Hawkes Point Process Research Articles

Efficient non-parametric estimation of variable productivity Hawkes processes

Several approaches to estimating the productivity function for a Hawkes point process with variable productivity are discussed, improved upon, and compared in terms of their root-mean-squared error and computational efficiency for various data sizes, and for binned as well as unbinned data. We find that for unbinned data, a regularized version of the analytic maximum likelihood estimator proposed by Schoenberg is the most accurate but is computationally burdensome. The unregularized version of the estimator is faster to compute but has lower accuracy, though both estimators outperform empirical or binned least squares estimators in terms of root-mean-squared error, especially when the mean productivity is 0.2 or greater. For binned data, binned least squares estimates are highly efficient both in terms of computation time and root-mean-squared error. An extension to estimating transmission time density is discussed, and an application to estimating the productivity of Covid-19 in the United States as a function of time from January 2020 to July 2022 is provided.

An Inhomogeneous Weibull–Hawkes Process to Model Underdispersed Acoustic Cues

AbstractA Hawkes point process describes self-exciting behaviour where event arrivals are triggered by historic events. These models are increasingly becoming a popular choice in analysing event-type data. Like all other inhomogeneous Poisson point processes, the waiting time between events in a Hawkes process is derived from an exponential distribution with mean one. However, as with many ecological and environmental data, this is an unrealistic assumption. We, therefore, extend and generalise the Hawkes process to account for potential under- or overdispersion in the waiting times between events by assuming the Weibull distribution as the foundation of the waiting times. We apply this model to the acoustic cue production times of sperm whales and show that our Weibull–Hawkes model better captures the inherent underdispersion in the interarrival times of echolocation clicks emitted by these whales.

Some statistical problems involved in forecasting and estimating the spread of SARS-CoV-2 using Hawkes point processes and SEIR models

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.

Simultaneous monitoring of multivariate time between events and their magnitude using multivariate marked Hawkes point process

ABSTRACT In this paper, monitoring the time between events is discussed. The control chart of the time between events in multivariate mode discusses the conditions under which the Hawkes point process generates the time data. It is also necessary to consider that monitoring time data individually and regardless of the magnitude of an event is not very important. Therefore, in addition to monitoring time data, it is also essential to monitor the magnitude of an event. Because of that, the magnitude of each event is considered along with monitoring the time between multivariate events through data generation with the multivariate marked Hawkes point process. Three statistics are presented for simultaneous monitoring of the time between events and the magnitude. A Max-EWMA, two single EWMA that each of which are used to monitor TBE and magnitude (TSEWMA), and the third statistic, derived from existing literature, is a Shewhart statistic that has been integrated into an EWMA statistic framework (EWMA-Z). These charts have been used for a case study and compared according to different shifts.

Higher-order memory guided temporal random walk for dynamic heterogeneous network embedding

Network embedding (NE) aims at learning node embeddings via structure-based sampling. However, there are complex patterns in network structure (heterogeneity, higher-order dependence, dynamics) in the real world. The existing methods suffer from high dependence and constraints on manually designed higher-order structures and loss of fine-grained temporal information. To solve the above challenges, we propose a novel higher-order memory guided temporal random walk for dynamic heterogeneous network embedding (HoMo-DyHNE). The proposed model is a two-stage architecture consisting of a meta-structure-independent random walk algorithm namely HoMo-TRW with transition vectors and higher-order memory, and a Hawkes-based featured Skip-gram (HFSG) incorporating a multivariate Hawkes point process to measure the history-current association intensity. Extensive experiments demonstrate the superior effectiveness of our proposed method.

SentiHawkes: a sentiment-aware Hawkes point process to model service quality of public transport using Twitter data

SentiHawkes: a sentiment-aware Hawkes point process to model service quality of public transport using Twitter data

A fractional Hawkes process II: Further characterization of the process

We characterize a Hawkes point process with kernel proportional to the probability density function of Mittag-Leffler random variables. This kernel decays as a power law with exponent β+1∈(1,2]. Several analytical results can be proved, in particular for the expected intensity of the point process and for the expected number of events of the counting process. These analytical results are used to validate algorithms that numerically invert the Laplace transform of the expected intensity as well as Monte Carlo simulations of the process. Finally, Monte Carlo simulations are used to derive the full distribution of the number of events. The algorithms used for this paper are available at https://github.com/habyarimanacassien/Fractional-Hawkes.

Asymptotic solutions to nonlinear Hawkes processes: A systematic classification of the steady-state solutions

Hawkes point processes are first-order non-Markovian stochastic models of intermittent bursty dynamics with applications to physical, seismic, epidemic, biological, financial, and social systems. While accounting for positive feedback loops that may lead to critical phenomena in complex systems, the standard linear Hawkes process only describes excitative phenomena. To describe the co-existence of excitatory and inhibitory effects (or negative feedbacks), extensions involving nonlinear dependences of the intensity as a function of past activity are needed. However, such nonlinear Hawkes processes have been found hitherto to be analytically intractable due to the interplay between their non-Markovian and nonlinear characteristics, with no analytical solutions available. Here, we present various exact and robust asymptotic solutions to nonlinear Hawkes processes using the field master equation approach. We report explicit power law formulas for the steady state intensity distributions $P_{\mathrm{ss}}(\lambda)\propto \lambda^{-1-a}$, where the tail exponent $a$ is expressed analytically as a function of parameters of the nonlinear Hawkes models. We present three robust interesting characteristics of the nonlinear Hawkes process: (i) for one-sided positive marks (i.e., in the absence of inhibitory effects), the nonlinear Hawkes process can exhibit any power law relation either as intermediate asymptotics ($a\leq 0$) or as true asymptotics ($a>0$) by appropriate model selection; (ii) for distribution of marks with zero mean (i.e., for balanced excitatory and inhibitory effects), the Zipf law ($a\approx 1$) is universally observed for a wide class of nonlinear Hawkes processes with fast-accelerating intensity map; (iii) for marks with a negative mean, the asymptotic power law tail becomes lighter with an exponent increasing above 1 ($a>1$) as the mean mark becomes more negative.

Self-exciting point process modelling of crimes on linear networks

Although there are recent developments for the analysis of first and second-order characteristics of point processes on networks, there are very few attempts in introducing models for network data. Motivated by the analysis of crime data in Bucaramanga (Colombia), we propose a spatiotemporal Hawkes point process model adapted to events living on linear networks. We first consider a non-parametric modelling strategy, for which we follow a non-parametric estimation of both the background and the triggering components. Then we consider a semi-parametric version, including a parametric estimation of the background based on covariates, and a non-parametric one of the triggering effects. Our model can be easily adapted to multi-type processes. Our network model outperforms a planar version, improving the fitting of the self-exciting point process model.

How Can Our Tweets Go Viral? Point-Process Modelling of Brand Content

People create and share content via online social networks, which provide an unparalleled opportunity for brands to gain visibility, promote products or services and drive revenue growth. Much research has focused on why, how, or what social content is popular, trending and “hype”. One central challenge is to forecast the spread (cascades) of information that leads to the popularity of content throughout a social network. Online content tends to have bursts and spikes, experiencing a different cascading pattern depending on the viral propagation. In this paper, we propose and test a flexible framework capable of modelling such patterns and trends. We take temporal and network perspectives and develop a model based on the multivariate Hawkes processes that account for social behaviour and network elements such as follower counts, and activity variation observed in collective re-sharing behaviour. We focus on Twitter as the most widely used micro-blogging online social network and measure the popularity of a brand's tweet by analysing the time-series path of three types of subsequent activities (retweets (RTs), replies (REs) and likes (LKs)). The specific model that we propose in this paper is the multidimensional epidemic-type aftershock sequence (METAS) model, a particular case of the multivariate Hawkes process. It consists of a power-law relaxation governing the timing of activities. It also includes an exponential boost as a reinforcement mechanism for the response amplitude to model the impact of influential users on their followers. Earlier attempts to model online cascades have treated all online responses as one type of activity. Rather than aggregating all the activities into one stream, and therefore, ignoring exciting effects among different types of activities, we incorporate the activity variation into the predictive models of content popularity, explicitly accounting for such excitation effects. We develop epidemic-type mutually exciting Hawkes point processes models to quantify such effects and to predict more accurately the number of follow-up activities (i.e., RTs, REs and LKs) on a brand tweet after it is posted. Our results suggest that the proposed model outperforms the state-of-the-art models in terms of prediction accuracy, as it is able to account for mutual excitations and cross-interactions between sequences of users’ activities from one type to another. These results are relevant for developing and executing a plan for online activities by the brand owners.

Popularity prediction for social media over arbitrary time horizons

Predicting the popularity of social media content in real time requires approaches that efficiently operate at global scale. Popularity prediction is important for many applications, including detection of harmful viral content to enable timely content moderation. The prediction task is difficult because views result from interactions between user interests, content features, resharing, feed ranking, and network structure. We consider the problem of accurately predicting popularity both at any given prediction time since a content item's creation and for arbitrary time horizons into the future. In order to achieve high accuracy for different prediction time horizons, it is essential for models to use static features (of content and user) as well as observed popularity growth up to prediction time. We propose a feature-based approach based on a self-excited Hawkes point process model, which involves prediction of the content's popularity at one or more reference horizons in tandem with a point predictor of an effective growth parameter that reflects the timescale of popularity growth. This results in a highly scalable method for popularity prediction over arbitrary prediction time horizons that also achieves a high degree of accuracy, compared to several leading baselines, on a dataset of public page content on Facebook over a two-month period, covering billions of content views and hundreds of thousands of distinct content items. The model has shown competitive prediction accuracy against a strong baseline that consists of separately trained models for specific prediction time horizons.

Is the recent surge in violence in American cities due to contagion?

Firearm violence rates have increased in U.S. cities in 2020 and into 2021. We investigate contagious and non-contagious space-time clustering in shooting events in four U.S. cities (Chicago, Los Angeles, New York and Philadelphia) from 2016 to 2020. We estimate the dynamic reproduction number (Rt) of shootings, a measure of contagion, using a Hawkes point process. We also measure concentration over time using a spatial Gini index. We find that the contagious spread of violence increased in 2020 in several, but not all, of the cities we considered. In all four cities, non-contagious (Poisson) events comprised the majority of shootings across time (including 2020). We also find that the spatial location and concentration of shooting hot spots remained stable across all years. We discuss the implications of our findings and directions for future research.

Nonparametric estimation of recursive point processes with application to mumps in Pennsylvania.

The self-exciting Hawkes point process model (Hawkes, 1971) has been used to describe and forecast communicable diseases. A variant of the Hawkes model, called the recursive model, was proposed by Schoenberg etal. (2019) and has been shown to fit well to various epidemic disease datasets. Unlike the Hawkes model, the recursive model allows the productivity to vary as the overall rate of incidence of the disease varies. Here, we extend the data-driven nonparametric expectation-maximization method of Marsan and Lengliné (2008) in order to fit the recursive model without assuming a particular functional form for the productivity. The nonparametric recursive model is trained to fit to weekly reported cases of mumps in Pennsylvania during the January 1970-September 1990 time frame and then assessed using one week forecasts for the October 1990-December 2001 time period. Both its training and predictive ability are evaluated compared to that of other candidate models, such as Hawkes and SVEILR (susceptible, vaccinated, exposed, infected, lightly infected, recovered) compartmentalmodels.

Point Process Models for the Spread of Coccidioidomycosis in California.

Coccidioidomycosis is an infectious disease of humans and other mammals that has seen a recent increase in occurrence in the southwestern United States, particularly in California. A rise in cases and risk to public health can serve as the impetus to apply newly developed methods that can quickly and accurately predict future caseloads. The recursive and Hawkes point process models with various triggering functions were fit to the data and their goodness of fit evaluated and compared. Although the point process models were largely similar in their fit to the data, the recursive point process model offered a slightly superior fit. We explored forecasting the spread of coccidioidomycosis in California from December 2002 to December 2017 using this recursive model, and we separated the training and testing portions of the data and achieved a root mean squared error of just 3.62 cases/week.

The Co-Production of Service: Modeling Service Times in Contact Centers Using Hawkes Processes

In customer support contact centers, a successful service interaction involves a messaging dialogue between a customer and an agent. Both parties depend on one another for information and problem solving, and this interaction defines a co-produced service process. In this paper, we propose, develop, and compare new stochastic models for service co-production in a contact center. A key observation is that the customer and agent’s co-produced service has cross- and self-exciting dynamics within each conversation. The cross-excitation stems from the two parties responding to one another, and the self-excitation captures one party sending follow-ups to their own prior message. Hence, messages beget messages, and we capture this phenomenon by introducing Hawkes point process models of the conversational services. These models distinguish between the role of the customer and of the agent, reflect the service process's dynamic evolution over time based on its own history, and include additional behavioral and operational aspects, including the agent's number of simultaneous assignments and measures of the amount of information and sentiment each message contains. To evaluate our service co-production models, we apply them to an industry contact center dataset containing nearly 5 million messages. We show that the Hawkes models better represent the service dynamics than do the classic Poisson and phase-type models. Indeed, we find that service interactions are characterized by strong agent-customer dependency and the centrality of the process's cross- and self-excitation attributes. Finally, we use the proposed models to improve upon routing algorithms used in contact centers. We show how an activity-based dynamic routing based on predicted information easily computed from our models can outperform well-known and widely used concurrency-based routing rules and substantially reduce customer waiting time, demonstrating how these history-dependent stochastic models can improve operational decision making in practice.

Excess financial volatility explained by endogenous excitations revealed by EM calibrations of a generalized Hawkes point process

Puzzling deviations from the predictions of rational finance theory have been extensively documented empirically. In this paper, we offer an explanation for one of these anomalies, the “excess volatility puzzle”, i.e. the observation that prices fluctuate more than fundamentally justified. Based on Expectation Maximization (EM) calibrations of a generalized Hawkes point process model to price changes of major currency pairs and equity futures, we construct a decomposition of the variance of high frequency price changes into an exogenous (and thus efficient) component and an endogenous (and thus inefficient) excess component. The endogenously induced excess volatility is found to be substantial, largely stable at longer time scales and thus provides a plausible ex-planation for the excess volatility puzzle. Furthermore, strong endogenous variations at shorter scales are found to lead to major temporary inefficiencies. For example, during the “flash crash” in the GBP/USD exchange rate on October 7, 2016, we document a significant breakdown of market efficiency and an excessive burst in volatility, almost entirely explained by endogenous feedback. Conversely, the shock to EUR/ USD volatility in response to the 2016 Brexit referendum was not accompanied by such a deterioration in market efficiency. These results underline that a more solid understanding of the microstructural origins of financial fluctuations also bears important lessons for neo-classical concepts, like market efficiency, which are fundamental to financial and economic theory.

A non-parametric Hawkes model of the spread of Ebola in west Africa

ABSTRACT Recently developed methods for the non-parametric estimation of Hawkes point process models facilitate their application for describing and forecasting the spread of epidemic diseases. We use data from the 2014 Ebola outbreak in West Africa to evaluate how well a simple Hawkes point process model can forecast the spread of Ebola virus in Guinea, Sierra Leone, and Liberia. For comparison, SEIR models that fit previously to the same data are evaluated using identical metrics. To test the predictive power of each of the models, we simulate the ability to make near real-time predictions during an actual outbreak by using the first 75% of the data for estimation and the subsequent 25% of the data for evaluation. Forecasts generated from Hawkes models more accurately describe the spread of Ebola in each of the three countries investigated and result in a 38% reduction in RMSE for weekly case estimation across all countries when compared to SEIR models (total RMSE of 59.8 cases/week using SEIR compared to 37.1 for Hawkes). We demonstrate that the improved fit from Hawkes modeling cannot be attributed to overfitting and evaluate the advantages and disadvantages of Hawkes models in general for forecasting the spread of epidemic diseases.

Survival analysis of failures based on Hawkes process with Weibull base intensity

In this paper, we construct a Hawkes process with time-varying base intensity to model the sequence of failure, i.e., failure events of the compressor station, and we combine survival analysis and point process model on various failure events of the compressor station based on Hawkes process. To our best knowledge, until now, nearly all relevant literature of the Hawkes point processes assumes that the base intensity of the conditional intensity function is time-invariant. This assumption is apparently too harsh to be verified. For example, in the practical application, including financial analysis, reliability analysis, survival analysis and social network analysis, the truth variation of the base intensity of the failure occurrence over time is not constant. The constant base intensity will not reflect the base intensity trend of the failure occurring over time. Thus, in order to solve this problem, in this paper, we propose a new time-varying base intensity, e.g. which is treated as obeying Weibull distribution. First, we introduce the base intensity into a Hawkes process that obeys the Weibull distribution, and then we propose an effective learning algorithm based on the maximum likelihood estimator. Experiments on the constant base intensity synthetic data, time-varying base intensity synthetic data, and real-world data show that our method can learn the triggering patterns of the Hawkes processes and the time-varying base intensity simultaneously and robustly. Experiments on real-world data also reveal the Granger causality of different types of failures and the base probability of failure varying over time. We put forward some suggestions for practical production based on the experimental results.

Truncated Hawkes point process modeling: System theory and system identification

Point processes have a growing list of applications, including, neuroscience, high frequency finance, genomics and more recently, streaming data. Perhaps the most used parameterized dynamic (aka self-exciting i.e. history dependent) model is the (linear) Hawkes model. But the Hawkes model assumes the infinite past is available and has to be truncated for empirical use. Here we develop, for the first time, the system theory for the truncated Hawkes process. This includes: necessary and sufficient conditions for stability; a Wold decomposition; and an asymptotic spectrum. We then discuss various parameterizations (Exponential, Laguerre, Binomial-autoregression) and their properties. Using the Laguerre parameterization, we next develop a new continuous time multiplicative EM algorithm for maximum likelihood estimation and develop some of its properties. It turns out that the multiplicative EM algorithm does not guarantee stability and so we develop a modified multiplicative EM that does. We also derive, for the first time for self-exciting processes, the Cramer–Rao bound as well as a BIC model selection criterion. We illustrate with simulations, comparing the Laguerre approach to the popular binomial-autoregression and then with an application to neural data recorded from a cat’s brain under visual stimuli.

Real-time predictions of the 2018–2019 Ebola virus disease outbreak in the Democratic Republic of the Congo using Hawkes point process models

As of June 16, 2019, an Ebola virus disease (EVD) outbreak has led to 2136 reported cases in the northeastern region of the Democratic Republic of the Congo (DRC). As this outbreak continues to threaten the lives and livelihoods of people already suffering from civil strife and armed conflict, relatively simple mathematical models and their short-term predictions have the potential to inform Ebola response efforts in real time. We applied recently developed non-parametrically estimated Hawkes point processes to model the expected cumulative case count using daily case counts from May 3, 2018, to June 16, 2019, initially reported by the Ministry of Health of DRC and later confirmed in World Health Organization situation reports. We generated probabilistic estimates of the ongoing EVD outbreak in DRC extending both before and after June 16, 2019, and evaluated their accuracy by comparing forecasted vs. actual outbreak sizes, out-of-sample log-likelihood scores and the error per day in the median forecast. The median estimated outbreak sizes for the prospective thee-, six-, and nine-week projections made using data up to June 16, 2019, were, respectively, 2317 (95% PI: 2222, 2464); 2440 (95% PI: 2250, 2790); and 2544 (95% PI: 2273, 3205). The nine-week projection experienced some degradation with a daily error in the median forecast of 6.73 cases, while the six- and three-week projections were more reliable, with corresponding errors of 4.96 and 4.85 cases per day, respectively. Our findings suggest the Hawkes point process may serve as an easily-applied statistical model to predict EVD outbreak trajectories in near real-time to better inform decision-making and resource allocation during Ebola response efforts.