高斯函数拟合参数选取标准的确定
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摘要
在水深测量中由于水深数据的离散化特征,需要进行数据拟合,求得未知区域的海底地形。针对高斯函数与协方差函数之间的密切联系,对高斯函数移位的线性组合拟合的基本形式进行了较全面的分析。首先,利用协方差函数旋转面等概念分析了高斯函数的拟合条件及其参数的初步确定;其次,利用高斯函数衰减较快的特性,提出了比值函数准则确定参数的数学方法,并从理论上剖析了参数选择的数学本质,得到了参数选取的有效区间。最后通过大量的实际数据计算,得到了与理论相一致的结果,提高了高斯函数拟合的效率和精度。
In water depth measurement,due to the discretization of water depth data,data fitting is needed to obtain seabed topography in unknown area. In the light of the close relation between Gauss ' s function and covariance function,the basic form of linear combination fitting of Gauss function shift is comprehensively analyzed. First of all,using the concept of the covariance function of rotation surface,analyze the fitting conditions of Gauss function and parameters are preliminarily determined; secondly,using Gauss function attenuation characteristics,puts forward a mathematical method to determine the ratio of function parameters,and theoretically analyzes the mathematical essence of parameter selection,achieving the effective intervals of parameter selection.Finally,a lot of practical data are calculated,and the result is consistent with the theory.which improves the efficiency and precision of Gauss function fitting.
In water depth measurement,due to the discretization of water depth data,data fitting is needed to obtain seabed topography in unknown area. In the light of the close relation between Gauss ' s function and covariance function,the basic form of linear combination fitting of Gauss function shift is comprehensively analyzed. First of all,using the concept of the covariance function of rotation surface,analyze the fitting conditions of Gauss function and parameters are preliminarily determined; secondly,using Gauss function attenuation characteristics,puts forward a mathematical method to determine the ratio of function parameters,and theoretically analyzes the mathematical essence of parameter selection,achieving the effective intervals of parameter selection.Finally,a lot of practical data are calculated,and the result is consistent with the theory.which improves the efficiency and precision of Gauss function fitting.
引文
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[2]刘经南,赵建虎.多波束测深系统的现状和发展趋势[J].海洋测绘,2002,22(5):4-6.
[3]刘雁春.海洋测深空间结构及其数据处理[M].北京:测绘出版社,2002.
[4]Deheuvels P,Martynov GV.A Karhunen-Loeve Decomposition of A Gaussian Process Generated by Independent Pairs of Exponential Random Variables[J].Journal of Functional Analysis,2008(255):2363-2394.
[5]Williams,C K I.,Barber,D.Bayesian classification with Gaussian processes[J].IEEE Transactions on Pattern Analysis and M achine Intelligence,1998(20):1342-1351.
[6]Mackay D J C.Bayesian Methods for Backpropagation Netw orks[M].M odels of Neural Netw orks III.Springer New York,1996:211-254.
[7]Neal R M.Bayesian Learning for Neural Networks[M].Bayesian learning for neural networks.Springer,1996:456-456.
[8]Hardy R L.Research results in the application ofmultiquadric equations to surveying and mapping problems[J].Surveying&M apping,1975(35):321-332.
[9]Wahba G.Spline models for observational data[M].Society for industrial and applied mathematics,1990.
[10]姚春晓.关于一类具有特殊协方差函数的高斯过程的极大似然估计方法[D].长春:东北师范大学,2016.
[11]何昀锋.高斯过程协方差函数的正定性与KarhunenLoeve展开[D].哈尔滨:哈尔滨工业大学,2011.
[12]边少锋,许江宁.计算机代数系统与大地测量数学分析[M].北京:国防工业出版社,2004.
[13]李厚朴,边少锋,钟斌.地理坐标系计算机代数精密分析理论[M].北京:国防工业出版社,2015.
[14]边少锋.配置法、多面法、移动法在坐标变换中的应用[J].测绘科学技术学报,1987(2):26-32.
[15]Sünkel H.Cardinal interpolation[R].OHIO STATE UNIV COLUMBUS DEPT OF GEODETIC SCIENCE,1981.
[16]Unser M,Aldroubi A.Polynomial Splines and WaveletsA Signal Processing Perspective[J].Wavelets,1992,20(8):91-123.