Abstract
Large-scale event time data recorded with high temporal resolutions have become increasingly available with the recent advancement of data collection technologies. Analyzing such type of data is challenging as the data are often large in size and/or have complex structures. We propose a novel multilevel functional data analysis procedure for temporal point processes. The proposed procedure can be used to model repeatedly observed temporal point patterns. A nonparametric approach is developed to consistently estimate the covariance kernels of the latent component processes at all levels. To predict the functional principal component scores, we propose a consistent estimation procedure by maximizing the conditional likelihoods of super-positions of point processes. We further extend our procedure to the bivariate point process case where potential correlations between the processes can be assessed. Asymptotic properties of the proposed estimators are investigated, and the effectiveness of our procedures is illustrated by a simulation study and an application to a stock trading dataset. Our nonparametric method can be used to model the latent intensity of user related or product related pattern, and provides insights on the variations and correlations in different activities. Furthermore, the proposed modeling framework is flexible and can be extended to more complex settings and other applications.