Point processes with event time uncertainty
This work addresses the challenge of incorporating time uncertainty into point process modeling for applications like network data analysis, offering incremental improvements over existing methods.
The authors tackled the problem of modeling point processes with uncertain event times, introducing a framework that handles non-stationary processes and includes stationary cases like the Hawkes process as special cases. They demonstrated that their approach outperforms previous GLM baselines on simulated and real data, achieving a parameter recovery guarantee with an O(1/k) convergence rate using k SGD steps.
Point processes are widely used statistical models for uncovering the temporal patterns in dependent event data. In many applications, the event time cannot be observed exactly, calling for the incorporation of time uncertainty into the modeling of point process data. In this work, we introduce a framework to model time-uncertain point processes possibly on a network. We start by deriving the formulation in the continuous-time setting under a few assumptions motivated by application scenarios. After imposing a time grid, we obtain a discrete-time model that facilitates inference and can be computed by first-order optimization methods such as Gradient Descent or Variation inequality (VI) using batch-based Stochastic Gradient Descent (SGD). The parameter recovery guarantee is proved for VI inference at an $O(1/k)$ convergence rate using $k$ SGD steps. Our framework handles non-stationary processes by modeling the inference kernel as a matrix (or tensor on a network) and it covers the stationary process, such as the classical Hawkes process, as a special case. We experimentally show that the proposed approach outperforms previous General Linear model (GLM) baselines on simulated and real data and reveals meaningful causal relations on a Sepsis-associated Derangements dataset.