Detecting Parameter Instabilities in Functional Concurrent Linear Regression

TL;DR

This work develops a CUSUM-based framework to detect parameter instabilities in the slope of a concurrent functional linear regression for $C[0,1]$-valued time series. It establishes a functional strong invariance principle, yielding null distributions as functionals of a $C[0,1]$-valued Brownian bridge and provides two complementary tests, $\widehat{\mathbb Q}_{\text{sup}}$ and $\widehat{\mathbb Q}_{L^2}$, with practical long-run covariance estimation and quantile computation. The methodology is validated via simulations, showing the $L^2$-based statistic favors global changes while the sup-norm is more powerful for spike-like, localized shifts. An application to fatigue-affected gait data from body-worn sensors demonstrates interpretable, phase-specific insights into inter-joint coupling and symmetry, highlighting potential for targeted biomechanical interventions. Overall, the paper delivers a rigorous, implementable toolkit for inference on slope stability in functional time series with clear biomechanical relevance.

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 Hölder 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.

Figures (5)

• Figure 1: Top panel: A snapshot of a few cycles from the knee angle data, denoted hereafter by $\{X_i (t)\}_{i \in \mathbb Z }$. Bottom panel: Hip angle data, denoted by $\{Y_i (t)\}_{i \in \mathbb Z }$, for a single runner, where $i \in \mathbb Z$ denotes cycles and $t$ captures the internal cyclic structure within a single gait cycle.
• Figure 2: Top panels: knee angle data (left) and hip angle data (right) from a short run where fatigue does not set in. Bottom panels: concurrent slope $\gamma_i(t)$ (left) and intercept $\alpha_i(t)$ (right) functions under a no-change setting. Collectively, these plots visually demonstrate the coupling structure between the lower-extremity joint angles within the gait cycle for a specific runner.
• Figure 3: Left: Scaled alternative \ref{['scaled_transformation']}. Right: Spiked alternative \ref{['spiked_transform']}. The pre-change slope $\gamma_0$ is shown as a solid curve and the post-change slope $\gamma_1$ as a dashed curve.
• Figure 4: Left: Slope comparison before and after the detected change point for Runner D based on the $L^2$-norm. Right: Corresponding comparison using the sup-norm. The change point detected under the sup-norm occurs slightly earlier than that obtained with the $L^2$-norm.
• Figure 5: Top row: Coordination profiles of most experienced athlete (Runner E in this work). Bottom row: profiles of least experienced athlete (Runner C). Both: Slope functions with Hip-Knee (left panel), Hip-Ankle (middle panel) and Ankle-Knee (right panel) with the latter being the regressor for each combination.

Theorems & Definitions (15)

• Theorem 3.1
• Theorem 3.2
• Corollary 3.1
• Lemma A.1
• Lemma A.2
• Lemma A.3: fan2023, Lemma S.2
• Lemma A.4: vandervaart2023, Theorem 2.2.4
• Lemma A.5
• proof
• Lemma A.6