Online Change Rate Learning for Functional Data
提出一种在线估计函数型数据均值和协方差变化率的方法，能实时更新并保持高统计效率，通过理论证明和模拟验证其与离线方法相当的精度，适用于电力消耗和交通事故等实时监测场景。
With advances in modern data collection technologies, functional data are increasingly received in a streaming manner. Although numerous methods have been developed to model such data, most focus on estimating the mean and covariance functions, while giving limited attention to their derivatives. In this paper, we introduce an online method for estimating the change rates of the mean and covariance functions, enabling real-time updates with high statistical efficiency. The method dynamically updates by incorporating incoming data and updating summary statistics of historical data. We establish the asymptotic normality of the estimators, classify functional data into three types based on the distributional properties (non-dense, semi-dense and ultra-dense), and moreover provide a data-driven procedure for the online bandwidth selection. By minimizing the loss of relative efficiency, we show that the proposed online change rate estimators perform comparably to the offline kernel estimators using the full data. We further compare our new estimators with representative offline and online methods through extensive simulations, demonstrating that they achieve an effective balance between statistical accuracy and computational time. Two real applications to hourly power consumption and traffic accident data, both capturing the changing trends in real time, further validate the broad applicability and substantial practical benefits of our new method.