一种基于Tikhonov正则化的改进多面函数拟合法
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摘要
多面函数拟合法的平滑系数取值问题一直没有得到很好的解决,为此,提出一种基于Tikhonov正则化的改进多面函数拟合法。该方法引入正则化替代平滑系数,根据泛化误差极小化原则确定正则化系数,规避了平滑系数的不确定性,并去除了原方法核函数个数的约束条件。通过GPS水平速度场拟合的实例对改进方法进行验证,并与原方法的结果进行比较。结果表明,改进方法拟合效果稳定,拟合精度和泛化能力较原方法均有明显提高。
In the multiquadric function fitting method, the determination of the smoothing coefficient has not been well solved, affecting the application and promotion of the method. This paper proposes a refined multiquadric function fitting method, based on Tikhonov regularization. To avoid the uncertainty of the smoothing coefficient and remove the constraint condition of the number of the kernel functions of the conventional method, the refined method employs the regularization parameter, determined by the error minimization method, instead of the smoothing coefficient. We apply the refined method to GPS horizontal velocity fitting, and compare the results of the refined with the conventional method. The statistical results of residuals show that the refined can achieve a more accurate result than the conventional method.
In the multiquadric function fitting method, the determination of the smoothing coefficient has not been well solved, affecting the application and promotion of the method. This paper proposes a refined multiquadric function fitting method, based on Tikhonov regularization. To avoid the uncertainty of the smoothing coefficient and remove the constraint condition of the number of the kernel functions of the conventional method, the refined method employs the regularization parameter, determined by the error minimization method, instead of the smoothing coefficient. We apply the refined method to GPS horizontal velocity fitting, and compare the results of the refined with the conventional method. The statistical results of residuals show that the refined can achieve a more accurate result than the conventional method.
引文
[1] Hardy R L. The Application of Multi-Quadric Equations and Point Mass Anomaly Models to Crustal Movement Studies[R]. NOAA Technical Report, 1978
[2] Hardy R L.Hardy’s Multiquadric-Biharmonic Method for Gravity Field Predictions II[J]. Computers and Mathematics with Applications,2001, 41 (7):1 043-1 048
[3] 杨国华,黄立人.速率面拟合法中多面函数几个特性的初步数值研究[J].地壳形变与地震,1990(4):70-82(Yang Guohua,Huang Liren.Preliminary Numerical Study on Several Characteristics of Multi-Faceted Functions in Rate Surface Fitting Method[J].Crustal Deformation and Earthquake,1990(4):70-82)
[4] 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 Multi-Quadric Collocation Model[J]. Geomatics and Information Science of Wuhan University, 2003,28(5): 547-550)
[5] 刘经南,施闯,姚宜斌,等.多面函数拟合法及其在建立中国地壳平面运动速度场模型中的应用研究[J].武汉大学学报:信息科学版,2001, 26 (6):500-503(Liu Jingnan, Shi Chuang, Yao Yibin, et al. Hardy Function Interpolation and Its Applications to the Establishment of Crustal Movement Speed Field[J]. Geomatics and Information Science of Wuhan University, 2001, 26(6): 500-503)
[6] 马洪滨,董仲宇.多面函数GPS水准高程拟合中光滑因子求定方法[J].东北大学学报:自然科学版,2008(8):1 176-1 178(Ma Hongbin,Dong Zhongyu.A Method for Determining Smoothing Factor in GPS Level Height Fitting of Multi-Faceted Functions[J].Journal of Northeastern University:Natural Science,2008(8):1 176-1 178)
[7] 彭钊,陈志遥,吕品姬.多面函数法在钻孔应变时序资料处理中的应用[J].中国地震,2015,31(2):382-389(Peng Zhao,Chen Zhiyao,Lü Pinji. Method of Multi-Kernel Function and Its Application in Borehole Strain Timing Data Processing[J].Earthquake Research in China,2015, 31(2):382 -389)
[8] Neumaier A. Solving Ill-Conditioned and Singular Linear Systems: A Tutorial on Regularization[J]. SIAM Review, 1998, 40:636-666
[9] 徐新禹,李建成,王正涛,等.Tikhonov正则化方法在GOCE重力场求解中的模拟研究[J].测绘学报,2010,39(5):465-470(Xu Xinyu,Li Jiancheng,Wang Zhengtao,et al.The Simulation Research on the Tikhonov Regularization Applied in Gravity Field Determination of GOCE Satellite Mission[J].Acta Geodaetica et Cartographica Sinica,2010,39(5):466-470)
[10] Andrew Y N. Feature Selection, L1 vs. L2 Regularization, and Rotational Invariance[C]. ACM, 2004
[11] Massini M, Fortunato M, Mancini S, et al. L2 Regularization for Learning Kernels[J]. Conference on Uncertainty in Artificial Intellige, 2012, 62(4):109-116
[12] Fei H, Huan J. L2 Norm Regularized Feature Kernel Regression for Graph Data[C]. 18th ACM Conference on Information and Knowledge Management, 2009
[2] Hardy R L.Hardy’s Multiquadric-Biharmonic Method for Gravity Field Predictions II[J]. Computers and Mathematics with Applications,2001, 41 (7):1 043-1 048
[3] 杨国华,黄立人.速率面拟合法中多面函数几个特性的初步数值研究[J].地壳形变与地震,1990(4):70-82(Yang Guohua,Huang Liren.Preliminary Numerical Study on Several Characteristics of Multi-Faceted Functions in Rate Surface Fitting Method[J].Crustal Deformation and Earthquake,1990(4):70-82)
[4] 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 Multi-Quadric Collocation Model[J]. Geomatics and Information Science of Wuhan University, 2003,28(5): 547-550)
[5] 刘经南,施闯,姚宜斌,等.多面函数拟合法及其在建立中国地壳平面运动速度场模型中的应用研究[J].武汉大学学报:信息科学版,2001, 26 (6):500-503(Liu Jingnan, Shi Chuang, Yao Yibin, et al. Hardy Function Interpolation and Its Applications to the Establishment of Crustal Movement Speed Field[J]. Geomatics and Information Science of Wuhan University, 2001, 26(6): 500-503)
[6] 马洪滨,董仲宇.多面函数GPS水准高程拟合中光滑因子求定方法[J].东北大学学报:自然科学版,2008(8):1 176-1 178(Ma Hongbin,Dong Zhongyu.A Method for Determining Smoothing Factor in GPS Level Height Fitting of Multi-Faceted Functions[J].Journal of Northeastern University:Natural Science,2008(8):1 176-1 178)
[7] 彭钊,陈志遥,吕品姬.多面函数法在钻孔应变时序资料处理中的应用[J].中国地震,2015,31(2):382-389(Peng Zhao,Chen Zhiyao,Lü Pinji. Method of Multi-Kernel Function and Its Application in Borehole Strain Timing Data Processing[J].Earthquake Research in China,2015, 31(2):382 -389)
[8] Neumaier A. Solving Ill-Conditioned and Singular Linear Systems: A Tutorial on Regularization[J]. SIAM Review, 1998, 40:636-666
[9] 徐新禹,李建成,王正涛,等.Tikhonov正则化方法在GOCE重力场求解中的模拟研究[J].测绘学报,2010,39(5):465-470(Xu Xinyu,Li Jiancheng,Wang Zhengtao,et al.The Simulation Research on the Tikhonov Regularization Applied in Gravity Field Determination of GOCE Satellite Mission[J].Acta Geodaetica et Cartographica Sinica,2010,39(5):466-470)
[10] Andrew Y N. Feature Selection, L1 vs. L2 Regularization, and Rotational Invariance[C]. ACM, 2004
[11] Massini M, Fortunato M, Mancini S, et al. L2 Regularization for Learning Kernels[J]. Conference on Uncertainty in Artificial Intellige, 2012, 62(4):109-116
[12] Fei H, Huan J. L2 Norm Regularized Feature Kernel Regression for Graph Data[C]. 18th ACM Conference on Information and Knowledge Management, 2009