16 research outputs found
Multivariate Spatiotemporal Hawkes Processes and Network Reconstruction
There is often latent network structure in spatial and temporal data and the
tools of network analysis can yield fascinating insights into such data. In
this paper, we develop a nonparametric method for network reconstruction from
spatiotemporal data sets using multivariate Hawkes processes. In contrast to
prior work on network reconstruction with point-process models, which has often
focused on exclusively temporal information, our approach uses both temporal
and spatial information and does not assume a specific parametric form of
network dynamics. This leads to an effective way of recovering an underlying
network. We illustrate our approach using both synthetic networks and networks
constructed from real-world data sets (a location-based social media network, a
narrative of crime events, and violent gang crimes). Our results demonstrate
that, in comparison to using only temporal data, our spatiotemporal approach
yields improved network reconstruction, providing a basis for meaningful
subsequent analysis --- such as community structure and motif analysis --- of
the reconstructed networks
Parameter estimation of binned Hawkes processes
A key difficulty that arises from real event data is imprecision in the recording of event time-stamps. In many cases, retaining event times with a high precision is expensive due to the sheer volume of activity. Combined with practical limits on the accuracy of measurements, binned data is common. In order to use point processes to model such event data, tools for handling parameter estimation are essential. Here we consider parameter estimation of the Hawkes process, a type of self-exciting point process that has found application in the modeling of financial stock markets, earthquakes and social media cascades. We develop a novel optimization approach to parameter estimation of binned Hawkes processes using a modified Expectation-Maximization algorithm, referred to as Binned Hawkes Expectation Maximization (BH-EM). Through a detailed simulation study, we demonstrate that existing methods are capable of producing severely biased and highly variable parameter estimates and that our novel BH-EM method significantly outperforms them in all studied circumstances. We further illustrate the performance on network flow (NetFlow) data between devices in a real large-scale computer network, to characterize triggering behavior. These results highlight the importance of correct handling of binned data
Latent Gaussian dynamic factor modeling and forecasting for multivariate count time series
This work considers estimation and forecasting in a multivariate count time
series model based on a copula-type transformation of a Gaussian dynamic factor
model. The estimation is based on second-order properties of the count and
underlying Gaussian models and applies to the case where the model dimension is
larger than the sample length. In addition, novel cross-validation schemes are
suggested for model selection. The forecasting is carried out through a
particle-based sequential Monte Carlo, leveraging Kalman filtering techniques.
A simulation study and an application are also considered
Dynamics and Inference for Voter Model Processes
We consider a discrete-time voter model process on a set of nodes, each being
in one of two states, either 0 or 1. In each time step, each node adopts the
state of a randomly sampled neighbor according to sampling probabilities,
referred to as node interaction parameters. We study the maximum likelihood
estimation of the node interaction parameters from observed node states for a
given number of realizations of the voter model process. In contrast to
previous work on parameter estimation of network autoregressive processes,
whose long-run behavior is according to a stationary stochastic process, the
voter model is an absorbing stochastic process that eventually reaches a
consensus state. This requires developing a framework for deriving parameter
estimation error bounds from observations consisting of several realizations of
a voter model process. We present parameter estimation error bounds by
interpreting the observation data as being generated according to an extended
voter process that consists of cycles, each corresponding to a realization of
the voter model process until absorption to a consensus state. In order to
obtain these results, consensus time of a voter model process plays an
important role. We present new bounds for all moments and a bound that holds
with any given probability for consensus time, which may be of independent
interest. In contrast to most existing work, our results yield a consensus time
bound that holds with high probability. We also present a sampling complexity
lower bound for parameter estimation within a prescribed error tolerance for
the class of locally stable estimators