约束下考虑坐标分量误差相关性的直线拟合

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. 

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

Straight-line fitting has received extensive attention both in curve fitting research and engineering practice. The methods of ordinary least squares and orthogonal least squares fitting ignore the existence of the observation error correlation. The coordinate pairs of surveying points, obtained by a total station in railway realignment, not only have different levels of precision but also have correlated noise. Meanwhile, straight-line fitting is usually under constraints in the realignment. Thus, a straight-line fitting model was derived based on the maximum likelihood estimation and Lagrange conditional extremum theory, considering constraints and correlated noise between coordinate components, and a Gauss-Newton algorithm was presented to search for the optimum. The method was tested with the field surveying data. Experimental results show that the proposed fitting method is capable of estimating straight-line parameters and their precisions in all circumstances by specifying stochastic models. When considering correlated noise, the precision of estimated parameters improve 9.2% with a constraint and improve 2.7% without constraints, respectively. The Gauss-Newton algorithm takes only 6 and 3 iteration times with a constraint and without constraints respectively, for locating the optimum straight-line.