High-Dimensional Markov-switching Ordinary Differential Processes
This work addresses the challenge of inferring underlying systems from observations in domains like neuroscience, though it appears incremental as it builds on existing frameworks with new algorithmic and theoretical contributions.
The study tackled the problem of recovering parameters from discrete observations of Markov-switching ordinary differential processes, developing a two-stage algorithm with theoretical guarantees and applying it to identify differences in brain network transition rates between ADHD and control groups.
We investigate the parameter recovery of Markov-switching ordinary differential processes from discrete observations, where the differential equations are nonlinear additive models. This framework has been widely applied in biological systems, control systems, and other domains; however, limited research has been conducted on reconstructing the generating processes from observations. In contrast, many physical systems, such as human brains, cannot be directly experimented upon and rely on observations to infer the underlying systems. To address this gap, this manuscript presents a comprehensive study of the model, encompassing algorithm design, optimization guarantees, and quantification of statistical errors. Specifically, we develop a two-stage algorithm that first recovers the continuous sample path from discrete samples and then estimates the parameters of the processes. We provide novel theoretical insights into the statistical error and linear convergence guarantee when the processes are $β$-mixing. Our analysis is based on the truncation of the latent posterior processes and demonstrates that the truncated processes approximate the true processes under mixing conditions. We apply this model to investigate the differences in resting-state brain networks between the ADHD group and normal controls, revealing differences in the transition rate matrices of the two groups.