建立回归模型常采用最小二乘方法并忽略自变量观测误差。尽管同时顾及自变量和因变量观测误差的总体最小二乘方法近年来得到了广泛研究，但在模型预测时，依然忽略了待预测自变量的观测误差。对此，本文提出了一种严格考虑所有变量观测误差的无缝线性回归和预测模型，该模型将回归模型的建立和因变量预测联合处理，在建立回归模型过程中对待预测自变量的观测误差进行估计并修正，从而提高了模型预测效果。理论证明，现有的几种线性回归模型都是无缝线性回归和预测模型的特例。试验结果表明，无缝线性回归和预测模型的预测效果优于现有的几种模型，尤其在变量观测误差相关性较大时，无缝模型对预测效果的改善更为显著。
The regression model was traditionally established by using the least squares (LS) method where the errors of independent variables were ignored. Although the weighted total least squares (TLS) method that captures errors of both dependent and independent variables was extensively studied for regression analysis in recent years, it still neglects the errors of independent variables when predicting the corresponding dependent variables.This paper puts forward a seamless linear regression and prediction model which estimates regression parameters and predicts dependent variables simultaneously by considering the errors of all variables.In the seamless model, the errors of independent variables in the prediction model are predicted and corrected to improve the prediction accuracy.The several existing regression models are theoretically proved to be the special cases of the proposed seamless model. The experimental results show that the proposed seamless model outperforms the other existing models in the sense of prediction accuracy, especially when the error correlation of variables is significant.
[1] 邓勃. 分析测试数据的统计处理方法[M]. 北京:清华大学出版社, 1995. DENG Bo. Statistical Processing Method for Data of Analytic and Test[M]. Beijing:Tsinghua University Press, 1995.
[2] CHAMBERS J M, CLEVELAND W S, KLEINER B, et al. Graphical Methods for Data Analysis[M]. Belmont:Duxbury Press, 1983.
[3] CAMPBELL D, CAMPBELL S. Introduction to Regression and Data Analysis[R]. New Haven:StatLab Workshop Series, Yale University Center for Science and Social Science Information, 2008.
[4] SYKES A O. An Introduction to Regression Analysis[R]. Chicago:The Inaugural Coase Lecture, Law School, University of Chicago, 1993.
[5] SCHAFFRIN B, WIESER A. On Weighted Total Least-squares Adjustment for Linear Regression[J]. Journal of Geodesy, 2008, 82(7):415-421.
[6] SHEN Yunzhong, LI Bofeng, CHEN Yi. An Iterative Solution of Weighted Total Least-squares Adjustment[J]. Journal of Geodesy, 2011, 85(4):229-238.
[7] LI Bofeng, SHEN Yunzhong, LI Weixiao. The Seamless Model for Three-dimensional Datum Transformation[J]. Science China Earth Sciences, 2012, 55(12):2099-2108.
[8] XU Peiling, LIU Jingnan, SHI Chuang. Total Least Squares Adjustment in Partial Errors-in-variables Models:Algorithm and Statistical Analysis[J]. Journal of Geodesy, 2012, 86(8):661-675.
[9] 刘经南, 曾文宪, 徐培亮. 整体最小二乘估计的研究进展[J]. 武汉大学学报(信息科学版), 2013, 38(5):505-512. LIU Jingnan, ZENG Wenxian, XU Peiliang. Overview of Total Least Squares Methods[J]. Geomatics and Information Science of Wuhan University, 2013, 38(5):505-512.
[10] 曾文宪, 方兴, 刘经南, 等. 附有不等式约束的加权整体最小二乘算法[J]. 测绘学报, 2014, 43(10):1013-1018. DOI:10.13485/j.cnki.11-2089.2014.0173. ZENG Wenxian, FANG Xing, LIU Jingnan, et al. Weighted Total Least Squares Algorithm with Inequality Constraints[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(10):1013-1018. DOI:10.13485/j.cnki.11-2089.2014.0173.
[11] 王乐洋, 赵英文, 陈晓勇, 等. 多元总体最小二乘问题的牛顿解法[J]. 测绘学报, 2016, 45(4):411-417. DOI:10.11947/j.AGCS.2016.20150246. WANG Leyang, ZHAO Yingwen, CHEN Xiaoyong, et al. A Newton Algorithm for Multivariate Total Least Squares Problems[J]. Acta Geodaetica et Cartographica Sinica, 2016, 45(4):411-417. DOI:10.11947/j.AGCS.2016.20150246.
[12] LI Bofeng, WANG Miaomiao, YANG Yuanxi. Multiple Linear Regression with Correlated Explanatory Variables and Responses[J]. Survey Review, 2015. DOI:10.1179/1752270615Y.0000000006.
[13] 王苗苗, 李博峰, 沈云中. 顾及自变量与因变量误差及相关性的线性回归[J]. 同济大学学报(自然科学版), 2016, 44(3):446-453. WANG Miaomiao, LI Bofeng, SHEN Yunzhong. Linear Regression with Corrected Errors of Independent and Dependent Variables[J]. Journal of Tongji University (Natural Science), 2016, 44(3):446-453.
[14] KOCH K R. Least Squares Adjustment and Collocation[J]. Bulletin Géodésique, 1977, 51(2):127-135.
[15] KOCH K R. Parameter Estimation and Hypothesis Testing in Linear Models[M]. 2nd ed. Berlin Heidelberg:Springer, 1999.
[16] LI Bofeng, SHEN Yunzhong, ZHANG Xingfu, et al. Seamless Multivariate Affine Error-in-variables Transformation and Its Application to Map Rectification[J]. International Journal of Geographical Information Science, 2013, 27(8):1572-1592.
[17] FANG Xing. Weighted Total Least Squares Solution for Application in Geodesy[D]. Hanover:Leibniz University Hanover, 2011.
[18] FANG Xing. Weighted Total Least Squares:Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy, 2013, 87(8):733-749.
[19] FANG Xing. Weighted Total Least-squares with Constraints:A Universal Formula for Geodetic Symmetrical Transformations[J]. Journal of Geodesy, 2015, 89(5):459-469.
[20] SNOW K. Topics in Total Least-squares Adjustment within the Errors-in-variables Model:Singular Cofactor Matrices and Priori Information[D]. Columbus:The Ohio State University, 2012.
[21] TEUNISSEN P G J. Adjustment Theory:An Introduction (Mathematical Geodesy and Positioning)[M]. Delft, The Netherlands:VSSD, Delft University Press, 2000.
