病态方程基于Liu估计的一种迭代估计新方法

姜兆英, 刘国林, 于胜文

姜兆英, 刘国林, 于胜文. 病态方程基于Liu估计的一种迭代估计新方法[J]. 武汉大学学报 ( 信息科学版), 2017, 42(8): 1172-1178. DOI: 10.13203/j.whugis20150218
引用本文: 姜兆英, 刘国林, 于胜文. 病态方程基于Liu估计的一种迭代估计新方法[J]. 武汉大学学报 ( 信息科学版), 2017, 42(8): 1172-1178. DOI: 10.13203/j.whugis20150218
JIANG Zhaoying, LIU Guolin, YU Shengwen. A New Iterative Estimator Method Based on Liu Estimator for Ill-Posed Equations[J]. Geomatics and Information Science of Wuhan University, 2017, 42(8): 1172-1178. DOI: 10.13203/j.whugis20150218
Citation: JIANG Zhaoying, LIU Guolin, YU Shengwen. A New Iterative Estimator Method Based on Liu Estimator for Ill-Posed Equations[J]. Geomatics and Information Science of Wuhan University, 2017, 42(8): 1172-1178. DOI: 10.13203/j.whugis20150218

病态方程基于Liu估计的一种迭代估计新方法

基金项目: 

国家自然科学基金 41274007

国家自然科学基金 41404003

山东省自然科学基金 ZR2012DM001

高等学校博士学科点专项科研基金 20123718110001

泰山学者建设工程 

详细信息
    作者简介:

    姜兆英, 博士, 主要从事InSAR数据处理和模型算法研究。15964287012@163.com

    通讯作者:

    刘国林, 博士, 教授。gliu@sdust.edu.cn

  • 中图分类号: P207

A New Iterative Estimator Method Based on Liu Estimator for Ill-Posed Equations

Funds: 

The National Natural Science Foundation of China 41274007

The National Natural Science Foundation of China 41404003

Shandong Province Natural Science Foundation of China ZR2012DM001

Specialized Research Fund for the Doctoral Program of Higher Education 20123718110001

Shandong Taishan Scholar Construction Project Under Special Funding 

More Information
    Author Bio:

    JIANG Zhaoying, PhD, specializes in InSAR data processing and model algorithm. E-mail:15964287012@163.com

    Corresponding author:

    LIU Guolin, PhD, professor. E-mail: gliu@sdust.edu.cn

  • 摘要: 当线性回归模型的设计矩阵病态时,最小二乘(least square,LS)估值方差大且不稳定,已不是一种优良估计。为了减弱病态性,许多有偏估计法如岭估计、主成分估计、Liu估计等被提出。基于Liu估计,引入迭代的思想,提出了一种新的有偏估计法—迭代估计法。借助对称正定矩阵的谱分解,将迭代公式转化为便于解算的解析表达式,并证明迭代公式在修正因子d∈[-1,1]是收敛的。基于Liu估计中修正因子d的确定方法,在均方误差最小的情况下给出最优修正因子d的确定公式。最后,分别利用LS估计、岭估计、Liu估计和提出的迭代估计对两个算例进行计算并给出实验结果。在第一个算例中,对观测向量添加不同的扰动,结果表明迭代估计法具有更强的抗干扰能力;第二个算例的结果表明,迭代估计法所得结果更接近于真值,即迭代估计法在均方误差意义下优于LS估计、岭估计和Liu估计。
    Abstract: In the presence of the design matrix's collinearity (which is equivalent to ill-conditioning) in the linear regression model, the least squares (LS) estimator has large variances and its solution is rather unstable, so the LS estimator is not the precise estimation any more. In order to weaken the ill-conditioning, many biased estimator methods are introduced, such as ridge estimator, the principal components estimator, the Liuestimator and so on. In this paper, based on the famous Liuestimator, we present a new biased estimator which is called a biased iterative estimator method. With the aid of spectral decomposition of the symmetric and positive matrix, the iterative formula is converted to a simple analytical expression conveniently for calculating. And the iterative formula is proved to be convergent in the condition of modified parameterd∈[-1, 1]. Following the deter mination method of modified parameter in the Liuestimator, we give a formula of the optimal modified parameter to minimize the mean squared error (MSE). Finally, we use the proposed biased iterative estimator, LS estimator, ridge estimator and the Liuestimator to calculate two numerical examples and give their experimental results. In the first example, we respectively add different perturbations to the observation vector. The simulation results show that compared with other three methods, the biased iterative estimator is more stable under the perturbation. Comparison results of the second example show that our new biased iterative estimator is more closed to the real value, that is superior, in the mean squared error sense, to the LS estimator, ridge estimator and the Liuestimator.
  • 表  1   ε=0.1时不同方法所得结果

    Table  1   Results of Different Methods when ε=0.1

    较值 真值 LS估计 岭估计 Liu估计 本文方法
    1 1.606 5 1.489 1 1.050 6 1.050 6
    1 -16.160 0 -5.304 3 1.053 7 1.053 9
    1 69.522 0 12.243 6 1.050 9 1.050 2
    1 294.290 0 12.176 4 1.049 1 1.046 1
    $\mathit{\boldsymbol{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X}}}$ 1 -2 082.400 0 -6.846 8 1.021 0 1.042 3
    1 3 765.700 0 -19.525 6 1.077 7 1.039 1
    1 -1 717.100 0 -16.402 1 1.018 6 1.036 2
    1 -1 428.500 0 2.268 8 1.019 1 1.033 8
    1 1 128.000 0 32.849 2 1.043 2 1.031 6
    ‖Δ$\mathit{\boldsymbol{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X}}}$‖ 0 4.986 9×103 45.749 3 0.139 6 0.129 9
    MSE($\mathit{\boldsymbol{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X}}}$) 0 2.486 9×107 20.930 0.019 48 0.016 8
    下载: 导出CSV

    表  2   ε=0.01时不同方法所得结果

    Table  2   Results of Different Methods when ε=0.01

    较值 真值 LS估计 岭估计 Liu估计 本文方法
    1 1.060 7 1.048 9 1.005 1 1.005 1
    1 -7.160 3 -3.695 7 1.005 4 1.005 4
    1 7.852 2 2.124 4 1.005 1 1.005 0
    1 30.329 0 2.117 6 1.004 9 1.004 6
    $\mathit{\boldsymbol{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X}}}$ 1 -207.340 0 2.153 2 1.002 1 1.004 2
    1 377.470 0 -1.052 6 1.007 8 1.003 9
    1 -170.810 0 -7.402 1 1.001 9 1.003 6
    1 -141.950 0 1.126 9 1.001 9 1.003 4
    1 113.700 0 4.184 9 1.004 3 1.003 2
    ‖Δ$\mathit{\boldsymbol{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X}}}$‖ 0 4.986 9×102 4.574 9 0.013 97 0.012 99
    MSE($\mathit{\boldsymbol{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X}}}$) 0 2.486 9×105 2.093 0 1.951 1×10-4 1.686 8×10-4
    下载: 导出CSV

    表  3   ε=0.001时不同方法所得结果

    Table  3   Results of Different Methods when ε=0.001

    较值 真值 LS估计 岭估计 Liu估计 本文方法
    1 1.006 1 1.004 9 1.000 5 1.000 5
    1 -0.828 4 0.936 9 1.000 5 1.000 5
    1 1.685 2 1.002 4 1.000 5 1.000 5
    1 3.932 9 1.111 8 1.000 5 1.000 5
    $\mathit{\boldsymbol{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X}}}$ 1 -1.983 4 0.921 5 1.000 2 1.000 4
    1 3.864 7 0.794 7 1.000 8 1.000 4
    1 -1.618 1 0.825 9 1.000 2 1.000 4
    1 -1.329 5 1.012 7 1.000 2 1.000 3
    1 1.227 0 1.318 5 1.000 4 1.000 3
    ‖Δ$\mathit{\boldsymbol{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X}}}$‖ 0 4.986 9 0.457 5 1.406 5×10-3 1.298 8×10-3
    MSE($\mathit{\boldsymbol{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X}}}$) 0 2.486 9×103 0.209 30 1.978 2×10-6 1.686 8×10-6
    下载: 导出CSV

    表  4   控制点的坐标和观测距离

    Table  4   Coordinate and Observation Distance of Control Points

    点号 坐标/m 观测距离/m
    x y z di, 10 di, 11
    P1 23.000 10.000 0.010 25.078 69 16.765 17
    P2 10.000 9.990 0.000 14.134 51 17.719 65
    P3 35.000 10.010 -0.010 36.415 88 28.442 94
    P4 100.000 19.990 0.005 101.479 43 93.168 39
    P5 -36.000 10.005 0.000 37.364 22 43.299 05
    P6 0.000 10.010 -0.005 10.010 04 8.600 60
    P7 56.000 9.995 0.010 56.996 06 49.256 18
    P8 -15.000 10.105 -0.010 18.035 90 22.559 66
    P9 -1.7000 10.008 -0.015 10.150 63 10.043 82
    下载: 导出CSV

    表  5   LS估计、岭估计、Liu估计以及本文方法的结果比较

    Table  5   Comparison of the Results Among LS Estimator、Ridge Estimator、Liu Estimator and Our Method

    较值 真值 LS估计 岭估计 Liu估计 本文方法
    0 -0.036 8 -0.041 1 -0.038 0 -0.038 4
    0 0.051 8 0.027 5 0.022 7 0.022 0
    0 9.363 1 0.086 0 0.375 9 0.055 9
    $\mathit{\boldsymbol{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X}}}$ 7 7.048 8 6.974 6 6.975 9 6.973 3
    10 5.596 3 10.036 3 9.866 2 10.019 6
    -5 -4.648 2 -4.873 5 -4.904 4 -4.911 7
    参数 0 0 0.482 7 0.137 5 0.137 5
    ‖Δ$\mathit{\boldsymbol{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X}}}$‖ 0 10.353 3 0.166 7 0.214 2 0.118 3
    MSE($\mathit{\boldsymbol{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{X}}}$) 0 18.281 3 0.219 6 0.170 9 0.014 0
    下载: 导出CSV
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  • 收稿日期:  2015-09-28
  • 发布日期:  2017-08-04

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