2 research outputs found
Network Estimation from Point Process Data
Consider observing a collection of discrete events within a network that
reflect how network nodes influence one another. Such data are common in spike
trains recorded from biological neural networks, interactions within a social
network, and a variety of other settings. Data of this form may be modeled as
self-exciting point processes, in which the likelihood of future events depends
on the past events. This paper addresses the problem of estimating
self-excitation parameters and inferring the underlying functional network
structure from self-exciting point process data. Past work in this area was
limited by strong assumptions which are addressed by the novel approach here.
Specifically, in this paper we (1) incorporate saturation in a point process
model which both ensures stability and models non-linear thresholding effects;
(2) impose general low-dimensional structural assumptions that include
sparsity, group sparsity and low-rankness that allows bounds to be developed in
the high-dimensional setting; and (3) incorporate long-range memory effects
through moving average and higher-order auto-regressive components. Using our
general framework, we provide a number of novel theoretical guarantees for
high-dimensional self-exciting point processes that reflect the role played by
the underlying network structure and long-term memory. We also provide
simulations and real data examples to support our methodology and main results.Comment: Submitted to IEEE Transactions on Information Theor
Context-dependent self-exciting point processes: models, methods, and risk bounds in high dimensions
High-dimensional autoregressive point processes model how current events
trigger or inhibit future events, such as activity by one member of a social
network can affect the future activity of his or her neighbors. While past work
has focused on estimating the underlying network structure based solely on the
times at which events occur on each node of the network, this paper examines
the more nuanced problem of estimating context-dependent networks that reflect
how features associated with an event (such as the content of a social media
post) modulate the strength of influences among nodes. Specifically, we
leverage ideas from compositional time series and regularization methods in
machine learning to conduct network estimation for high-dimensional marked
point processes. Two models and corresponding estimators are considered in
detail: an autoregressive multinomial model suited to categorical marks and a
logistic-normal model suited to marks with mixed membership in different
categories. Importantly, the logistic-normal model leads to a convex negative
log-likelihood objective and captures dependence across categories. We provide
theoretical guarantees for both estimators, which we validate by simulations
and a synthetic data-generating model. We further validate our methods through
two real data examples and demonstrate the advantages and disadvantages of both
approaches