Principal component analysis of spatially indexed functions

We develop an expansion, similar in some respects to the Karhunen–Loève expansion, but which is more suitable for functional data indexed by spatial locations on a grid. Unlike the traditional Karhunen–Loève expansion, it takes into account the spatial dependence between the functions. By doing so, it provides a more efficient dimension reduction tool, both theoretically and in finite samples, for functional data with moderate spatial dependence. For such data, it also possesses other theoretical and practical advantages over the currently used approach. The paper develops complete asymptotic theory and estimation methodology. The performance of the method is examined by a simulation study and data analysis. The new tools are implemented in an R package.

本文提出了一种展开方法，其在若干维度上与卡亨南-洛埃夫展开（Karhunen–Loève expansion）相仿，但更适配于以网格空间位置为索引的函数型数据。与传统卡亨南-洛埃夫展开不同，该方法充分考虑了函数间的空间相关性。借此优势，针对具备中等空间相关性的函数型数据，该方法在理论层面与有限样本情境下均能提供更为高效的降维工具。针对这类数据，相较于当前主流的处理方案，该方法还兼具其他理论与实践层面的优势。本文构建了完整的渐近理论与估计方法体系，并通过模拟实验与实际数据分析对该方法的性能开展了验证与评估。上述新工具已封装为R包供使用者调用。