Comparative Study of Probabilistic and Least-Squares Methods for Developing Predictive Models

This article explores the comparison between the probability method and the least squares method in the design of linear predictive models. It points out that these two approaches have distinct theoretical foundations and can lead to varied or similar results in terms of precision and performance under certain assumptions. The article underlines the importance of comparing these two approaches to choose the one best suited to the context, available data and modeling objectives.

Least square method based on Haar wavelet to solve multi-dimensional stochastic Ito-Volterra integral equations

This paper proposes a method combining blue the Haar wavelet and the least square to solve the multi-dimensional stochastic Ito-Volterra integral equation.This approach is to transform stochastic integral equations into a system of algebraic equations.Meanwhile,the error analysis is proven.Finally,the effectiveness of the approach is verified by two numerical examples.

运用Four Square Writing Method促进高中英语读写教学——以“Theme Parks—Fun and More Than Fun”为阅读文本

读写教学可以使学生通过综合性语言实践活动,实现语言能力、文化意识、思维品质和学习能力的融合发展。运用Four Square Writing Method促进高中英语读写教学可以采用引导学生先搭建图示化的阅读文本结构和内容框架,然后内化所学框架,进而迁移运用所学框架解决实际生活中的新问题这一路径,以培养学生的核心素养。

GPS short-delay multipath estimation and mitigation based on least square method

The GPS multipath signal model is presented, which indicates that the coherent DLL outputs in multipath environment are the convolution between the ideal DLL outputs and the channel responses. So the channel responses can be estimated by a least square method using the observed curve of the DLL discriminator. In terms of the estimated multipath channels, two multipath mitigation methods are discussed, which are equalization filtering and multipath subtracting, respectively. It is shown, by computer simulation, that the least square method has a good performance in channels estimation and the multipath errors can be mitigated almost completely by either of the methods. However, the multipath subtracting method has relative small remnant errors than equalization filtering.

Online Detection of Broken Rotor Bar Fault in Induction Motors by Combining Estimation of Signal Parameters via Min-norm Algorithm and Least Square Method

Current research in broken rotor bar （BRB） fault detection in induction motors is primarily focused on a high-frequency resolution analysis of the stator current. Compared with a discrete Fourier transformation, the parametric spectrum estimation technique has a higher frequency accuracy and resolution. However, the existing detection methods based on parametric spectrum estima- tion cannot realize online detection, owing to the large computational cost. To improve the efficiency of BRB fault detection, a new detection method based on the min-norm algorithm and least square estimation is proposed in this paper. First, the stator current is filtered using a band-pass filter and divided into short overlapped data windows. The min-norm algorithm is then applied to determine the fre- quencies of the fundamental and fault characteristic com- ponents with each overlapped data window. Next, based on the frequency values obtained, a model of the fault current signal is constructed. Subsequently, a linear least squares problem solved through singular value decomposition is designed to estimate the amplitudes and phases of the related components. Finally, the proposed method is applied to a simulated current and an actual motor, the results of which indicate that, not only parametric spectrum estimation technique.

A STUDY ON THE WEIGHT FUNCTION OF THE MOVING LEAST SQUARE APPROXIMATION IN THE LOCAL BOUNDARY INTEGRAL EQUATION METHOD

The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.

Estimating canopy closure density and above-ground tree biomass using partial least square methods in Chinese boreal forests

Boreal forests play an important role in global environment systems. Understanding boreal forest ecosystem structure and function requires accurate monitoring and estimating of forest canopy and biomass. We used partial least square regression （PLSR） models to relate forest parameters, i.e. canopy closure density and above ground tree biomass, to Landsat ETM＋ data. The established models were optimized according to the variable importance for projection （VIP） criterion and the bootstrap method, and their performance was compared using several statistical indices. All variables selected by the VIP criterion passed the bootstrap test （p〈0.05）. The simplified models without insignificant variables （VIP 〈1） performed as well as the full model but with less computation time. The relative root mean square error （RMSE%） was 29% for canopy closure density, and 58% for above ground tree biomass. We conclude that PLSR can be an effective method for estimating canopy closure density and above ground biomass.

The least square particle finite element method for simulating large amplitude sloshing flows

Large amplitude sloshing in tanks is simulated by the least square particle finite element method （LSPFEM） in this paper. The least square finite element method （LSFEM） is employed to spatially discrete the Navier-Stokes equations, and to avoid the stabilization issues due to the incompressibility condition for equal-order interpolation of the velocity and the pressure, as usually used in Galerkin method to satisfy the well-known LBB condition. The LSPFEM also uses the Lagrangian description to model the motion of nodes （particles）. A mesh which connects these nodes is constructed by a triangulation algorithm to avoid the mesh distortion. A quasi a-shapes algorithm is used to identify the free surface boundary. The nodes are viewed as particles which can freely move and even separate from the main fluid domain. Finally this method is used to study the large amplitude sloshing evolution in two dimensional tanks. The results are compared with those obtained by Flow-3d with good agreement.

ELASTO-PLASTICITY ANALYSIS BASED ON COLLOCATION WITH THE MOVING LEAST SQUARE METHOD

A meshless approach based on the moving least square method is developed for elasto-plasticity analysis,in which the incremental formulation is used.In this approach,the dis- placement shape functions are constructed by using the moving least square approximation,and the discrete governing equations for elasto-plastic material are constructed with the direct collo- cation method.The boundary conditions are also imposed by collocation.The method established is a truly meshless one,as it does not need any mesh,either for the purpose of interpolation of the solution variables,or for the purpose of construction of the discrete equations.It is simply formu- lated and very efficient,and no post-processing procedure is required to compute the derivatives of the unknown variables,since the solution from this method based on the moving least square approximation is already smooth enough.Numerical examples are given to verify the accuracy of the meshless method proposed for elasto-plasticity analysis.

Improving Middle Square Method RNG Using Chaotic Map

One of the classic approaches in PRNGs is the middle square method in which with a simple mathe-matical model generating pseudorandom numbers in high speed and minimum correlation between output numbers. Despite these unique characteristics, the method contains weaknesses that a broader application of this algo- rithm will face. In this paper is studied middle square method and then a logistic chaotic map is introduced with its specific features and its improved weaknesses via using these characteristics. Finally the NIST tests suite s are presented, in order to detect the specific characteristics expected from truly random sequences.

ADAPTIVE FUSION ALGORITHMS BASED ON WEIGHTED LEAST SQUARE METHOD

Weighted fusion algorithms, which can be applied in the area of multi-sensor data fusion, are advanced based on weighted least square method. A weighted fusion algorithm, in which the relationship between weight coefficients and measurement noise is established, is proposed by giving attention to the correlation of measurement noise. Then a simplified weighted fusion algorithm is deduced on the assumption that measurement noise is uncorrelated. In addition, an algorithm, which can adjust the weight coefficients in the simplified algorithm by making estimations of measurement noise from measurements, is presented. It is proved by emulation and experiment that the precision performance of the multi-sensor system based on these algorithms is better than that of the multi-sensor system based on other algorithms.

Assessing dynamic response of multispan viscoelastic thin beams under a moving mass via generalized moving least square method

Dynamic response of multispan viscoelastic thin beams subjected to a moving mass is studied by an efficient numerical method in some detail. To this end, the unknown parameters of the problem are discretized in spatial domain using generalized moving least square method （GMLSM） and then, discrete equations of motion based on Lagrange＇s equation are obtained. Maximum deflection and bending moments are considered as the important design parameters. The design parameter spectra in terms of mass weight and velocity of the moving mass are presented for multispan viscoelastic beams as well as various values of relaxation rate and beam span number. A reasonable good agreement is achieved between the results of the proposed solution and those obtained by other researchers. The results indicate that, although the load inertia effects in beams with higher span number would be intensified for higher levels of moving mass velocity, the maximum values of design parameters would increase either. Moreover, the possibility of mass separation is shown to be more critical as the span number of the beam increases. This fact also violates the linear relation between the mass weight of the moving load and the associated design parameters, especially for high moving mass velocities. However, as the relaxation rate of the beam material increases, the load inertia effects as well as the possibility of moving mass separation reduces.

Preconditioned iterative methods for solving weighted linear least squares problems

A class of preconditioned iterative methods, i.e., preconditioned generalized accelerated overrelaxation （GAOR） methods, is proposed to solve linear systems based on a class of weighted linear least squares problems. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned iterative methods is better than that of the original methods. Furthermore, the effectiveness of the proposed methods is shown in the numerical experiment.

Penalized total least squares method for dealing with systematic errors in partial EIV model and its precision estimation

When the total least squares(TLS)solution is used to solve the parameters in the errors-in-variables(EIV)model,the obtained parameter estimations will be unreliable in the observations containing systematic errors.To solve this problem,we propose to add the nonparametric part(systematic errors)to the partial EIV model,and build the partial EIV model to weaken the influence of systematic errors.Then,having rewritten the model as a nonlinear model,we derive the formula of parameter estimations based on the penalized total least squares criterion.Furthermore,based on the second-order approximation method of precision estimation,we derive the second-order bias and covariance of parameter estimations and calculate the mean square error(MSE).Aiming at the selection of the smoothing factor,we propose to use the U curve method.The experiments show that the proposed method can mitigate the influence of systematic errors to a certain extent compared with the traditional method and get more reliable parameter estimations and its precision information,which validates the feasibility and effectiveness of the proposed method.

Stability of the MGS-like elimination method for equality constrained least squares problems

This paper proves that the weighting method via modified Gram-Schmidt（MGS） for solving the equality constrained least squares problem in the limit is equivalent to the direct elimination method via MGS（MGS-elimination method）. By virtue of this equivalence, the backward and forward roundoff error analysis of the MGS-elimination method is proved. Numerical experiments are provided to verify the results.

Research on Vibration Suppression of the Finite Plate with Square Steel Beams Using Traveling Wave Method

The vibration suppression of the finite plate with square steel beams is studied using traveling wave method. The finite plate with square beams is modeled as the coupling systems between the plate flexural motion and the flexural and torsional motions for the square beams. The vibration response at any position of the coupling structure can be obtained by wave method. Numerical results show that comparing to finite element method (FEM), not only the low frequency but also the medium-high frequency vibration response of the finite plate with square beam can be effectively calculated by wave method. The suppression effect can be increased as the square beam is located at one-third of the length of plate or increasing the height of the beam. The study provides reference for arranged square beams applying to vibration suppression of ship and train structures.

Least Squares Method from the View Point of Deep Learning

The least squares method is one of the most fundamental methods in Statistics to estimate correlations among various data. On the other hand, Deep Learning is the heart of Artificial Intelligence and it is a learning method based on the least squares. In this paper we reconsider the least squares method from the view point of Deep Learning and we carry out the computation thoroughly for the gradient descent sequence in a very simple setting. Depending on the values of the learning rate, an essential parameter of Deep Learning, the least squares methods of Statistics and Deep Learning reveal an interesting difference.

Least Squares Method from the View Point of Deep Learning II: Generalization

The least squares method is one of the most fundamental methods in Statistics to estimate correlations among various data. On the other hand, Deep Learning is the heart of Artificial Intelligence and it is a learning method based on the least squares method, in which a parameter called learning rate plays an important role. It is in general very hard to determine its value. In this paper we generalize the preceding paper [K. Fujii: Least squares method from the view point of Deep Learning: Advances in Pure Mathematics, 8, 485-493, 2018] and give an admissible value of the learning rate, which is easily obtained.

Comparison of Calibration Curve Method and Partial Least Square Method in the Laser Induced Breakdown Spectroscopy Quantitative Analysis

The Laser Induced Breakdown Spectroscopy (LIBS) is a fast, non-contact, no sample preparation analytic technology;it is very suitable for on-line analysis of alloy composition. In the copper smelting industry, analysis and control of the copper alloy concentration affect the quality of the products greatly, so LIBS is an efficient quantitative analysis tech- nology in the copper smelting industry. But for the lead brass, the components of Pb, Al and Ni elements are very low and the atomic emission lines are easily submerged under copper complex characteristic spectral lines because of the matrix effects. So it is difficult to get the online quantitative result of these important elements. In this paper, both the partial least squares (PLS) method and the calibration curve (CC) method are used to quantitatively analyze the laser induced breakdown spectroscopy data which is obtained from the standard lead brass alloy samples. Both the major and trace elements were quantitatively analyzed. By comparing the two results of the different calibration method, some useful results were obtained: both for major and trace elements, the PLS method was better than the CC method in quantitative analysis. And the regression coefficient of PLS method is compared with the original spectral data with background interference to explain the advantage of the PLS method in the LIBS quantitative analysis. Results proved that the PLS method used in laser induced breakdown spectroscopy was suitable for simultaneous quantitative analysis of different content elements in copper smelting industry.

Prediction of Logistics Demand via Least Square Method and Multi-Layer Perceptron

To implement the prediction of the logistics demand capacity of a certain region,a comprehensive index system is constructed,which is composed of freight volume and other eight relevant economic indices,such as gross domestic product(GDP),consumer price index(CPI),total import and export volume,port's cargo throughput,total retail sales of consumer goods,total fixed asset investment,highway mileage,and resident population,to form the foundation for the model calculation.Based on the least square method(LSM)to fit the parameters,the study obtains an accurate mathematical model and predicts the changes of each index in the next five years.Using artificial intelligence software,the research establishes the logistics demand model of multi-layer perceptron(MLP)neural network,makes an empirical analysis on the logistics demand of Quanzhou City,and predicts its logistics demand in the next five years,which provides some references for formulating logistics planning and development strategy.