多元线性模型回归系数的主成分估计

本文对多元线性模型回归系数提出了主成分估计，并证明了主成分估计优于最小二乘估计。进一步，对最小二乘估计的任一线性变换，给出了均方误差的一个无偏估计，并应用极小化均方误差的无偏估计的方法，给出了确定偏参数的公式。     

多元线性正态SRM参数估计的渐近协方差阵及其估计

针对多元线性正态结构关系模型（SRM）,本文给出了真值向量协方差阵的广义逆构造,并在三种常见误差协方差阵假定下应用标准5法分别给出了模型结构参数约束最小二乘估计的渐近协方差阵及其相合估计。     

PC准则下生长曲线模型回归参数阵岭估计的优良性

将 PC准则推广应用于生长曲线模型参数阵的最小二乘估计和岭估计优劣性的比较 ,给出了岭估计在PC准则下优于最小二乘估计的条件     

混合回归模型参数MM估计的迭代算法及其收敛性

本文在[1]与[2]的基础上,给出混合回归模型参数两步M估计的一个迭代算法,并证明了这种算法的收敛性,最后通过随机模拟结果和一个实例,比较了两步M估计与两步最小二乘估计的优劣。     

PC准则下生长曲线模型回归参数阵岭估计的估计优良性

将ＰＣ准则推广应用于生长曲线模型参数阵的最小二乘估计和岭估计优劣性的比较,给出了岭估计在ＰＣ准则下优于最小二乘估计的条件。     

具有先验信息的线性模型参数估计精度影响的度量

从广义方差比，协方差迹之经以及均方误差这三个不同的度量指标刻划了用最小二乘估计代替具有先验信息的线性模型的混合估计后对估计精度的影响问题。     

回归系数的局部根方有偏估计

线性回归模型中回归系数β的估计常用最小二乘估计(LSE)．当自变量间存在多重共线性关系时，最小二乘估计就失去了它的优良性．文提出了一种局部根方估计．证明了它的种种优良性．如容许性、相合性、Ф优良性、优效性及其对最小二乘估计抗干扰性的改进．结出了在均方误差(MSE)准则和Pitman靠近准则下该估计对通常根方估计和LSE改进的范围。     

约束条件下的线性贝叶斯估计

本文提出等式约束下线性模型中回归参数的线性贝叶斯估计,证明其在均方误差矩阵准则下相对于约束最小二乘估计的优越性,并采用蒙特卡洛模拟和数值算例验证其优越性．     

半相依回归模型参数的协方差改进估计

本文研究了两个半相依回归系统的未知回归系数的估计问题。本文首先给出一种基于方差分量限定估计的两步协方差改进估计,并且给出了均方误差意义下优于最小二乘估计的条件。对于基于方差分量非限定估计的两步协方差改进估计,利用服从Wishart分布随机变量的可加性,本文给出了一种全新的估计形式,并且证明了该估计较文献中给出的两步协方差改进估计更加有效。     

任意四元非平面阵的定向性能分析

定向精度是被动声纳系统重要技术指标,合理的阵结构可以提高其定向性能。建立空间任意四元阵模型,利用矩阵变换推导出定向的最小二乘解并从理论上给出方位角估计的误差公式,求得方位角估计方差的均值最小和不受方位角影响应满足的条件。借助计算机仿真给出俯仰角的估计误差的定性分析。最后得到空间四元阵在满足一定的空间结构时具有最佳的定向性能,为四元阵设计提供了参考依据。     

块对角占优块三对角方程组的块重叠分割无通信并行求解方法

基于并行计算的分治思想,对块三对角线性方程组的求解提出了一个块重叠分割无通信的高效可扩展并行算法(PBOPUC算法)。当系统严格块对角占优时,在机器精度内,得到与精确解等价的近似解。通过精度分析,得到子方程组的阶数与精度的关系,并用它来控制精度和并行效率。本文的算法已经在上海大学的高性能并行计算机"自强3000"上实现,结果说明,并行计算效率接近100%,加速比几乎是线性的。     

Simplified Estimation of Parameters in a Double Exponential Distribution

Simplified estimators of the location and scale parameters of a double exponential distribution are given. Complete and symmetrically censored samples are considered. The high efficiency of these estimators relative to the best linear unbiased estimator (BLUE) make them useful in practice.     

线性互补问题的二级多重分裂并行算法

运用二级迭代方法与矩阵多重分裂理论,同时考虑并行计算,提出了求解线性互补问题的二级多重分裂并行算法。在一定条件下证明了算法的收敛性,分析了算法的收敛速度。该算法与已有算法相比,具有计算量小、计算速度快等特点,因而特别适于求解大规模问题。数值结果表明,该算法是十分有效的。     

关于贝叶斯估计的进一步讨论

文章则从绝对差损失函数与相对差损失函数出发 ,先导出两种新的贝叶斯解——中值估计和积分比估计 ;然后再对实际问题进行具体应用 ;在选定同类先验分布条件下将以上结果与传统的条件期望估计、最大后验估计相比较 ,探讨了各类贝叶斯解的优良性 .     

Simultaneous determination of bubble diameter and relative refractive index using glare circles

In the shadow image of a spherical gas bubble, high intensity rings are visible, i.e., glare circles. These can be used to obtain a more precise estimate of the bubble diameter than can be obtained from the shadow contour. The glare circle diameter can also be used to determine the relative refractive index by comparing it with the shadow diameter. The precision of this refractive index measurement reaches the third decimal, which is demonstrated experimentally. Thus, one can simultaneously determine the bubble diameter (from the shadow contour) and the relative refractive index (from the glare circle).     

矩阵方程X-A~TX~(-1)A=Q的牛顿迭代解法

非线性矩阵方程X-A~TX~(-1)A=Q在控制理论、动态规划、插值理论和随机滤波等领域中具有广泛的应用.本文给出了该矩阵方程的等价形式并利用牛顿法对该等价矩阵方程进行求解.通过定义一类用牛顿法求根时产生的矩阵序列与用牛顿法求解矩阵方程时产生的矩阵序列相同的矩阵函数,证明了由牛顿迭代法求解矩阵方程时产生的矩阵序列包含在具有唯一解的闭球内,并收敛到闭球内的唯一解.给出了该方程近似解与真解的误差估计式,并给出了说明牛顿算法对该方程求解有效性的数值例子.     

Numerical study of the three-dimensional wave equation using the mesh-free kp-Ritz method

This paper deals with numerical modeling of three-dimensional linear wave propagation based on the mesh-free kp-Ritz method. The mesh-free kernel particle estimate is employed to approximate the 3D displacement field. A system of discrete equations is obtained through application of the Ritz minimization procedure to the energy expressions. Convergence analysis and error estimates of the kp-Ritz method for three-dimensional wave equation are also presented in the paper. From the error analysis, we found that the error bound between the numerical and the exact solution is directly related to the radii of weight functions and the time step length. Effectiveness of the kp-Ritz method for three-dimensional wave equation is investigated by three numerical examples.     

Quantitative analysis of error bounds in the recovery of depth from defocused images

Depth from defocus involves estimating the relative blur between a pair of defocused images of a scene captured with different lens settings. When a priori information about the scene is available, it is possible to estimate the depth even from a single image. However, experimental studies indicate that the depth estimate improves with multiple observations. We provide a mathematical underpinning to this evidence by deriving and comparing the theoretical bounds for the error in the estimate of blur corresponding to the case of a single image and for a pair of defocused images. A new theorem is proposed that proves that the Cramér-Rao bound on the variance of the error in the estimate of blur decreases with an increase in the number of observations. The difference in the bounds turns out to be a function of the relative blurring between the observations. Hence one can indeed get better estimates of depth from multiple defocused images compared with those using only a single image, provided that these images are differently blurred. Results on synthetic as well as real data are given to further validate the claim.     

Motion-free superresolution and the role of relative blur

Motion-free superresolution refers to the process of generating a high-resolution image from a set of defocused and downsampled observations blurred to different extents. We first examine the theory of motion-free superresolution and derive analytical expressions in the discrete Fourier transform domain for obtaining a superresolved image from its blurred and downsampled versions. The analysis leads to the construction of an appropriate system matrix whose inverse yields the desired reconstruction filters. Next, we address the effect of relative blurring among the defocused observations on the quality of the reconstructed image. To get a quantitative perspective of this effect, we derive the Cramér-Rao lower bound (CRLB) for the covariance of the error in the estimate of the superresolved image. The CRLB depends on the system matrix, and the condition number of this matrix is studied as a function of the blur kernels. It is shown that the stability of the problem and the estimate of the superresolved image become better as the relative blur increases. Adding more observations without enriching the blur span does not improve the quality of reconstruction. Several simulation results are given for the purpose of validation.     

Modified nonlocal boundary value problem method for an ill-posed problem for the biharmonic equation

In this paper, we propose a modified nonlocal boundary value problem method for an homogeneous biharmonic equation in a rectangular domain. We show that the considered problem is ill-posed in the sense of Hadamard, i.e. the solution does not depend continuously on the given data. Convergence estimates for the regularized solution are obtained under a priori bound assumptions for the exact solution. Some numerical results are given to show the effectiveness of the proposed method.