Modeling and estimation of multi-source clustering in crime and security data

While the presence of clustering in crime and security event data is well established, the mechanism(s) by which clustering arises is not fully understood. Both contagion models and history independent correlation models are applied, but not simultaneously. In an attempt to disentangle contagion from other types of correlation, we consider a Hawkes process with background rate driven by a log Gaussian Cox process. Our inference methodology is an efficient Metropolis adjusted Langevin algorithm for filtering of the intensity and estimation of the model parameters. We apply the methodology to property and violent crime data from Chicago, terrorist attack data from Northern Ireland and Israel, and civilian casualty data from Iraq. For each data set we quantify the uncertainty in the levels of contagion vs. history independent correlation.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS647 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Latent Self-Exciting Point Process Model for Spatial-Temporal Networks

We propose a latent self-exciting point process model that describes geographically distributed interactions between pairs of entities. In contrast to most existing approaches that assume fully observable interactions, here we consider a scenario where certain interaction events lack information about participants. Instead, this information needs to be inferred from the available observations. We develop an efficient approximate algorithm based on variational expectation-maximization to infer unknown participants in an event given the location and the time of the event. We validate the model on synthetic as well as real-world data, and obtain very promising results on the identity-inference task. We also use our model to predict the timing and participants of future events, and demonstrate that it compares favorably with baseline approaches.Comment: 20 pages, 6 figures (v3); 11 pages, 6 figures (v2); previous version appeared in the 9th Bayesian Modeling Applications Workshop, UAI'1

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

Multivariate Hawkes Processes for Large-scale Inference

In this paper, we present a framework for fitting multivariate Hawkes processes for large-scale problems both in the number of events in the observed history $n$ and the number of event types $d$ (i.e. dimensions). The proposed Low-Rank Hawkes Process (LRHP) framework introduces a low-rank approximation of the kernel matrix that allows to perform the nonparametric learning of the $d^2$ triggering kernels using at most $O(ndr^2)$ operations, where $r$ is the rank of the approximation ($r \ll d,n$). This comes as a major improvement to the existing state-of-the-art inference algorithms that are in $O(nd^2)$. Furthermore, the low-rank approximation allows LRHP to learn representative patterns of interaction between event types, which may be valuable for the analysis of such complex processes in real world datasets. The efficiency and scalability of our approach is illustrated with numerical experiments on simulated as well as real datasets.Comment: 16 pages, 5 figure