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约束下考虑坐标分量误差相关性的直线拟合

宋占峰 郭捷佳 李军

宋占峰, 郭捷佳, 李军. 约束下考虑坐标分量误差相关性的直线拟合[J]. 西南交通大学学报, 2021, 56(6): 1283-1289. doi: 10.3969/j.issn.0258-2724.20200120
引用本文: 宋占峰, 郭捷佳, 李军. 约束下考虑坐标分量误差相关性的直线拟合[J]. 西南交通大学学报, 2021, 56(6): 1283-1289. doi: 10.3969/j.issn.0258-2724.20200120
SONG Zhanfeng, GUO Jiejia, LI Jun. Fitting a Straight-Line to Data Points with Correlated Noise Between Coordinate Components under Constraints[J]. Journal of Southwest Jiaotong University, 2021, 56(6): 1283-1289. doi: 10.3969/j.issn.0258-2724.20200120
Citation: SONG Zhanfeng, GUO Jiejia, LI Jun. Fitting a Straight-Line to Data Points with Correlated Noise Between Coordinate Components under Constraints[J]. Journal of Southwest Jiaotong University, 2021, 56(6): 1283-1289. doi: 10.3969/j.issn.0258-2724.20200120

约束下考虑坐标分量误差相关性的直线拟合

doi: 10.3969/j.issn.0258-2724.20200120
基金项目: 国家自然科学基金(51678574)
详细信息
    作者简介:

    宋占峰(1973—),男,副教授,博士,研究方向为道路与铁道线路优化设计方法,E-mail:songzhanfeng@csu.edu.cn

    通讯作者:

    李军(1973—),男,副教授,博士,研究方向为道路与铁道工程信息化及优化,E-mail:lijun_csu@csu.edu.cn

  • 中图分类号: U212.3

Fitting a Straight-Line to Data Points with Correlated Noise Between Coordinate Components under Constraints

  • 摘要:

    直线拟合在曲线拟合研究及工程实践中受到广泛关注,常用的普通最小二乘和正交最小二乘忽略了坐标分量误差相关性的存在. 基于此,首先论证了在铁路线路整正中全站仪测量坐标点的纵横坐标间存在误差相关性,同时线路中直线的拟合受到相邻线元的约束;然后,基于极大似然估计及拉格朗日条件极值原理,推导出了顾及约束和坐标分量误差相关性的直线拟合通用模型,并给出了高斯-牛顿迭代算法搜索最优解;最后,采用了实测的数据进行了验证及测试. 试验结果表明:该方法能在任何误差分布情况下考虑约束估计直线参数及其精度;考虑坐标相关误差时,参数估计精度在约束及无约束下分别提高了9.2%和2.7%;高斯-牛顿算法在约束及无约束情况下分别仅6次及3次迭代就搜索出最优直线.

     

  • 图 1  全站仪测量原理

    Figure 1.  Measuring principle

    图 2  拟合原理

    Figure 2.  Fitting principle

    图 3  高斯-牛顿迭代算法

    Figure 3.  Gauss-Newton iteration algorithm

    图 4  约束及无约束的直线拟合

    Figure 4.  Straight-line fitting with a constraint and without constraint

    表  1  实地观测点坐标及采用的3种随机模型

    Table  1.   Coordinate pairs of field surveying data and three stochastic models for fitting

    点号x/my/mC 中非零元素P1 中非零元素P2 中非零元素P3 中非零元素
    $\sigma _x^2$/mm2$\sigma _y^2$/mm2${\sigma _{xy}}$/mm2pxpypxy${p_x}/{p_y}$${p_x}/{p_y}$
    1 688.639 1398.869 7.3371 9.7881 −0.8902 0.1378 0.1033 0.0125 1 10000
    2 701.467 1383.525 7.3289 9.3014 −0.9080 0.1381 0.1088 0.0135 1 10000
    3 714.294 1368.180 7.3340 8.8888 −0.9115 0.1381 0.1140 0.0142 1 10000
    4 727.121 1352.835 7.3547 8.5485 −0.9080 0.1378 0.1185 0.0146 1 10000
    5 739.953 1337.495 7.3948 8.2767 −0.9044 0.1371 0.1225 0.0150 1 10000
    6 752.783 1322.152 7.4602 8.0684 −0.9071 0.1359 0.1257 0.0153 1 10000
    7 765.609 1306.806 7.5584 7.9166 −0.9210 0.1342 0.1281 0.0156 1 10000
    8 778.434 1291.460 7.6977 7.8126 −0.9487 0.1319 0.1299 0.0160 1 10000
    9 791.262 1276.115 7.8867 7.7477 −0.9909 0.1289 0.1312 0.0165 1 10000
    10 804.088 1260.770 8.1340 7.7132 −1.0462 0.1251 0.1320 0.0170 1 10000
    11 816.915 1245.425 8.4471 7.7012 −1.1111 0.1207 0.1324 0.0174 1 10000
    12 829.740 1230.078 8.8323 7.7054 −1.1805 0.1156 0.1325 0.0177 1 10000
    13 842.564 1214.731 9.2939 7.7206 −1.2485 0.1100 0.1324 0.0178 1 10000
    14 855.389 1199.384 9.8346 7.7437 −1.3087 0.1040 0.1321 0.0176 1 10000
    15 868.213 1184.036 10.4556 7.7729 −1.3546 0.0979 0.1316 0.0171 1 10000
    16 881.043 1168.694 11.1562 7.8077 −1.3803 0.0916 0.1309 0.0162 1 10000
    17 893.875 1153.353 11.9355 7.8489 −1.3806 0.0855 0.1301 0.0150 1 10000
    18 906.703 1138.009 12.7917 7.8981 −1.3511 0.0796 0.1289 0.0136 1 10000
    19 919.537 1122.670 13.7218 7.9569 −1.2880 0.0740 0.1276 0.0120 1 10000
    20 932.371 1107.331 14.7236 8.0277 −1.1886 0.0687 0.1261 0.0102 1 10000
    下载: 导出CSV

    表  2  顾及约束和相关误差的直线拟合过程

    Table  2.   Process of straight-line fitting with both a constraint and correlated noise

    迭代数/次$\hat a$$\hat b$/m${\hat \sigma _0}$/mm${\sigma _a}$${\sigma _b}$/mmw/mm
    0−0.6816542642210.15348059093.3188806.14777 × 10−293044.287650−479524.603400
    1−0.9351651062013.4987785844.0968086.07990 × 10−39201.714082−59428.755230
    2−1.1468870432183.687057448.7437959.66425 × 10−41148.244207−9789.990571
    3−1.1951307842221.77980510.3629463.19907 × 10−535.137584−394.045005
    4−1.1972013922223.40350916.4920035.52543 × 10−559.714483−0.691388
    5−1.1972048862223.40620516.5250025.55595 × 10−560.003229−0.000002
    6−1.1972048862223.40620516.5250025.55598 × 10−560.0035150
    下载: 导出CSV

    表  3  约束下3种随机模型拟合直线的参数估值及其精度

    Table  3.   Parameter estimation of fitting line and their precisions of three stochastic models with constraints

    随机模型$\hat a$$\hat b$/m${\hat \sigma _0}$/mm${\sigma _a}$${\sigma _b}$/mmxZ/myZ/m迭代数/次耗时/s
    P1−1.197204892223.406216.5255.55598 × 10−560.01079.9809930.447860.494
    P2−1.197222362223.425149.0316.12012 × 10−566.11079.9772930.452260.503
    P3−1.197222332223.425076.4786.12012 × 10−566.11079.9772930.452260.496
    下载: 导出CSV

    表  4  无约束下3种随机模型拟合直线的参数估值及其精度

    Table  4.   Parameter estimation of line fitting and their precisions of three stochastic models without constraint

    随机模型$\hat a$$\hat b$/m${\hat \sigma _0}$/mm${\sigma _a}$${\sigma _b}$/mm迭代数/次耗时/s
    P1−1.1962693222222.6720002.2936553.10233 × 10−525.00885230.461
    P2−1.1962590742222.6638796.7107393.16316 × 10−525.74394330.414
    P3−1.1962590652222.6638720.4624883.16316 × 10−525.74394230.438
    下载: 导出CSV
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出版历程
  • 收稿日期:  2020-03-24
  • 修回日期:  2020-08-03
  • 网络出版日期:  2020-08-24
  • 刊出日期:  2020-08-24

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