高维时变模型的通用自适应联合学习框架
A Generalized Adaptive Joint Learning Framework for High-Dimensional Time-Varying Models
AI总结：
本文提出了一种高维时变模型的通用自适应联合学习框架，通过两阶段筛选和细化程序，结合自适应分组惩罚与确信筛选原理，实现变量选择与突变点检测，确保在超高维情况下具有Oracle性质。
摘要：
在现代生物医学和计量经济学研究中，纵向过程往往由复杂的时变关联和突然的制度转变所刻画，这些特征在相关结果之间共享。标准的函数数据分析（FDA）方法，其优先考虑平滑性，往往无法捕捉这些动态结构特征，特别是在高维设置中。本文介绍了自适应联合学习（AJL），一种分层正则化框架，旨在将功能变量选择与结构突变点检测整合到多变量时变系数模型中。与标准的同时估计方法不同，我们提出了一种理论基础的两阶段筛选和细化程序。该框架首先将自适应分组惩罚与确信筛选原理相结合，以稳健地识别活跃预测变量，随后进行精细化融合正则化步骤，有效在多个结果之间借力以检测局部制度转变。我们为在超高维情况下（p >> n）的估计器提供了严谨的理论分析。关键点在于我们建立了第一阶段的确信筛选一致性，这为证明细化估计器具有Oracle性质（即在事先不知真活跃集和突变点位置时表现相同）奠定了基础。一个关键的理论贡献是通过欠平滑条件显式处理近似偏差以确保有效的渐近推断。所提出的方法通过全面的模拟和对睡眠-EDF数据的应用得到了验证，揭示了生理状态中的新动态模式。
英文摘要：
In modern biomedical and econometric studies, longitudinal processes are often characterized by complex time-varying associations and abrupt regime shifts that are shared across correlated outcomes. Standard functional data analysis (FDA) methods, which prioritize smoothness, often fail to capture these dynamic structural features, particularly in high-dimensional settings. This article introduces Adaptive Joint Learning (AJL), a hierarchical regularization framework designed to integrate functional variable selection with structural changepoint detection in multivariate time-varying coefficient models. Unlike standard simultaneous estimation approaches, we propose a theoretically grounded two-stage screening-and-refinement procedure. This framework first synergizes adaptive group-wise penalization with sure screening principles to robustly identify active predictors, followed by a refined fused regularization step that effectively borrows strength across multiple outcomes to detect local regime shifts. We provide a rigorous theoretical analysis of the estimator in the ultra-high-dimensional regime (p >> n). Crucially, we establish the sure screening consistency of the first stage, which serves as the foundation for proving that the refined estimator achieves the oracle property-performing as well as if the true active set and changepoint locations were known a priori. A key theoretical contribution is the explicit handling of approximation bias via undersmoothing conditions to ensure valid asymptotic inference. The proposed method is validated through comprehensive simulations and an application to Sleep-EDF data, revealing novel dynamic patterns in physiological states.
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