Change point localisation and inference in fragmented functional data

TL;DR

This paper introduces a new method for detecting change points in fragmented functional data, providing consistent localization and distributional results, supported by theoretical analysis and numerical experiments.

Contribution

It proposes the FFDP algorithm for efficient change point detection in fragmented data with theoretical guarantees and new asymptotic distribution results.

Findings

Consistent change point localization rates achieved.

Derived limiting distributions for refined estimators.

Non-asymptotic covariance estimation error bounds.

Abstract

We study the problem of change point localisation and inference for sequentially collected fragmented functional data, where each curve is observed only over discrete grids randomly sampled over a short fragment. The sequence of underlying covariance functions is assumed to be piecewise constant, with changes happening at unknown time points. To localise the change points, we propose a computationally efficient fragmented functional dynamic programming (FFDP) algorithm with consistent change point localisation rates. With an extra step of local refinement, we derive the limiting distributions for the refined change point estimators in two different regimes where the minimal jump size vanishes and where it remains constant as the sample size diverges. Such results are the first time seen in the fragmented functional data literature. As a byproduct of independent interest, we also present a non-asymptotic result on the estimation error of the covariance function estimators over intervals with change points inspired by Lin et al. (2021). Our result accounts for the effects of the sampling grid size within each fragment under novel identifiability conditions. Extensive numerical studies are also provided to support our theoretical results.