Online Change Rate Learning for Functional Data

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.