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Functional data analysis is typically performed in two steps: first,functionally representing discrete observations, and then applying functionalmethods to the so-represented data. The initial choice of a functionalrepresentation may have a significant impact on the second phase of theanalysis, as shown in recent research, where data-driven spline basesoutperformed the predefined rigid choice of functional representation. Themethod chooses an initial functional basis by an efficient placement of theknots using a simple machine-learning algorithm. The approach does not applydirectly when the data are defined on domains of a higher dimension than onesuch as, for example, images. The reason is that in higher dimensions theconvenient and numerically efficient spline bases are obtained as tensor basesfrom 1D spline bases that require knots that are located on a lattice. Thisdoes not allow for a flexible knot placement that was fundamental for the 1Dapproach. The goal of this research is to propose two modified approaches thatcircumvent the problem by coding the irregular knot selection into theirdensities and utilizing these densities through the topology of the spaces ofsplines. This allows for regular grids for the knots and thus facilitates usingthe spline tensor bases. It is tested on 1D data showing that its performanceis comparable to or better than the previous methods.

Empirically Driven Spline Bases for Functional Principal Component Analysis in 2D