约束最小二乘问题的几个算法

讨论了具有线性等式与不等式最小二乘问题及等式约束带权最小二乘问题,运用矩阵Householder正交分解及有效约束集法等技巧,分别给出了几个有效实用的算法,最后进行了数值检验.     

等式约束对病态问题的影响及约束正则化方法

有效利用参数间已知的等式约束信息能够提高最小二乘解的精度,消除秩亏,但是等式约束能否消除或减弱平差模型的病态性尚不明了,由此提出了一种通过消除部分参数将等式约束病态问题转化为无约束问题的方法。然后分析了等式约束对病态问题的影响,用简单实例证明了加入约束后,系统可能呈现良态或病态,它的性态由原设计阵和等式约束共同决定,并提出了求解等式约束病态问题的诊断-正则化两步方法。最后用一个数值实例验证了该方法的可行性。     

等式约束EIV模型的Newton-Gauss迭代解法及其精度评定

利用平差参数间的合理等式约束能够提高解的稳定性。针对变量误差模型EIV(errors-in-variables)引入等式约束,分别针对系数阵良态和病态两种情形建立了约束总体最小二乘准则。基于非线性最小二乘问题的常用解法Newton-Gauss法,由约束准则构建了拉格朗日极值函数并由欧拉-拉格朗日必要条件导出了等式约束EIV模型的Newton-Gauss迭代解。针对精度评定时未考虑参数估值偏差所带来的影响这一不足,基于蒙特卡罗模拟法提出了一种估计约束EIV模型单位权方差和参数估值的协方差阵的数值方法。算例分析结果表明,约束总体最小二乘解严格满足先验等式约束条件;当系数阵病态时,约束条件能够提升解的稳定性和精度。此外,基于蒙特卡罗的数值方法能够获得稳定且合理的精度评定结果。     

附加一次和二次等式约束的Partial-EIV模型及相应算法

研究了附加一次和二次等式约束的Partial-EIV模型,推导了加权整体最小二乘估计准则下相应的计算公式,并讨论了仅附加一次等式约束的Partial-EIV模型和仅附加二次等式约束的Partial-EIV模型。通过正交线性回归和平面坐标转换两个算例进行实验,将新算法与已有的附加等式约束的EIV模型的方法进行了对比,发现文中方法计算效率更高,且适用于结构化EIV模型的求解。     

改进的约束总体最小二乘法

针对传统的约束最小二乘模型和总体最小二乘模型的局限性,该文提出了一种改进的约束总体最小二乘法。假设约束总体最小二乘问题中约束方程系数矩阵也存在误差,然后构造函数模型的广义拉格朗日函数,采用最小二乘法迭代求解非线性的法方程,最终获得了改进的约束总体最小二乘法的牛顿-高斯迭代公式和平差模型精度的无偏估计。该算法采用了更接近实际的平差模型,能够获得更加接近真值的估计参数,同时平差模型的精度更加接近模拟数据加入的噪声水平。实验结果表明,本文算法可有效解决对参数进行约束时的数据处理问题。     

融合正交几何信息的非线性等式约束整体最小二乘平差及迭代算法

正交距离最小二乘和加权整体最小二乘是解自变量含误差拟合问题的两种独立准则。加权整体最小二乘与正交距离最小二乘不同,它不考虑测量点与拟合点之间的连线垂直于拟合对象的几何信息,不能确保测量点到拟合对象的距离的平方和为极小值。针对该问题,本文将正交几何信息作为约束条件融入加权整体最小二乘,提出一种约束方程带有误差改正数的非线性等式约束整体最小二乘平差法。首先,把加权整体最小二乘平差的函数式看作是非线性方程,连同正交几何约束方程一并线性化,得到线性的平差函数方程;然后,采用拉格朗日乘数法推导其参数估计及精度评定公式,并给出迭代计算算法;最后,以平面直线拟合为例,对本文方法和计算算法进行验证。试验结果表明:①本文方法和算法具有可行性;②与加权最小二乘和加权整体最小二乘相比,本文方法计算的测量点到拟合直线的垂直距离平方和最小;③本文方法计算的测量点到拟合直线的距离与测量点到拟合点的距离相等。     

不等式约束最小二乘问题的解及其统计性质

对不等式约束最小二乘平差问题,借助非线性规划中的凝聚约束方法把多个不等式约束转化为一个等式约束,采用拉格朗日极值法求解,解与贝叶斯解或单纯形解一致。其优点在于该解能够表示为观测的明显表达式,由此解的统计性质与最优性可以确定。给出演示该方法的GPS单点定位算例。     

带不等式约束的最小二乘方法比较

解决带不等式约束的平差问题已有较多方法,但不同方法之间的对比分析不足。本文讨论三种带不等式约束的最小二乘算法:简单迭代算法、虚拟误差方程法、迭代乘子法,三种算法都是基于有效约束理论将不等式约束转换为等式约束,通过迭代求解。采用同一组数据和蒙特卡罗仿真实验方法对三种算法计算结果进行对比分析,实验结果分析表明:迭代乘子法算法效率最高,推荐使用该算法。     

基于最小二乘支持向量机回归的GPS高程转换模型

探讨了将最小二乘估计引入到支持向量机中,给出最小二乘支持向量机回归(LS-SVR)算法,该算法用等式约束代替传统支持向量机回归中的不等式约束,使得求解过程更为简单。将构建的LS-SVR模型应用到GPS高程转换中,提出该回归模型的实现步骤。实例表明,在有限样本情况下,LS-SVR完全可以达到与传统GPS高程拟合方法相同的精度,但实现起来更简单,而且在泛化性能方面具有理论上的保证。     

等式约束病态模型的截断奇异值解及其统计性质

利用平差模型间合理的先验信息能够显著提高解的稳定性和精度。本文基于病态模型引入等式约束条件,并采用截断奇异值法重构了系数阵以削弱其病态性;建立了修正等式约束模型,导出了病态模型的约束截断奇异值解及其偏差、方差以及均方误差公式;分析了截掉奇异值所引起解的偏差引入量与方差下降量的关系,得到了确定截断参数的条件。数值算例和病态测边网算例分析结果表明,最小二乘解严重偏离真值,500次模拟实验的平均RMSE为6.693 5,正则化解和截断奇异值解精度较最小二乘解有所提高,平均RMSE分别为0.365 8和0.365 2;本文提出的约束截断奇异值解的精度最高,与约束正则化解精度相当,其平均RMSE仅为0.057 3。     

Convex optimization under inequality constraints in rank-deficient systems

Many geodetic applications require the minimization of a convex objective function subject to some linear equality and/or inequality constraints. If a system is singular (e.g., a geodetic network without a defined datum) this results in a manifold of solutions. Most state-of-the-art algorithms for inequality constrained optimization (e.g., the Active-Set-Method or primal-dual Interior-Point-Methods) are either not able to deal with a rank-deficient objective function or yield only one of an infinite number of particular solutions. In this contribution, we develop a framework for the rigorous computation of a general solution of a rank-deficient problem with inequality constraints. We aim for the computation of a unique particular solution which fulfills predefined optimality criteria as well as for an adequate representation of the homogeneous solution including the constraints. Our theoretical findings are applied in a case study to determine optimal repetition numbers for a geodetic network to demonstrate the potential of the proposed framework.     

Area-to-point Kriging with inequality-type data

In practical applications of area-to-point spatial interpolation, inequality constraints, such as non-negativity or more general constraints on the maximum and/or minimum attribute value, should be taken into account. The geostatistical framework proposed in this paper deals with the spatial interpolation problem of downscaling areal data under such constraints, while: (1) explicitly accounting for support differences between sample data and unknown values, (2) guaranteeing coherent (mass-preserving) predictions, and (3) providing a measure of reliability (uncertainty) for the resulting predictions. The formal equivalence between Kriging and spline interpolation allows solving constrained area-to-point interpolation problems via quadratic programming (QP) algorithms, after accounting for the support differences between various constraints involved in the problem formulation. In addition, if inequality constraints are enforced on the entire set of points discretizing the study domain, the numerical algorithms for QP problems are applied only to selected locations where the corresponding predictions violate such constraints. The application of the proposed method of area-to-point spatial interpolation with inequality constraints in one and two dimension is demonstrated using realistically simulated data.     

基于约束条件的线要素化简算法质量评估

随着制图综合质量评价越来越受重视,对地图要素中所占比例最大的线要素的化简算法进行质量评估变得十分必要。结合线化简算法自身的特点,从几何和语义两方面总结了线要素在化简过程容易出现的问题,在分析线要素化简应满足的约束条件的基础上,提出对算法实施评估的评价指标,并选择水平中误差等三种评价指标对三种线化简算法具体实施质量评估。试验结果证明了本文提出的评估方法的科学性。     

等式约束的效率分析和一致性检验

利用正定二次型矩阵极值的方法进一步分析等式约束对平差结果的影响,推导出等式约束线性模型的验后单位权方差(中误差)最小值,验证了该值和无等式约束的方差一致性,并将等式约束线性模型的验后单位权中误差最小值作为等式约束的验后单位权中误差限差的基准值。     

线性混合模型的光谱解混算法综述

混合像元的存在不仅影响了基于高光谱图像的地物识别和分类精度,而且已经成为遥感科学向定量化发展的主要障碍。目前的混合像元分解算法大多采用线性混合模型,其关键步骤为端元提取。文中从线性混合模型的定义出发,总结了近年来提出的端元提取算法,并重点对SMACC、VCA、SGA等算法进行了深入的分析,最后总结了混合像元分解的发展趋势。     

A new algorithm for single receiver DCB estimation using IGS TEC maps

A new algorithm for single receiver DCB estimation using GIM vertical TEC gridded values is proposed. It estimates receiver DCB and vertical residual ionospheric delays using the least squares approach with linear constraints. The performance of the proposed algorithm was assessed by comparing estimated receiver DCBs with those provided by the IGS. The same comparisons were done using two other algorithms for receiver DCB estimation. It is demonstrated that the proposed algorithm is capable of reproducing IGS DCB values at the level of 0.1?C0.3?ns, which is better than the level of agreement observed for the other two algorithms. For our tests, we considered data from more than 100 IGS stations, daily, such that all major regions of the world were covered. Besides, both ionospherically quiet and disturbed days were considered. It provides some evidence that the aforementioned level of agreement with IGS receiver DCB values does not significantly dependent on geographical region and the state of the ionosphere. The algorithm is easy to implement and can be considered for online use.     

一种基于线特征的道路网变化检测算法

提出一种基于线特征的道路网变化检测算法。首先根据边缘的梯度信息从多时相遥感图像中提取变化的线特征;然后根据变化线特征的局部特性,检测出与道路模型相符合的变化道路段;最后通过道路网的全局约束条件,进行变化道路段的连接,实现变化道路的检测。提出的道路网变化检测算法将边缘的相位和幅度信息作为变化检测的判定依据,从而避免了道路的匹配与比较工作,降低了变化检测算法的复杂度,具有很强的实用性。将本文提出的方法用于多时相遥感图像的道路网变化检测,从实验结果可以看出该方法的有效性。     

基于非线性优化的摄像机2D标定法

基于平面靶标,利用不同的非线性优化方法精确标定摄像机的线性参数以及畸变参数。基于绝对二次曲线的图像（IAC）构成约束方程,线性求解线性模型内外参数的初始值,在此基础上充分考虑非线性模型中的径向畸变和切向畸变,利用Rodrigues旋转公式减小优化参数的个数,分别采用最速下降法和LM最优化方法求解精确参数。实验结果表明,基于非线性优化的2D标定法能够简化初始值的计算,获得精确的非线性参数,最速下降法具有较快的收敛速度,LM算法能够获得更小的投影误差,且利用优化工具箱可以显著减小计算难度。可以根据实际需要选择不同的优化算法,实现摄像机的快速精确标定。     

基于线性混合模型的端元提取方法综述

混合像元是遥感领域研究的热点,而基于线性混合模型的光谱解混合技术正在越来越广泛地应用在光谱数据分析和遥感地物量化中,这项技术的关键就在于确定端元光谱。本文归纳了目前几种比较成熟的端元提取算法,分析了它们的主要思想和存在的优缺点,最后介绍了端元提取技术的应用及其发展趋势。     

Propagation algorithms for attitude-related ambiguity resolution

A new approach for ambiguity resolution is presented and tested by means of Monte-Carlo simulations. This approach applies to both single and double phase differences sampled at a given instant, and to any number of baselines and lines of sight, provided the number of lines of sight exceeds a certain threshold. The algorithms are based on the combined global geometry of the baselines and lines of sight and thereby consider all geometric constraints that are inherent in the problem. The algorithms have been programmed and applied to a large number of different configurations of GPS sight lines, and limited to baselines of approximately 1 m in length and the GPS L1 frequency only. The cause of the occurrence of multiple solutions has been investigated. To check the adequacy and limitations of the new algorithms for practical application, it was assessed in which proportion the algorithms either failed to provide a result, provided a false result, or provided a good result as a function of number of baselines, lines of sight and accuracy of the ambiguous phase difference measurements. Electronic Publication