The Analysis of Two-Way Functional Data Using Two-Way Regularized Singular Value Decompositions
The Analysis of Two-Way Functional Data Using Two-Way Regularized Singular Value Decompositions
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DOI：
10.1198/jasa.2009.tm08024
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发表时间：
2009-12-01
影响因子：
3.7
通讯作者：
Buja, Andreas
中科院分区：
文献类型：
--
作者：
Huang, Jianhua Z.;Shen, Haipeng;Buja, Andreas
双向函数数据由数据矩阵组成，该数据矩阵的行域和列域都是结构化的，例如在时间上或空间上，例如当数据是在空间中不同位置收集的时间序列时。通过在数据矩阵的奇异值分解(SVD)中引入左右奇异向量的正则化，我们将单向泛函主成分分析(PCA)推广到双向泛函数据。我们将重点放在惩罚方法上，并解决了从单向回归惩罚构造适当的双向惩罚这一非常重要的问题。我们引入了条件交叉验证平滑参数选择，即左奇异向量在右奇异向量的条件下是交叉验证的，反之亦然。这个概念可以作为交替优化算法的一部分来实现。除了惩罚方法之外，我们还简要地考虑了基扩展的双向正则化。通过一个仿真算例和两个实际数据算例说明了所提方法的有效性。网上提供的补充材料表明，几种针对受惩罚的SVD的“自然”方法是有缺陷的，并解释了原因。
Two-way functional data consist of a data matrix whose row and column domains are both structured, for example, temporally or spatially, as when the data are time series collected at different locations in space. We extend one-way functional principal component analysis (PCA) to two-way functional data by introducing regularization of both left and right singular vectors in the singular value decomposition (SVD) of the data matrix. We focus oil a penalization approach and solve the nontrivial problem of constructing proper two-way penalties from one-way regression penalties. We introduce conditional cross-validated smoothing parameter selection whereby left-singular vectors are cross-validated conditional on right-singular vectors, and vice versa. The concept can be realized as part of an alternating optimization algorithm. In addition to the penalization approach, we briefly consider two-way regularization with basis expansion. The proposed methods are illustrated with one simulated and two real data examples. Supplemental materials available online show that several "natural" approaches to penalized SVDs are flawed and explain why so.