Joint Modeling of Conditional Mean and Variability in Multilevel Functional Data: Application to Postprandial Glucose Trajectories in the AEGIS Study

Heteroscedasticity -- where the variance of a variable changes with other variables -- is pervasive in real data, and elucidating why it arises from the perspective of statistical moments is crucial in scientific knowledge discovery and decision-making. However, standard causal discovery does not reveal which causes act on the mean versus the variance, as it returns a single moment-agnostic graph, limiting interpretability and downstream intervention design. We propose a Bayesian, moment-driven causal discovery framework that infers separate mean and variance causal graphs from observational heteroscedastic data. We first derive the identification results by establishing sufficient conditions under which these two graphs are separately identifiable. Building on this theory, we develop a variational inference method that learns a posterior distribution over both graphs, enabling principled uncertainty quantification of structural features (e.g., edges, paths, and subgraphs). To address the challenges of parameter optimization in heteroscedastic models with two graph structures, we take a curvature-aware optimization approach and develop a prior incorporation technique that leverages domain knowledge on node orderings, improving sample efficiency. Experiments on synthetic, semi-synthetic, and real data show that our approach accurately recovers mean and variance structures and outperforms state-of-the-art baselines.

Continuous glucose monitors, smartwatches, meal logs, and health profiles form a rich multimodal personal context that can significantly enhance glucose forecasting. However, existing methods struggle to effectively utilize these signals to improve glucose prediction because the relevance of different modalities varies drastically across individuals and time. We present GlucoAgent, an LLM-based agentic system that addresses this challenge through dynamic, case-by-case analysis at inference time. For each forecast, GlucoAgent writes and runs code over an individual's multimodal context and returns predictions with a reason and supporting evidence. The generated programs are highly diverse, ranging from retrieving similar past meals to fitting and back-testing physiological and statistical models. On CGMacros, a multimodal glucose dataset, GlucoAgent outperforms time series foundation models on every forecasting horizon, achieving an average RMSE of 22.5 mg/dL compared to 23.5 mg/dL of Chronos-2, the leading multimodal time series foundation model. Furthermore, GlucoAgent is highly effective at leveraging multimodal data. The addition of multimodal data reduces its average RMSE by 1.4 mg/dL (24.0 mg/dL to 22.5 mg/dL), compared to a marginal 0.3 mg/dL (23.8 mg/dL to 23.5 mg/dL) improvement for Chronos-2. This advantage is most pronounced in challenging post-meal windows, where GlucoAgent uses multimodal context to reduce 60-minute forecast RMSE by 4.6 mg/dL (31.0 mg/dL to 26.5 mg/dL), far exceeding the 0.8 mg/dL (31.9 mg/dL to 31.1 mg/dL) reduction achieved by Chronos-2.

Longitudinal studies often collect data at sparse, irregular, and unequally spaced time points. Such heterogeneity is often driven by subject-specific covariates, yet existing methods have been restricted to a scalar endpoint value, completely neglecting the underlying response trajectories. We propose a novel Longitudinal Random Forest (LRF) framework that leverages tree-based ensemble machine learning with adaptive node-wise longitudinal trajectory estimation. The LRF framework makes five methodological contributions. it captures each subject's individual response trajectory while simultaneously accommodating within-node correlation, between-node heterogeneity, and nonlinear and interactive covariate effects. It introduces a novel trajectory-based splitting criterion that maximizes trajectory separation while incorporating a size-weighted penalty; it provides two variants, Principal Analysis by Conditional Expectation (LRF-PACE) and adaptive linear mixed-effects models (LRF-adaptiveLMM), which employ nonparametric and semiparametric node-wise smoothers, respectively, while learning covariate effects in a data-driven manner. It provides a comprehensive interpretation of covariates using both the classical trajectory-based permutation variable importance measure (PVIM) and a newly proposed finite-way interaction frequency count, and it not only predicts entire trajectories for new subjects but also forecasts future trajectories for existing subjects. Extensive simulation studies demonstrate that LRF achieves superior performance over several competing methods, even under severe sparsity. The practical significance of the LRF framework lies in its ability to address five important clinical questions.