Selecting the number of knots for penalized splines
Selecting the number of knots for penalized splines
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DOI：
10.1198/106186002853
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发表时间：
2002-12-01
影响因子：
2.4
通讯作者：
Ruppert, D
中科院分区：
文献类型：
--
作者：
Ruppert, D
惩罚样条线或 P 样条线是通过带有粗糙度惩罚的最小二乘法拟合的回归样条线。 P 样条曲线与平滑样条曲线有很多共同点，但 P 样条曲线使用的惩罚类型比平滑样条曲线更通用。此外，P 样条的节点数量和位置并不像平滑样条那样固定。一般来说，P 样条的结点位于自变量的固定分位数处，唯一需要选择的调整参数是结点数量和惩罚参数。在本文中，研究了节点数量对 P 样条性能的影响。提出了两种自动选择结数的算法。当发现广义交叉验证统计量（GCV）随着节点数量的最后增加而没有改善时，近视算法就会停止。完整搜索以可能的结数的固定序列检查所有候选者，并选择使 GCV 最小化的候选者。近视算法在许多情况下效果很好，但可能会过早停止。完整搜索算法在所有检查的示例中都运行良好。描述了用于计算单变量和加性 P 样条的 Demmler-Reinsch 型对角化。 Demmler-Reinsch 基础对于平滑样条线无效，因为平滑样条线有太多节点。然而，对于 P 样条，Demmler-Reinsch 基础对于超快速广义交叉验证非常有用。
Penalized splines, or P-splines, are regression splines fit by least-squares with a roughness penalty. P-splines have much in common with smoothing splines, but the type of penalty used with a P-spline is somewhat more general than for a smoothing spline. Also, the number and location of the knots of a P-spline is not fixed as with a smoothing spline. Generally, the knots of a P-spline are at fixed quantiles of the independent variable and the only tuning parameters to choose are the number of knots and the penalty parameter. In this article, the effects of the number of knots on the performance of P-splines are studied. Two algorithms are proposed for the automatic selection of the number of knots. The myopic algorithm stops when no improvement in the generalized cross-validation statistic (GCV) is noticed with the last increase in the number of knots. The full search examines all candidates in a fixed sequence of possible numbers of knots and chooses the candidate that minimizes GCV. The myopic algorithm works well in many cases but can stop prematurely. The full-search algorithm worked well in all examples examined. A Demmler-Reinsch type diagonalization for computing univariate and additive P-splines is described. The Demmler-Reinsch basis is not effective for smoothing splines because smoothing splines have too many knots. For P-splines, however, the Demmler-Reinsch basis is very useful for super-fast generalized cross-validation.