Spline Based Methods for Functional Data on Multivariate Domains

TL;DR

This paper introduces two modified spline-based methods for functional data analysis on multivariate domains, enabling flexible knot placement and tensor basis construction, with demonstrated improved performance on 1D data.

Contribution

It proposes novel approaches that incorporate irregular knot densities into spline tensor bases, overcoming limitations of traditional methods in higher-dimensional domains.

Findings

Performance comparable or better than existing methods on 1D data

Enables flexible knot placement in multivariate spline bases

Facilitates analysis of higher-dimensional functional data

Abstract

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 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 bases are obtained as tensor bases from 1D spline bases that require knots that are located on a lattice. This does not allow for a flexible knot placement that was fundamental for the 1D approach. The goal of this research is to propose two modified approaches that circumvent the problem by coding the irregular knot selection into their densities and utilizing these densities through the topology of the spaces of splines. This allows for regular grids for the knots and thus facilitates using the spline tensor bases. It is tested on 1D data showing that its performance is comparable to or better than the previous methods.