Functional Estimation of Manifold-Valued Diffusion Processes with Applications

Abstract

As an initial objective, we aimed to fuse the outputs of different electrocardiogram-derived respiration (EDR) algorithms to create one EDR signal that is of higher quality. For our procedure, we viewed each EDR algorithm as a software sensor that recorded breathing activity from a different vantage point, identified high-quality software sensors based on the respiratory signal quality index, aligned the highest-quality EDRs with a phase synchronization technique based on the graph connection Laplacian, and finally fused those aligned, high-quality EDRs; we refer to the output as the sync-ensembled EDR signal. The proposed algorithm was evaluated on two large-scale databases of whole-night polysomnograms by multiple evaluation metrics, and we found that the sync-ensembled EDR provides robust respiratory information from electrocardiogram. Such results inspire the modeling of nonstationary high-dimensional time series as located on, or well-approximated by, a low-dimensional manifold, owing to the homeostatic nature of physiological systems. In addition to the motivating application of EDR algorithms, the modeling of such a dataset by manifold-valued diffusion processes has been shown to provide valuable insights and to guide the design of algorithms for clinical applications. In this dissertation, we propose Nadaraya-Watson-type nonparametric estimators for the drift vector field and diffusion matrix. Assuming a time-homogeneous stochastic differential equation on a smooth complete manifold without boundary, we show that as the sampling interval and kernel bandwidth vanish with increasing trajectory length, recurrence of the process yields asymptotic consistency and normality of the drift and diffusion estimators, as well as the associated occupation density. Analysis of the diffusion estimator further produces a tangent space estimator for dependent data, which has its own interest and is essential for drift estimation. Numerical experiments across a range of manifold configurations support the theoretical results.