Inference for High-Dimensional Self-Exciting Point Processes
Abstract
In a variety of settings, our only glimpse at a networkÕs structure is through the lens of time series observations. For instance, in a social network, we may observe a time series of membersÕ activities, such as posts on social media. In electrical systems, cascading chains of power failures reveal critical information about the underlying power distribution network. During epidemics, networks among computers or a population are reflected by the time at which each node becomes infected. In biological neural networks, firing neurons can trigger or inhibit the firing of their neighbors, so that information about the network structure is embedded within spike train observations. This proposal focuses on the setting in which a networkÕs functional structure (modeled as directed edge weights of a graph) corresponds to the extent to which one nodeÕs activity stimulates or inhibits activity in another node. For instance, the network structure may indicate who is influencing whom within a social network or the connectivity of neurons. The interactions between nodes are thus critical to a fundamental understanding of the underlying functional network structure and accurate predictions of likely future events. The above processes are self-exciting in that the likelihood of future events depends on past events (i.e., a particular type of autoregressive process). This temporal dependence among events can make accurate inference particularly challenging and requires the development of new theory and novel algorithms. The primary goal of the proposed work is to develop fundamental theory and algorithms for functional network inference using self-exciting point process observations. We focus on two particular challenges: (a) accurate estimation of functional network edges and their weights and (b) robust testing of a networkÕs structure. Network estimation allows us to infer strong influences among nodes (suggesting potential future intervention mechanisms) to better model cascading chains of interactions within a network, and to assess network vulnerabilities. Network testing allows us to determine whether functional networks are the same in the face of different external conditions or detect when a networkÕs structure changes. We assume that we monitor M network nodes and record the identities of the associated node and time of each event. A node and event may represent a person ÒlikingÓ a photo or article shared by another person in a social network, a neuron firing in the brain, a power failure, or the incidence of a disease. Classical methods for analyzing autoregressive processes and other dynamical systems have focussed on settings where (a) the parameter space is finite or low-dimensional (i.e., M is small relative to the number of events recorded over time) or (b) the data generated is Gaussian. Both of these approaches are ill-suited to the data of interest here: our data is best modeled with point processes, and modern networks are large (i.e., large M). Our approach will build upon state-of-the-art methods and novel theoretical frameworks developed by both PIs for Poisson data (cf. [1, 2]), and generalize those notions to the far more challenging tasks of analyzing self-exciting point processes. In addition, the proposed work will address context specific network inference. In some settings (e.g., social networks) each event may be associated with a feature vector (e.g., whether a social media post is political, humorous, or personal). Many systems have different functional networks corresponding to different categories of events, which we will model in our theory and methods. The proposed work will result in computational tools that can be used in a wide variety of applications domains, novel insights based on theory into how data should be collected or interventions managed, and graduate students trained in high-dimensional statistics, dynamical systems, point processes, and low-dimensional and sparse models.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Sep 11, 2018
- Source ID
- W911NF1710357
Entities
People
- Garvesh Raskutti
Organizations
- Army Contracting Command
- United States Army
- University of Wisconsin–Madison