SMOOTHING BY SPLINE FUNCTIONS
SMOOTHING BY SPLINE FUNCTIONS
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DOI：
10.1007/bf02162161
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发表时间：
1967-01-01
影响因子：
2.1
通讯作者：
REINSCH, CH
中科院分区：
文献类型：
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作者：
REINSCH, CH
2.公式令xi，Yi，i= 0. n被给定，并假设x 0 <x1 <.（处理可以容易地扩展到汇合横坐标的情况）。要构造的平滑函数[1]（x）应使x~（1）在所有函数g（x）中最小化fg ~（x）Zdxx 0，使得（2）~（1）。(gI.. n和S_0是给定的数。常数S是多余的，只是为了方便而引入。它允许控制平滑程度的量Sy ~的隐式重新缩放。S的推荐值取决于相对权重By（*.如果可以的话，应该使用纵坐标y~的标准偏差的估计值来表示~ Yi。在这种情况下，S的自然值位于对应于（2）的左侧的置信区间内：
2. Formulation Let xi, Yi, i= 0..... n be given and assume that x0< xl<...< x~,(the treatment may easily be extended to the case of confluent abscissae). The smoothing function](x) to be constructed shall x~(l) minimize fg"(x) Zdx x0 among all functions g (x) such that (2)~.(gI~)--Yq~ s, geC*[xo,,~].Here, 8y~> 0, i= 0..... n and S~ _0 are given numbers. The constant S is redundant and is introduced only for convenience. It allows for an implicit rescaling of the quantities 8y~ which control the extent of smoothing. Recommended values for S depend on the relative weights By (*. If available, one should use for~ Yi an estimate of the standard deviation of the ordinate y~. In this case, natural values of S lie within the confidence interval corresponding to the lefthand side of (2):