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Existing methods for heterogeneous network representation learning mainly focus on static networks, overlooking the significant impact of temporal attributes on node representations. However, real heterogeneous information networks are very dynamic, and even minor changes in nodes and edges can affect the entire structure and semantics. In this context, a dynamic heterogeneous network representation learning method based on Hawkes process was proposed. Firstly, the vector representation of nodes was obtained by utilizing the relational rotation encoding method and attention mechanism, where the attention coefficients of adjacent nodes were learned. Secondly, the optimal weighted combination of different meta-paths was learned to better captures the structural and semantic information of the network. Finally, leveraging the time decay effect, time features were introduced into node representations through the formation of neighborhood sequences, resulting in the ultimate embedding representation of nodes. Experimental results on various benchmark datasets indicate that the proposed method significantly outperforms baseline methods. In node classification tasks, Macro-F1 average is increased by 0.15% to 3.45%, and NMI value in node clustering tasks is improved by 1.08% to 3.57%.

AbstractFueled in part by recent applications in neuroscience, the multivariate Hawkes process has become a popular tool for modeling the network of interactions among high-dimensional point process data. While evaluating the uncertainty of the network estimates is critical in scientific applications, existing methodological and theoretical work has primarily addressed estimation. To bridge this gap, we develop a new statistical inference procedure for high-dimensional Hawkes processes. The key ingredient for the inference procedure is a new concentration inequality on the first- and second-order statistics for integrated stochastic processes, which summarize the entire history of the process. Combining recent martingale central limit theorem with the new concentration inequality, we then characterize the convergence rate of the test statistics in a continuous time domain. Finally, to account for potential non-stationarity of the process in practice, we extend our statistical inference procedure to a flexible class of Hawkes processes with time-varying background intensities and unknown transition functions. The finite sample validity of the inferential tools is illustrated via extensive simulations and further applied to a neuron spike train dataset. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work. [ABSTRACT FROM AUTHOR]

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Connected and autonomous vehicles (CAVs) are advancing at a fast speed with the improvement of the automotive industry, which opens up new possibilities for different attacks. A Distributed Denial-of-Service (DDoS) attacker floods the in-vehicle network with fake messages, resulting in the failure of driving assistance systems and impairment of vehicle control functionalities, seriously disrupting the normal operation of the vehicle. In this paper, we propose a novel DDoS attack detection method for in-vehicle Ethernet Scalable service-Oriented Middleware over IP (SOME/IP), which integrates the Hawkes process with Long Short-Term Memory networks (LSTMs) to capture the dynamic behavioral features of the attacker. Specifically, we employ the Hawkes process to capture features of the DDoS attack, with its parameters reflecting the dynamism and self-exciting properties of the attack events. Subsequently, we propose a novel deep learning network structure, an HP-LSTM block, inspired by the Hawkes process, while employing a residual attention block to enhance the model's detection efficiency and accuracy. Additionally, due to the scarcity of publicly available datasets for SOME/IP, we employed a mature SOME/IP generator to create a dataset for evaluating the validity of the proposed detection model. Finally, extensive experiments were conducted to demonstrate the effectiveness of the proposed DDoS attack detection method. [ABSTRACT FROM AUTHOR]

Classic diffusion processes fail to explain asset return volatility. Many empirical findings on asset return time series, such as heavy tails, skewness and volatility clustering, suggest decomposing the volatility of an asset's return into two components, one caused by a Brownian motion and another by a jump process. We analyze the sensitivity of European call options to memory and self-excitation parameters, underlying price, volatility and jump risks. We expand Heston's stochastic volatility model by adding to the instantaneous asset prices, a jump component driven by a Hawkes process with a kernel function or memory kernel that is a Fourier transform of a probability measure. This kernel function defines the memory of the asset price process. For instance, if it is fast decreasing, the contagion effect between asset price jumps is limited in time. Otherwise, the processes remember the history of asset price jumps for a long period. To investigate the impact of different rates of decay or types of memory, we consider four probability measures: Laplace, Gaussian, Logistic and Cauchy. Unlike Hawkes processes with exponential kernels, the Markov property is lost but stationarity is preserved; this ensures that the unconditional expected arrival rate of the jump does not explode. In the absence of the Markov property, we use the Fourier transform representation to derive a closed form expression of a European call option price based on characteristic functions. A numerical illustration shows that our extension of the Heston model achieves a better fit of the Euro Stoxx 50 option data than the standard version. [ABSTRACT FROM AUTHOR]

Modelling spatial and temporal patterns in ecology is imperative to understand the complex processes inherent in ecological phenomena. Log‐Gaussian Cox processes are a popular choice among ecologists to describe the spatiotemporal distribution of point‐referenced data. In addition, point pattern models where events instigate others nearby (i.e., self‐exciting behaviour) are becoming increasingly popular to infer the contagious nature of events (e.g., animal sightings). While there are existing R packages that facilitate fitting spatiotemporal point processes and, separately, self‐exciting models, none incorporate both. We present an R package, stelfi, that fits spatiotemporal self‐exciting and log‐Gaussian Cox process models using Template Model Builder through a range of custom‐written C++ templates. We illustrate the use of stelfi's functions fitting models to Sasquatch (bigfoot) sightings data within the USA. The structure of these data is typical of many seen in ecology studies. We show, from a temporal Hawkes process to a spatiotemporal self‐exciting model, how the models offered by the package enable additional insights into the temporal and spatial progression of point pattern data. We present extensions to these well‐known models that include spatiotemporal self‐excitation and joint likelihood models, which are better suited to capture the complex mechanisms inherent in many ecological data. The package stelfi offers user‐friendly functionality, is open source, and is available from CRAN. It offers the implementation of complex spatiotemporal point process models in R for applications even beyond the field of ecology. [ABSTRACT FROM AUTHOR]

Event sequences with spatiotemporal characteristics have been rapidly produced in various domains, such as earthquakes in seismology, electronic medical records in health care, and transactions in the financial market. These data are discrete events that are continuous and often continue for weeks, months, or years, and past events may trigger subsequent events. In this context, modeling spatiotemporal event sequences and forecasting the occurrence time and marker of the next event has become a hot topic. However, existing models either fail to capture the long-term temporal dependencies or ignore the essential spatial information between sequences. Moreover, existing models learn the influence of past events and predicted future events in Euclidean space, which has been shown to cause a significant distortion in hierarchical structure data. To correctly predict future events from historical events and design interventions and controls to guide the event dynamics to the desired results and inspired by the high capacity of modeling data in hyperbolic space, we proposed a novel hyperbolic graph transformer Hawkes process (HGTHP) model to capture the long-term temporal dependencies and spatial information from historical events with a hierarchical structure. The core concept of the HGTHP is to integrate the learned spatial information into the event embedding as auxiliary information and capture long-short term temporal dependencies from event sequences in non-Euclidean space by a hyperbolic self-attention mechanism. Numerous experiments on synthetic and real-world datasets proved that the proposed model obtains spatiotemporal information from hyperbolic space, and its predictions outperform those of state-of-the-art baselines in both time and marker, proving the proposed model's effectiveness. [ABSTRACT FROM AUTHOR]

This article reviews some of the statistical issues involved with modeling SARS-CoV02 (Covid-19) in Los Angeles County, California, using Hawkes point process models and SEIR models. The two types of models are compared, and their pros and cons are discussed. We also discuss particular statistical decisions, such as where to place the upper limits on y-axes, and whether to use a Bayesian or frequentist version of the model, how to estimate seroprevalence, and fitting the density of transmission times in the Hawkes model. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of learning temporal graph representations, which capture the changing nature of complex evolving networks. Existing approaches mainly focus on adding new nodes and edges to capture dynamic graph structures. However, to achieve more accurate representation of graph evolution, we consider both the addition and deletion of nodes and edges as events. These events occur at irregular time scales and are modeled using temporal point processes. Our goal is to learn the conditional intensity function of the temporal point process to investigate the influence of deletion events on node representation learning for link-level prediction. We incorporate network entropy, a measure of node and edge significance, to capture the effect of node deletion and edge removal in our framework. Additionally, we leveraged the characteristics of a generalized temporal Hawkes process, which considers the inhibitory effects of events where past occurrences can reduce future intensity. This framework enables dynamic representation learning by effectively modeling both addition and deletion events in the temporal graph. To evaluate our approach, we utilize autonomous system graphs, a family of inhomogeneous sparse graphs with instances of node and edge additions and deletions, in a link prediction task. By integrating these enhancements into our framework, we improve the accuracy of dynamic link prediction and enable better understanding of the dynamic evolution of complex networks. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of learning temporal graph representations, which capture the changing nature of complex evolving networks. Existing approaches mainly focus on adding new nodes and edges to capture dynamic graph structures. However, to achieve more accurate representation of graph evolution, we consider both the addition and deletion of nodes and edges as events. These events occur at irregular time scales and are modeled using temporal point processes. Our goal is to learn the conditional intensity function of the temporal point process to investigate the influence of deletion events on node representation learning for link-level prediction. We incorporate network entropy, a measure of node and edge significance, to capture the effect of node deletion and edge removal in our framework. Additionally, we leveraged the characteristics of a generalized temporal Hawkes process, which considers the inhibitory effects of events where past occurrences can reduce future intensity. This framework enables dynamic representation learning by effectively modeling both addition and deletion events in the temporal graph. To evaluate our approach, we utilize autonomous system graphs, a family of inhomogeneous sparse graphs with instances of node and edge additions and deletions, in a link prediction task. By integrating these enhancements into our framework, we improve the accuracy of dynamic link prediction and enable better understanding of the dynamic evolution of complex networks.

Abstract Hawkes process is a class of self-exciting point processes with clustering effect whose jump rate relies on their entire past history. This process is usually defined as a continuous-time setting and has been widely applied in several fields, including insurance, finance, queueing theory, and statistics. The Hawkes model is generally non-Markovian because the future development of a self-exciting point process is determined by the timing of past events. However, it can be Markovian in special cases such as when the exciting function is an exponential function or a sum of exponential functions. Difficulty arises when the exciting function is not an exponential function or a sum of exponentials, in which case the process can be non-Markovian. The inverse Markovian case for Hawkes processes was introduced by Seol (Stat. Probab. Lett. 155:108580, 2019) who studied some asymptotic behaviors. An extended version of the inverse Markovian Hawkes process was also studied by Seol (J. Korean Math. Soc. 58(4):819–833, 2021). In the current work, we propose a class of Markovian self-exciting processes that interpolates between the Hawkes process and the inverse Hawkes process. We derived limit theorems for the newly considered class of Markovian self-exciting processes. In particular, we established both the law of large numbers (LLN) and central limit theorems (CLT) with some key results.

Abstract Modelling spatial and temporal patterns in ecology is imperative to understand the complex processes inherent in ecological phenomena. Log‐Gaussian Cox processes are a popular choice among ecologists to describe the spatiotemporal distribution of point‐referenced data. In addition, point pattern models where events instigate others nearby (i.e., self‐exciting behaviour) are becoming increasingly popular to infer the contagious nature of events (e.g., animal sightings). While there are existing R packages that facilitate fitting spatiotemporal point processes and, separately, self‐exciting models, none incorporate both. We present an R package, stelfi, that fits spatiotemporal self‐exciting and log‐Gaussian Cox process models using Template Model Builder through a range of custom‐written C++ templates. We illustrate the use of stelfi's functions fitting models to Sasquatch (bigfoot) sightings data within the USA. The structure of these data is typical of many seen in ecology studies. We show, from a temporal Hawkes process to a spatiotemporal self‐exciting model, how the models offered by the package enable additional insights into the temporal and spatial progression of point pattern data. We present extensions to these well‐known models that include spatiotemporal self‐excitation and joint likelihood models, which are better suited to capture the complex mechanisms inherent in many ecological data. The package stelfi offers user‐friendly functionality, is open source, and is available from CRAN. It offers the implementation of complex spatiotemporal point process models in R for applications even beyond the field of ecology.

In the era of big data, mathematical and scientific methods significantly contribute to educational reform, continuously revitalizing precision teaching. In this paper, we use the new Hawkes process to collect and vectorize data on how students learn. Then, we use the unique thermal coding method to combine the behavioral feature vector with the teaching feature vector. Finally, we classify the data and use the softmax function to predict the learning effect. Based on the prediction results, a mathematical model for precise teaching optimization of vocational education English has been established, and the decision function is used to solve the problem and optimize teaching methods. It was found that all the indicators of learning behavior characteristics of students in the experimental class applying the teaching method optimization model were significantly improved, and there was a significant difference at the level of 0.05 (P = 0.000

Exploring the digital development of art education can effectively enhance students’ creativity in art design. This paper constructs a digital teaching platform for art education based on big data technology and analyzes student learning behavior data in the teaching platform. For student learning behavior with a new Hawkes process for data extraction, the decomposition of the student learning effect is realized by data encoder, event-level decoder and course-level decoder. Performance tests and art teaching effect analysis were conducted for the platform built in this paper. From the performance test, the throughput of the test interface is 580 when the server CPU is maintained at about 26%, memory is maintained at about 48.62%, and load is maintained at about 4.7. From the teaching effect, the four dimensions of students’ creativity changes had significant differences; namely, the p-values of adventurousness, curiosity, imagination, and challenge were 0.189, 0.247, 0.076, and 0.425, respectively. This indicates that the digital teaching platform for art education built by using big data technology can effectively enhance students’ creativity and also provide a new direction for the digital development of art education.

Connected and autonomous vehicles (CAVs) are advancing at a fast speed with the improvement of the automotive industry, which opens up new possibilities for different attacks. A Distributed Denial-of-Service (DDoS) attacker floods the in-vehicle network with fake messages, resulting in the failure of driving assistance systems and impairment of vehicle control functionalities, seriously disrupting the normal operation of the vehicle. In this paper, we propose a novel DDoS attack detection method for in-vehicle Ethernet Scalable service-Oriented Middleware over IP (SOME/IP), which integrates the Hawkes process with Long Short-Term Memory networks (LSTMs) to capture the dynamic behavioral features of the attacker. Specifically, we employ the Hawkes process to capture features of the DDoS attack, with its parameters reflecting the dynamism and self-exciting properties of the attack events. Subsequently, we propose a novel deep learning network structure, an HP-LSTM block, inspired by the Hawkes process, while employing a residual attention block to enhance the model’s detection efficiency and accuracy. Additionally, due to the scarcity of publicly available datasets for SOME/IP, we employed a mature SOME/IP generator to create a dataset for evaluating the validity of the proposed detection model. Finally, extensive experiments were conducted to demonstrate the effectiveness of the proposed DDoS attack detection method.

Rampant terrorism poses a serious threat to the national security of many countries worldwide, particularly due to separatism and extreme nationalism. This paper focuses on the development and application of a temporal self-exciting point process model to the terror data of three countries: the US, Turkey, and the Philippines. To account for occurrences with the same time-stamp, this paper introduces the order mark and reward term in parameter selection. The reward term considers the triggering effect between events in the same time-stamp but different order. Additionally, this paper provides comparisons between the self-exciting models generated by day-based and month-based arrival times. Another highlight of this paper is the development of a model to predict the number of terror events using a combination of simulation and machine learning, specifically the random forest method, to achieve better predictions. This research offers an insightful approach to discover terror event patterns and forecast future occurrences of terror events, which may have practical application towards national security strategies. [ABSTRACT FROM AUTHOR]

Modeling Hawkes process using deep learning is superior to traditional statistical methods in the goodness of fit. However, methods based on RNN or self-attention are deficient in long-time dependence and recursive induction, respectively. Universal Transformer (UT) is an advanced framework to integrate these two requirements simultaneously due to its continuous transformation of self-attention in the depth of the position. In addition, migration of the UT framework involves the problem of effectively matching Hawkes process modeling. Thus, in this paper, an iterative convolutional enhancing self-attention Hawkes process with time relative position encoding (ICAHP-TR) is proposed, which is based on improved UT. First, the embedding maps from dense layers are carried out on sequences of arrival time points and markers to enrich event representation. Second, the deep network composed of UT extracts hidden historical information from event expression with the characteristics of recursion and the global receptive field. Third, two designed mechanics, including the relative positional encoding on the time step and the convolution enhancing perceptual attention are adopted to avoid losing dependencies between relative and adjacent positions in the Hawkes process. Finally, the hidden historical information is mapped by Dense layers as parameters in Hawkes process intensity function, thereby obtaining the likelihood function as the network loss. The experimental results show that the proposed methods demonstrate the effectiveness of synthetic datasets and real-world datasets from the perspective of both the goodness of fit and predictive ability compared with other baseline methods. [ABSTRACT FROM AUTHOR]

Hawkes process is a class of self-exciting point processes with clustering effect whose jump rate relies on their entire past history. This process is usually defined as a continuous-time setting and has been widely applied in several fields, including insurance, finance, queueing theory, and statistics. The Hawkes model is generally non-Markovian because the future development of a self-exciting point process is determined by the timing of past events. However, it can be Markovian in special cases such as when the exciting function is an exponential function or a sum of exponential functions. Difficulty arises when the exciting function is not an exponential function or a sum of exponentials, in which case the process can be non-Markovian. The inverse Markovian case for Hawkes processes was introduced by Seol (Stat. Probab. Lett. 155:108580, 2019) who studied some asymptotic behaviors. An extended version of the inverse Markovian Hawkes process was also studied by Seol (J. Korean Math. Soc. 58(4):819–833, 2021). In the current work, we propose a class of Markovian self-exciting processes that interpolates between the Hawkes process and the inverse Hawkes process. We derived limit theorems for the newly considered class of Markovian self-exciting processes. In particular, we established both the law of large numbers (LLN) and central limit theorems (CLT) with some key results. [ABSTRACT FROM AUTHOR]

Mobile phones contain a wealth of private information, so we try to keep them secure. We provide large-scale evidence that the psychological profiles of individuals and their relations with their peers can be predicted from seemingly anonymous communication traces–calling and texting logs that service providers routinely collect. Based on two extensive longitudinal studies containing more than 900 college students, we use point process modeling to describe communication patterns. We automatically predict the peer relationship type and temporal dynamics, and assess user personality based on the modeling. For some personality traits, the results are comparable to the gold-standard performances obtained from survey self-report data. Findings illustrate how information usually residing outside the control of individuals can be used to reconstruct sensitive information.

The Hawkes process, which is generally defined for the continuous-time setting, can be described as a self-exciting simple point process with a clustering effect, whose jump rate depends on its entire history. Due to past events determining future developments of self-exciting point processes, the Hawkes model is generally not Markovian. In certain special circumstances, it can be Markovian with a generator of the model if the exciting function is an exponential function or the sum of exponential functions. In the case of non-Markovian processes, difficulties arise when the exciting function is not an exponential function or a sum of exponential functions. The intensity of the Hawkes process is given by the sum of a baseline intensity and other terms that depend on the entire history of the point process, as compared to a standard Poisson process. It is one of the main methods used for studying the dynamical properties of general point processes, and is highly important for credit risk studies. The baseline intensity, which is instrumental in the Hawkes model, is usually defined for deterministic cases. In this paper, we consider a linear Hawkes model where the baseline intensity is randomly defined, and investigate the asymptotic results of the large deviations principle for the newly defined model. The Hawkes processes with randomized baseline intensity, dealt with in this paper, have wide applications in insurance, finance, queue theory, and statistics. [ABSTRACT FROM AUTHOR]

The dynamic recommender system realizes the real-time recommendation for users by learning the dynamic interest characteristics, which is especially suitable for the scenarios of rapid transfer of user interests, such as e-commerce and social media. The dynamic recommendation model mainly depends on the user-item history interaction sequence with timestamp, which contains historical records that reflect changes in the true interests of users and the popularity of items. Previous methods usually model interaction sequences to learn the dynamic embedding of users and items. However, these methods can not directly capture the excitation effects of different historical information on the evolution process of both sides of the interaction, i.e., the ability of events to influence the occurrence of another event. In this work, we propose a Dynamic Graph Hawkes Process based on Linear complexity Self-Attention (DGHP-LISA) for dynamic recommender systems, which is a new framework for modeling the dynamic relationship between users and items at the same time. Specifically, DGHP-LISA is built on dynamic graph and uses Hawkes process to capture the excitation effects between events. In addition, we propose a new self-attention with linear complexity to model the time correlation of different historical events and the dynamic correlation between different update mechanisms, which drives more accurate modeling of the evolution process of both sides of the interaction. Extensive experiments on three real-world datasets show that our model achieves consistent improvements over state-of-the-art baselines.

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 numerous reasons. For example, the most common algorithms just return point estimates with little quantification of uncertainty. At the same time, Bayesian Markov chain Monte Carlo implementations remain slow to run and do not scale well, and few have been extended to include spatial structure. This makes it difficult to explore the effects of stochastic uncertainty. Here, we present a new approach to ETAS modeling using an alternative Bayesian method, the integrated nested Laplace approximation (INLA). We have implemented this model in a new R-Package called ETAS.inlabru, which is built on the R packages R-INLA and inlabru. Our study has included extending these packages, which provided tools for modeling log-Gaussian Cox processes, to include the self-exciting Hawkes process that ETAS is a special case of. While we just present the temporal component here, the model scales to a spatio-temporal model and may include a variety of spatial covariates. This is a fast method that returns joint posteriors on the ETAS background and triggering parameters. Using a series of synthetic case studies, we explore the robustness of ETAS inversions using this method of inversion. We also included runnable notebooks to reproduce the figures in this article as part of the package's GitHub repository. We demonstrate that reliable estimates of the model parameters require that the catalog data contain periods of relative quiescence, as well as triggered sequences. We explore the robustness of the method under stochastic uncertainty in the training data and show that the method is robust to a wide range of starting conditions. We show how the inclusion of historic earthquakes prior to the modeled time window affects the quality of the inversion. Finally, we show that rate-dependent incompleteness of earthquake catalogs after large earthquakes have a significant and detrimental effect on the ETAS posteriors. We believe that the speed of the inlabru inversion, which includes a rigorous estimation of uncertainty, will enable a deeper exploration of how to use ETAS robustly for seismicity modeling and operational earthquake forecasting.

The Japan Electric Power Exchange (JEPX) provides a platform for the trading of electric energy in a manner similar to more traditional financial markets. As the number of market agents increase, there is an increasing need for effective price-forecasting models. Electricity price data are observed to exhibit periods of relatively stable, i.e., low-magnitude, low-variance prices interspersed with periods of higher prices accompanied by larger uncertainty. The price data time series therefore exhibits a temporal non-stationarity characteristic that is difficult to capture with typical time series modeling frameworks. In this paper, we implement models for the occurrence of price spike events where spikes are defined as observing prices above a predefined threshold set here at 25 JPY/kWh. This value corresponds to about the 90th percentile of observed prices during peak trading periods. The price spikes time series is observed to be rare events that occur in clusters. We therefore propose to model the data as a Hawkes process whereby the occurrence of a spike event increases the probability of observing more spikes in the period immediately following a price spike event. We test two variations of the classical Hawkes model: the first variation models the change in the magnitude of the underlying intensity as a function of the magnitude of the price spike while the second variation models the change in the decay rate of the underlying intensity as a function of the magnitude of the price spike. An analysis of the performance of the models based on the mean absolute error (MAE) of the spike occurrence probability, a weighted accuracy index, and the Matthews correlation coefficient (MCC) metrics shows the effectiveness of the variable magnitude variation of the Hawkes model in generating short-term forecasts of the occurrence of price spike events. The modified Hawkes model especially outperforms other candidate models as the length of the forecasting horizon increases. [ABSTRACT FROM AUTHOR]

Hawkes Process has been used to model Limit Order Book (LOB) dynamics in several ways in the literature however the focus has been limited to capturing the inter-event times while the order size is usually assumed to be constant. We propose a novel methodology of using Compound Hawkes Process for the LOB where each event has an order size sampled from a calibrated distribution. The process is formulated in a novel way such that the spread of the process always remains positive. Further, we condition the model parameters on time of day to support empirical observations. We make use of an enhanced non-parametric method to calibrate the Hawkes kernels and allow for inhibitory cross-excitation kernels. We showcase the results and quality of fits for an equity stock's LOB in the NASDAQ exchange and compare them against several baselines. Finally, we conduct a market impact study of the simulator and show the empirical observation of a concave market impact function is indeed replicated. • We propose a novel formulation of the Hawkes Process to maintain positive spread. • We augment the non-parametric methods proposed in Kirchner 2017 to work with slow decaying kernels and to be more stable. • We make use of calibrated distributions for sampling the order sizes of the events rather than assuming unit order-size. • We formulate the in-spread order arrivals as a function of the current spread — utilizing the well known fact that spreads are mean-reverting. • We perform a MI study on this simulator along with testing against several baselines and comparing against empirical data. [ABSTRACT FROM AUTHOR]

Online videos have become the most popular method to obtain information for the public in recent years, such as TikTok and YouTube, and regional sites like Bilibili and Douyin. Compared with its growing influence, the analysis of user behavior on video sites is still less investigated. Herein, we fetch the video data from Bilibili.com and analyze the video views, comments, and other behaviors on the website. We found that the description model based on the Hawkes process can accurately predict the video views, which suggests that on the Bilibili website, the self-incentive mechanism of information cascade diffusion plays a decisive role in online views. Meanwhile, we also found that the view increment of the videos during the same period of time conforms to the general power-law distribution.

Social platforms make information propagation anywhere anytime. Large quantity data recording information spreading is available for further understanding the intrinsic mechanism within these stochastic processes. Based on the empirical spreading trees of tweets on Twitter, the heterogeneity of Twitter users is explored, turning out the burstiness in the spreading process. A few super spreaders can significantly change the trends of information spreading. Moreover, an improved Hawkes process is designed in this study to better investigate users’ heterogeneity during information propagation. Verification is provided for accuracy and stability of the improved Hawkes model in simulating propagation patterns revealed in empirical sequential data, predicting spreading trends, and predicting probability of information outbreaks. Our improved Hawkes model is an effective spreading model for detecting and quantifying super spreaders during the propagation process, which may shed light on the control and prediction of information spreading in social media.

Rampant terrorism poses a serious threat to the national security of many countries worldwide, particularly due to separatism and extreme nationalism. This paper focuses on the development and application of a temporal self-exciting point process model to the terror data of three countries: the US, Turkey, and the Philippines. To account for occurrences with the same time-stamp, this paper introduces the order mark and reward term in parameter selection. The reward term considers the triggering effect between events in the same time-stamp but different order. Additionally, this paper provides comparisons between the self-exciting models generated by day-based and month-based arrival times. Another highlight of this paper is the development of a model to predict the number of terror events using a combination of simulation and machine learning, specifically the random forest method, to achieve better predictions. This research offers an insightful approach to discover terror event patterns and forecast future occurrences of terror events, which may have practical application towards national security strategies.

In this paper, following Nourdin-Peccati's methodology, we combine the Malliavin calculus and Stein's method to provide general bounds on the Wasserstein distance between the law of functionals of a compound Hawkes process and the one of a Gaussian random variable. To achieve this, we rely on the Poisson imbedding representation of a Hawkes process to provide a Malliavin calculus for the Hawkes processes, and more generally for compound Hawkes processes. As an application, we close a gap in the literature by providing a quantitative Central Limit Theorem for the compound Hawkes process. [ABSTRACT FROM AUTHOR]

It has been clarified that words in written texts are classified into two groups called Type-I and Type-II words. The Type-I words are words that exhibit long-range dynamic correlations in written texts while the Type-II words do not show any type of dynamic correlations. Although the stochastic process of yielding Type-II words has been clarified to be a superposition of Poisson point processes with various intensities, there is no definitive model for Type-I words. In this study, we introduce a Hawkes process, which is known as a kind of self-exciting point process, as a candidate for the stochastic process that governs yielding Type-I words; i.e., the purpose of this study is to establish that the Hawkes process is useful to model occurrence patterns of Type-I words in real written texts. The relation between the Hawkes process and an existing model for Type-I words, in which hierarchical structures of written texts are considered to play a central role in yielding dynamic correlations, will also be discussed. [ABSTRACT FROM AUTHOR]

Despite a vast application of temporal point processes in infectious disease diffusion forecasting, ecommerce, traffic prediction, preventive maintenance, etc, there is no significant development in improving the simulation and prediction of temporal point processes in real-world environments. With this problem at hand, we propose a novel methodology for learning temporal point processes based on one-dimensional numerical integration techniques. These techniques are used for linearising the negative maximum likelihood (neML) function and enabling backpropagation of the neML derivatives. Our approach is tested on two real-life datasets. Firstly, on high frequency point process data, (prediction of highway traffic) and secondly, on a very low frequency point processes dataset, (prediction of ski injuries in ski resorts). Four different point process baseline models were compared: second-order Polynomial inhomogeneous process, Hawkes process with exponential kernel, Gaussian process, and Poisson process. The results show the ability of the proposed methodology to generalize on different datasets and illustrate how different numerical integration techniques and mathematical models influence the quality of the obtained models. The presented methodology is not limited to these datasets and can be further used to optimize and predict other processes that are based on temporal point processes. [ABSTRACT FROM AUTHOR]

Knowledge Tracing (KT) is a method that seeks to forecast students' future performance based on their historical interactions with intelligent tutoring systems. Various KT techniques have been developed from distinct perspectives, such as probability, logic, and deep learning, to scrutinize interaction sequences. However, most current methods predominantly concentrate on temporal or structural information, considering simple question sequences or interaction sequences, but fail to describe the topological dependencies between sequences, which can lead to changes in causal structures,as referred to as dynamic structural information in the paper. The dynamic information includes the clustering effect among concepts and questions, as well as the group effect among students and questions: Questions with the same knowledge concepts tend to present increasingly similar response patterns in the sequence, and concurrently, students who answer similar questions often exhibit closer knowledge states. We introduce the STHKT method, which employs a tripartite heterogeneous graph to capture relationships among concepts, questions, and students. By incorporating temporal information through a topological graph, we employ two Graph Convolution Networks (GCNs) to model the two effects separately. Following this, we apply the Hawkes Process to introduce a spatiotemporal attention mechanism that combines both effects. Through comparative experiments, ablation studies and case studies, we validate the efficacy of key components in the model. Experiment results indicate that the STHKT model can effectively model the two major effects, integrate dynamic structural information, and to some extent improve the accuracy of predicting students' response probabilities. It is the first time that these two effects have been considered in the process of knowledge tracing, and it is adaptable to large-scale online education platforms. • We construct a tripartite heterogeneous graph of concepts–questions–students. • We topologize to connect temporal and structural information in the KT process. • We use two GCN networks to model clustering effect and group effect. • Intensity function in the topological Hawkes process enhances the attention score. • Our method achieves more precise result. [ABSTRACT FROM AUTHOR]

The Hawkes process, which is generally defined for the continuous-time setting, can be described as a self-exciting simple point process with a clustering effect, whose jump rate depends on its entire history. Due to past events determining future developments of self-exciting point processes, the Hawkes model is generally not Markovian. In certain special circumstances, it can be Markovian with a generator of the model if the exciting function is an exponential function or the sum of exponential functions. In the case of non-Markovian processes, difficulties arise when the exciting function is not an exponential function or a sum of exponential functions. The intensity of the Hawkes process is given by the sum of a baseline intensity and other terms that depend on the entire history of the point process, as compared to a standard Poisson process. It is one of the main methods used for studying the dynamical properties of general point processes, and is highly important for credit risk studies. The baseline intensity, which is instrumental in the Hawkes model, is usually defined for deterministic cases. In this paper, we consider a linear Hawkes model where the baseline intensity is randomly defined, and investigate the asymptotic results of the large deviations principle for the newly defined model. The Hawkes processes with randomized baseline intensity, dealt with in this paper, have wide applications in insurance, finance, queue theory, and statistics.

Hawkes processes are a class of self-exciting point processes with a clustering effect whose jump rate is determined by its past history. They are generally regarded as continuous-time processes and have been widely applied in a number of fields, such as insurance, finance, queueing, and statistics. The Hawkes model is generally non-Markovian because its future development depends on the timing of past events. However, it can be Markovian under certain circumstances. If the exciting function is an exponential function or a sum of exponential functions, the model can be Markovian with a generator of the model. In contrast to the general Hawkes processes, the inverse Hawkes process has some specific features and self-excitation indicates severity. Inverse Markovian Hawkes processes were introduced by Seol, who studied some asymptotic behaviors. An extended version of inverse Markovian Hawkes processes was also studied by Seol. With this paper, we propose a non-Markovian inverse Hawkes process, which is a more general inverse Hawkes process that features several existing models of self-exciting processes. In particular, we established both the law of large numbers (LLN) and Central limit theorems (CLT) for a newly considered non-Markovian inverse Hawkes process. [ABSTRACT FROM AUTHOR]

Background defining features of the COVID-19 pandemic in many countries were the tragic extent to which care home residents were affected and the difficulty in preventing the introduction and subsequent spread of infection. Management of risk in care homes requires good evidence on the most important transmission pathways. One hypothesised route at the start of the pandemic, prior to widespread testing, was the transfer of patients from hospitals that were experiencing high levels of nosocomial events. Methods we tested the hypothesis that hospital discharge events increased the intensity of care home cases using a national individually linked health record cohort in Wales, UK. We monitored 186,772 hospital discharge events over the period from March to July 2020, tracking individuals to 923 care homes and recording the daily case rate in the homes populated by 15,772 residents. We estimated the risk of an increase in case rates following exposure to a hospital discharge using multi-level hierarchical logistic regression and a novel stochastic Hawkes process outbreak model. Findings in regression analysis, after adjusting for care home size, we found no significant association between hospital discharge and subsequent increases in care home case numbers (odds ratio: 0.99, 95% CI: 0.82, 1.90). Risk factors for increased cases included care home size, care home resident density and provision of nursing care. Using our outbreak model, we found a significant effect of hospital discharge on the subsequent intensity of cases. However, the effect was small and considerably less than the effect of care home size, suggesting the highest risk of introduction came from interaction with the community. We estimated that approximately 1.8% of hospital discharged patients may have been infected. Interpretation there is growing evidence in the UK that the risk of transfer of COVID-19 from the high-risk hospital setting to the high-risk care home setting during the early stages of the pandemic was relatively small. Although access to testing was limited to initial symptomatic cases in each care home at this time, our results suggest that reduced numbers of discharges, selection of patients and action taken within care homes following transfer all may have contributed to the mitigation. The precise key transmission routes from the community remain to be quantified. [ABSTRACT FROM AUTHOR]

Asynchronous 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. [ABSTRACT FROM AUTHOR]

Interval-censored data solely records the aggregated counts of events during specific time intervals - such as the number of patients admitted to the hospital or the volume of vehicles passing traffc loop detectors - and not the exact occurrence time of the events. It is currently not understood how to fit the Hawkes point processes to this kind of data. Its typical loss function (the point process log-likelihood) cannot be computed without exact event times. Furthermore, it does not have the independent increments property to use the Poisson likelihood. This work builds a novel point process, a set of tools, and approximations for fitting Hawkes processes within interval-censored data scenarios. First, we define the Mean Behavior Poisson process (MBPP), a novel Poisson process with a direct parameter correspondence to the popular self-exciting Hawkes process. We fit MBPP in the interval-censored setting using an interval-censored Poisson log-likelihood (IC-LL). We use the parameter equivalence to uncover the parameters of the associated Hawkes process. Second, we introduce two novel exogenous functions to distinguish the exogenous from the endogenous events. We propose the multi-impulse exogenous function - for when the exogenous events are observed as event time - and the latent homogeneous Poisson process exogenous function - for when the exogenous events are presented as interval-censored volumes. Third, we provide several approximation methods to estimate the intensity and compensator function of MBPP when no analytical solution exists. Fourth and finally, we connect the interval-censored loss of MBPP to a broader class of Bregman divergence-based functions. Using the connection, we show that the popularity estimation algorithm Hawkes Intensity Process (HIP) (Rizoiu et al., 2017b) is a particular case of the MBPP. We verify our models through empirical testing on synthetic data and real-world data. We find that our MBPP outperforms HIP on real-world datasets for the task of popularity prediction. This work makes it possible to effciently fit the Hawkes process to interval-censored data. [ABSTRACT FROM AUTHOR]

It is an important research point of social network analysis to predict future propagation range of infor-mation based on its early propagation characteristics. DeepHawkes model combines Hawkes model with deep learning, which not only inherits clear interpretability of Hawkes model to characterize and model the information diffusion process, but also carries on the high prediction power of end-to-end deep learning by automatically learning the latent representations of the input data, bridging the gap between prediction and understanding of information cascades. However, DeepHawkes model ignores the effect of the text content on the propagation. The LDA-Deep-Hawkes model takes cascade factors as well as text content into account, and models the process of information diffusion in a more comprehensive way, so as to further improve the prediction accuracy while inheriting the high interpretability of DeepHawkes model. The prediction accuracy of LDA-DeepHawkes model is compared with other models on two real data sets from Sina Weibo, and the influence of parameters of the model on the prediction accuracy is analyzed. The experimental results show that the LDA-DeepHawkes model has better prediction accuracy, indicating that the text content of information is also an important factor affecting the information diffusion.

The Japan Electric Power Exchange (JEPX) provides a platform for the trading of electric energy in a manner similar to more traditional financial markets. As the number of market agents increase, there is an increasing need for effective price-forecasting models. Electricity price data are observed to exhibit periods of relatively stable, i.e., low-magnitude, low-variance prices interspersed with periods of higher prices accompanied by larger uncertainty. The price data time series therefore exhibits a temporal non-stationarity characteristic that is difficult to capture with typical time series modeling frameworks. In this paper, we implement models for the occurrence of price spike events where spikes are defined as observing prices above a predefined threshold set here at 25 JPY/kWh. This value corresponds to about the 90th percentile of observed prices during peak trading periods. The price spikes time series is observed to be rare events that occur in clusters. We therefore propose to model the data as a Hawkes process whereby the occurrence of a spike event increases the probability of observing more spikes in the period immediately following a price spike event. We test two variations of the classical Hawkes model: the first variation models the change in the magnitude of the underlying intensity as a function of the magnitude of the price spike while the second variation models the change in the decay rate of the underlying intensity as a function of the magnitude of the price spike. An analysis of the performance of the models based on the mean absolute error (MAE) of the spike occurrence probability, a weighted accuracy index, and the Matthews correlation coefficient (MCC) metrics shows the effectiveness of the variable magnitude variation of the Hawkes model in generating short-term forecasts of the occurrence of price spike events. The modified Hawkes model especially outperforms other candidate models as the length of the forecasting horizon increases.

In this paper, the insurance reserve period will be classified according to the claim frequency domain, such as high- or low-frequency periods. We use the clustering method to create and group claims data according to their frequency period. Meanwhile, we use a risk process to mimic and predict the movement of the reserve from time to time in each group of claim period that is formed. The risk process model used here is the Hawkes process, which is a one-dimensional simple point process and a special type of self-exciting process. Based on this process, we will estimate the reserve at a certain date in the future and the average historical reserve for each group period.

Multicellular organisms rely on spatial signaling among cells to drive their organization, development, and response to stimuli. Several models have been proposed to capture the behavior of spatial signaling in multicellular systems, but existing approaches fail to capture both the autonomous behavior of single cells and the interactions of a cell with its neighbors simultaneously. We propose a spatiotemporal model of dynamic cell signaling based on Hawkes processes--self-exciting point processes--that model the signaling processes within a cell and spatial couplings between cells. With this cellular point process (CPP), we capture both the single-cell pathway activation rate and the magnitude and duration of signaling between cells relative to their spatial location. Furthermore, our model captures tissues composed of heterogeneous cell types with different bursting rates and signaling behaviors across multiple signaling proteins. We apply our model to epithelial cell systems that exhibit a range of autonomous and spatial signaling behaviors basally and under pharmacological exposure. Our model identifies known drug-induced signaling deficits, characterizes signaling changes across a wound front, and generalizes to multichannel observations. [ABSTRACT FROM AUTHOR]

It has been clarified that words in written texts are classified into two groups called Type-I and Type-II words. The Type-I words are words that exhibit long-range dynamic correlations in written texts while the Type-II words do not show any type of dynamic correlations. Although the stochastic process of yielding Type-II words has been clarified to be a superposition of Poisson point processes with various intensities, there is no definitive model for Type-I words. In this study, we introduce a Hawkes process, which is known as a kind of self-exciting point process, as a candidate for the stochastic process that governs yielding Type-I words; i.e., the purpose of this study is to establish that the Hawkes process is useful to model occurrence patterns of Type-I words in real written texts. The relation between the Hawkes process and an existing model for Type-I words, in which hierarchical structures of written texts are considered to play a central role in yielding dynamic correlations, will also be discussed.

We use point processes to analyze market order arrivals on the intraday market for hourly electricity deliveries in Germany in the second quarter of 2015. As we distinguish between buys and sells, we work in a multivariate setting. We model the arrivals with a Hawkes process whose baseline intensity comprises either only an exponentially increasing component or a constant in addition to the exponentially increasing component, and whose excitation decays exponentially. Our goodness-of-fit tests indicate that the models where the intensity of each market order type is excited at least by events of the same type are the most promising ones. Based on the Akaike information criterion, the model without a constant in the baseline intensity and only self-excitation is selected in almost 50% of the cases on both market sides. The typical jump size of intensities in case of the arrival of a market order of the same type is quite large, yet rather short lived. Diurnal patterns in the parameters of the baseline intensity and the branching ratio of self-excitation are observable. Contemporaneous relationships between different parameters such as the jump size and decay rate of self and cross-excitation are found. [ABSTRACT FROM AUTHOR]

Hawkes processes are a class of self-exciting point processes with a clustering effect whose jump rate is determined by its past history. They are generally regarded as continuous-time processes and have been widely applied in a number of fields, such as insurance, finance, queueing, and statistics. The Hawkes model is generally non-Markovian because its future development depends on the timing of past events. However, it can be Markovian under certain circumstances. If the exciting function is an exponential function or a sum of exponential functions, the model can be Markovian with a generator of the model. In contrast to the general Hawkes processes, the inverse Hawkes process has some specific features and self-excitation indicates severity. Inverse Markovian Hawkes processes were introduced by Seol, who studied some asymptotic behaviors. An extended version of inverse Markovian Hawkes processes was also studied by Seol. With this paper, we propose a non-Markovian inverse Hawkes process, which is a more general inverse Hawkes process that features several existing models of self-exciting processes. In particular, we established both the law of large numbers (LLN) and Central limit theorems (CLT) for a newly considered non-Markovian inverse Hawkes process.

Motivated by the recent outbreak of coronavirus (COVID-19), we propose a stochastic model of epidemic temporal growth and mitigation based on a time-modulated Hawkes process. The model is sufficiently rich to incorporate specific characteristics of the novel coronavirus, to capture the impact of undetected, asymptomatic and super-diffusive individuals, and especially to take into account time-varying counter-measures and detection efforts. Yet, it is simple enough to allow scalable and efficient computation of the temporal evolution of the epidemic, and exploration of what–if scenarios. Compared to traditional compartmental models, our approach allows a more faithful description of virus specific features, such as distributions for the time spent in stages, which is crucial when the time-scale of control (e.g., mobility restrictions) is comparable to the lifetime of a single infection. We apply the model to the first and second wave of COVID-19 in Italy, shedding light onto several effects related to mobility restrictions introduced by the government, and to the effectiveness of contact tracing and mass testing performed by the national health service. [ABSTRACT FROM AUTHOR]