  Modeling event patterns is a central task in a wide range of disciplines. In
applications such as studying human activity patterns, events often arrive
clustered with sporadic and long periods of inactivity. Such heterogeneity in
event patterns poses challenges for existing point process models. In this
article, we propose a new class of clustered point processes that alternate
between active and inactive states. The proposed model is flexible, highly
interpretable, and can provide useful insights into event patterns. A composite
likelihood approach and a composite EM estimation procedure are developed for
efficient and numerically stable parameter estimation. We study both the
computational and statistical properties of the estimator including
convergence, consistency, and asymptotic normality. The proposed method is
applied to Donald Trump's Twitter data to investigate if and how his behaviors
evolved before, during, and after the presidential campaign. Additionally, we
analyze large-scale social media data from Sina Weibo and identify interesting
groups of users with distinct behaviors.
