Multivariate Spatiotemporal Hawkes Processes and Network Reconstruction

Abstract

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.

Modeling of Spatio-Temporal Hawkes Processes With Randomized Kernels

“…A significant amount of research has been conducted in signal processing, applied mathematics, and machine learning literatures to learn and apply spatio-temporal point processes [1], [4], [7], [11]. The approach of spatio-temporal modeling with point processes has been applied to various real-world scenarios such as seismological modeling of earthquakes and aftershocks [13], [27]- [29], criminological modeling of the dynamics of illegal incidents [7], [26], forecasting of disease outbreaks [16], network analysis [11], [18], [30]- [32], and so on. When carefully analyzed, the behavior of underlying systems can vary among different contexts.…”

“…Furthermore, we investigate the fitting performance of the introduced method through an extensive set of experiments involving synthetic and real-life datasets. The results show that our method provides significant improvements compared to the EM algorithm and stochastic declustering, which are commonly favored in the point process literature [11], [18]- [20], [27], [28]. Finally, we perform event analysis over real-life datasets by interpreting the inferred parameters.…”

“…In our problem, events are described by their locations, times and types. Therefore, we consider a multi-dimensional form of point processes called as spatio-temporal point processes [11], [15], [16]. We particularly work on spatio-temporal Hawkes processes that have a self-exciting nature by their default form, in which the intensity value is triggered by past events.…”

“…We particularly work on spatio-temporal Hawkes processes that have a self-exciting nature by their default form, in which the intensity value is triggered by past events. In this approach, the excitation between events is usually modeled as to be decaying exponentially in time with an exponential kernel, and in space with a Gaussian kernel [11], [16].…”

“…Therefore, we define our evaluation metric in terms of negative log-likelihood per event, which directly expresses the fitting performance. In order to learn the parameters of a spatio-temporal Hawkes process, there exist several inference methods in the point process literature, most notably, expectation-maximization (EM) [11], [18], [19] algorithm and stochastic declustering [20]- [23]. Recently, gradient descent-based optimization methods has also been preferred in the context of temporal point processes [24]- [26].…”