Structured penalties for functional linear models-partially empirical eigenvectors for regression.
Structured penalties for functional linear models-partially empirical eigenvectors for regression.
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DOI：
10.1214/12-ejs676
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发表时间：
2012-01-01
影响因子：
1.1
通讯作者：
Feng Z
中科院分区：
文献类型：
--
作者：
Randolph TW;Harezlak J;Feng Z
函数数据的挑战之一是将几何结构或局部相关性纳入分析。这种结构在越来越多的生物医学技术的输出中是固有的，并且通常使用函数线性模型来估计预测函数和标量响应之间的关系。估计系数函数的问题的常见方法通常涉及两个阶段：正则化和估计。正则化通常通过降维来完成，投影到预定义的基函数跨度或减少的特征向量（主成分）集合上。相比之下，我们提出了一个统一的方法，直接将几何结构的估计过程中，利用联合特征的预测和线性惩罚算子。从这个意义上说，回归中的成分是“部分经验的”，框架由广义奇异值分解（GSVD）提供。惩罚估计的形式不是新的，但GSVD澄清了过程，并通过明确惩罚和预测因子对估计系数函数的偏差，方差和性能的联合影响来通知惩罚的选择。实验室光谱数据和模拟被用来说明的概念。
One of the challenges with functional data is incorporating geometric structure, or local correlation, into the analysis. This structure is inherent in the output from an increasing number of biomedical technologies, and a functional linear model is often used to estimate the relationship between the predictor functions and scalar responses. Common approaches to the problem of estimating a coefficient function typically involve two stages: regularization and estimation. Regularization is usually done via dimension reduction, projecting onto a predefined span of basis functions or a reduced set of eigenvectors (principal components). In contrast, we present a unified approach that directly incorporates geometric structure into the estimation process by exploiting the joint eigenproperties of the predictors and a linear penalty operator. In this sense, the components in the regression are ‘partially empirical’ and the framework is provided by the generalized singular value decomposition (GSVD). The form of the penalized estimation is not new, but the GSVD clarifies the process and informs the choice of penalty by making explicit the joint influence of the penalty and predictors on the bias, variance and performance of the estimated coefficient function. Laboratory spectroscopy data and simulations are used to illustrate the concepts.