Thin Plate Regression Splines
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薄板回归样条
- 研究问题
- 如何为一维及多维数据构造可用于大样本的低秩平滑器，同时减少薄板样条计算负担与样条结点选择问题？
- 方法与思路
- 作者变换并截断薄板样条问题产生的基，按给定基维度最小化对原平滑问题的扰动，并用Lanczos迭代提高效率。
- 材料陈述的贡献
- 摘要称，低秩平滑器避免结点布置问题，支持广义加性模型交互项等扩展，并改善含薄板样条模型的计算效率。
- 阅读关注点
- 应核对最小扰动标准、基维度与惩罚参数选择，以及近似误差、计算复杂度和多维交互项的具体构造。
- 阅读边界
- 仅据Crossref摘要；摘要中的最优性针对给定维度下的平滑问题扰动，未说明预测误差优势或具体计算节省幅度。
原文摘要
Summary I discuss the production of low rank smoothers for d ≥ 1 dimensional data, which can be fitted by regression or penalized regression methods. The smoothers are constructed by a simple transformation and truncation of the basis that arises from the solution of the thin plate spline smoothing problem and are optimal in the sense that the truncation is designed to result in the minimum possible perturbation of the thin plate spline smoothing problem given the dimension of the basis used to construct the smoother. By making use of Lanczos iteration the basis change and truncation are computationally efficient. The smoothers allow the use of approximate thin plate spline models with large data sets, avoid the problems that are associated with ‘knot placement’ that usually complicate modelling with regression splines or penalized regression splines, provide a sensible way of modelling interaction terms in generalized additive models, provide low rank approximations to generalized smoothing spline models, appropriate for use with large data sets, provide a means for incorporating smooth functions of more than one variable into non-linear models and improve the computational efficiency of penalized likelihood models incorporating thin plate splines. Given that the approach produces spline-like models with a sparse basis, it also provides a natural way of incorporating unpenalized spline-like terms in linear and generalized linear models, and these can be treated just like any other model terms from the point of view of model selection, inference and diagnostics.