Detecting changes in functional linear models

TL;DR

This paper proposes a method to detect changes in functional linear models by projecting data onto finite-dimensional spaces and analyzing residuals, applicable to models like autoregressive processes in Hilbert spaces.

Contribution

It introduces a novel change detection procedure for functional linear models, including autoregressive models, using residual-based test statistics and covariance estimation techniques.

Findings

Effective change detection in functional linear models.

Inclusion of autoregressive models in Hilbert spaces.

Analysis of covariance estimators for residuals.

Abstract

We observe two sequences of curve which are connected via an integral operator. Our model includes linear models as well as autoregressive models in Hilbert spaces. We wish to test the null hypothesis that the operator did not change during the observation period. Our method is based on projecting the observations onto a suitably chosen finite dimensional space. The testing procedure is based on functionals of the weighted residuals of the projections. Since the quadratic form is based on estimating the long-term covariance matrix of the residuals, we also provide some results on Bartlett-type estimators.