Choosing the Right Norm for Change Point Detection in Functional Data

TL;DR

This paper introduces an $L^1$ norm-based method for change point detection in functional data, demonstrating its superior performance over existing $L^2$ and supremum norm methods through theoretical analysis and empirical validation.

Contribution

It proposes a novel $L^1$ norm approach for change point detection in functional data, with theoretical validation and a power enhancement component for sparse alternatives.

Findings

$L^1$ norm method outperforms $L^2$ and supremum norm methods in various scenarios.

The proposed method is validated both theoretically and empirically.

Power enhancement improves detection of sparse change points.

Abstract

We consider the problem of detecting a change point in a sequence of mean functions from a functional time series. We propose an $L^1$ norm based methodology and establish its theoretical validity both for classical and for relevant hypotheses. We compare the proposed method with currently available methodology that is based on the $L^2$ and supremum norms. Additionally we investigate the asymptotic behaviour under the alternative for all three methods and showcase both theoretically and empirically that the $L^1$ norm achieves the best performance in a broad range of scenarios. We also propose a power enhancement component that improves the performance of the $L^1$ test against sparse alternatives. Finally we apply the proposed methodology to both synthetic and real data.