Conventional Least-squares Method Research Articles

Investigating a Fractal–Fractional Mathematical Model of the Third Wave of COVID-19 with Vaccination in Saudi Arabia

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is responsible for coronavirus disease-19 (COVID-19). This virus has caused a global pandemic, marked by several mutations leading to multiple waves of infection. This paper proposes a comprehensive and integrative mathematical approach to the third wave of COVID-19 (Omicron) in the Kingdom of Saudi Arabia (KSA) for the period between 16 December 2022 and 8 February 2023. It may help to implement a better response in the next waves. For this purpose, in this article, we generate a new mathematical transmission model for coronavirus, particularly during the third wave in the KSA caused by the Omicron variant, factoring in the impact of vaccination. We developed this model using a fractal-fractional derivative approach. It categorizes the total population into six segments: susceptible, vaccinated, exposed, asymptomatic infected, symptomatic infected, and recovered individuals. The conventional least-squares method is used for estimating the model parameters. The Perov fixed point theorem is utilized to demonstrate the solution’s uniqueness and existence. Moreover, we investigate the Ulam–Hyers stability of this fractal–fractional model. Our numerical approach involves a two-step Newton polynomial approximation. We present simulation results that vary according to the fractional orders (γ) and fractal dimensions (θ), providing detailed analysis and discussion. Our graphical analysis shows that the fractal-fractional derivative model offers more biologically realistic results than traditional integer-order and other fractional models.

Ambient noise tomography of a linear seismic array based on an improved Voronoi tessellation

Ambient noise tomography, when applied to a dense linear seismic array, has the capability to provide detailed insights into the fine velocity structures across diverse tectonic settings. The linear station arrangement naturally generates parallel and concentrated ray paths along the array trend. This unique geometry requires specific optimization of the inversion methodology and model parameterization. The Bayesian-based transdimensional inversion method, characterized by its fully non-linear nature and high degree of freedom in parameter settings, offers a powerful tool for ambient noise inversion. To effectively adapt this method to a linear array layout, we propose a modification to the Voronoi cell tessellation built in the transdimensional method. By introducing spatial priority to the Voronoi kernels, we strategically increased the density of Voronoi cells along the direction of the array. We then applied the modified approach to a linear seismic array in the North China Craton and validated its robustness through phase velocity images and resolution tests. Our improved non-uniform sampling technique in the 2-D model space accelerates convergence while simultaneously enhancing model accuracy. Compared with the conventional damped least-squares method, the proposed algorithm revealed a shear-wave velocity map with notable low-velocity anomalies situated in the middle and lower crust beneath the borders of the Ordos block and its surrounding orogenic belt. Aligned with the crustal structures revealed by receiver function and electrical imaging, our findings indicated that the western and eastern margins of the Ordos block had experienced intensive crustal wedge deformation and re-melting, respectively.

Identification of time-varying inertia tensor parameters of a space solar power satellite based on a robust weighted recursive algorithm under impulsive disturbance

Considering changes in the structural size and mass caused by the on-orbit assembly process in a planar space solar power satellite, this study proposes an improved robust recursive algorithm to identify the time-varying inertia tensor parameters of the system. In this algorithm, to decrease the effect of impulsive disturbance on inertial parameter identification during the on-orbit assembly and improve computational accuracy, robust weighted factor and multi-innovation theory are involved. Then, the time-varying inertia tensor parameters of this large flexible structure are identified based on the distributed measurement results of multiple attitude sensors. The simulation results indicate that the proposed algorithm improves the identification accuracy and has better robustness than the original method under the impulsive disturbance. The relative values of the average relative error caused by the impulsive disturbance under the proposed algorithm are approximately 27.8–73.0% less than those under the original method. Additionally, the results also demonstrate that the proposed algorithm has higher noise immunity than the classical recursive least-squares method in a time-varying system. When the impulsive disturbance and zero-mean Gaussian white noise are applied to the system, compared with the conventional recursive least-squares method, the proposed algorithm yields lower absolute and relative values of the average relative error by approximately 4.2% and 44.9%, respectively.

On the detection of upper mantle discontinuities with radon-transformed receiver functions (CRISP-RF)

SUMMARYSeismic interrogation of the upper mantle from the base of the crust to the top of the mantle transition zone has revealed discontinuities that are variable in space, depth, lateral extent, amplitude and lack a unified explanation for their origin. Improved constraints on the detectability and properties of mantle discontinuities can be obtained with P-to-S receiver function (Ps-RF) where energy scatters from P to S as seismic waves propagate across discontinuities of interest. However, due to the interference of crustal multiples, uppermost mantle discontinuities are more commonly imaged with lower resolution S-to-P receiver function (Sp-RF). In this study, a new method called CRISP-RF (Clean Receiver-function Imaging using SParse Radon Filters) is proposed, which incorporates ideas from compressive sensing and model-based image reconstruction. The central idea involves applying a sparse Radon transform to effectively decompose the Ps-RF into its underlying wavefield contributions, that is direct conversions, multiples, and noise, based on the phase moveout and coherence. A masking filter is then designed and applied to create a multiple-free and denoised Ps-RF. We demonstrate, using synthetic experiment, that our implementation of the Radon transform using a sparsity-promoting regularization outperforms the conventional least-squares methods and can effectively isolate direct Ps conversions. We further apply the CRISP-RF workflow on real data, including single station data on cratons, common-conversion-point stack at continental margins and seismic data from ocean islands. The application of CRISP-RF to global data sets will advance our understanding of the enigmatic origins of the upper mantle discontinuities like the ubiquitous mid-lithospheric discontinuity and the elusive X-discontinuity.

Genetic-evolutionary adaptive moment estimation-based semisupervised deep sequential convolution network for seismic impedance inversion: Application and uncertainty analysis

Recent developments in deep learning have shown significant improvement in the accuracy of acoustic impedance inversion results. However, the conventional gradient-based optimizers such as the root-mean-square propagation, the momentum, the adaptive moment estimation (ADAM), etc., used in the deep learning framework, inherently tend to converge at the nearest optimum point, thereby compromising the solution by not attaining the global minimum. We apply a hybrid global optimizer, genetic-evolutionary ADAM (GADAM) to address the issue of convergence at a local optimum in a semisupervised deep sequential convolution network-based learning framework to solve the nonconvex seismic impedance inversion problem. GADAM combines the advantages of adaptive learning of ADAM and genetic evolution of genetic algorithm, which facilitates faster convergence, and avoids sinking into the local minima. The efficacy of GADAM is tested on synthetic benchmark data and field examples. The results are compared with that obtained from a widely used ADAM optimizer and conventional least-squares method. In addition, uncertainty analysis is performed to check the implication of the optimizer’s choice in obtaining efficient and accurate seismic impedance values. Results find that the level of uncertainty and minima of loss function attained using the GADAM optimizer are comparatively lower than that for ADAM. Thus, we demonstrate that the hybrid optimizer, i.e., GADAM is more efficient than the extensively used ADAM optimizer in impedance estimation from seismic data in a deep learning framework.

Fault Current Phasor Estimation Below One Cycle Using Fourier Analysis of Decaying DC Component

The full-cycle discrete Fourier transform (FCDFT) is commonly used to estimate phasors in numerical relays because of its immunity to harmonics and accurate identification of a decaying dc component (DDC) in a fault current. For fast protection of power systems, it is necessary to speed up the phasor estimation by using a data window below one cycle. The conventional least squares (LS) methods below one cycle approximate the DDC as the first several terms of the Taylor series expansion, which degrades the phasor-estimation accuracy. This paper proposes an LS method that uses Fourier analysis for approximation of the DDC to speed up phasor estimation without accuracy degradation. The phasor-estimation accuracy of the proposed method is investigated for the data-window length in simulations using synthetic data of fault currents containing DDC, harmonics, and measurement noise. The phasor-estimation speed is also investigated with respect to the delay of the antialiasing filter in the numerical relay. The proposed method is compared with the conventional LS and FCDFT methods in simulations using synthetic fault-current data. The feasibility of the proposed method is demonstrated through the phasor estimation of simulated transmission-line faults.

Time-Series InSAR Dynamic Analysis With Robust Sequential Adjustment

Nowadays, SAR instruments such as ESA’s Sentinel-1 constellation, ICEYE’s constellation of small and agile radar satellites, and the upcoming ALOS-4 and NASA/ISRO SAR missions provide new opportunities for near-real-time monitoring of geohazards with enhanced spatiotemporal resolution. Sequential dynamic adjustment model is regarded as an effective way to rapidly update time-series InSAR measurements. However, the accuracy of geophysical parameters of interest estimated from the conventional sequential least squares is greatly sensitive to the anomalous observations and/or anomalous prior parameter information. This letter aims to introduce the robust sequential adjustment method based on the M-estimation principle into near-real-time InSAR deformation monitoring to mitigate the effect of anomalous errors. Using both synthetic and real Sentinel-1 SAR datasets over Echigo plain in Japan, we fully evaluate the performance of the robust sequential estimation approach with respect to unwrapping errors in the SAR data stack. Measurements at 9 GPS stations located in the study area are used to validate the results. We find that the averaged RMSE of robust sequential adjustment is reduced by 15% in comparison with that of the conventional sequential least-squares method.

Improved Multi-Echo Gradient-Echo-Based Myelin Water Fraction Mapping Using Dimensionality Reduction.

Multi-echo gradient-echo (mGRE)-based myelin water fraction (MWF) mapping is a promising myelin water imaging (MWI) modality but is vulnerable to noise and artifact corruption. The linear dimensionality reduction (LDR) method has recently shown improvements with regard to these challenges. However, the magnitude value based low rank operators have been shown to misestimate the MWF for regions with [Formula: see text] anisotropy. This paper presents a nonlinear dimensionality reduction (NLDR) method to estimate the MWF map better by encouraging nonlinear low dimensionality of mGRE signal sources. Specifically, we implemented a fully connected deep autoencoder to extract the low-dimensional features of complex-valued signals and incorporated a sparse regularization to separate the anomaly sources that do not reside in the low-dimensional manifold. Simulations and in vivo experiments were performed to evaluate the accuracy of the MWF map under various situations. The proposed NLDR-based MWF improves the accuracy of the MWF map over the conventional nonlinear least-squares method and the LDR-based MWF and maintains robustness against noise and artifact corruption.

Control of managed pressure drilling systems using nonlinear predictive generalized minimum variance approach based on a Volterra model

This paper proposes a nonlinear predictive generalized minimum variance (NPGMV) control scheme for automatic control of the managed pressure drilling (MPD) systems in the presence of disturbances. Since the exact model of the system is not usually available in practice, the hydraulic flow model of MPD is described by an autoregressive second-order Volterra model. The conventional least-squares method is applied to input–output data, thereby identifying the Volterra model. Bottom-hole pressure regulation and kick handling are achieved through the control scheme. To deal with a reservoir kick, the proposed method switches to flow control mode automatically, and prevents the reservoir fluid influx into the well surface. The proposed method also has the capability to keep the bottom-hole pressure above the reservoir pressure during a formidable scenario such as pipe connection. In addition, the robustness of the controller in the face of heave disturbance and uncertainty is investigated. To show the effectiveness of the proposed method, the comparison study of several scenarios with a switching PI controller is provided and demonstrates the NPGMV control outperforms other approaches with respect to steady-state performance.

Design of nonlinear predictive generalized minimum variance control for performance monitoring of nonlinear control systems

In this paper, a nonlinear predictive generalized minimum variance (NPGMV) controller is proposed and explicitly formulated for a class of nonlinear systems modeled by autoregressive second-order Volterra series, applying the polynomial approach. Hence, a new benchmark controller for performance assessment is introduced to improve the achievable control performance. Furthermore, to have an efficient control assessment, a data-driven algorithm based on the NPGMV control is presented that uses only the closed-loop operating data. In the design procedure, a multi-step cost function is defined to incorporate predictive action. Exploiting the predictive control concept enables the control scheme to handle constrained problems. Also, the proposed control algorithm utilizes an inherent integrating effect, which is essential for practical purposes. Volterra series are employed for modeling and identification of the nonlinear processes, using conventional least-squares methods. To show the effectiveness of the proposed methodology, simulation results and comparison studies are provided on a cascade Wiener model and a continuous stirred tank reactor (CSTR) chemical pilot plant. Finally, an experimental study on a pressure pilot plant is used to demonstrate the applicability of the proposed control scheme. The simulation and experimental results indicate satisfactory performance of the proposed controller.

A dual response surface-based efficient fragility analysis approach of offshore structures under random wave load

This paper presents an efficient fragility analysis approach for offshore structures subjected to random wave load considering parameter uncertainty. Fragility analysis of structures under random dynamic load is generally accomplished by the direct Monte Carlo simulation (MCS), which requires extensive computational time. Hence, in the present study, a new cumulative distribution function matching-based fragility analysis approach is proposed in the framework of dual response surface method to reduce this computational burden. The proposed approach does not require MCS during failure probability computations once the two response surfaces of mean and standard deviation of response are obtained. Thereby, a substantial amount of computational time is saved. The proposed approach is generic in nature and can be applied to any offshore structure, with different probability distributions of system parameters and higher uncertainty levels, as well. The record-to-record variation of random wave force time-histories (that occurs even for the same set up of hazard parameters) can be easily considered by the proposed approach. The accuracy of the proposed approach is maintained by applying the moving least-squares method, in place of the conventional least-squares method, which is often reported to be a source of error. Two illustrative examples (one simple benchmark problem and another practical steel offshore platform structure) are presented to demonstrate the efficiency of the proposed fragility analysis approach. The results indicate that the present moving least-squares method yields more accurate solutions than the conventional least-squares method. Also, the computational efficiency by the proposed approach over the usual MCS-based approach can be clearly envisaged from the numerical studies.

Bayesian estimation of equivalent circuit parameters of photovoltaic cells

The equivalent circuit model of a photovoltaic cell is useful for understanding device loss mechanisms as well as the design and analysis of systems using photovoltaic cells. Although nonlinear least-squares methods are routinely applied for the extraction of equivalent circuit parameters using current–voltage characteristic curves, they only yield a point estimation sensitive to the initial value and do not provide an estimation error. In this study, a data-driven approach using Bayesian estimation is proposed to extract equivalent circuit parameters. Contrary to the conventional least-squares method, this method does not require tedious initial value adjustments and provides an estimation error.

An approach to the analysis of the impedance spectra of solid oxide fuel cell using the DRT technique

Determination of the polarization resistance of the anode and cathode in a solid oxide fuel cell (SOFC) with a thin-film electrolyte is a complicated task due to the difficulty of the manufacturing the reference electrode. In this case, only the total polarization resistance of the electrodes is available for analysis. When the relaxation times of the electrode processes are close, the separation of the electrode reactions, in most cases, is impossible by the conventional nonlinear least-squares method due to а several reaction processes in anode and cathode. In this paper, using a fuel cell with a supporting cathode as an example it is shown that the distribution of relaxation times (DRT) technique for the analysis of the impedance spectra of the fuel cell can be used to determine the polarization resistance of the cathode and anode. This becomes possible at comparing and analyzing the DRT functions obtained by varying the composition of gas mixtures at the cathode and anode of the fuel cell, using an electrocatalyst to accelerate the cathode reaction as well as comparing the DRT functions of the fuel cell with the supporting cathode to the DRT functions of cells with supporting electrolyte and symmetric electrodes.

Polynomial least squares multiple-model estimation: simple, optimal, adaptive, practical

This paper presents a polynomial least squares (LS) method of overcoming numerous weaknesses and deficiencies in the interacting multiple model (IMM), which comprises multiple Kalman filter models as the state-of-the-art algorithm for tracking maneuvering targets. The paper also addresses several polynomial LS misunderstandings and flaws applied in econometrics. These aims are achieved by first uniquely deriving a very simple version of conventional LS, which fits discrete deterministic data by minimizing the sum of squared deviations of data from an assumed polynomial (often called data-fitting). The contrasting contemporary polynomial LS method filters out corrupting statistical noise from already existing known polynomials, as in target tracking. Contemporary polynomial LS is developed from the conventional LS method derived by Gauss. Polynomial LS conventionally applied to data-fitting and contemporarily applied to noise filtering are shown to be different problems that are not to be conflated as done in econometrics, where data-fitting is described as regression using a 1st degree polynomial called ordinary LS. Most significantly, analogous to the IMM, the polynomial LS multiple model (LSMM) is derived. Contrary to the IMM, which is not optimized; the LSMM is optimized with a trade-off between the statistical variance and the deterministic bias-squared to minimize the mean-square-error (variance plus bias-squared). A sequence of optimal LSMMs matched to accelerations covering the spectrum of acceleration between zero and the assumed maximum are derived and applied in designing an adaptive algorithm. Results demonstrate improved target tracking and accuracy by the LSMM of maneuvering targets compared with published IMM performances.

Feature‐oriented channel estimation in reconfigurable intelligent surface‐assisted wireless communication systems

In this6study, the channel estimation problem is investigated for a wireless communication system assisted by a reconfigurable intelligent surface (RIS). The RIS thus creates an assistant channel, which has the features of positivity and dominance. Owing to these features, the channel estimation problem is formulated as a constrained residual sum of squares minimisation problem, which differs radically from the traditional channel estimation issue. An efficient Lagrange multiplier and dual ascent-based estimation scheme is then designed to obtain an iterative solution for the estimator. Moreover, the Cramer–Rao lower bounds are deduced as a performance benchmark. Simulation results show that the authors' designed scheme improves the estimation accuracy up to 33%, compared with the conventional least-square method in the low signal-to-noise ratio regime.

Artificial neural network for multi-echo gradient echo-based myelin water fraction estimation.

To demonstrate robust myelin water fraction (MWF) mapping using an artificial neural network (ANN) with multi-echo gradient-echo (GRE) signal. Multi-echo gradient-echo signals simulated with a three-pool exponential model were used to generate the training data set for the ANN, which was designed to yield the MWF. We investigated the performance of our proposed ANN for various conditions using both numerical simulations and in vivo data. Simulations were conducted with various SNRs to investigate the performance of the ANN. In vivo data with high spatial resolutions were applied in the analyses, and results were compared with MWFs derived by the nonlinear least-squares algorithm using a complex three-pool exponential model. The network results for the simulations show high accuracies against noise compared with nonlinear least-squares MWFs: RMS-error value of 5.46 for the nonlinear least-squares MWF and 3.56 for the ANN MWF at an SNR of 150 (relative gain=34.80%). These effects were also found in the in vivo data, with reduced SDs in the region-of-interest analyses. These effects of the ANN demonstrate the feasibility of acquiring high-resolution myelin water images. The simulation results and in vivo data suggest that the ANN facilitates more robust MWF mapping in multi-echo gradient-echo sequences compared with the conventional nonlinear least-squares method.

Geological Structure-Guided Initial Model Building for Prestack AVO/AVA Inversion

Reconstructing an accurate and high-resolution subsurface model is attractive in the fields of both geology and seismology. However, due to the band-limited characteristics of seismic data, the inversion greatly depends on the reliability of the initial model. A fairly acceptable initial model could lay a good foundation for seismic inversion. In this article, we first introduce a well-log interpolation method with the local slope as a constraint for building a high-fidelity starting model in prestack amplitude versus offset/angle (AVO/AVA) inversion. First, we briefly review the basic theory of general seismic inversion. Then, instead of using the conventional preconditioned least-squares method, we introduce shaping regularization theory into the geological structure-guided well-log interpolation to accelerate the convergence. We use the plane-wave destruction (PWD) algorithm to extract the slope attribute from seismic data, images, or velocity models. The slope is used as the constraint to solve the inverse problem based on the shaping regularization method. Numerical examples demonstrate that the proposed initial model building method performs better than the conventional ones. It greatly improves the accuracy of inversion results. Furthermore, we apply the proposed model building method to the inverse problems of AVO/AVA inversion and reservoir parameter estimation of several field data sets for the first time, which demonstrate encouraging performance.

SAR Interferometric Baseline Refinement Based on Flat-Earth Phase without a Ground Control Point

Interferometric baseline estimation is a key procedure of interferometric synthetic aperture radar (SAR) data processing. The error of the interferometric baseline affects not only the removal of the flat-earth phase, but also the transformation coefficient between the topographic phase and elevation, which will affect the topographic phase removal for differential interferometric SAR (D-InSAR) and the accuracy of the final generated digital elevation model (DEM) product for interferometric synthetic aperture (InSAR). To obtain a highly accurate interferometric baseline, this paper firstly investigates the geometry of InSAR imaging and establishes a rigorous relationship between the interferometric baseline and the flat-earth phase. Then, a baseline refinement method without a ground control point (GCP) is proposed, where a relevant theoretical model and resolving method are developed. Synthetic and real SAR datasets are used in the experiments, and a comparison with the conventional least-square (LS) baseline refinement method is made. The results demonstrate that the proposed method exhibits an obvious improvement over the conventional LS method, with percentages of up to 51.5% in the cross-track direction. Therefore, the proposed method is effective and advantageous.

Optimized Pseudo-Padé Fourier Migrator in Terms of Propagation Angles

A wide-angle Fourier migrator is proposed to more accurately image complex media with strong lateral velocity contrasts. The broadband wave-equation migrator is developed based on the pseudo-Pade approximation, where the Pade coefficients are independent of spatial coordinates, leading to a pure Fourier transform-based matching solution for one-way wavefield extrapolation. We use genetic algorithms to estimate the constant Pade coefficients more accurately than is feasible with conventional least-squares methods. Because of the global features of pure Fourier migrators, we present an angle-partitioning optimization scheme with dip focusing to improve the performance of the Fourier migrator for super-wide-angle waves and strong velocity contrasts. The wavefield gradient is used to calculate propagation angles during dual-domain wavefield extrapolation. Particular attention is paid to the first-order optimized pseudo-Pade Fourier (OPF1) migrator, which significantly improves the split-step Fourier (SSF) method for strong lateral variations at the cost of one additional Fourier transform in each step. Wavefield extrapolation based on the OPF1 method actually constitutes linear interpolation in the wavenumber domain between two split-step terms. We benchmark the OPF1 migrator with other typical migrators based on the exact dispersion equation. Numerical experiments with impulse responses, the SEG/EAGE salt model and 3D field data demonstrate the excellent performance and efficiency of seismic imaging with the OPF1 migrator.

Implementation and benchmarking of a crosstalk-free method for wavefront Zernike coefficients reconstruction using Shack-Hartmann sensor data

In wavefront characterization, often the combination of a Shack-Hartmann sensor and a reconstruction method utilizing the Cartesian derivatives of Zernike circle polynomials (the least-squares method, to be called here Method A) is used, which is known to introduce crosstalk. In \citep{janssen2014zernike} a crosstalk-free analytic expression of the LMS estimator of the wavefront Zernike coefficients is given in terms of wavefront partial derivatives (leading to what we call Method B). Here, we show an implementation of this analytic result where the derivative data are obtained using the Shack-Hartmann sensor and compare it with the conventional least-squares method.