Spline-based methods for functional data on multivariate domains

Functional data analysis is typically performed in two steps: first, functionally representing discrete observations, and then applying functional methods to the so-represented data. The initial choice of a functional representation may have a significant impact on the second phase of the analysis, as shown in recent research, where data-driven spline bases outperformed the predefined rigid choice of functional representation. The method chooses an initial functional basis by an efficient placement of the knots using a simple machine-learning algorithm. The knot selection approach does not apply directly when the data are defined on domains of a higher dimension than one such as, for example, images. The reason is that in higher dimensions the convenient and numerically efficient spline spaces use tensor bases that require knots located on a lattice. This fundamentally limits flexible knot placement which is fundamental for the approach. The goal of this research is two-fold: first, to propose modified approaches that circumvent the issue by coding the irregular knot selection into the topology of the spaces of tensor-based splines; second, to apply the approach to a classification problem workflow for functional data that utilizes knot selection. The performance is preliminarily accessed on a benchmark dataset and shown to be comparable to or better than the previous methods.