Nonstationary high-dimensional time series are increasingly encountered in biomedical research as measurement technologies advance. Owing to the homeostatic nature of physiological systems, such datasets are often located on, or can be well approximated by, a low-dimensional manifold. Modeling such datasets by manifold-valued Itô diffusion processes has been shown to provide valuable insights and to guide the design of algorithms for clinical applications. In this paper, we propose Nadaraya-Watson type nonparametric estimators for the drift vector field and diffusion matrix of the process from one trajectory. 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.

翻译：随着测量技术的进步，生物医学研究中日益频繁地遇到非平稳高维时间序列。由于生理系统的稳态特性，此类数据集通常位于低维流形上，或可被其良好近似。研究表明，通过流形值伊藤扩散过程对这些数据集进行建模，能够提供有价值的洞见，并指导临床应用的算法设计。本文针对单条轨迹，提出了漂移向量场和扩散矩阵的纳达拉亚-沃森型非参数估计量。假设在无边界光滑完备流形上定义的时间齐次随机微分方程成立，我们证明：当采样间隔和核带宽随轨迹长度增加而趋于零时，过程的递归性保证了漂移和扩散估计量以及相关占据密度的渐近一致性与正态性。对扩散估计量的分析进一步衍生出相依数据的切空间估计量，该估计量本身具有独立研究价值，且对漂移估计至关重要。涵盖多种流形构型的数值实验验证了理论结果。