Daniel López-Montero, Antonio Álvarez-López, Marcos Matabuena
原文摘要
Modern datasets across many disciplines increasingly consist of time-evolving, potentially infinite-dimensional random objects, such as dynamic functional data, which are naturally modeled in Hilbert spaces. In these settings, characterizing probability measures, for example, through densities, can be ill-defined or technically challenging. Motivated by clustering applications, we propose a Gaussian mixture framework for Hilbert-space-valued data based on kernel mean embeddings and develop efficient optimization algorithms for estimation. We establish theoretical guarantees showing that the proposed algorithm is well-defined and that the model yields a dense class of approximations in infinite-dimensional spaces. We evaluate the framework through extensive experiments on diverse data geometries, including $L^2$-functional data and graph signals arising in modern medical applications.
Functional data analysis provides a framework for studying time-varying associations between covariates and high-frequency biological signals. In digital health, participants often contribute repeated functional trajectories, yet existing multilevel regression methods largely focus on the conditional mean, overlooking variability and its role in biological heterogeneity. We propose a heteroskedastic functional mixed-effects regression framework that jointly models the conditional mean and variance of repeated trajectories, with participant-specific random effects in both components. We also develop a participant-level bootstrap to quantify uncertainty in estimated functional parameters. We apply the method to postprandial glucose trajectories from the AEGIS study. Premeal glucose, carbohydrate intake, and meal type show distinct time-dependent associations with glucose variability, whereas HbA1c is more strongly associated with the conditional mean than with variability. Substantial between-participant differences in residual variability remain even after accounting for observed covariates. These findings show that modeling conditional variability alongside the mean provides insights into glucose regulation and individual heterogeneity that mean-focused analyses may miss. More broadly, the proposed framework enables joint characterization of average responses, variability, and participant-specific heterogeneity in repeated biological signals.