High-Dimensional Markov-switching Ordinary Differential Processes

TL;DR

This paper develops a novel two-stage algorithm for recovering parameters of high-dimensional Markov-switching ordinary differential processes from discrete data, with applications in brain network analysis.

Contribution

It introduces a new method with theoretical guarantees for parameter recovery in nonlinear additive models with Markov switching, addressing a gap in existing research.

Findings

Algorithm successfully recovers process parameters from discrete observations.

Provides statistical error bounds and convergence guarantees under mixing conditions.

Applied to brain data, reveals differences in transition dynamics between ADHD and controls.

Abstract

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 $\beta$-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.