Mean and covariance estimation of functional data streams

Mean and covariance estimation is a critical problem in functional data analysis. With the emergence of new types of functional data, there is a growing need for more sophisticated methodologies to effectively analyze them. In this article, we consider the online estimation for a new type of data named functional data streams, where functional data are continuously loaded into computer memory and cleared once processed, in this situation a real-time and efficient method is urgently needed. To tackle this issue, we propose an online estimation approach based on the basis expansion method. In addition to the update of sufficient statistics and dynamically tuning parameters, we also develop strategies to increase basis functions and generate robust estimators. The results of numerical experiments and two real datasets further demonstrate the effectiveness of the proposed method.