We estimate the general influence functions for spatio-temporal Hawkes processes using a tensor recovery approach by formulating the location dependent influence function that captures the influence of historical events as a tensor kernel. We assume a low-rank structure for the tensor kernel and cast the estimation problem as a convex optimization problem using the Fourier transformed nuclear norm (TNN). We provide theoretical performance guarantees for our approach and present an algorithm to solve the optimization problem. Moreover, we demonstrate the efficiency of our estimation with numerical simulations.
Bilkent University（比尔肯特大学）；DataBoss A. S.（DataBoss股份公司）
We investigate spatio-temporal event analysis using point processes. Inferring the dynamics of event sequences spatiotemporally has many practical applications including crime prediction, social media analysis, and traffic forecasting. In particular, we focus on spatio-temporal Hawkes processes that are commonly used due to their capability to capture excitations between event occurrences. We introduce a novel inference framework based on randomized transformations and gradient descent to learn the process. We replace the spatial kernel calculations by randomized Fourier feature-based transfor…
Imperial College London（伦敦帝国理工学院）；University of Copenhagen（哥本哈根大学）；Boston College（波士顿学院）；University of Oxford（牛津大学）
Hawkes processes are point process models that have been used to capture self-excitatory behavior in social interactions, neural activity, earthquakes and viral epidemics. They can model the occurrence of the times and locations of events. Here we develop a new class of spatiotemporal Hawkes processes that can capture both triggering and clustering behavior and we provide an efficient method for performing inference. We use a log-Gaussian Cox process (LGCP) as prior for the background rate of the Hawkes process which gives arbitrary flexibility to capture a wide range of underlying background …
Hawkes process is one of the most commonly used models for investigating the self-exciting nature of earthquake occurrences. However, seismicity patterns have complicated characteristics due to heterogeneous geology and stresses, for which existing methods with Hawkes process cannot fully capture. This study introduces novel nonparametric Hawkes process models that are flexible in three distinct ways. First, we incorporate the spatial inhomogeneity of the self-excitation earthquake productivity. Second, we consider the anisotropy in aftershock occurrences. Third, we reflect the space-time inte…
Indiana University - Purdue University Indianapolis（印第安纳大学-普渡大学印第安纳波利斯分校）
We propose a novel framework for integrating Hawkes processes with multi-armed bandit algorithms to solve spatio-temporal event forecasting and detection problems when data may be undersampled or spatially biased. In particular, we introduce an upper confidence bound algorithm using Bayesian spatial Hawkes process estimation for balancing the tradeoff between exploiting geographic regions where data has been collected and exploring geographic regions where data is unobserved. We first validate our model using simulated data and then apply it to the problem of disaster search and rescue using c…
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 recoverin…
Graphon Hawkes Processes: Sparse Estimation and Network Recovery
University of Amsterdam（阿姆斯特丹大学）；Eurandom, Eindhoven University of Technology（埃因霍温理工大学 Eurandom）；CentER, Tilburg University（蒂尔堡大学 CentER）；Mathematical Institute, Leiden University（莱顿大学数学研究所）；Korteweg-de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学柯特韦格-德弗里斯数学研究所）；Amsterdam Business School, University of Amsterdam（阿姆斯特丹大学阿姆斯特丹商学院）
A Spatio-Temporal Online Robust Tensor Recovery Approach for Streaming Traffic Data Imputation
Center for Intelligent Decision-making and Machine Learning, School of Management, Xi’an Jiaotong University（智能决策与机器学习中心，管理学院，西安交通大学）；Department of Marketing, School of Management, Xi’an Jiaotong University（市场营销系，管理学院，西安交通大学）
A multivariate Hawkes process enables self- and cross-excitations through a triggering matrix that behaves like an asymmetrical covariance structure, characterizing pairwise interactions between the event types. Full-rank estimation of all interactions is often infeasible in empirical settings. Models that specialize on a spatiotemporal application alleviate this obstacle by exploiting spatial locality, allowing the dyadic relationships between events to depend only on separation in time and relative distances in real Euclidean space. Here we generalize this framework to any multivariate Hawke…
We study the spatio-temporal prediction problem, which has attracted the attention of many researchers due to its critical real-life applications. In particular, we introduce a novel approach to this problem. Our approach is based on the Hawkes process, which is a non-stationary and self-exciting point process. We extend the formulations of a standard point process model that can represent time-series data to represent a spatio-temporal data. We model the data as nonstationary in time and space. Furthermore, we partition the spatial region we are working on into subregions via an adaptive deci…
Mutations sometimes increase contagiousness for evolving pathogens. During an epidemic, scientists use viral genome data to infer a shared evolutionary history and connect this history to geographic spread. We propose a model that directly relates a pathogen's evolution to its spatial contagion dynamics -- effectively combining the two epidemiological paradigms of phylogenetic inference and self-exciting process modeling -- and apply this \emph{phylogenetic Hawkes process} to a Bayesian analysis of 23,422 viral cases from the 2014-2016 Ebola outbreak in West Africa. The proposed model is able …
Crimes emerge out of complex interactions of human behaviors and situations. Linkages between crime incidents are highly complex. Detecting crime linkage given a set of incidents is a highly challenging task since we only have limited information, including text descriptions, incident times, and locations. In practice, there are very few labels. We propose a new statistical modeling framework for {\it spatio-temporal-textual} data and demonstrate its usage on crime linkage detection. We capture linkages of crime incidents via multivariate marked spatio-temporal Hawkes processes and treat embed…
A self-exciting spatio-temporal point process is fitted to incident data from the UK National Traffic Information Service to model the rates of primary and secondary accidents on the M25 motorway in a 12-month period during 2017-18. This process uses a background component to represent primary accidents, and a self-exciting component to represent secondary accidents. The background consists of periodic daily and weekly components, a spatial component and a long-term trend. The self-exciting components are decaying, unidirectional functions of space and time. These components are determined via…
We consider online monitoring of the network event data to detect local changes in a cluster when the affected data stream distribution shifts from one point process to another with different parameters. Specifically, we are interested in detecting a change point that causes a shift of the underlying data distribution that follows a multivariate Hawkes process with exponential decay temporal kernel, whereby the Hawkes process is considered to account for spatio-temporal correlation between observations. The proposed detection procedure is based on scan score statistics. We derive the asymptoti…
Georgia Institute of Technology（佐治亚理工学院）；Duke University（杜克大学）
Self- and mutually-exciting point processes are popular models in machine learning and statistics for dependent discrete event data. To date, most existing models assume stationary kernels (including the classical Hawkes processes) and simple parametric models. Modern applications with complex event data require more general point process models that can incorporate contextual information of the events, called marks, besides the temporal and location information. Moreover, such applications often require non-stationary models to capture more complex spatio-temporal dependence. To tackle these …
Technical University of Munich（慕尼黑工业大学）；LMU Klinikum Grosshadern（慕尼黑大学总医院格罗斯哈登分院）
In recent years, Generative Adversarial Networks (GAN) have emerged as a powerful method for learning the mapping from noisy latent spaces to realistic data samples in high-dimensional space. So far, the development and application of GANs have been predominantly focused on spatial data such as images. In this project, we aim at modeling of spatio-temporal sensor data instead, i.e. dynamic data over time. The main goal is to encode temporal data into a global and low-dimensional latent vector that captures the dynamics of the spatio-temporal signal. To this end, we incorporate auto-regressive …
We develop a new family of marked point processes by focusing the characteristic properties of marked Hawkes processes exclusively to the space of marks, providing the freedom to specify a different model for the occurrence times. This is possible through the decomposition of the joint distribution of marks and times that allows to separately specify the conditional distribution of marks given the filtration of the process and the current time. We develop a Bayesian framework for the inference and prediction from this family of marked point processes that can naturally accommodate process and …
Department of Mathematics, Duke University（杜克大学数学系）；H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院H. Milton Stewart工业与系统工程学院）
University at Buffalo（布法罗大学）；INRIA（法国国家信息与自动化研究所）；Université Côte d’ Azur（蔚蓝海岸大学）；Stanford University（斯坦福大学）；Purdue University（普渡大学）
This paper puts forth a novel bi-linear modeling framework for data recovery via manifold-learning and sparse-approximation arguments and considers its application to dynamic magnetic-resonance imaging (dMRI). Each temporal-domain MR image is viewed as a point that lies onto or close to a smooth manifold, and landmark points are identified to describe the point cloud concisely. To facilitate computations, a dimensionality reduction module generates low-dimensional/compressed renditions of the landmark points. Recovery of the high-fidelity MRI data is realized by solving a non-convex minimizati…
Idiap Research Institute（伊迪亚普研究所）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
We tackle the multi-party speech recovery problem through modeling the acoustic of the reverberant chambers. Our approach exploits structured sparsity models to perform room modeling and speech recovery. We propose a scheme for characterizing the room acoustic from the unknown competing speech sources relying on localization of the early images of the speakers by sparse approximation of the spatial spectra of the virtual sources in a free-space model. The images are then clustered exploiting the low-rank structure of the spectro-temporal components belonging to each source. This enables us to …
Inria Saclay（法国国家信息与自动化研究所萨克雷中心）；Université de Montpellier（蒙彼利埃大学）；CEA（法国原子能和替代能源委员会）
Detecting where and when brain regions activate in a cognitive task or in a given clinical condition is the promise of non-invasive techniques like magnetoencephalography (MEG) or electroencephalography (EEG). This problem, referred to as source localization, or source imaging, poses however a high-dimensional statistical inference challenge. While sparsity promoting regularizations have been proposed to address the regression problem, it remains unclear how to ensure statistical control of false detections. Moreover, M/EEG source imaging requires to work with spatio-temporal data and autocorr…
Source imaging based on magnetoencephalography (MEG) and electroencephalography (EEG) allows for the non-invasive analysis of brain activity with high temporal and good spatial resolution. As the bioelectromagnetic inverse problem is ill-posed, constraints are required. For the analysis of evoked brain activity, spatial sparsity of the neuronal activation is a common assumption. It is often taken into account using convex constraints based on the l1-norm. The resulting source estimates are however biased in amplitude and often suboptimal in terms of source selection due to high correlations in…
