Detecting Parameter Instabilities in Functional Concurrent Linear Regression

TL;DR

This paper introduces a new statistical method to detect structural breaks in the slope function of functional linear regression models for time series data, with applications in biomechanics.

Contribution

It develops CUSUM-based tests using $L^2$ and sup-norms for detecting global and local changes in functional regression models, with theoretical guarantees and real-data application.

Findings

Tests are consistent against various break types.

Simulation studies confirm good finite-sample performance.

Application to sports data reveals movement pattern changes with fatigue.

Abstract

We develop methodology to detect structural breaks in the slope function of a concurrent functional linear regression model for functional time series in $C[0,1]$. Our test is based on a CUSUM process of regressor-weighted OLS residual functions. To accommodate both global and local changes, we propose $L^2$- and sup-norm versions, with the sup-norm particularly sensitive to spike-like changes. Under H\"older regularity and weak dependence conditions, we establish a functional strong invariance principle, derive the asymptotic null distribution, and show that the resulting tests are consistent against a broad class of alternatives with breaks in the slope function. Simulation studies illustrate finite-sample size and power. We apply the method to sports data obtained via body-worn sensors from running athletes, focusing on hip and knee joint-angle trajectories recorded during a fatiguing run. As fatigue accumulates, runners adapt their movement patterns, and sufficiently pronounced adjustments are expected to appear as a change point in the regression relationship. In this manner, we illustrate how the proposed tests support interpretable inference for biomechanical functional time series.