基于U曲线法的半参数模型中正则化参数确定
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摘要
针对半参数模型补偿最小二乘估计中正则化参数合理确定的问题,研究一种正则化参数确定方法即U曲线法,基于该方法确定合适的正则化参数,能够有效地控制残差范数与信号范数之间的平衡,得到较准确的参数估值;通过仿真算例分析,将基于U曲线法确定正则化参数的半参数模型的参数估计解和其他方法进行比较,研究结果表明:模拟的系统误差为周期性时,应用L曲线法、U曲线法确定的正则化参数进而求得的参数估值与其真值差值向量的范数分别为4.632 4×10-4和3.497 0×10-4;当模拟的系统误差呈线性周期性时,应用L曲线法和U曲线法确定的正则化参数进而求得的参数估值与其真值差值向量的范数分别为7×10-4和4×10-4,故采用U曲线法确定的正则化参数所求得的参数估值的精度比L曲线法的高,能较好地将观测值中的系统误差分离出来。
To solve the problem of the regularization parameter in the semiparametric model, the U curve method was researched as a regularization parameter selection method. By determining the appropriate regularization parameters, the balance between the residual part and the smoothness part was better controlled, and more accurate parameter estimation was obtained. Through the simulation examples and real examples, the parametric estimation solution of the semiparametric model based on the U curve method to determine the regularization parameters was compared with other methods. The results show that when the simulated system error is periodic, the parameter estimation is obtained by using the regularized parameters determined by U-curve method and L-curve method respectively, and the norm of the difference vector between it and its true value is respectively 4.632 4×10-4 and 3.497 0×10-4. When the simulated system error is linear periodicity, the parameter estimation is also obtained by using the regularized parameters determined by U-curve method and L-curve method respectively,and the norm of the difference vector between it and its true value is respectively 7×10-4 and 4×10-4. By comparison, the accuracy of parameter estimation by using the regularized parameters determined by U-curve method is better than that by L-curve method, and the U-curve method can better separate the systematic errors from the observed values.
To solve the problem of the regularization parameter in the semiparametric model, the U curve method was researched as a regularization parameter selection method. By determining the appropriate regularization parameters, the balance between the residual part and the smoothness part was better controlled, and more accurate parameter estimation was obtained. Through the simulation examples and real examples, the parametric estimation solution of the semiparametric model based on the U curve method to determine the regularization parameters was compared with other methods. The results show that when the simulated system error is periodic, the parameter estimation is obtained by using the regularized parameters determined by U-curve method and L-curve method respectively, and the norm of the difference vector between it and its true value is respectively 4.632 4×10-4 and 3.497 0×10-4. When the simulated system error is linear periodicity, the parameter estimation is also obtained by using the regularized parameters determined by U-curve method and L-curve method respectively,and the norm of the difference vector between it and its true value is respectively 7×10-4 and 4×10-4. By comparison, the accuracy of parameter estimation by using the regularized parameters determined by U-curve method is better than that by L-curve method, and the U-curve method can better separate the systematic errors from the observed values.
引文
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[12]朱建军,田玉淼,陶肖静.带准则参数的平差准则及其统一与解算[J].测绘学报, 2012, 41(1):8-13.ZHU Jianjun, TIAN Yumiao, TAO Xiaojing. United expression and solution of adjustment criteria with parameters[J]. Acta Geodaetica et Cartographica Sinica, 2012, 41(1):8-13.
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[15] HANSEN P C. Analysis of discrete ill-posed problems by means of the L-curve[J]. SIAM Review, 1992, 34(4):561-580.
[16] FISCHER B, HEGLAND M. Collocation, filtering and non parametric regression. Part I[J]. Zeitschrift für Vermessungswesen, 1999, 124(1):17-24.
[17] FISCHER B, HEGLAND M. Collocation, filtering and non Parametric regression. PartⅡ[J]. Zeitschrift für Vermessungswesen, 1999, 124(2):46-52.
[18] KRAWCZYK-STAHDO D, RUDNICKI M. Regularization parameter selection in discrete ill-posed problems:the use of the U-curve[J]. International Journal of Applied Mathematics and Computer Science, 2007, 17(2):157-164.
[19]陶本藻,姚宜斌.基于多面核函数配置型模型的参数估计[J].武汉大学学报(信息科学版), 2003, 28(5):547-550.TAO Benzao, YAO Yibin. Parameter estimation based on multiquadric collocation model[J]. Editorial Board of Geomatics and Information Science of Wuhan University, 2003, 28(5):547-550.
[20]丁士俊.测量数据的建模和半参数估计[D].武汉:武汉大学测绘学院, 2005:25-44.DING Shijun. Survey data modeling and semiparametric estmating[D]. Wuhan:Wuhan University. School of Geodesy and Geomaties, 2005:25-44.
[2] GREEN P J, SILVERMAN B W. Nonpararnetric regression and generalized linear models:a roughness penalty approach[M].London:Chapman and Hall, 1994:128-134.
[3] GREEN P J. Penalized likelihood for general semi-parametric regression models[J]. International Statistical Review, 1987, 55(3):245-259.
[4]左廷英,曾磊.顾及未知系统误差的变形监测滤波算法[J].中南大学学报(自然科学版), 2011, 42(3):738-743.ZUO Tingying, ZENG Lei. Filter algorithm in consideration of unknown systematic errors for deformation monitoring[J].Journal of Central South University(Science and Technology),2011, 42(3):738-743.
[5]熊立新,罗周全,谭浪浪,等.采空区激光探测精度影响因素分析及误差修正[J].中南大学学报(自然科学版), 2014, 45(4):1244-1250.XIONG Lixin, LUO Zhouquan, TAN Langlang, et al. Impact factors analysis and error correction of laser scanning for goaf[J]. Journal of Central South University(Science and Technology), 2014, 45(4):1244-1250.
[6]陶肖静.半参数平差模型的平差准则研究[D].长沙:中南大学地球科学与信息物理学院, 2011:14-31.TAO Xiaojing. Study on the adjustment criterion of the semiparametric adjustment model[D]. Changsha:Central South University. School of Geosciences and Info-physics, 2011:14-31.
[7]陶肖静,朱建军,田玉淼.半参数模型中影响正则化参数的因素分析[J].武汉大学学报(信息科学版), 2012, 37(3):298-301.TAO Xiaojing, ZHU Jianjun, TIAN Yumiao. Analysis of factors influencing smoothing parameter in semiparametric model[J].Geomatics and Information Science of Wuhan University, 2012,37(3):298-301.
[8]丁士俊,朱留洋,姜卫平.一种计算补偿最小二乘正则化参数的最优化方法[J].大地测量与地球动力学报, 2015, 35(1):115-117.DING Shijun, ZHU Liuyang, JIANG Weiping. An optimization method for calculating regularized parameter of panalized least squares[J]. Journal of Geodesy and Geodynamics, 2015, 35(1):115-117.
[9]张俊,独知行,张显云.测量平差双正则化参数解算半参数模型的研究[J].测绘科学, 2014, 39(5):96-98.ZHANG Jun, DU Zhixing, ZHANG Xianyun. Solution method of semi-parametric model by adding double-smoothing parameters[J]. Science of Surveying and Mapping, 2014, 39(5):96-98.
[10] SEVERINI T A, STANISWALIS J G. Quasi-likelihood estimation in semiparametric models[J]. Journal of the American Statistical Association, 1994, 89(426):501-511.
[11]刘晓刚,李迎春,肖云,等.重力与磁力测量数据向下延拓中最优正则化参数确定方法[J].测绘学报, 2014, 43(9):881-887.LIU Xiaogang, LI Yingchun, XIAO Yun, et al. Optimal regularization parameter determination method in downward continuation of gravimetric and geomagnetic data[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(9):881-887.
[12]朱建军,田玉淼,陶肖静.带准则参数的平差准则及其统一与解算[J].测绘学报, 2012, 41(1):8-13.ZHU Jianjun, TIAN Yumiao, TAO Xiaojing. United expression and solution of adjustment criteria with parameters[J]. Acta Geodaetica et Cartographica Sinica, 2012, 41(1):8-13.
[13] GOLUB G H, HEATH M, WAHBA G. Generalized crossvalidation as a method for choosing a good ridge parameter[J].Technometrics, 1979, 21(2):215-223.
[14] HANSEN P C, O’LEARY D P. The use of the L-curve in the regularization of discrete ill-posed problems[J]. SIAM Journal on Scientific Computing, 1993, 14(6):1487-1503.
[15] HANSEN P C. Analysis of discrete ill-posed problems by means of the L-curve[J]. SIAM Review, 1992, 34(4):561-580.
[16] FISCHER B, HEGLAND M. Collocation, filtering and non parametric regression. Part I[J]. Zeitschrift für Vermessungswesen, 1999, 124(1):17-24.
[17] FISCHER B, HEGLAND M. Collocation, filtering and non Parametric regression. PartⅡ[J]. Zeitschrift für Vermessungswesen, 1999, 124(2):46-52.
[18] KRAWCZYK-STAHDO D, RUDNICKI M. Regularization parameter selection in discrete ill-posed problems:the use of the U-curve[J]. International Journal of Applied Mathematics and Computer Science, 2007, 17(2):157-164.
[19]陶本藻,姚宜斌.基于多面核函数配置型模型的参数估计[J].武汉大学学报(信息科学版), 2003, 28(5):547-550.TAO Benzao, YAO Yibin. Parameter estimation based on multiquadric collocation model[J]. Editorial Board of Geomatics and Information Science of Wuhan University, 2003, 28(5):547-550.
[20]丁士俊.测量数据的建模和半参数估计[D].武汉:武汉大学测绘学院, 2005:25-44.DING Shijun. Survey data modeling and semiparametric estmating[D]. Wuhan:Wuhan University. School of Geodesy and Geomaties, 2005:25-44.
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