Selecting the Number of Knots for Penalized Splines
研究了惩罚样条中节点数量对性能的影响，提出了两种自动选择节点数量的算法，并描述了用于快速广义交叉验证的Demmler-Reinsch对角化方法。
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.