Structural adaptation and rate accelerated estimation in bivariate functional data

TL;DR

This paper introduces directional regularity to characterize anisotropy in multivariate functional data, enabling faster smoothing convergence through basis adaptation, with algorithms, bounds, simulations, and real data application.

Contribution

It proposes a new anisotropy concept, directional regularity, and develops an algorithm for basis adaptation in bivariate functional data analysis.

Findings

Faster convergence rates achieved via basis change based on anisotropy.

Algorithm with non-asymptotic bounds for estimating the change-of-basis matrix.

Successful application to rainfall data demonstrating practical utility.

Abstract

We introduce directional regularity, a new definition of anisotropy for multivariate functional data. Instead of taking the conventional view, which determines anisotropy as a notion of smoothness along a dimension, directional regularity additionally views anisotropy through the lens of directions. We show that faster rates of convergence for smoothing can be obtained through a change-of-basis by adapting to the anisotropy of a bivariate process. An algorithm for the estimation and identification of the change-of-basis matrix is constructed, made possible due to the replication structure of functional data. Non-asymptotic bounds are provided for our algorithm, supplemented by numerical evidence from an extensive simulation study. Finally, a real-world rainfall measurement dataset is analyzed with our methods.