The Least Square Solutions to the Quaternion Matrix Equation AX=B

A norm of a quaternion matrix is defined. The expressions of the least square solutions of the quaternion matrix equation AX = B and the equation with the constraint condition DX = E are given.

Rail Detection Based on LSD and the Least Square Curve Fitting

It is necessary to rely on the rail gauge to determine whether the object beside the track will affect train operation safety or not.A convenient and fast method based on line segment detector(LSD)and the least square curve fitting to identify the rail in the image is proposed in this paper.The image in front of the train can be obtained through the camera on-board.After preprocessing,it will be divided equally along the longitudinal axis.Utilizing the characteristics of the LSD algorithm,the edges are approximated into multiple line segments.After screening the terminals of the line segments,it can generate the mathematical model of the rail in the image based on the least square.Experiments show that the algorithm in this paper can fit the rail curve accurately and has good applicability and robustness.

Evaluation of dimension of fractal time series with the least square method

Properties of fractional Brownian motions （fBms） have been investigated by researchers in different fields, e.g. statistics, hydrology, biology, finance, and public transportation, which has helped us better understand many complex time series observed in nature [1-4]. The Hurst exponent H （0 〈 H 〈 1） is the most important parameter characterizing any given time series F（t）, where t represents the time steps, and the fractal dimension D is determined via the relation D = 2 - H.

Pattern synthesis optimization of 3-D ODAR based on improved GA using LSFE method

Pattern synthesis in 3-D opportunistic digital array radar(ODAR) becomes complex when a multitude of antennas are considered to be randomly distributed in a three dimensional space.In order to obtain an optimal pattern,several freedoms must be constrained.A new pattern synthesis approach based on the improved genetic algorithm(GA) using the least square fitness estimation(LSFE) method is proposed.Parameters optimized by this method include antenna locations,stimulus states and phase weights.The new algorithm demonstrates that the fitness variation tendency of GA can be effectively predicted after several "eras" by the LSFE method.It is shown that by comparing the variation of LSFE curve slope,the GA operator can be adaptively modified to avoid premature convergence of the algorithm.The validity of the algorithm is verified using computer implementation.

Analytical Structure Matching and Very Precise Approach to the Coulombic Quantum Three—Body Problem

A powerful approach to solve the Coulombic quantum three-body problem is proposed. The approach is exponentially convergent and more efficient than the hyperspherical coordinate method and the correlation-function hyperspherical harmonic method. This approach is numerically competitive with the variational methods, such as that using the Hylleraas-type basis functions. Numerical comparisons are made to demonstrate the efficiency of this approach, by calculating the nonrelativistic and infinite-nuclear-mass limit of the ground state energy of the helium atom. The exponential convergency of this approach is due to the full matching between the analytical structure of the basis functions that are used in this paper and the true wavefunction. This full matching was not reached by most other methods. For example, the variational method using the Hylleraas-type basis does not reflects the logarithmic singularity of the true wavefunction at the origin as predicted by Bartlett and Fock. Two important approaches are proposed in this work to reach this full matching: the coordinate transformation method and the asymptotic series method. Besides these, this work makes use of the least square method to substitute complicated numerical integrations in solving the Schr?dinger equation without much loss of accuracy, which is routinely used by people to fit a theoretical curve with discrete experimental data, but here is used to simplify the computation.

Analysis of Dynamic Characteristics of Crustal Horizontal Deformation in the South Segment of Longmenshan Fault Zone Revealed by GPS Data after the Wenchuan Earthquake

In order to obtain deformation parameters in the south segment of Longmenshan fault zone,Euler datum transformation and the least square collocation for data interpolation and smoothing are used to process GPS displacement time series data in the south segment of Longmenshan fault zone,and the rigid and elastic-plastic block motion model is used to calculate the strain parameters in each subarea. Conjoint analysis of displacement,velocity of each station and strain parameters of each subarea reveals that the influence of the Wenchuan earthquake on the south segment of Longmenshan fault zone increases from southeast to northwest,causing a highest deformation rate 6 times the background value and heightening the influence of the hidden faults on the difference of the earth surface along its two sides,which leads to the seismic risk of the southern segment increasing from north to south. The comparison of seismic risk among subareas based on the tectonic and seismicity background indicates that the most dangerous area is on the southeast of Longmenshan faults,and the background strain accumulation and the promoting effect of the Wenchuan earthquake advanced the occurrence of Lushan earthquake and the sinistral strike-slip on the rupture plane. The Wenchuan earthquake also caused a slight two-year long continuous strain release in the south segment of Xianshuihe fault,but the influence is far less than the effect of the compressive strain caused by the Sichuan-Yunnan block.

A Computational Model for Velocity Separation in Shallow Sea

Based on the hydrodynamical feature and the theoretical velocity profiles of tidal flow and wind-induced flow in shallow sea, a computational model is established for the first time, which can separate observed velocity into tidal velocity and wind-induced velocity by use of the least square method. With the model, not only the surface. velocities of tidal. flow and wind-induced flow are obtained, but also the bed roughness height is determined and the wind velocity above the water surface is estimated. For verification of the model, the observed velocity in the Yellow River Estuary and the laboratory test is separated, then it is applied to the Yangtze River Estuary. All the results are satisfactory. The research results show that the model is simple in method, feasible in process and reasonable in result. The model is a valid approach to analysis and computation of field data, and can be applied to separate the observed velocity in shallow sea; at the same time, reasonable boundary conditions of the surface and bottom can be obtained for two- and three-dimensional numerical simulation.

A New Model for Velocity Profiles in Estuarine Waters

Water motion in estuarine waters is the result of the action of various dynamic factors. Firstly, based on the hydro- dynamic characteristics in estuarine waters, neglecting the nonlinear effects of various flow hydrodynamic factors, the logarithm velocity profile of tidal current and the cubic velocity profile of Hansen and Rattray （1965） made for linear super- position at a sense of first order, a new model for velocity profile in estuarine waters is established. Then, by introducing the least square method combination of enumeration, the velocity profile data of wind-driven current measured in the laboratory and that observed at the North and the South Branches of the Yangtze Estuary are verified and compared with other formulas, all with satisfactory results. The results show that the new model not only considers the influences of various dynamic factors, such as tide, wind force, run-off and density pressure with high accuracy, but also provides reasonable boundary conditions on the bottom for hydrodynamics numerical simulation in estuarine waters. Thereby, the accuracy and credibility of numerical computation and prediction of water flow are improved. The research is theoretically important for the estuarine hydrodynamics.

Rough Set Based Fuzzy Neural Network for Pattern Classification

A rough set based fuzzy neural network algorithm is proposed to solve the problem of pattern recognition. The least square algorithm (LSA) is used in the learning process of fuzzy neural network to obtain the performance of global convergence. In addition, the numbers of rules and the initial weights and structure of fuzzy neural networks are difficult to determine. Here rough sets are introduced to decide the numbers of rules and original weights. Finally, experiment results show the algorithm may get better effect than the BP algorithm.