流形值扩散过程的功能估计
Functional Estimation of Manifold-Valued Diffusion Processes
AI总结：
本文提出基于单条轨迹的非参数估计方法，用于估计流形值扩散过程的漂移向量场和扩散矩阵，通过分析扩散估计器进一步得到依赖数据的切空间估计器，支持理论结果的数值实验。
摘要：
非平稳高维时间序列在生物医学研究中日益增多，由于生理系统的稳态性质，此类数据集通常位于或可近似为低维流形上。通过流形值Itô扩散过程建模此类数据集已被证明能提供有价值的见解并指导临床算法设计。本文提出Nadaraya-Watson类型的非参数估计器，假设在光滑完备流形上的时间齐次随机微分方程，证明当采样间隔和核带宽随着轨迹长度增加而趋于零时，过程的复发性使漂移和扩散估计器以及相关占据密度具有渐近一致性和正态性。扩散估计器的分析进一步产生依赖数据的切空间估计器，具有自身价值并对于漂移估计至关重要。跨多种流形配置的数值实验支持了理论结果。
英文摘要：
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.
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