Title: Statistical Estimation, Uncertainty Quantification, and Detection for Hawkes Processes
Date: Tuesday, November 19th, 2024
Time: 1:45 – 3:00 PM EST
Location: ISYE Studio 108
Zoom link: https://gatech.zoom.us/j/4307309361?pwd=OGE1dXhsV0Nsa2d1WFMyTzMrMDVUZz09
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Committee:
Dr. Yao Xie (Advisor), H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology
Dr. Simon Mak, Department of Statistical Science, Duke University
Dr. Ashwin Pananjady, H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology
Dr. Kamran Paynabar, H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology
Dr. Sen Na, H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology
Abstract:
Spatio-temporal point processes as generalization of the one-dimensional point process which only contains a temporal element, is now widely adopted to model event data which contains the events' occurrence time and other information such as location, category, and text description etc. A natural way to model the event data would be aggregating the number of events by discretizing the time and location, but sometimes the events can be too sparse if we divide the space into fine grids and would make the statistical analysis difficult. Point process as an alternative approach allows us to model the events using its exact information.
In this thesis, we focus on the self-exciting point process, also known as the Hawkes process, where historical events can trigger others by adding to the intensity function at a later time. We discuss the statistical estimation, uncertainty quantification and detection problems of spatio-temporal self-exciting point processes, via theoretical analysis and numerical examples.
The first study considers the uncertainty quantification of the network Hawkes process. In this model the network structure is equivalent to the Granger causality between time series. Finding confidence intervals for the edge weights leads to a more precise understanding of the underlying network structure and reduce false positive edges. This chapter focuses on the non-asymptotic analysis based on the log-likelihood function, and provide coverage guarantee of the proposed confidence intervals.
The second study is about the sequential detection problem of the network Hawkes process, when both the pre-change and post-change models are known. We use an adaptation of the classic CUSUM test, which is known to be optimal in minimizing the detection delay while controlling false alarm rate. Due to the highly dynamic nature of the Hawkes process, we discuss the computational aspects of the CUSUM test, and compare the performance via numerical experiments.
The third study is on the recovery guarantee of self-exciting point process with a more general kernel function, which is needed to capture the non-homogeneous triggering effect in the spatio-temporal space in real-life examples. We first discuss the recovery proproties in a general function space, and then the effects of a finite-dimensional representation.