Jianbin Tan

Jianbin Tan, Anru R. Zhang

Functional data, i.e., smooth random functions observed over a continuous domain, are increasingly available in areas such as biomedical research, health informatics, and epidemiology. However, effective statistical analysis for functional data is often hindered by challenges such as privacy constraints, sparse and irregular sampling, infinite dimensionality, and non-Gaussian structures. To address these challenges, we introduce a novel framework named Smooth Flow Matching (SFM), tailored for generative modeling of functional data to enable statistical analysis without exposing sensitive real data. Built upon flow-matching ideas, SFM constructs a semiparametric copula flow to generate infinite-dimensional functional data, free from Gaussianity or low-rank assumptions. It is computationally efficient, handles irregular observations, and guarantees the smoothness of the generated functions, offering a practical and flexible solution in scenarios where existing deep generative methods are not applicable. Through extensive simulation studies, we demonstrate the advantages of SFM in terms of both synthetic data quality and computational efficiency. We then apply SFM to generate clinical trajectory data from the MIMIC-IV patient electronic health records (EHR) longitudinal database. Our analysis showcases the ability of SFM to produce high-quality surrogate data for downstream statistical tasks, highlighting its potential to boost the utility of EHR data for clinical applications.

Jiaqiang Li, Jianbin Tan, Xueqin Wang

Equation discovery is a fundamental learning task for uncovering the underlying dynamics of complex systems, with wide-ranging applications in areas such as brain connectivity analysis, climate modeling, gene regulation, and physical system simulation. However, many existing approaches rely on accurate derivative estimation and are limited to first-order dynamical systems, restricting their applicability to real-world scenarios. In this work, we propose sparse equation matching (SEM), a unified framework that encompasses several existing equation discovery methods under a common formulation. SEM introduces an integral-based sparse regression method using Green's functions, enabling derivative-free estimation of differential operators and their associated driving functions in general-order dynamical systems. The effectiveness of SEM is demonstrated through extensive simulations, benchmarking its performance against derivative-based approaches. We then apply SEM to electroencephalographic (EEG) data recorded during multiple oculomotor tasks, collected from 52 participants in a brain-computer interface experiment. Our method identifies active brain regions across participants and reveals task-specific connectivity patterns. These findings offer valuable insights into brain connectivity and the underlying neural mechanisms.