Principal Component Analysis of Two-Dimensional Functional Data
Principal Component Analysis of Two-Dimensional Functional Data
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DOI：
10.1080/10618600.2013.827986
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发表时间：
2014-09-01
影响因子：
2.4
通讯作者：
Pan, Huijun
中科院分区：
文献类型：
--
作者：
Zhou, Lan;Pan, Huijun
本文提出并比较了两种对可能不规则区域上的二维泛函数据进行主成分分析的方法。第一种方法应用从二维函数的精细离散化获得的数据矩阵的奇异值分解。当仅在可能稀疏且可能因函数不同而不同的离散点观察到函数时，该方法在奇异值分解之前结合了初始平滑步骤。第二种方法采用混合效应模型，将PC函数指定为三角剖分上的二元样条线，将PC分数指定为随机效果。我们将薄板罚用于正则化函数估计，并提出了一种有效的期望最大化算法来计算参数的惩罚似然估计。基于混合效应模型的方法在一个统一的框架中集成了散点图平滑和功能PC分析，仿真研究表明，该方法比分别执行平滑和PC分析的两步法更有效。利用德克萨斯州气象站记录的100年来的气温数据，应用所提出的方法分析了德克萨斯州的气温变化。这篇文章的补充材料可以在网上找到。
This article presents and compares two approaches of principal component (PC) analysis for two-dimensional functional data on a possibly irregular domain. The first approach applies the singular value decomposition of the data matrix obtained from a fine discretization of the two-dimensional functions. When the functions are only observed at discrete points that are possibly sparse and may differ from function to function, this approach incorporates an initial smoothing step prior to the singular value decomposition. The second approach employs a mixed effects model that specifies the PC functions as bivariate splines on triangulations and the PC scores as random effects. We apply the thin-plate penalty for regularizing the function estimation and develop an effective expectation-maximization algorithm for calculating the penalized likelihood estimates of the parameters. The mixed effects model-based approach integrates scatterplot smoothing and functional PC analysis in a unified framework and is shown in a simulation study to be more efficient than the two-step approach that separately performs smoothing and PC analysis. The proposed methods are applied to analyze the temperature variation in Texas using 100 years of temperature data recorded by Texas weather stations. Supplementary materials for this article are available online.