Newton-Raphson Algorithm Research Articles

Order-restricted statistical inference and optimal censoring scheme for Gompertz distribution based on joint type-II progressive censored data

This paper considers the order-restricted statistical inference for two populations based on the joint type-II progressive censoring scheme. The lifetime distributions of the two populations are supposed to follow the Gompertz distribution with the same shape parameter but different scale parameters. The maximum likelihood estimates of the unknown parameters are derived by employing the Newton-Raphson algorithm and the expectation-maximization algorithm, respectively. The Fisher information matrix is then employed to construct asymptotic confidence intervals. For Bayes estimation, we assume an ordered Beta-Gamma prior for the scale parameters and a Gamma prior for the common shape parameter. Bayes estimations and the highest posterior density credible intervals for unknown parameters under two different loss functions are obtained with the importance sampling technique. To evaluate the performance of order-restricted inference, extensive Monte Carlo simulations are performed and two air-conditioning systems datasets are used to illustrate the proposed inference methods. In addition, the results are compared with the case when there is no order restriction on the parameters. Finally, the optimal censoring scheme is obtained by four optimality criteria.

Excitation and manipulation of super cavity solitons in multi-stable passive Kerr resonators

We report on the theoretical analysis as well as the numerical simulations about the nonlinear dynamics of cavity solitons in a passive Kerr resonator operating in the multistable regime under the condition of a sufficiently strong pump. In this regime, the adjacent tilted cavity resonances might overlap, thus leading to the co-existence of combinatory states of temporal cavity solitons and the extended modulation instability patterns. The cavity in the regime of multistablity may sustain distinct families of cavity solitons, vividly termed as super cavity solitons with much higher intensity and broader spectra if compared with those in the conventional bi-stable regime. The description of such complex cavity dynamics in the multstable regime requires either the infinite-dimensional Ikeda map, or the derived mean-field coupled Lugiato-Lefever equations by involving multiple contributing cavity resonances. With the latter model, we revealed the existence of different orders of super cavity solitons, whose stationary solutions were obtained by using the Newton-Raphson algorithm. Along this line, with the continuation calculation, we have plotted the Hopf / saddle-node bifurcation curves, thus identifying the existing map of the stable and breathing (super) cavity solitons. With this defined parameter space, we have proposed an efficient method to excite and switch the super cavity solitons by adding an appropriate intensity (or phase) perturbation on the pump. Such deterministic cavity soliton manipulation technique is demonstrated to underpin the multi-level coding, which may enable the large capacity all-optical buffering based on the passive fiber ring cavities.

Tighter ‘uniform bounds for Black–Scholes implied volatility’ and the applications to root-finding

Using the option delta systematically, we derive tighter lower and upper bounds of the Black–Scholes implied volatility than those in Tehranchi (2016) [11]. As an application, we propose a Newton–Raphson algorithm on the log price that converges rapidly for all price ranges when using a new lower bound as an initial guess. Our new algorithm is a better alternative to the widely used naive Newton–Raphson algorithm, whose convergence is slow for extreme option prices.

Determination of the plastic J integral of ductile material using the XFEM with only Heaviside function and variable-node elements

In this paper, the extended finite element method (XFEM) with only Heaviside function is proposed for the elastic–plastic fracture mechanics (EPFM) modeling. The proposed method removes tip enrichment functions depending on the polar coordinates at the crack front, and a step function only determined by the level set function is utilized to track the crack front. To alleviate the volumetric locking phenomena caused by the plastic incompressibility, the B-bar method is incorporated into the three dimensional (3D) XFEM program. Therefore, the fully integration scheme can be chosen to ensure the accuracy when addressing large plastic deformation in EPFM analysis. Additionally, the material tangent stiffness matrix of Ramberg-Osgood constitutive is given, and the local refinement technique using variable-node elements is adopted to reduce the number of elements and nodes for efficient analysis. A Newton-Raphson iterative algorithm is developed to solve the nonlinear algebraic equations caused by material nonlinearity. Several numerical examples including the determination of crack opening displacement, and the fully plastic J integral in the ductile materials are presented to test the performance of the proposed method. Comparisons with the results from the existing methodologies show that the new enrichment scheme can save computational cost and obtain sufficient accuracy even in the case of 3D curved crack.

Optimizing algorithm for high-precision imaging stitching systems based on spline surfaces

As optical systems continue to advance, non-uniform rational B-spline (NURBS) surfaces increasingly being considered in asymmetric optical systems due to their localized control characteristics. However, the representation of NURBS surfaces has complicated the analysis of these systems, leading to a significant computational burden. To address this challenge, we propose an optimizing algorithm for imaging optical systems based on high-precision ray tracing of NURBS surfaces. This method initiates with getting a knot grid as prior information, in conjunction with the Newton-Raphson algorithm, to obtain high-precision numerical solutions for the intersection of rays with a NURBS surface. Building upon this methodology, we introduce an optimization technique that includes shape evaluation to generate an evaluation function specific to NURBS surfaces. This approach is then applied within a rapid optimization process that accounted for the region of ray influence. Under consistent control point grids and sampling ray conditions, we present an off-axis four-mirror system to showcase that our algorithm has achieved a computational efficiency improvement of approximately 14 times compared to the previous method. This high-precision imaging design based on spline surfaces fulfills the need for efficient and accurate algorithms for NURBS surface applications in various imaging systems, providing guidance for practical applications.

Estimation method of the Homodyned K-distribution based on the Improved Shuffled Frog Leaping Algorithm for ultrasound tissue characterization

Abstract Aiming at promoting the accuracy of the integral value of the probability distribution function (PDF) of Homodyned K-distribution (HK), a new maximum likelihood estimation method for HK distribution parameters is proposed by using the improved shuffled frog leaping algorithm (ADM-SFLA) and Newton-Raphson iterative algorithm (NR). To improve the accuracy of the global search of the traditional SFLA, two variation factors of Gauss and Cauchy are first used to find the global optimal solution in a convergence state. Secondly, to obtain the integral node with optimal segmentation, the ADM-SFLA is used to perform the non-equidistant adaptive segmentation of the integral interval. Then, the PDF of the HK distribution is calculated between each numerical integration cell based on Simpson integral formula. Subsequently, the integral value of the PDF for HK distribution is obtained by summing the integration values from each numerical integration cell. Finally, the optimal estimation of HK distribution parameters is obtained based on NR algorithm. The performance of the proposed method is evaluated and compared through simulated data as well as the envelope signals obtained from the ultrasonic simulation model with different scattering components. The results show that the ADM-SFLA based method can improve the estimation accuracy of HK distribution parameters effectively.

Newton-Raphson Meets Sparsity: Sparse Learning Via a Novel Penalty and a Fast Solver.

In machine learning and statistics, the penalized regression methods are the main tools for variable selection (or feature selection) in high-dimensional sparse data analysis. Due to the nonsmoothness of the associated thresholding operators of commonly used penalties such as the least absolute shrinkage and selection operator (LASSO), the smoothly clipped absolute deviation (SCAD), and the minimax concave penalty (MCP), the classical Newton-Raphson algorithm cannot be used. In this article, we propose a cubic Hermite interpolation penalty (CHIP) with a smoothing thresholding operator. Theoretically, we establish the nonasymptotic estimation error bounds for the global minimizer of the CHIP penalized high-dimensional linear regression. Moreover, we show that the estimated support coincides with the target support with a high probability. We derive the Karush-Kuhn-Tucker (KKT) condition for the CHIP penalized estimator and then develop a support detection-based Newton-Raphson (SDNR) algorithm to solve it. Simulation studies demonstrate that the proposed method performs well in a wide range of finite sample situations. We also illustrate the application of our method with a real data example.

A Comprehensive Method for Computing Non-Linear Elastokinematic Properties of Passenger Car Suspension Systems: Double Wishbone Case Study

Abstract Suspension and steering design play a major role in ensuring the correct dynamic behavior of road vehicles. Passenger cars are especially demanding from this point of view: NVH and ride comfort requirements often collide with active safety-related requirements such as road holding in steady-state conditions and stability in transients. Driving pleasure is also important for market success, therefore accurate steering feedback and predictable handling properties are additional priorities. Since flexible bushings are used as interface between the suspension arms and the chassis, extra degrees-of-freedom make the design process a complex task. While the use of a multibody software is common practice in the industry, a dedicated computational tool can be more practical and straightforward, especially when undertaking the design of a new suspension concept ground-up. The paper presents a computational methodology for the design of an independent suspension with the associated kinematic and compliance attributes. Typical elastokinematic properties like toe, camber, wheelbase, and track variations versus tyre forces and moments can be computed by means of a dedicated software tool. A sort of validation was performed either by means of a comparison with a MathWorks Simscape® Multibody based model. Finally, a sensitivity analysis is given as an example. Computationally, the method proposed is intuitively based on the equilibrium equations. The nonlinear equations are then solved with Newton–Raphson algorithm. The method can be also optimized for computational efficiency and is thoroughly described so that the reader can easily replicate it in the desired programing environment.

An Optimal Strategy for Estimating Weibull distribution Parameters by Using Maximum Likelihood Method

Several methods have been used to estimate the Weibull parameters such as least square method (LSM), weighted least square method (WLSM), method of moments (MOM), and maximum likelihood (MLE). The maximum likelihood method is the most popular method (MLE). Newton-Raphson method has been applied to solve the normal equations of MLE’s in order to estimate the Weibull parameters. The method was used to find the optimal values of the Weibull distribution parameters for which the log-likelihood function is maximized. We tried to find the approximation solution to the normal equations of the MLE’s because there is no close form for get analytical solution. In this work, we tried to carry out a study that show the difference between two strategies to solve the MLE equations using Newton-Raphson algorithm. Both two strategies are provided an optimal solution to estimate the Weibull distribution parameters but which one more easer and which one converges faster. Therefore, we applied both strategies to estimate the Weibull’s shape and scale parameters using two different types of data (Real and simulation). We compared between the results that we got by applying the two strategies. Two studies have been done for comparing and selecting the optimal strategy to estimate Weibull distribution parameters using maximum likelihood method. We used some measurements to compare between the results such as number of steps for convergence (convergence condition), the estimated values for AIC, BIC and the RMSE value. The results show the numerical solution that we got by applying first strategy convergence faster than the solution that we got by applying second strategy. Moreover, the MRSE estimated by applying the first strategy is lower than the MRSE estimated by applying second strategy for the simulation study with different noise levels and different samples size.

Coexisting phenomena and antimonotonic evolution in a memristive Shinriki circuit

This paper is aimed at investigating the dynamical behaviors of the Shinriki circuit with a grounded asymmetric diode-bridge-based memristor through the implicit mapping method, including coexisting phenomena and antimonotonicity. To elucidate these behaviors, the basic properties, including equilibrium points and symmetry, of a Shinriki circuit incorporating a grounded asymmetric diode-bridge-based memristor are first illustrated. The discretization scheme for the circuit is established through the implicit mapping method, and the relatively complete bifurcation trees of each periodic motion by tuning a variable resistor can be analyzed through the Newton-Raphson algorithm. The unique unstable periodic bifurcation routes are also revealed. The evolution of antimonotonicity triggered by period-doubling or saddle-node bifurcations, along with the formed unstable motions, are comprehensively discussed. Furthermore, the simulation results are verified via a Field Programmable Gate Array (FPGA), by which both the asymmetric stable and unstable coexisting periodic attractors can be captured. The results will contribute to explaining the dynamical behaviors of numerous memristive circuits from a novel semi-analytical way, thus opening the possibility of analyzing some phenomena, such as unstable bifurcation routes, that cannot be directly observed via the numerical method.

Inference on exponentiated Rayleigh distribution with constant stress partially accelerated life tests under progressive type-II censoring

This study aims to explore the issues of evaluating the parameters and the accelerating factor based on constant stress for partially accelerating life tests when the potential failure times have an exponentiated Rayleigh distribution. Within the framework of progressive Type-II censoring schemes, we employ the Newton-Raphson algorithm as an iterative methodology to gain the maximum likelihood estimates, accompanied by proof of the existence of these point estimators. We also construct asymptotic confidence intervals for interested parameters and acceleration factors by utilizing the asymptotical characteristics of the maximum likelihood estimators. The Bayesian estimations of unknown parameters are derived by using the independent gamma priors and dependent Gamma-Dirichlet prior on the basis of square error and relatively smooth LINEX loss functions, respectively. Furthermore, we adopt the importance sampling method to compute Bayesian point estimates and the credible intervals with the highest posterior density. To validate the effectiveness of the suggested approaches, a series of simulated experiments are carried out. Lastly, we conduct analyzes on two actual datasets to show the applicability of the suggested techniques.

Thermomechanical responses of steel frames under aftershock after fires

A novel advanced fiber beam-column element, which incorporates stability functions and the fiber distributed plasticity model to capture geometric and material nonlinearities, respectively, has been successfully developed to predict the nonlinear inelastic thermo-mechanical behaviors of steel frames subjected to aftershock after fires. To address the nonlinear dynamic problems resulting from aftershocks after fires, a sequential nonlinear thermal incremental-iterative solution scheme has been developed, which is based on the Newton-Raphson algorithm and Newmark-β method. In addition, a transcendental probability-based method is employed to express the statistical characteristics of aftershock intensity relative to the mainshock. The proposed method is applied to analyze three framed structures: a steel portal frame, a two-story plane steel frame, and a four-story asymmetric space steel frame. This analysis shows the accuracy, computational performance, and applicability of the proposed method. Remarkably, the results obtained from the code work, using just one or two elements per member, are in close agreement with all the results analyzed by Abaqus. This emphasizes that the proposed method effectively predicts the response of framed structures under aftershock after fires with excellent computational efficiency, significantly surpassing traditional finite element approaches in commercial software like Abaqus. The successful results of the dynamic analysis of the frames under aftershocks following fires underscore the importance of considering aftershocks after a fire incident. It is demonstrated that the occurrence of an aftershock accelerates the failure of the frame compared to the case when the frame is subjected to fire alone. This highlights the critical need to account for aftershocks when assessing the safety and structural integrity of framed structures after a fire event.

Simulation Study for Estimating the Parameters and Reliability Function of Weighted Exponential Distribution with Fuzzy Data

This paper investigates the estimation of the two unknown parameters and the reliability function of the weighted exponential distribution. It explores Bayesian and non-Bayesian (maximum likelihood) estimation methods when the information availableisin the form of fuzzy data. The Newton-Raphson algorithm is used to obtain the maximum likelihood estimates. In Bayes estimation, the symmetric squared error loss function is used. This loss function linksequalimportance to the losses due to overestimating and underestimating equal magnitude. Lindley approximation procedure in Bayesian estimation theory is used to evaluate the ratio of integrals. A comparative analysis using simulation is carried out to evaluate the performance of the obtained parameters estimators using mean squared error criteria and the performance of the obtained reliability estimators using integrated mean squared error criteria. The simulation results demonstrate that, for different sample sizes, the performance of Bayes estimates surpasses the maximum likelihood, and that all estimators perform consistently

Nonlinear inelastic behaviors of concrete-filled steel tubular beam-column structures

A new advanced method combining stability function and distributed plasticity model has been developed using Fortran programming language to predict the nonlinear inelastic behavior of concrete-filled steel tubalar (CFST) under static loading. The advantage of this method is the ability to accurately study the nonlinear behavior using only one or two beam-column elements per member instead of using solid and shell elements as traditional methods, thereby improving the model analysis time. The Generalized Displacement Control (GDC) algorithm, capable of analyzing beyond the limit point, will be used to solve the nonlinear equilibrium equations instead of the traditional Newton-Raphson algorithm. The element stiffness matrix is integrated through the Gauss-Lobatto numerical integration framework, while the nonlinear geometric effects P-Δ and P-δ are considered using stability functions and a corresponding geometric matrix. The reliability and accuracy of the proposed method are verified by comparing the analysis results with experimental data. The obtained results have demonstrated that using beam-column elements for simulation, the proposed method still provides accurate results while significantly reducing computational resources. Therefore, this new method holds promise as a useful tool for practical design and analysis of statically loaded CFST structures.

A Partial Maximum Likelihood Method to Estimate Cox Model for Competing Risk Data with Application

Many Research and clinical studies have addressed the occurrence of failure (death). As a result of other (external) factors, which may introduce additional risks that compete with the event under study. The resulting data refers to the complete data on Competing risks, which are affected by time differences. The exact times of occurrence of these risks are known, meaning the times of failure are observed for all observations with certainty. The impact of these risks on the hazard function is estimated based on the Cox proportional hazards model, which is estimated using the partial maximum likelihood method and numerical algorithms for parameter estimation. This includes the effect of variables on the Cox hazard function and the nonparametric part, estimated by assessing the effect of time on the hazard function using the Kaplan-Meier formula and calculating competing risks through the cumulative hazard function. These methods were applied to experimental data through large-scale simulations of different sizes and parameters and for several arithmetic means and standard deviations models. Moreover, applied to real data from a sample of 80 individuals with breast cancer. Analyzing the simulation results and real data revealed that the Downhill algorithm outperforms the Newton-Raphson algorithm in terms of estimation accuracy and efficiency, based on the statistical criterion for comparison, the root mean square error. In addition, competitive risks explained the effect of common variables in increasing competitive risks beyond the hazard function of the Cox model of the Newton-Raphson estimator. While it is converging to the hazard function of the Cox model for the Downhill estimator . Paper type: Research paper

Mapping QTL controlling count traits with excess zeros and ones using a zero-and-one-inflated generalized Poisson regression model.

The research on the quantitative trait locus (QTL) mapping of count data has aroused the wide attention of researchers. There are frequent problems in applied research that limit the application of the conventional Poisson model in the analysis of count phenotypes, which include the overdispersion and excess zeros and ones. In this article, a novel model, that is, the zero-and-one-inflated generalized Poisson (ZOIGP) model, is proposed to deal with these problems. Based on the proposed model, a score test is performed for the inflation parameter, in which the ZOIGP model with a constant proportion of excess zeros and ones is compared with a standard generalized Poisson model. To illustrate the practicability of the ZOIGP model, we extend it to the QTL interval mapping application that underpins count phenotype with excess zeros and excess ones. The genetic effects are estimated utilizing the expectation-maximization algorithm embedded with the Newton-Raphson algorithm, and the genome-wide scan and likelihood ratio test is performed to map and test the potential QTLs. The statistical properties exhibited by the proposed method are investigated through simulation. Finally, a real data analysis example is used to illustrate the utility of the proposed method for QTLmapping.

Extended Newton–Raphson Algorithm for Accurate Power-Flow Calculation of Industrial Facilities Based on Induction Motor Catalog Information

Extended Newton–Raphson Algorithm for Accurate Power-Flow Calculation of Industrial Facilities Based on Induction Motor Catalog Information

High spatial resolution diffuse optical tomography based on cross-correlation of chaotic light.

A diffuse optical tomography system with chaotic laser is proposed for the three-dimensional optical phantom. The high signal-to-noise ratio is beneficial to improve the spatial resolution of diffuse optical tomography. It is essential to drive the chaotic laser as the incident light into the optical phantom. The transmitted light emitted from phantom as the detection light and a part of the incident light as the reference light to carry out cross-correlation analysis. The high-density source-detector configuration in parallel plate structure is designed for detecting targets in the phantom. The propagation of chaotic laser in the phantom is studied theoretically and experimentally based on the diffusion equation. Image reconstruction is achieved by the cross-correlation analysis of chaotic laser and the Newton-Raphson nonlinear algorithm. The performance of the proposed system has been assessed by reconstruction localization accuracy and contrast-noise-ratio. The results show that the spatial resolution of the proposed system can reach 1.5 mm and the localization error is less than 1 mm.

Statistical inference for a two-parameter Rayleigh distribution under generalized progressive hybrid censoring scheme

The two-parameter Rayleigh distribution, as an extended distribution of the Rayleigh distribution, has been widely applied in reliability analysis. With the introduction of the location parameter, two-parameter Rayleigh distribution becomes more flexible in fitting real-data. In this paper, based on generalized progressively hybrid censored(GPHC) sample from the two-parameter Rayleigh distribution, Classical and Bayesian inferences are discussed. The Newton-Raphson(NR) and Expectation-Maximization(EM) algorithms are used to compute the maximum likelihood estimates(MLEs). As well as the asymptotic confidence interval(ACI) estimation is obtained through the asymptotic distribution theory of maximum likelihood estimation(MLE) and computation of the observed Fisher information matrix. In Bayesian frame, the estimation of unknown parameters and prediction of future observable are taken into consideration. Due to Bayesian estimation is challenging to compute precisely and for the purpose of comparison, the Lindley's approximation, the Tierney-Kadane(TK) approximation and Markov chain Monte Carlo(MCMC) method are employed to obtain Bayesian estimates. Then, combining the MCMC algorithm mentioned in the article, the one- and two- samples Bayesian prediction are obtained. Finally, the simulation results are provided and a real-life data set is used for illustration purpose.

Evaluation of FACTS Contributions Using Branch Flow Model and Newton–Raphson Algorithm

Flexible alternating current transmission systems (FACTSs) have been widely incorporated in electric power systems in order to control system parameters. This paper proposes the modeling of four FACTS devices, using the Branch Flow Model (BF) as an optimization problem to reduce the complexity of the Newton–Raphson (NR) load flow code with FACTS devices. The devices are represented as variable impedances, as a function of a firing angle, and as voltage source converters (VSCs) located on the buses and transmission lines. This proposed model solves the problem associated with the selection of appropriate initial conditions of the parameters of each device that guarantee convergence. The model is validated by evaluating its percentage deviation with respect to the NR method, using the standard test systems, IEEE 5-bus, IEEE 14-bus, IEEE 30-bus, and IEEE 57-bus systems.