Koopman Spectral Analysis of Intermittent Dynamics in Complex Systems: A Case Study in Pathophysiological Processes of Obstructive Sleep Apnea

TL;DR

This paper introduces a spectral analysis method combining Koopman operator theory and wavelet analysis to characterize intermittent dynamics in complex systems, validated through heart rate variability data in sleep apnea.

Contribution

It proposes a novel approach to analyze intermittent modes and chaotic bursts in complex systems using spectral decomposition and wavelet analysis, applied to sleep apnea data.

Findings

Fat-tailed distribution of intermittent forcing in sleep apnea

Quantitative measures of burst durations and inter-burst intervals

Identification of dominant frequencies in apneic events

Abstract

Complex systems, such as pathophysiological processes, commonly exhibit chaotic, nonlinear, and intermittent phenomena. Koopman operator theory and Hankel alternative view of Koopman (HAVOK) model have been widely used to decompose the chaos of the complex system dynamics into an intermittent forced linear system. Although the statistics of the intermittent forcing have been proposed to characterize intermittencies in the HAVOK model, they were not adequate to attribute for the mode switching of nonlinear dynamics and the fat-tailed non-Gaussian distribution originated from high-frequency bursts and rarely-observed intermittent forcing. The paper proposed a new intermittency dynamics analysis approach to characterize the intermittent phases, chaotic bursts, and local spectral-temporal properties of various intermittent dynamics modes using spectral decomposition and wavelet analysis. To validate our methods, the intermittency behavior of apneic events in obstructive sleep apnea disorder was selected as the case, in which heart rate variability (HRV) features were extracted. Next, we constructed the Hankel matrix from the HRV features and obtained the last eigen time-delay coordinate by singular value decomposition of the Hankel matrix, which was modeled as an intermittent forcing input. The statistics of the forcing in OSA demonstrated the fat-tailed distribution of the intermittent forcing, which correspond to the intermittency of the underlying OSA pathophysiological process. The pooled means and standard deviations of the burst duration and the inter-burst duration across OSA patients were also calculated to be minutes and minutes. Scalogram amplitude and spectral decomposition of the wavelet transform exhibited various predominant frequencies and dynamics modes associated with apneic events.