有偏估计若干问题的研究
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摘要
病态性在常规大地测量网平差、GPS快速定位、大地测量反演以及变形监测网的数据处理中是大量存在的，其危害性十分严重。当确知测量平差模型存在病态性时，必须设法消除或减弱病态性的影响，以提高参数估计和平差成果的精度。事实上，分析测量平差系统病态性的实质、克服或减弱测量平差系统病态性的影响、采用有偏估计等方法提高参数估计和平差成果的精度，是当前GPS等重大测量工程数据处理中所面临的一个重要课题，它已被国际大地测量协会(IAG)确立为现代测量误差理论及数据处理研究中的一项重要内容。
本文针对岭估计、Stein均匀压缩估计和主成分估计存在的缺陷，构造了两种新的有偏估计：岭—压缩组合估计和岭—主成分组合估计，讨论了两种新估计在均方误差意义下和数值稳定性方面的优良性质，讨论了有偏估计中偏参数的选取问题，通过理论分析和算例分析说明这是两种很有潜力的有偏估计。
在均方误差准则下对目前应用最广泛的两种有偏估计——岭估计和主成分估计与LS估计进行了比较研究，得到了岭估计、主成分估计优于LS估计的条件；然后运用统计方法对这些条件的成立进行了检验，从假设检验的角度解决了有偏估计与LS估计之间的选择问题。
针对粗差和病态性同时存在问题，把拟准检定法的优点和有偏估计的优点结合在一起，提出了基于拟准检定的抗差化有偏估计。通过数值实验说明了基于拟准检定的抗差化有偏估计确实是一种很好的估计，对消除粗差和病态性的不良影响非常有效。
对岭估计进行了影响分析研究，在Cook距离下阐述了观测数据之间的交叉影响、联合影响、掩盖、提升和降低、增大等作用，揭示了观测数据之间的内在本质联系，为粗差探测开辟新的途径。
最后，总结了本文的研究成果和有待进一步研究和解决的几个问题。
As we know, the ill-conditioning problem is present in data processing of geodesy networks adjustment, GPS (Global Positioning System) fast orientation, geodesy inversion and distortional inspecting networks. Moreover, its damage is serious. If the ill-conditioning is really existence in adjustment model, we must take measures to remove or weaken its influence in order to gain good precisions of parameters estimator and adjustment production. In order to get this, some useful explorations on how to analysis and solve the problem of the ill-conditioning have been made and several biased estimators have been put forward. In fact, analyzing the essence, overcoming the effect of ill-conditioning and obtaining more accurate and stable parameters estimator is an new task in GPS surveying data processing, which have been determined as an important studying field in contemporary surveying error theory and engineering data processing by the International Association of Geodesy (IAG).
In order to reduce the deficiencies of ridge estimator, Stein shrunken estimator and principal component estimator, two new biased estimators, so-called combining ridge and shrunken estimator, and, combining ridge and principal component estimator, are constructed respectively. Their good properties in the mean squared error and the numerical value stability are investigated, the determination of biased parameter of the estimators are discussed, and some important conclusions are obtained, respectively. Theory analysis and the computational results demonstrate that the two estimators potential estimator in surveying adjustment.
The comparisons between the two most important biased estimators, ordinary ridge estimator and principal components estimator, and LS estimator are conducted by using the criterion of mean squared error; and the conditions to show the superiority of each of these two estimators over the LS estimator have been obtained. Then, the tests have been suggested to verify whether or not these conditions hold in given situations by using the statistical method. Finally, the computational results demonstrate that the problem of selection between biased estimator and least squares estimator can be solved effectively by using the hypothesis testing approach.
In order to combat the influences of both outlier and ill-conditioning on geodetic adjustments, a new robust-biased estimation method is proposed by combining quasi-accurate detection (QUAD) of gross error and biased estimation. Several selection schemes of the biased parameters included in the biased estimators based on QUAD are given in detail. A numerical example illustrates that the new robust-biased estimation method not only can resist the bad influence of outlier and effectively overcome the difficulty caused by ill-conditioning simultaneously, but also is far more accurate than LS estimation, biased estimation, robust estimation and generalized shrunken type-robust estimation.
The influence analysis is studied in the ordinary ridge estimator. The cross influence and the
action of masking, boosting, reducing, enhancing, etc, among the surveying data are discussed under the Cook distance. The essence relationship among the observation data is shown out and a new approach to outlier identification is found.
At last the paper summarizes the research productions and the next development direction.
本文针对岭估计、Stein均匀压缩估计和主成分估计存在的缺陷，构造了两种新的有偏估计：岭—压缩组合估计和岭—主成分组合估计，讨论了两种新估计在均方误差意义下和数值稳定性方面的优良性质，讨论了有偏估计中偏参数的选取问题，通过理论分析和算例分析说明这是两种很有潜力的有偏估计。
在均方误差准则下对目前应用最广泛的两种有偏估计——岭估计和主成分估计与LS估计进行了比较研究，得到了岭估计、主成分估计优于LS估计的条件；然后运用统计方法对这些条件的成立进行了检验，从假设检验的角度解决了有偏估计与LS估计之间的选择问题。
针对粗差和病态性同时存在问题，把拟准检定法的优点和有偏估计的优点结合在一起，提出了基于拟准检定的抗差化有偏估计。通过数值实验说明了基于拟准检定的抗差化有偏估计确实是一种很好的估计，对消除粗差和病态性的不良影响非常有效。
对岭估计进行了影响分析研究，在Cook距离下阐述了观测数据之间的交叉影响、联合影响、掩盖、提升和降低、增大等作用，揭示了观测数据之间的内在本质联系，为粗差探测开辟新的途径。
最后，总结了本文的研究成果和有待进一步研究和解决的几个问题。
As we know, the ill-conditioning problem is present in data processing of geodesy networks adjustment, GPS (Global Positioning System) fast orientation, geodesy inversion and distortional inspecting networks. Moreover, its damage is serious. If the ill-conditioning is really existence in adjustment model, we must take measures to remove or weaken its influence in order to gain good precisions of parameters estimator and adjustment production. In order to get this, some useful explorations on how to analysis and solve the problem of the ill-conditioning have been made and several biased estimators have been put forward. In fact, analyzing the essence, overcoming the effect of ill-conditioning and obtaining more accurate and stable parameters estimator is an new task in GPS surveying data processing, which have been determined as an important studying field in contemporary surveying error theory and engineering data processing by the International Association of Geodesy (IAG).
In order to reduce the deficiencies of ridge estimator, Stein shrunken estimator and principal component estimator, two new biased estimators, so-called combining ridge and shrunken estimator, and, combining ridge and principal component estimator, are constructed respectively. Their good properties in the mean squared error and the numerical value stability are investigated, the determination of biased parameter of the estimators are discussed, and some important conclusions are obtained, respectively. Theory analysis and the computational results demonstrate that the two estimators potential estimator in surveying adjustment.
The comparisons between the two most important biased estimators, ordinary ridge estimator and principal components estimator, and LS estimator are conducted by using the criterion of mean squared error; and the conditions to show the superiority of each of these two estimators over the LS estimator have been obtained. Then, the tests have been suggested to verify whether or not these conditions hold in given situations by using the statistical method. Finally, the computational results demonstrate that the problem of selection between biased estimator and least squares estimator can be solved effectively by using the hypothesis testing approach.
In order to combat the influences of both outlier and ill-conditioning on geodetic adjustments, a new robust-biased estimation method is proposed by combining quasi-accurate detection (QUAD) of gross error and biased estimation. Several selection schemes of the biased parameters included in the biased estimators based on QUAD are given in detail. A numerical example illustrates that the new robust-biased estimation method not only can resist the bad influence of outlier and effectively overcome the difficulty caused by ill-conditioning simultaneously, but also is far more accurate than LS estimation, biased estimation, robust estimation and generalized shrunken type-robust estimation.
The influence analysis is studied in the ordinary ridge estimator. The cross influence and the
action of masking, boosting, reducing, enhancing, etc, among the surveying data are discussed under the Cook distance. The essence relationship among the observation data is shown out and a new approach to outlier identification is found.
At last the paper summarizes the research productions and the next development direction.
引文
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[2] Baarda, W. A. (1968). A testing procedure for use in geodetic networks. Netherlands Geodetic Commission, Publ. On Geodesy, New Series, 2(5): 1-60.
[3] Baye, M. & Parker, D. (1984). Combining ridge and principal component regression: a money demand illustration. Commun.Statist.-Theor.Meth., 2:197-205.
[4] Belsley, D. A. (1991). Conditioning Diagnostics: Collinearity and Weak Data in Regression. Wiley: New York.
[5] Berberan, A. (1992). Outlier detection and heterogeneous observations. Austral. J. Geod. Photogr. Surv., 56: 49-61.
[6] Caspary, W. & Haen, W. (1990). Simultaneous estimation of location and scale parameters in the context of robust M-estimation. Manuscr. Geod., 15: 273-283.
[7] 陈希孺，王松桂．近代回归分析．合肥：安徽教育出版社．1987．
[8] Cross, P. A. & Price, D. R. (1985). A strategy for the distinction between single and multiple gross errors in geodetic networks. Manuscr. Geod., 10: 172-178.
[9] Ethrog, U. (1991). Statistical test of significance for testing outlying observations. Survey Review, 31 : 62-70.
[10] 樊功瑜．主成分与主成分估计．测绘工程．1995，4(4)：1-7．
[11] 方开泰，全辉，陈庆云．实用回归分析．北京：科学出版社．1988
[12] Gruber, M. H. J. (1998). Improving Efficiency by Shrinkage. Dekker: New York.
[13] Gui, Q. M. (1993). A unified expression of linear biased estimators. Presented Paper of the IAG General Meeting in Beijing.
[14] Gui, Q.M. & Dong, X.R. (1993). A class of biased estimators for LN estimation problem. Manuscr. Geod., 18: 1-9.
[15] Gui, Q. M. (1997). A new class of biased estimators in free net adjustment model. Geomatics Research Australasia, 66: 27-46.
[16] Gui, Q.M. & Zhang, J.J. (1998a). Least squares estimation and biased estimation for
variance covariance components. Geomatics Research Australasia, 68:91-104.
[17] Gui, Q. M. & Zhang, J.J. (1998b): Robust biased estimation and its applications in geodetic adjustments. J. of Geodesy, 72: 430-435.
[18] Gui, Q. M. & Liu, J. S. (1999).Generalized shrunken type robust estimation. J. of Surveying Engineering, Vol. 125.
[19] Gui, Q.M., Duan, Q.T. & Zhu, J.Q. (1999). Robust latent root estimation and its application in geodetic adjustments. Allg.Verm.-Nachr., 106:134-138.
[20] Gui, Q.M., & Liu, J.S. (2000). Biased estimation in Gauss-Markov model. Allg.Verm.-Nachr., 107: 104-108.
[21] Gui, Q. M., Guo, J.F. & Yang, Y.X. (2001). Biased estimation in Gauss-Markov model with constraints. Allg.Verm.-Nachr., 108: 32-36.
[22] Gui, Q.M., Guo, J.F., Yang, Y.X. & Sui, L.F. (2001). Biased estimation in Gauss-Markov model not of full rank. Allg.Verm.-Nachr., 108: 390-393.
[23] 归庆明，郭建锋．G-M模型参数的岭型广义逆估计及其优良性．测绘科学．2001，26(3)：20-24
[24] Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. & Stahel, W. A. (1986). Robust Statistics. John Wiley: New York.
[25] Huang, Y. C. & Mertikas, S. P. (1995). On the design of robust regression estimators. Manuscr. Geod., 20: 145-160.
[26] 黄维彬．近代平差理论及其应用．北京：解放军出版社．1992．
[27] 黄幼才．岭估计及其应用．武汉测绘科技大学学报．1987,12(4)：64-73．
[28] 靳奉祥．影响分析理论及其在变形测量中的应用[学位论文]．长沙：中南工业大学资源环境与建筑工程学院．1994．
[29] Kaminski, W. & Wisniewski, Z. (1994). The method of growing rigor for the adjustment of geodetic observations contaminated by gross errors. Manuscr. Geod., 19: 55-61.
[30] Keating, J. P., Mason, R. L. & Sen, P. K. (1993). Pitman's Measure of Closeness: A Comparison of Statistical Estimators. SIAM.
[31] Koch, K. R. (1987). Parameter Estimation and Hypothesis Testing in Linear Models. Spring-Verlag: New York.
[32] Koch, K. R. (1996). Robuste parameterschaetzung. Allg.Verm.-Nachr., 103: 1-18.
[33] Kubik, K. & Wang, Y. (1991). Comparison of different principles for outlier detection. Austral. J. Geod. Photogr. Surv., 54: 67-80.
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