia groups and between the subgroups with and without Sjögren's syndrome. The correlation between FF and s-SFR was analyzed by Pearson's correlation. Excellent intra- and interobserver agreement during the measurement of FFs by IDEAL-IQ method (ICC>0.99, respectively). FF value in the xerostomia group was statistically significantly higher than the value in the normal group (p < 0.05). Within the xerostomia group, the average FF value of females with Sjögren's syndrome was higher than that of females without Sjögren's syndrome. However, the difference was not statistically significant (p > 0.05). Within the xerostomia group, FF value correlated negatively with s-SFR (p < 0.05). The FF of the parotid gland was higher in the xerostomia group than in the normal group and FF value and s-SFR showed a negative correlation. Analyses of the FF using IDEAL-IQ in menopausal females can be helpful for the quantitative diagnosis of xerostomia.

Quantification of lumbar vertebral fat deposition: Correlation with menopausal status, non-alcoholic fatty liver disease and subcutaneous adipose tissue.

To assess abdominal fat deposition and lumbar vertebra with iterative decomposition of water and fat with echo asymmetry and least-squares estimation (IDEAL-IQ) and investigate their correlation with menopausal status. Two hundred forty women who underwent routine abdominal MRI and IDEAL-IQ between January 2016 and April 2021 were divided into two cohorts (first cohort: 120 pre- or postmenopausal women with severe fatty livers or without fatty livers; second cohort: 120 pre- or postmenopausal women who were obese or normal weight). The fat fraction (FF) values of the liver (FFliver) and lumbar vertebra (FFlumbar) in the first group and the FF values of subcutaneous adipose tissue (SAT) (FFSAT) and FFlumbar in the second group were measured and compared using IDEAL-IQ. Two hundred forty women were evaluated. FFlumbar was significantly higher in both pre- and postmenopausal women with severe fatty liver than in patients without fatty livers (premenopausal women: p < 0.001, postmenopausal women: p < 0.001). No significant difference in the FFlumbar was observed between obese patients and normal-weight patients among pre- and postmenopausal women (premenopausal women: p = 0.113, postmenopausal women: p = 0.092). Significantly greater lumbar fat deposition was observed in postmenopausal women than in premenopausal women with or without fatty liver and obesity (p < 0.001 for each group). A high correlation was detected between FFliver and FFlumbar in women with severe fatty liver (premenopausal women: r=0.76, p<0.01; postmenopausal women: r=0.82, p<0.01). Fat deposition in the vertebral marrow was significantly associated with liver fat deposition in postmenopausal women.

Protection Algorithm for Fault Identification and Isolation in DC Microgrid

In renewable energy dominated distributed ring configuration direct current (DC) networks, the protection philosophy is one of the critical challenging task. It is due to the existence of power electronic converters and erratic attributes of distributed energy sources. Consequently, conventional current direction based as well as over current protection strategies is not applicable for DC microgrids. In this paper, protection algorithm for fault recognition and isolation of faulty line is presented based on the polarity of change in inductance immediately after fault inception. The voltage and current sample information is used to determine the parameter by employing the least square estimation (LSE) technique. The efficiency of the proposed method is tested for internal and external faults, the impact of fault resistance and fault location, different system configurations, and load change conditions in MATLAB/Simulink simulation. It is noted that proposed method would categorize internal and external faults perfectly. The operating time of the proposed protection method is comparatively less than the existing methods. It also improves selectivity, security, and reliability under above mentioned abnormal cases.

Reliability Estimation for the Inverse Weibull Distribution Under Adaptive Type-II Progressive Hybrid Censoring: Comparative Study

The aim of this study is to investigate different methods of estimating the stress-strength reliability parameter, $\theta =P(Y&lt;X)$, when the strength (X) and the stress (Y) are independent random variables taken from the inverse Weibull distribution (IWD), with the same shape parameter and different scale parameters. Based on Adaptive Type-II Hybrid progressive censored samples, we employ classical and Bayesian approaches. In the classical approach, we use the maximum likelihood estimator (MLE), the approximate maximum likelihood estimator (AMLE), and the least squares estimator (LSE). In contrast, the Bayesian approach utilizes symmetric and asymmetric loss functions. Due to the absence of explicit forms for Bayes estimators, we propose using Lindley's approximation method for computing the Bayes estimators. We compare these estimators using extensive simulations and two criteria: the bias and the mean square error (MSE). Finally, two real-life data examples are provided for illustrations.

Estimation of a Structural Break Point in Linear Regression Models

This study proposes a point estimator of the break location for a one-time structural break in linear regression models. If the break magnitude is small, the least-squares estimator of the break date has two modes at the ends of the finite sample period, regardless of the true break location. To solve this problem, I suggest an alternative estimator based on a modification of the least-squares objective function. The modified objective function incorporates estimation uncertainty that varies across potential break dates. The new break point estimator is consistent and has a unimodal finite sample distribution under small break magnitudes. A limit distribution is provided under an in-fill asymptotic framework. Monte Carlo simulation results suggest that the new estimator outperforms the least-squares estimator. I apply the method to estimate the break date in U.S. and U.K. stock return prediction models.

ON THE CONSISTENCY OF THE LEAST SQUARES ESTIMATOR IN MODELS SAMPLED AT RANDOM TIMES DRIVEN BY LONG MEMORY NOISE: THE RENEWAL CASE

In this study, we prove the strong consistency of the least squares estimator in a random sampled linear regression model with long-memory noise and an independent set of random times given by renewal process sampling. Additionally, we illustrate how to work with a random number of observations up to time T = 1. A simulation study is provided to illustrate the behavior of the different terms, as well as the performance of the estimator under various values of the Hurst parameter H.

Indoor Visible Light Positioning Based on Improved Whale Optimization Method With Min-Max Algorithm

Recently, the Min-Max algorithm has been widely used in various indoor positioning systems due to its simplicity and robustness. In this article, an improved whale optimization algorithm (IWOA) associated with the Min-Max algorithm is proposed to improve the positioning accuracy concerning the original Min-Max algorithm for visible light positioning (VLP) systems. In the proposed algorithm, the three-dimensional (3-D) region of interest (RoI) affiliated with the target node (TN) is first obtained using the original Min-Max algorithm, and then the IWOA is employed to find the accurate position of the TN in the 3-D RoI with the lowest fitness value. In software simulations, the signal-to-noise ratio (SNR) value of the VLP system is 15 dB and the standard deviation of measured distance noise is 1 m. Our software simulation results show that in the case of two-dimensional (2-D) positioning, the averaged positioning error (APE) of the proposed IWOA-Min-Max algorithm is decreased by 84.6%, 52.1%, 1.9%, and 19.93%, respectively, for the original Min-Max algorithm, the extended Min-Max algorithms, the whale optimization algorithm (WOA), and the least square estimation (LSE). Furthermore, in the case of 3-D positioning, it is also reduced by 81.7%, 75.4%, 8.5%, and 63.9%, respectively, compared with that of the above four existing algorithms. Finally, hardware-in-the-loop simulation (HILS) results are also provided to verify the effectiveness of the proposed IWOA-Min-Max algorithm.

On estimating parameters of a multi-component Chirp Model with equal chirp rates

Multi-component chirp signal models with equal chirp rates appear in various radar applications, e.g., synthetic aperture radar, echo signal of a rapid mobile target, etc. Many sub-optimal estimators have been developed for such models, however, these suffer from the problem of either identifiability or error propagation effect. In this paper, we have developed theoretical properties of the least squares estimators (LSEs) of the parameters of multi-component chirp model with equal chirp rates, where the model is contaminated with linear stationary errors. We also propose two computationally efficient estimators as alternative to LSEs, namely sequential combined estimators and sequential plugin estimators. Strong consistency and asymptotic normality of these estimators have been derived. Interestingly, it is observed that sequential combined estimator of the chirp rate parameter is asymptotically efficient. Extensive numerical simulations have been performed, which validate satisfactory computational and theoretical performance of all three estimators. We have also analysed a simulated radar data with the help of our proposed estimators of multi-component chirp model with equal chirp rates, which performs efficiently in recovery of inverse synthetic aperture radar (ISAR) image of a target from a noisy data.

Composite model-reference adaptive control with least-squares estimator

This work presents a new design of a composite model-reference adaptive control (MRAC) with a least-squares parameter estimator. The objective of the design is to preserve the remarkable transient adaptation characteristic obtained by a modified MRAC algorithm recently introduced and, at the same time, enjoy the superior parameter convergence performance of a least-squares estimator. The algorithm employs two different estimators for the same controller parameter, one updated by a gradient law driven by the tracking error and the other updated by a least-squares algorithm driven by a prediction error. In this way, the performance of each one is kept unchanged. The existence of a Lyapunov function assures the global uniform stability of the proposed composite adaptive scheme and simulation results confirm and illustrate its properties.

Boundary adaptive observer design for heat PDEs with spatially varying parameters

We are considering the problem of observer design for parabolic partial differential equations (PDEs) that are subject to parameter uncertainty. The novelty lies in the fact that the unknown parameters are allowed to be spatially varying and the associated regressor is subject to a mild assumption. Then, the initial observer design problem, involving a scalar PDE with spatially-varying unknown parameters, is shown to be transformable into a new problem involving coupled PDEs with spatially-invariant parameters and a bounded modelling error. Invoking the high-gain principle and the so-called decoupling transformation technique, an adaptive observer is designed that provides online estimates of the state and parameter vectors. The original spatially varying parameters are recovered using least-squares estimators. All state and parameter estimation errors are shown, under appropriate persistent-excitation (PE) conditions, to be exponentially convergent to balls cantered on the origin with radiuses depending on the adaptive observer gain: the higher the observer gain the better the accuracy of estimates.

Adaptive boundary observer for ARZ traffic flow model with domain and boundary uncertainties

This paper studies the problem of the adaptive boundary observer design for the Aw-Rascle-Zhang (ARZ) traffic flow model, which is subject to both relaxation time uncertainty in the domain and boundary input disturbance. The boundary input disturbance comes from merging vehicles’ velocities at the downstream on-ramp, and we scale the disturbance by a low-pass filter based on the ordinary differential equation (ODE). Then, the ARZ model with the domain uncertainty and boundary input disturbance can be linearized to a coupled ODE-PDE system. Based on the swapping transformation, an adaptive boundary observer with least-squares type parameter estimation is designed to estimate the traffic states, the domain uncertainty, and the boundary input disturbance, simultaneously. The exponential convergence conditions w.r.t. the observer feedback gains are given by employing the Lyapunov technique. Finally, the simulation results are presented to illustrate the effectiveness of the designed adaptive boundary observer.

Extremum Seeking Control With an Adaptive Gain Based on Gradient Estimation Error

This article presents an extremum-seeking control (ESC) algorithm with an adaptive step size that adjusts the aggressiveness of the controller based on the quality of estimates obtained using a gradient estimator, which is intrinsic to many ESC algorithms. The adaptive step size ensures that the integral-action produced by the ESC control law does not destabilize the closed-loop system. To quantify the quality of the gradient estimate, we present a batch least-squares (BLS) estimator with a novel weighting term and guarantee that the gradient estimation error is bounded. The adaptive step size then maximizes the decrease of the combined plant and controller Lyapunov function for the worst-case estimation error. We also ensure that our ESC controller is input-to-state stable with respect to a class of dither signals. Finally, we demonstrate our ESC controller through benchmark examples and a practical application: leak detection with drones.

Enhanced Signal Detection and Constellation Design for Massive SIMO Communications With 1-Bit ADCs

In this paper, we investigate a transmitter and receiver design for a single-user massive SIMO (single-input multiple-output) system with 1-bit analog-to-digital converters (ADCs) at the base station (BS), where the user adopts higher-order modulation, e.g., 16-quadrature amplitude modulation (16-QAM), for the data transmission. For the channel estimation and the signal detection, linear least-squares (LS) estimation and maximum ratio combining (MRC) are respectively employed. In this context, we first introduce closed-form formulas for the mean of the estimated symbols and for the correlation matrix between their real and imaginary parts considering the effect of 1-bit quantization. The study of the distribution of the estimated symbols indicates that, in presence of 1-bit ADCs, the conventional 16-QAM detector and the typical square 16-QAM modulation are not adequate. In light of this, we propose three novel symbol detectors and re-design the 16-QAM modulation in order to improve the symbol error rate (SER). Furthermore, the upper bound on the SER is analyzed based on the pair-wise error probability and the boundary equation between two regions is also studied. Through numerical results, the proposed framework, i.e., the symbol detector and the transmit constellation design, shows a significant enhancement in the SER performance against the conventional detector and the typical square 16-QAM modulation.

Finite-Sample Analysis of Identification of Switched Linear Systems With Arbitrary or Restricted Switching

For the identification of switched systems with measured states and a measured switching signal, this letter aims to analyze the effect of switching strategies on the estimation error. The data is assumed to be collected from globally asymptotically or marginally stable switched systems under switches that are arbitrary or subject to an average dwell time constraint. Then the switched system is estimated by the least-squares (LS) estimator. To capture the effect of the parameters of the switching strategies on the LS estimation error, finite-sample error bounds are developed in this letter. The obtained error bounds show that the estimation error is logarithmic of the switching parameters when there are only stable modes; however, when there are unstable modes, the estimation error bound can increase linearly as the switching parameter changes. This suggests that in the presence of unstable modes, the switching strategy should be properly designed to avoid the significant increase of the estimation error.

An improved least squares (LS) channel estimation method based on CNN for OFDM systems

&lt;abstract&gt;&lt;p&gt;Least squares (LS) is a commonly used pilot-based channel estimation algorithm in orthogonal frequency division multiplexing (OFDM) systems. This algorithm is simple and easy to implement because of its low computation complexity. However, it has poor performance, especially at low signal-to-noise ratio (SNR). To solve this problem, an improved LS channel estimation method based on convolutional neural network (CNN) is proposed on the basis of analyzing the traditional LS channel estimation methods. A channel estimation compensated network is designed based on CNN, which can solve the problem of performance degradation of the mean square error (MSE) through the online and offline modules. By designing the input-output relations, training data set, and testing data set, a CNN network is iteratively trained to learn the relevant features of the channels, so that the traditional LS estimation value can be corrected to improve the accuracy. Simulation results show that the proposed method can achieve better performance in bit error rate (BER) and MSE, compared with the traditional channel estimation methods.&lt;/p&gt;&lt;/abstract&gt;

Efficient Least-Squares State Estimation Using Uniform Sampling

Formulating state estimation for a large-scale, discrete-time, linear time-invariant system as a least-squares problem can be computationally challenging as the problem dimensions increase with time. Recently, randomized sampling has demonstrated promising results in approximating this problem by using fewer rows, resulting in a polynomial-sized approximate problem. However, these algorithms necessitate calculating the statistical leverage scores of the rows, which can be challenging. In this paper, we propose an alternative approach to approximate leverage scores by uniformly sampling rows. We demonstrate that this method provides a sufficiently weak form of approximation for obtaining an estimate of each leverage score. Our approach delivers a reasonable approximation of the leverage scores, which is suitable for approximating and solving the least-squares estimation problem with proven theoretical guarantees.

An effective treatment of adding-up restrictions in the inference of a general linear model

&lt;abstract&gt;&lt;p&gt;This article offers a general procedure of carrying out estimation and inference under a linear statistical model $ {\bf y} = {\bf X} \pmb{\beta} + \pmb{\varepsilon} $ with an adding-up restriction $ {\bf A} {\bf y} = {\bf b} $ to the observed random vector $ {\bf y} $. We first propose an available way of converting the adding-up restrictions to a linear matrix equation for $ \pmb{\beta} $ and a matrix equality for the covariance matrix of the error term $ \pmb{\varepsilon} $, which can help in combining the two model equations in certain consistent form. We then give the derivations and presentations of analytic expressions of the ordinary least-squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of parametric vector $ {\bf K} \pmb{\beta} $ using various analytical algebraic operations of the given vectors and matrices in the model.&lt;/p&gt;&lt;/abstract&gt;

A geometric approach of gradient descent algorithms in linear neural networks

&lt;p style='text-indent:20px;'&gt;In this paper, we propose a geometric framework to analyze the convergence properties of gradient descent trajectories in the context of linear neural networks. We translate a well-known empirical observation of linear neural nets into a conjecture that we call the &lt;i&gt;overfitting conjecture&lt;/i&gt; which states that, for almost all training data and initial conditions, the trajectory of the corresponding gradient descent system converges to a global minimum. This would imply that the solution achieved by vanilla gradient descent algorithms is equivalent to that of the least-squares estimation, for linear neural networks of an arbitrary number of hidden layers. Built upon a key invariance property induced by the network structure, we first establish convergence of gradient descent trajectories to critical points of the square loss function in the case of linear networks of arbitrary depth. Our second result is the proof of the &lt;i&gt;overfitting conjecture&lt;/i&gt; in the case of single-hidden-layer linear networks with an argument based on the notion of normal hyperbolicity and under a generic property on the training data (i.e., holding for almost all training data).&lt;/p&gt;