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 interactions between aftershocks with a non-separable spatio-temporal triggering structure. For model estimation, we extend the model-independent stochastic declustering (MISD) algorithm and suggest substituting its histogram-based estimators with kernel methods. We demonstrate the utility of the proposed methods by applying them to the seismicity data in regions with active seismic activities.
Hawkes processes are a particularly interesting class of stochastic process that have been applied in diverse areas, from earthquake modelling to financial analysis. They are point processes whose defining characteristic is that they 'self-excite', meaning that each arrival increases the rate of future arrivals for some period of time. Hawkes processes are well established, particularly within the financial literature, yet many of the treatments are inaccessible to one not acquainted with the topic. This survey provides background, introduces the field and historical developments, and touches …
Yale University（耶鲁大学）；Princeton University（普林斯顿大学）；Georgia Institute of Technology（佐治亚理工学院）
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 numeri…
Epidemic-Type Aftershock Sequence (ETAS) models are point processes that have found prominence in seismological modeling. Its success has led to the development of a number of different versions of the ETAS model. Among these extensions is the RETAS model which has shown potential to improve the modeling capabilities of the ETAS class of models. The RETAS model endows the main-shock arrival process with a renewal process which serves as an alternative to the homogeneous Poisson process. Model fitting is performed using likelihood-based estimation by directly optimizing the exact likelihood. Ho…
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 …
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…
To analyse interaction in marked spatio-temporal point processes (MSTPPs), we introduce marked (cross) second-order reduced moment measures and K-functions for general inhomogeneous second-order intensity reweighted stationary MSTPPs. These summary statistics, which allow us to quantify dependence between different mark categories of the points, are depending on the specific mark space and mark reference measure chosen. We also look closer at how the summary statistics reduce under assumptions such as the MSTPP being multivariate and/or stationary. A new test for independent marking is devised…
The event sequence of many diverse systems is represented as a sequence of discrete events in a continuous space. Examples of such an event sequence are earthquake aftershock events, financial transactions, e-commerce transactions, social network activity of a user, and the user's web search pattern. Finding such an intricate pattern helps discover which event will occur in the future and when it will occur. A Hawkes process is a mathematical tool used for modeling such time series discrete events. Traditionally, the Hawkes process uses a critical component for modeling data as an intensity fu…
This article proposes a spatiotemporal point process model that enhances the classical Epidemic-Type Aftershock Sequence (ETAS) model by incorporating a renewal main-shock arrival process, which we term the renewal ETAS (RETAS) model. This modification is similar in spirit to the renewal Hawkes (RHawkes) process but the conditional intensity process supports a spatial domain. It empowers the main-shock intensity with the capability to reset upon the arrival of main-shocks and therefore allows for heavier clustering of earthquakes than the spatiotemporal ETAS model introduced by Ogata (1998). W…
Spatiotemporal Analysis for Frequency-Magnitude Distribution of Earthquakes Across Mainland China: Comparing Classical and b-positive b-values Across Tectonic Regimes
University of Chinese Academy of Sciences（中国科学院大学）；Harvard University（哈佛大学）
Because most natural phenomena exhibit dependence at multiple scales like locations of earthquakes or forest fire occurrences, spatio-temporal single-scale point process models are unrealistic in many applications. This motivates us to construct generalizations of classical Gibbs models. In this paper, we extend the Geyer saturation point process model to the spatio-temporal multi-scale framework. The simulation process is carried out through a birth-death Metropolis-Hastings algorithm. In a simulation study, we compare two common methods for statistical inference in Gibbs models: the pseudo-l…
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 …
The exact mechanisms leading to an earthquake are not fully understood and the space-time structural features are non-trivial. Previous studies suggest the seismicity of very low intensity earthquakes, known as micro-earthquakes, may contain information about the source process before major earthquakes, as they can quantify modifications to stress or strain across time that finally lead to a major earthquake. This work uses the history of seismic activity of micro-earthquakes to analyze the spatio-temporal statistical independence among the monitoring stations of a seismic network. Using point…
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…
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 …
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…
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）
Self-exciting spatio-temporal point process models predict the rate of events as a function of space, time, and the previous history of events. These models naturally capture triggering and clustering behavior, and have been widely used in fields where spatio-temporal clustering of events is observed, such as earthquake modeling, infectious disease, and crime. In the past several decades, advances have been made in estimation, inference, simulation, and diagnostic tools for self-exciting point process models. In this review, I describe the basic theory, survey related estimation and inference …
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…
The discovery of gas in Groningen in 1959 has been a massive boon to the Dutch economy. From the 1990s onwards though, gas production has led to induced seismicity. In this paper, we carry out a comprehensive exploratory analysis of the spatio-temporal earthquake catalogue. We develop a non-parametric adaptive kernel smoothing technique to estimate the spatio-temporal hazard map and to interpolate monthly well-based gas production statistics. Second- and higher-order inhomogeneous summary statistics are used to show that the state of the art rate-and-state models for the prediction of seismic …
We examine the applicability of modern neural network architectures to the midterm prediction of earthquakes. Our data-based classification model aims to predict if an earthquake with the magnitude above a threshold takes place at a given area of size $10 \times 10$ kilometers in $10$-$60$ days from a given moment. Our deep neural network model has a recurrent part (LSTM) that accounts for time dependencies between earthquakes and a convolutional part that accounts for spatial dependencies. Obtained results show that neural networks-based models beat baseline feature-based models that also acc…
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…
We study the spatio-temporal prediction problem and introduce a novel point-process-based prediction algorithm. Spatio-temporal prediction is extensively studied in Machine Learning literature due to its critical real-life applications such as crime, earthquake, and social event prediction. Despite these thorough studies, specific problems inherent to the application domain are not yet fully explored. Here, we address the non-stationary spatio-temporal prediction problem on both densely and sparsely distributed sequences. We introduce a probabilistic approach that partitions the spatial domain…
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…
Spatio-temporal interaction is inherent to cases of infectious diseases and occurrences of earthquakes, whereas the spread of other events, such as cancer or crime, is less evident. Statistical significance tests of space-time clustering usually assess the correlation between the spatial and temporal (transformed) distances of the events. Although appealing through simplicity, these classical tests do not adjust for the underlying population nor can they account for a distance decay of interaction. We propose to use the framework of an endemic-epidemic point process model to jointly estimate a…
Self-exciting Hawkes processes are used to model events which cluster in time and space, and have been widely studied in seismology under the name of the Epidemic Type Aftershock Sequence (ETAS) model. In the ETAS framework, the occurrence of the mainshock earthquakes in a geographical region is assumed to follow an inhomogeneous spatial point process, and aftershock events are then modelled via a separate triggering kernel. Most previous studies of the ETAS model have relied on point estimates of the model parameters due to the complexity of the likelihood function, and the difficulty in esti…
The spatio-temporal Epidemic Type Aftershock Sequence (ETAS) model is widely used to describe the self-exciting nature of earthquake occurrences. While traditional inference methods provide only point estimates of the model parameters, we aim at a full Bayesian treatment of model inference, allowing naturally to incorporate prior knowledge and uncertainty quantification of the resulting estimates. Therefore, we introduce a highly flexible, non-parametric representation for the spatially varying ETAS background intensity through a Gaussian process (GP) prior. Combined with classical triggering …
The Hawkes process and its extensions effectively model self-excitatory phenomena including earthquakes, viral pandemics, financial transactions, neural spike trains and the spread of memes through social networks. The usefulness of these stochastic process models within a host of economic sectors and scientific disciplines is undercut by the processes' computational burden: complexity of likelihood evaluations grows quadratically in the number of observations for both the temporal and spatiotemporal Hawkes processes. We show that, with care, one may parallelize these calculations using both c…
The Epidemic Type Aftershock Sequence (ETAS) model is widely used to model seismic sequences and underpins Operational Earthquake Forecasting (OEF). However, it remains challenging to assess the reliability of inverted ETAS parameters for a range of reasons. The most common algorithms just return point estimates with little quantification of uncertainty, and Bayesian Markov Chain Monte Carlo implementations remain slow to run, do not scale well and few have been extended to include spatial structure. Here we present a new approach to ETAS modelling using an alternative Bayesian method, the Int…
