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 article 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. Supplementary materials for this article are available online.