Modelling and smoothing parameter estimation with multiple quadratic penalties
Modelling and smoothing parameter estimation with multiple quadratic penalties
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DOI：
10.1111/1467-9868.00240
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发表时间：
2000-01-01
影响因子：
5.8
通讯作者：
Wood, SN
中科院分区：
文献类型：
--
作者：
Wood, SN
惩罚似然方法提供了一系列实用的建模工具，包括样条平滑、广义加性模型和岭回归的变体。选择正确的惩罚权重是使用这些方法的关键部分，在单一惩罚的情况下，分析师有几种有根据的技术可供选择。然而，许多建模问题建议采用多重惩罚的公式，但这里缺乏通用方法。通过广义岭回归问题最小化并行于 W-1/2 (Xp - y) 并行于 (2) rho + Sigma(i=1)(m)theta(i)p'S(i)p 的迭代解决方案，可以将具有多重惩罚的广泛模型拟合到数据（p 是参数向量，X 是设计矩阵，S-i 是一个非负定系数矩阵，定义具有相关平滑参数 theta(i) 的第 i 个惩罚，W 是对角权重矩阵， y 是数据或伪数据的向量，而 rho 是为了计算效率而包含的“整体”平滑参数）。本文展示了如何通过将广义交叉验证应用于该问题来有效地执行平滑参数选择，以及如何允许使用多重惩罚来拟合非线性、广义线性和线性模型，从而大大增加了惩罚建模方法的范围。给出了非线性建模、广义加性建模和各向异性平滑的示例。
Penalized likelihood methods provide a range of practical modelling tools, including spline smoothing, generalized additive models and variants of ridge regression. Selecting the correct weights for penalties is a critical part of using these methods and in the single-penalty case the analyst has several well-founded techniques to choose from. However, many modelling problems suggest a formulation employing multiple penalties, and here general methodology is lacking. A wide family of models with multiple penalties can be fitted to data by iterative solution of the generalized ridge regression problem minimize parallel to W-1/2 (Xp - y)parallel to(2) rho + Sigma(i=1)(m)theta(i)p'S(i)p (p is a parameter vector, X a design matrix, S-i a non-negative definite coefficient matrix defining the ith penalty with associated smoothing parameter theta(i), W a diagonal weight matrix, y a vector of data or pseudodata and rho an 'overall' smoothing parameter included for computational efficiency). This paper shows how smoothing parameter selection can be performed efficiently by applying generalized cross-validation to this problem and how this allows non-linear, generalized linear and linear models to be fitted using multiple penalties, substantially increasing the scope of penalized modelling methods. Examples of non-linear modelling, generalized additive modelling and anisotropic smoothing are given.