从参数估计的精度和算法的复杂度出发,对P范分布参数的估计方法进行了改进。根据误差分布的实际情况,引入实数阶和对数矩估计方法,建立了P范分布的参数估计的实数阶矩估计方法。首先,利用实数阶矩估计法,导出了形状参数p与实数阶阶数r的关系式,对形状参数的选取给出了相应的建议;其次,改进矩估计理论,利用对数矩估计方法导出了形状参数、期望及中误差的非线性估计公式,消除了函数截断误差对参数估计值计算的影响,并利用迭代算法给出了相应参数的解算方法和计算流程;最后,用一个模拟算例和两个实测算例分析了实数矩、对数矩和极大似然估计3种估计方法的稳定性和精度。结果说明,本文提出的矩估计方法在稳定性、精度和收敛速度等方面均优于极大似然估计方法,推广了现有的误差理论。
The estimation methods of P-norm distribution is improved in this paper from the perspective of the parameters estimation precision and algorithm complexity. The real order and logarithmic moment estimation is introduced and the real order moment estimation method of P-norm distribution is established based on the actual error distribution. First of all, the relation between the shape parameter p and the real order value r is derived by using the real order moment estimation, and corresponding suggestions are provided for shape parameter's selection. Then, the nonlinear estimation formula of shape parameter, expectations and mean square error is derived via logarithmic moment estimation, function truncation error on the calculation of parameter estimation is eliminated and the solving method of corresponding parameters and calculation process is given, leading an improvement of the theory. Finally, some examples are performed for analyzing the stability and precision of such three methods including real order moment, logarithmic moment and maximum likelihood estimation. The result shows that the stability, precision and convergence speed of the method in this paper are better than maximum likelihood estimation, which generalized the existing errors theory.
[1] 孙海燕. P范分布理论及其在现代测量数据处理中的应用[D]. 武汉:武汉测绘科技大学, 1995. SUN Haiyan. Theory of P-norm Distribution and Application of Surveying Data Processing[D]. Wuhan:Wuhan Technical University of Surveying and Mapping, 1995.
[2] 李克行. 基于LP估计的SLR数据处理与分析[D]. 上海:中国科学院研究生院, 2005. LI Kexing. SLR Data Processing and Analysis Based on LP Estimation[D]. Shanghai:Graduate School of the Chinese Academy of Sciences, 2005.
[3] 蓝悦明, 贾媛. GPS观测值误差分布的研究[J]. 测绘通报, 2008(4):12-13, 54. LAN Yueming, JIA Yuan. The Study of Error Distribution for GPS Observed Values[J]. Bulletin of Surveying and Mapping, 2008(4):12-13, 54.
[4] 潘雄, 王俊雷, 袁珊丽, 等. 一元P-范分布参数估计的改进方法[J]. 测绘科学, 2011, 36(2):48-49, 52. PAN Xiong, WANG Junlei, YUAN Shanli, et al. An Improvement Method of Parameters Estimation in P-norm Distribution[J]. Science of Surveying and Mapping, 2011, 36(2):48-49, 52.
[5] 胡宏昌, 孙海燕.P-范分布参数σ的估计[J]. 武汉大学学报(信息科学版), 2002, 27(5):483-485. DOI:10.3321/j.issn:1671-8860.2002.05.009. HU Hongchang, SUN Haiyan. Parameter σ Estimation of P-norm Distribution[J]. Geomatics and Information Science of Wuhan University, 2002, 27(5):483-485. DOI:10.3321/j.issn:1671-8860.2002.05.009.
[6] 李博峰, 沈云中. P-范分布混合整数模型极大似然估计[J]. 测绘学报, 2010, 39(2):141-145. LI Bofeng, SHEN Yunzhong. Maximum Likelihood Estimation in Mixed Integer Linear Model with P-norm Distribution[J]. Acta Geodaetica et Cartographica Sinica, 2010, 39(2):141-145.
[7] 潘雄, 付宗堂. 一元有界P范分布的参数自适应估计[J]. 武汉大学学报(信息科学版), 2007, 32(4):323-325, 335. PAN Xiong,FU Zongtang.Parameter Adaptive Estimation of Bounded P-norm Distribution[J]. Geomatics and Information Science of Wuhan University, 2007, 32(4):323-325, 335.
[8] 潘雄, 赵启龙, 王俊雷, 等. 一元非对称P范分布的极大似然平差[J]. 测绘学报, 2011, 40(1):33-36. PAN Xiong, ZHAO Qilong, WANG Junlei, et al. Maximum Likelihood Adjustment of the Monadic Unsymmetrical P-norm Distribution[J]. Acta Geodaetica et Cartographica Sinica, 2011, 40(1):33-36.
[9] 於宗俦, 孙海燕, 陈之中. 多元P-范极大似然平差[J]. 测绘学报, 1996, 25(4):241-246. YU Zongchou, SUN Haiyan, CHEN Zhizhong. The Maximum Likelihood Adjustment of the P-norm distribution[J]. Acta Geodaetica et Cartographica Sinica, 1996, 25(4):241-246.
[10] 潘雄, 孙海燕. 半参数P-范极大似然回归[J]. 测绘学报, 2005, 34(1):30-34. DOI:10.3321/j.issn:1001-1595.2005.01.006. PAN Xiong, SUN Haiyan. Semiparametric P-norm Maximum Likelihood Regression[J]. Acta Geodaetica et Cartographica Sinica, 2005, 34(1):30-34. DOI:10.3321/j.issn:1001-1595.2005.01.006.
[11] 吴杰, 李正心, 王家骥. 自适应LP估计及其在照相天体测量中的应用[J]. 天文学报, 1996, 37(2):132-139. WU Jie, Li Zhengxin, WANG Jiaji. Adaptive LP Estimation and Its Application in the Photographic Astrometry[J]. Acta Astronomica Sinica, 1996, 37(2):132-139.
[12] 潘雄, 程少杰, 赵春茹. 一元P范分布的参数快速估计方法[J]. 武汉大学学报(信息科学版), 2010, 35(2):189-192. PAN Xiong, CHENG Shaojie, ZHAO Chunru. A Fast Parameter Estimation in P-norm Distribution[J]. Geomatics and Information Science of Wuhan University, 2010, 35(2):189-192.
[13] 郑作亚, 卢秀山, 彭军还. LP估计在星载GPS运动学定轨中的应用及精度分析[J]. 天文学报, 2007, 48(2):210-219. ZHENG Zuoya, LU Xiushan, PENG Junhuan. The Application and Precision Analysis of Satellite-borne GPS Kinematic Orbit Determination with LP Estimation[J]. Acta Astronomica Sinica, 2007, 48(2):210-219.
[14] 吴杰, 李正心, 张忠萍. 自适应LP估计及其应用[J]. 测绘学报, 1996, 25(1):31-36. WU Jie, LI Zhengxin, ZHANG Zhongping. Adaptive LP Estimation and its Application[J]. Acta Geodaetica et Cartographica Sinica, 1996, 25(1):31-36.
[15] 潘晓刚,李强, 周海银. 基于自适应稳健最小P范数估计的轨道确定方法研究[J]. 天文学报, 2010, 51(3):285-298. PAN Xiaogang, LI Qiang, ZHOU Haiyin. The Research of Orbit Determination Method Based on the Adaptive Robust Least P-norm Estimation[J]. Acta Astronomica Sinica, 2010, 51(3):285-298.
[16] 刘大杰, 史文中, 童小华. GIS空间数据的精度分析与质量控制[M]. 上海:上海科学技术文献出版社, 1999. LIU Dajie, SHI Wenzhong, TONG Xiaohua. Accuracy Analysis and Quality Control of Spatial Data in GIS[M]. Shanghai:Shanghai Science and Technology Literature Press, 1999.
[17] 覃文忠. 地图数字化数据误差的分布检验[D]. 上海:同济大学, 1999. QIN Wenzhong. The Distribution Test of the Digital Map Data Error[D]. Shanghai:Tongji University, 1999.
[18] 胡文琳, 王永良, 王首勇. 基于zrlog(z)期望的K分布参数估计[J]. 电子与信息学报, 2008, 30(1):203-205. HU Wenlin, WANG Yongliang, WANG Shouyong. Estimation of the Parameter of K-distribution Based on zrlog(z) Expectation[J]. Journal of Electronics & Information Technology, 2008, 30(1):203-205.
[19] 胡文琳, 王永良, 王首勇. 基于矩方法的K分布杂波参数估计研究[J]. 雷达科学与技术, 2007, 5(3):194-198. HU Wenlin, WANG Yongliang, WANG Shouyong. Research on Estimation of Parameters of K-Distribution Based on the MOM Approach[J]. Radar Science and Technology, 2007, 5(3):194-198.
[20] ISKANDER D R, ZOUBIR A M. Estimation of the Parameters of the K-distribution Using Higher Order and Fractional Moments[Radar Clutter] [J]. IEEE Transactions on Aerospace and Electronic Systems, 1999, 35(4):1453-1457.
