Hawkes Process Model Research Articles

A fractional Hawkes process model for earthquake aftershock sequences

Abstract A new type of Hawkes process, known as the fractional Hawkes Process (FHP), has been recently introduced. This process uses a Mittag-Leffler density as the kernel function which is asymptotically a power law and so similar to the Omori–Utsu law, suggesting the FHP may be an appropriate earthquake model. However, it is currently an unmarked point process meaning it is independent of an earthquake’s magnitude. We extend the existing FHP, by incorporating Utsu’s aftershock productivity law and a time-scaling parameter from the fractional Zener Model to a marked version so that it may better model earthquake aftershock sequences. We call this model the ‘Seismic Fractional Hawkes Process’ (SFHP). We then estimate parameters via maximum likelihood and provide evidence for these estimates being consistent and asymptotically normal via a simulation study. The SFHP is then compared to the epidemic type aftershock sequence and FHP models on four aftershock sequences from Southern California and New Zealand. While it is inconclusive if the seismic fractional Hawkes process performs better in a retrospective predictive performance experiment, it does perform favourably against both models in terms of information criteria and residual diagnostics especially when the aftershock clustering is stronger.

The statistical and dynamic modeling of the first part of the 2013-2014 Euromaidan protests in Ukraine: The Revolution of Dignity and preceding times.

Ukraine's tug-of-war between Russia and the West has had significant and lasting consequences for the country. In 2013, Viktor Yanukovych, the Ukrainian president aligned with Russia, opted against signing an association agreement with the European Union. This agreement aimed to facilitate trade and travel between the EU and Ukraine. This decision sparked widespread protests that coalesced in Kyiv's Maidan Square, eventually becoming known as the Euromaidan protests. In this study, we analyze the protest data from 2013, sourced from Ukraine's Center for Social and Labor Research. Despite the dataset's limitations and occasional inconsistencies, we demonstrate the extraction of valuable insights and the construction of a descriptive model from such data. Our investigation reveals a pre-existing state of self-excitation within the system even before the onset of the Euromaidan protests. This self-excitation intensified during the Euromaidan protests. A statistical analysis indicates that the government's utilization of force correlates with increased future protests, exacerbating rather than quelling the protest movement. Furthermore, we introduce the implementation of Hawkes process models to comprehend the spatiotemporal dynamics of the protest activity. Our findings highlight that, while protest activities spread across the entire country, the driving force behind the dynamics of these protests was the level of activity in Kyiv. Furthermore, in contrast to prior research that emphasized geographical proximity as a key predictor of event propagation, our study illustrates that the political alignment among oblasts, which are the distinct municipalities comprising Ukraine, had a more profound impact than mere geographic distance. This underscores the significance of social and cultural factors in molding the trajectory of political movements.

Estimating high-dimensional Hawkes process with time-dependent covariates

The Hawkes process models have been recently become a popular tool for modeling and analysis of neural spike train data. Despite this popularity, existing methodological and theoretical work has mainly focused on models without assumption of time-varying covariates, such as stimulus drive. In this article, motivated by neuronal spike trains study in the presence of stimuli, we propose a new Hawkes process model, where covariates are included in the intensity function. We consider the problem of simultaneous variable selection and estimation for the Hawkes process in the high-dimensional regime. Estimation of the intensity function of the high-dimensional point process is considered within a nonparametric framework, applying B-splines and the SCAD penalty for matters of sparsity. We apply the Doob-Kolmogorov inequality to establish the consistency of the resulting estimators. Finally, we illustrate the performance of our proposal through simulation and demonstrate its utility by applying it to the neuron spike train data set.

Exogenous and endogenous factors affecting stock market transactions: A Hawkes process analysis of the Tokyo Stock Exchange during the COVID-19 pandemic.

Transactions in financial markets are not evenly spaced but can be concentrated within a short period of time. In this study, we investigated the factors that determine the transaction frequency in financial markets. Specifically, we employed the Hawkes process model to identify exogenous and endogenous forces governing transactions of individual stocks in the Tokyo Stock Exchange during the COVID-19 pandemic. To enhance the accuracy of our analysis, we introduced a novel EM algorithm for the estimation of exogenous and endogenous factors that specifically addresses the interdependence of the values of these factors over time. We detected a substantial change in the transaction frequency in response to policy change announcements. Moreover, there is significant heterogeneity in the transaction frequency among individual stocks. We also found a tendency where stocks with high market capitalization tend to significantly respond to external news, while their excitation relationship between transactions is weak. This suggests the capability of quantifying the market state from the viewpoint of the exogenous and endogenous factors generating transactions for various stocks.

A2DJP: A Two Graph-Based Component Fused Learning Framework for Urban Anomaly Distribution and Duration Joint-Prediction

Modern intelligent transportation system (ITS) has greatly benefitted people's daily life. However, the chanciness and suddenness of urban anomalies may greatly restrict the trouble-free operations of ITS. To be aware of future urban anomalies and their possible influences, great efforts have been achieved on these two aspects, but comprehensive predictions of urban anomalies including the predictions of distributions and durations, are still beingless. And the spatiotemporal cascade self/mutual exciting influences among anomalies have never been considered in previous studies. In this paper, we propose a novel Anomaly Distribution and Duration Joint-Prediction (A2DJP) algorithm to simultaneously filtrate urban subregions and estimate the durations of corresponding potential anomalies in the future. To capture the spatiotemporal correlations between urban traffics and anomalies, we use a modified Graph Convolution Network and Long Short-Term Memory integrated network. To learn the cascade correlations among anomalies themselves, we devise a novel Spatiotemporal neural Hawkes Process model, which contains a Hawkes Process (HP) based GCN and HP-based LSTM to extract the anomaly-wise spatiotemporal cascading correlations. By fusing the spatiotemporal correlations between traffics and anomalies, we then simultaneously predict the distributions and durations of future anomalies. Extensive experiments on real-world datasets demonstrate that our proposed method significantly outperforms the state-of-the-art solutions.

Characterizing military medical evacuation dispatching and delivery policies via a self-exciting spatio-temporal Hawkes process model

The military medical evacuation (MEDEVAC) system is the primary method to evacuate time-sensitive combat casualties from the battlefield to appropriately staffed and equipped medical treatment facilities (MTFs). The intent of this paper is to analyze how MTF capability and capacity impact MEDEVAC system performance. We develop a self-exciting spatio-temporal Hawkes process model as well as a notional, synthetically generated MEDEVAC scenario of the post-Bosnian War to evaluate and compare eight distinct MEDEVAC dispatching and delivery policies within a high-intensity combat environment. The results garnered from this paper highlight the substantial impact MTFs have on MEDEVAC systems and should be considered in future research.

Power Grid Fault Prediction System Collaborates with Power Supply Guarantee Command Platform for Smart Grid

The safe operation of power system is a pretty important part for the whole society and it is influencing everyone’s daily life. As the biggest power operation company, the State Grid Corporation of China has to ensure the normal operation of the power grid. However, the power system can be affected by several issues such as all kinds of extreme weather, aging of equipment, and with rapid development of the national economy and power industry, the electricity loads are becoming complicated and diversified. Due to the reasons above, pressure on the safe operation of the power grid is increasing significantly. In order to guarantee the safety of daily operation for power system, we propose a new system, including a fault predicting part and aftertreatment part. For the predicting part, machine learning technologies are used especially the Hawkes process model, to predict when and what kind of fault may happen, by which failures can be prevented effectively. In the aftertreatment part, a power supply guarantee command platform is established that can send abnormal information to headquarters at the very beginning and then our platform will search for the optimal maintainer intelligently to solve the problem immediately.

Flexible spatio-temporal Hawkes process models for earthquake occurrences

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.

Closed-form modeling of neuronal spike train statistics using multivariate Hawkes cumulants.

We derive exact analytical expressions for the cumulants of any orders of neuronal membrane potentials driven by spike trains in a multivariate Hawkes process model with excitation and inhibition. Such expressions can be used for the prediction and sensitivity analysis of the statistical behavior of the model over time and to estimate the probability densities of neuronal membrane potentials using Gram-Charlier expansions. Our results are shown to provide a better alternative to Monte Carlo estimates via stochastic simulations and computer codes based on combinatorial recursions are included.

Network Hawkes Process Models for Exploring Latent Hierarchy in Social Animal Interactions

AbstractGroup-based social dominance hierarchies are of essential interest in understanding social structure (DeDeo & Hobson in, Proceedings of the National Academy of Sciences 118(21), 2021). Recent animal behaviour research studies can record aggressive interactions observed over time. Models that can explore the underlying hierarchy from the observed temporal dynamics in behaviours are therefore crucial. Traditional ranking methods aggregate interactions across time into win/loss counts, equalizing dynamic interactions with the underlying hierarchy. Although these models have gleaned important behavioural insights from such data, they are limited in addressing many important questions that remain unresolved. In this paper, we take advantage of the observed interactions' timestamps, proposing a series of network point process models with latent ranks. We carefully design these models to incorporate important theories on animal behaviour that account for dynamic patterns observed in the interaction data, including the winner effect, bursting and pair-flip phenomena. Through iteratively constructing and evaluating these models we arrive at the final cohort Markov-modulated Hawkes process (C-MMHP), which best characterizes all aforementioned patterns observed in interaction data. As such, inference on our model components can be readily interpreted in terms of theories on animal behaviours. The probabilistic nature of our model allows us to estimate the uncertainty in our ranking. In particular, our model is able to provide insights into the distribution of power within the hierarchy which forms and the strength of the established hierarchy. We compare all models using simulated and real data. Using statistically developed diagnostic perspectives, we demonstrate that the C-MMHP model outperforms other methods, capturing relevant latent ranking structures that lead to meaningful predictions for real data.

Markov-modulated Hawkes processes for modeling sporadic and bursty event occurrences in social interactions

Modeling event dynamics is central to many disciplines. Patterns in observed social interaction events can be commonly modeled using point processes. Such social interaction event data often exhibit self-exciting, heterogeneous and sporadic trends which is challenging for conventional models. It is reasonable to assume that there exists a hidden state process that drives different event dynamics at different states. In this paper we propose a Markov modulated Hawkes process (MMHP) model for learning such a mixture of social interaction event dynamics and develop corresponding inference algorithms. Numerical experiments using synthetic data demonstrate that MMHP with the proposed estimation algorithms consistently recover the true hidden state process in simulations, while email data from a large university and data from an animal behavior study show that the procedure captures distinct event dynamics that reveal interesting social structures in the real data.

Beach-level 24-hour forecasts of Florida red tide-induced respiratory irritation

An accurate forecast of the red tide respiratory irritation level would improve the lives of many people living in areas affected by algal blooms. Using a decades-long database of daily beach conditions, two conceptually different models to forecast the respiratory irritation risk level one day ahead of time are trained. One model is wind-based, using the current days’ respiratory level and the predicted wind direction of the following day. The other model is a probabilistic self-exciting Hawkes process model. Both models are trained on beaches in Florida during 2011--2017 and applied to the red tide bloom during 2018-2019. For beaches where there is enough historical data to develop a model, the model which performs best depends on the beach. The wind-based model is the most accurate at half the beaches, correctly predicting the respiratory risk level on average about 84% of the time. The Hawkes model is the most accurate (81% accuracy) at nearly all of the remaining beaches.

Tri‐transformer Hawkes process via dot‐product attention operations with event type and temporal encoding

AbstractAsynchronous event sequences widely exist in the real world, such as social networks, electronic medical records, financial data, and genome analysis. For modeling asynchronous event sequences in the continuous time domain, point process has become the underpinning. In the initial research stage, Hawkes process is widely used because it can capture the self‐triggering and mutual triggering modes between different events in a variety of point process functions. In recent years, due to the development of neural networks, deep point process (also known as neural point process) can learn models with the stronger fitting ability and reduce the dependence on prior knowledge by using the powerful capacity of neural networks. The proposal of the transformer Hawkes process (THP) has led to a huge performance improvement, so a new climax of the transformer‐based deep Hawkes process is set off. However, THP does not make full use of the event and temporal information underlying the asynchronous event sequence, meanwhile, if we simply take the event type encoding and temporal encoding as the sequence encoding, a single transformer may suffer from learning bias. In order to circumvent these problems, we propose a tri‐transformer Hawkes process model (TTHP), in which the event and temporal information are introduced to the dot‐product attention operations as auxiliary information to form different multihead attention, respectively, and are utilized to build three heterogeneous learners. A series of well‐designed experiments on synthetic and real‐world datasets validate the effectiveness of the proposed TTHP.

Temporal attention augmented transformer Hawkes process

In recent years, mining the knowledge from asynchronous sequences by Hawkes process is a subject worthy of continued attention, and Hawkes processes based on the neural network have gradually become the most hotly researched fields, especially based on the recurrence neural network (RNN). However, these models still contain some inherent shortcomings of RNN, such as vanishing and exploding gradient and long-term dependency problems. Meanwhile, transformer-based on self-attention has achieved great success in sequential modeling like text processing and speech recognition. Although the Transformer Hawkes process (THP) has gained huge performance improvement, THPs do not effectively utilize the temporal information in the asynchronous events, for these asynchronous sequences, the event occurrence instants are as important as the types of events, while conventional THPs simply convert temporal information into position encoding and add them as the input of transformer. With this in mind, we come up with a new kind of Transformer-based Hawkes process model, temporal attention augmented transformer Hawkes Process (TAA-THP), we modify the traditional dot-product attention structure and introduce the temporal encoding into attention structure. We conduct numerous experiments on a wide range of synthetic and real-life datasets to validate the performance of our proposed TAA-THP model, a significant improvement compared with existing baseline models on the different measurements is achieved, including log-likelihood on the test dataset, and prediction accuracies of event types and occurrence times. In addition, through the ablation studies, we vividly demonstrate the merit of introducing additional temporal attention by comparing the performance of the model with and without temporal attention.

Towards Predictive Analysis on Disease Progression: A Variational Hawkes Process Model.

Massively available longitudinal data about long-term disease trajectories of patients provides a golden mine for the understanding of disease progression and efficient health service delivery. It calls for quantitative modeling of disease progression, which is a tricky problem due to the complexity of the disease progression process as well as the irregularity of time documented in trajectories. In this study, we tackle the problem with the goal of predictively analyzing disease progression. Specifically, we propose a novel Variational Hawkes Process (VHP) model to generalize disease progression and predict future patient states based on the clinical observational data of past disease trajectories. First, Hawkes Process captures the intensity of irregular visits in a trajectory documented to medical facilities and controls the aforementioned information flowing into future visits. Thereafter, the captured intensity is incorporated into a Variational Auto-Encoder to generate the representation of the future partial disease trajectory for a target patient in a predictive manner. To further improve the prediction performance, we equip the proposed model with a disease trajectory discriminator to distinguish the generated trajectories from real ones. We evaluate the proposed model on two public datasets from the MIMIC-III database pertaining to heart failure and sepsis patients, respectively, and one real-world dataset from a Chinese hospital pertaining to heart failure patients with multiple admissions. Experimental results demonstrate that the proposed model significantly outperforms state-of-the-art baselines, and may derive a set of practical implications that can benefit a wide spectrum of management and applications on disease progression.

Educators, Solicitors, Flamers, Motivators, Sympathizers: Characterizing Roles in Online Extremist Movements

Social media provides the means by which extremist social movements, such as white supremacy and anti-LGBTQ, thrive online. Yet, we know little about the roles played by the participants of such movements. In this paper, we investigate these participants to characterize their roles, their role dynamics, and their influence in spreading online extremism. Our participants-online extremist accounts-are 4,876 public Facebook pages or groups that have shared information from the websites of 289 Southern Poverty LawCenter (SPLC) designated extremist groups. Guided by theories of participatory activism, we map the information sharing features of these extremists accounts. By clustering the quantitative features followed by qualitative expert validation, we identify five roles surrounding extremist activism-educators, solicitors, flamers, motivators, sympathizers. For example, solicitors use links from extremist websites to attract donations and participation in extremist issues, whereas flamers share inflammatory extremist content inciting anger. We further investigate role dynamics such as, how stable these roles are over time and how likely will extremist accounts transition from one role into another. We find that roles core to the movement-educators and solicitors-are more stable, while flamers and motivators can transition to sympathizers with high probability. Finally, using a Hawkes process model, we test which roles are more influential in spreading various types of information. We find that educators and solicitors exert the most influence in triggering extremist link posts, whereas flamers are influential in triggering the spread of information from fake news sources. Our results help in situating various roles on the trajectory of deeper engagement into the extremist movements and understanding the potential effect of various counter-extremism interventions. Our findings have implications for understanding how online extremist movements flourish through participatory activism and how they gain a spectrum of allies for mobilizing extremism online.

Universal transformer Hawkes process with adaptive recursive iteration

Asynchronous events sequences are widely distributed in the natural world and human activities, such as earthquakes records, users’ activities in social media, and so on. How to distill the information from these seemingly disorganized data is a persistent topic that researchers focus on. One of the most useful models is the point process model, and on the basis, the researchers obtain many noticeable results. Moreover, in recent years, point process models on the foundation of neural networks, especially recurrent neural networks (RNN) are proposed and compare with the traditional models, their performance is greatly improved. Enlighten by transformer model, which can learn sequence data efficiently without recurrent and convolutional structure, transformer Hawkes process comes out, and achieves state-of-the-art performance. However, there is some research proving that the re-introduction of recursive calculations in transformer can further improve transformer’s performance. Thus, we come out with a new kind of transformer Hawkes process model, universal transformer Hawkes process (UTHP), which contains both recursive mechanism and self-attention mechanism, and to improve the local perception ability of the model, we also introduce convolutional neural network (CNN) in the position-wise-feed-forward part. We conduct experiments on several datasets to validate the effectiveness of UTHP and explore the changes after the introduction of the recursive mechanism. These experiments on multiple datasets demonstrate that the performance of our proposed new model has a certain improvement compared with the previous state-of-the-art models.

Comparison of prospective Hawkes and recursive point process models for Ebola in DRC

AbstractPoint process models, such as Hawkes and recursive models, have recently been shown to offer improved accuracy over more traditional compartmental models for the purposes of modeling and forecasting the spread of disease epidemics. To explicitly test the performance of these two models in a real‐world and ongoing epidemic, we compared the fit of Hawkes and recursive models to outbreak data on Ebola virus disease (EVD) in the Democratic Republic of the Congo in 2018–2020. The models were estimated, and the forecasts were produced, time‐stamped, and stored in real time, so that their prospective value can be assessed and to guard against potential overfitting. The fit of the two models was similar, with both models resulting in much smaller errors in the beginning and waning phases of the epidemic and with slightly smaller error sizes on average for the Hawkes model compared with the recursive model. Our results suggest that both Hawkes and recursive point process models can be used in near real time during the course of an epidemic to help predict future cases and inform management and mitigation strategies.

Feature Extraction of Sequence of Keystrokes in Fixed Text Using the Multivariate Hawkes Process

In this paper, we propose a new method of extracting the features of keystrokes. The Hawkes process based on exponential excitation kernel was used to model the sequence of keystrokes in fixed text, and the intensity function vector and adjacency matrix of the model obtained through training were regarded as the characteristics of the keystrokes. A visual analysis was carried out on the CMU keystroke raw data and the feature data extracted using the proposed method. We used one-class classifier to compare the classification effect of CMU keystroke raw data and the feature data extracted by the Hawkes process model and POHMM model. The experimental results show that the feature data extracted using the proposed method contains rich information to distinguish users. In addition, the feature data extracted using the proposed method has a slightly better classification performance than the original CMU keystroke data for some users who are not easy to distinguish.

Asymptotic distribution of the score test for detecting marks in hawkes processes

The score test is a computationally efficient method for determining whether marks have a significant impact on the intensity of a Hawkes process. This paper provides theoretical justification for use of this test. It is shown that the score statistic has an asymptotic chi-squared distribution under the null hypothesis that marks do not impact the intensity process. For local power, the asymptotic distribution against local alternatives is proved to be non-central chi-squared. A stationary marked Hawkes process is constructed using a thinning method when the marks are observations on a continuous time stationary process and the joint likelihood of event times and marks is developed for this case, substantially extending existing results which only cover independent and identically distributed marks. These asymptotic chi-squared distributions required for the size and local power of the score test extend existing asymptotic results for likelihood estimates of the unmarked Hawkes process model under mild additional conditions on the moments and ergodicity of the marks process and an additional uniform boundedness assumption, shown to be true for the exponential decay Hawkes process.