Separating now from before
Researchers have proposed a way to separate two possible contributions in regression with curve-shaped measurements: a predictor's value at the same moment as the response and the contribution of its earlier trajectory. The study asks whether those concurrent and historical coefficients—the model terms assigned to each contribution—can be separately identified and estimated.
At the heart of the paper is a conditional mathematical result. If the predictor's covariance eigenfunctions—the mathematical patterns used to describe how it varies—meet a stated condition, namely that they are not pointwise square-summable, the concurrent and historical components are separately identifiable. Under the theorem's assumptions, the two coefficients are also the unique population choices that minimize squared loss. That means the separation is guaranteed only within the model and assumptions used by the paper.
The estimator reflects that split. It first residualizes the predictor at each time, meaning it removes the part explained by the predictor's concurrent value; it then estimates the historical coefficient with smoothing-spline regularization and recovers the concurrent coefficient by correction.
The asymptotic analysis predicts a trade-off tied to smoothness: smoother predictor paths favor historical estimation, while rougher paths favor concurrent estimation. It also derives results for both in-sample error and conditional out-of-sample prediction error for the fitted conditional mean.
The finite-sample test
To see how the method behaved with finite samples, the authors ran two data-generating processes on 51 equally spaced grid points. They compared historical-only, concurrent-only and combined procedures using 50 and 500 curves.
In the first process, the combined model's mean concurrent-coefficient error was 0.07 with 50 curves and 0.02 with 500, compared with 2.02 and 2.01 for the concurrent-only model. Its historical-coefficient error fell from 0.85 to 0.40. In the second process, combined concurrent error was 0.08 and 0.03, versus 6.71 and 6.70 for concurrent-only, while combined historical error fell from 0.98 to 0.43.
Confidence-band coverage—the proportion of repeated samples in which a band contained its target—was closer to nominal in the larger simulations. With 500 curves, coverage was 0.89 and 0.90 for nominal 0.90 bands, and 0.95 for both processes for nominal 0.95 bands. With 50 curves, reported coverage ranged from 0.84 to 0.91.
The gait applications
In a gait-angle application involving 39 children, the combined and concurrent-only estimates of the concurrent coefficient were effectively identical, and the combined historical estimate was effectively zero. The historical-only model, however, produced strongly nonzero historical estimates, a pattern the paper interprets as consistent with bias from omitting the concurrent term.
For the angle data, the paper's 95% simultaneous confidence band for the reduced concurrent coefficient excluded zero over gait-cycle intervals 0 to 0.29, 0.57 to 0.73 and 0.90 to 1.00. That supported rejection of the null that the concurrent coefficient was zero under the stated no-history assumption.
In torque data from 120 amateur runners, the authors described both concurrent and historical components as present. They reported that the single-effect models distorted the estimates, while the combined model matched the expected biomechanical structure.
For a relevance test, the torque data were split into 40 pre-estimation observations and 80 test observations. The confidence and relevance bands overlapped across the gait cycle, so the study did not reject the null of no concurrent coefficient. The result depends on the chosen relevance limits and the data split.
A conditional result
The evidence is conditional. The identification and asymptotic results require the paper's stated assumptions; the simulations use only two processes; and the gait applications are observational case studies. Those applications illustrate model behavior in the supplied datasets but do not establish causal effects or show that the combined model is superior in every setting.
One further caution is that the simultaneous confidence band targets a reduced concurrent coefficient, not generally the structural concurrent coefficient, and full inference for the structural concurrent and historical coefficients is not supplied. The work is a version 1 arXiv preprint dated 20 August 2026, leaving questions about robustness beyond the stated assumptions, simulations and two gait applications.
Paper data and sources
Original title: Combining Concurrent and Historical Functional Linear Regression
Authors: Alois Kneip, Dominik Liebl, Sven Otto
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text