Abstract Functional principal component analysis for sparse longitudinal data usually proceeds by first smoothing the covariance surface, and then obtaining an eigendecomposition of the associated covariance operator. Here we consider the use of penalized tensor product splines for the initial smoothing step. Drawing on a result regarding finite-rank symmetric integral operators, we derive an explicit spline representation of the estimated eigenfunctions, and show that the effect of penalization can be notably disparate for alternative approaches to tensor product smoothing. The latter phenomenon is illustrated with two data sets derived from magnetic resonance imaging of the human brain.

中文翻译：

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张量积样条和功能主成分

摘要 稀疏纵向数据的函数主成分分析通常通过先平滑协方差表面，然后获得相关协方差算子的特征分解来进行。在这里，我们考虑在初始平滑步骤中使用惩罚张量积样条。利用关于有限秩对称积分算子的结果，我们推导出了估计特征函数的显式样条表示，并表明惩罚的效果对于张量积平滑的替代方法可能明显不同。后一种现象用来自人脑磁共振成像的两个数据集来说明。