Stable and Efficient Multiple Smoothing Parameter Estimation for Generalized Additive Models
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广义加性模型中稳定高效的多重平滑参数估计
- 研究问题
- 如何在广义加性模型拟合中稳定选择多个平滑参数，尤其应对随参数变化的数值秩亏和似然不确定性问题？
- 方法与思路
- 作者采用惩罚回归样条表示模型，以列主元 QR 分解和奇异值分解选择平滑参数，并允许加入岭惩罚以处理拟合困难。
- 材料陈述的贡献
- 摘要称方法能识别并处理数值秩亏，兼容指定与估计的平滑参数及其下界；计算效率和模拟表现与已有方法比较。
- 阅读关注点
- 应核对参数选择准则、秩判定与误差传播机制、岭惩罚对估计的影响，以及模拟中与混合模型方法比较的设定。
- 阅读边界
- 仅据所给 OpenAlex 摘要；算法细节、稳定性保证和模拟数值未说明，不能据此断言其在所有模型中更快或更准确。
原文摘要
Representation of generalized additive models (GAM's) using penalized regression splines allows GAM's to be employed in a straightforward manner using penalized regression methods. Not only is inference facilitated by this approach, but it is also possible to integrate model selection in the form of smoothing parameter selection into model fitting in a computationally efficient manner using well founded criteria such as generalized cross-validation. The current fitting and smoothing parameter selection methods for such models are usually effective, but do not provide the level of numerical stability to which users of linear regression packages, for example, are accustomed. In particular the existing methods cannot deal adequately with numerical rank deficiency of the GAM fitting problem, and it is not straightforward to produce methods that can do so, given that the degree of rank deficiency can be smoothing parameter dependent. In addition, models with the potential flexibility of GAM's can also present practical fitting difficulties as a result of indeterminacy in the model likelihood: Data with many zeros fitted by a model with a log link are a good example. In this article it is proposed that GAM's with a ridge penalty provide a practical solution in such circumstances, and a multiple smoothing parameter selection method suitable for use in the presence of such a penalty is developed. The method is based on the pivoted QR decomposition and the singular value decomposition, so that with or without a ridge penalty it has good error propagation properties and is capable of detecting and coping elegantly with numerical rank deficiency. The method also allows mixtures of user specified and estimated smoothing parameters and the setting of lower bounds on smoothing parameters. In terms of computational efficiency, the method compares well with existing methods. A simulation study compares the method to existing methods, including treating GAM's as mixed models.