Hawkes Model Research Articles

Topology sensing of FANET under missing data

The topological structure of a flying ad hoc network (FANET) is crucial to understand, explain, and predict the behavior of unmanned aerial vehicle (UAV) swarms. Most studies focusing on topology sensing use perfect observations and complete datasets. However, the received signal dataset, being non-cooperative, commonly encounters instances of missing data, causing the performance of the existing algorithms to degrade. We investigate the issue of topology sensing of FANET based on external observations and propose a topology sensing method for FANET with missing data while introducing link-prediction methods to correct the topology inference results. First, we employ multi-dimensional Hawkes processes to model the communication event sequence in the network. Subsequently, to solve the problem in which the binary decision threshold is difficult to determine and cannot be adapted to the application scenario, we propose an extended multi-dimensional Hawkes model suitable for FANET and use the maximum likelihood estimation method for topology inference. Finally, to solve the problem of the low accuracy of inference results owing to missing data, we perform community detection on the observation network and combine the community detection and inference results to construct a mixed connection probability matrix, based on which we perform topology correction. The results of the analysis show that the topology sensing method proposed in this study is robust against missing data, indicating that it is an effective solution for solving this problem.

Application of tests for contagion in point processes to measles, Chlamydia, Lyme disease, and suicide

A recently proposed likelihood-ratio test for identifying causal triggering in point process data is applied to a variety of case counts of diseases of varying infectiousness. The test, suggested by McGovern et al. (2023), involves comparing the likelihood of a fitted Hawkes model to that of a fitted Poisson cluster model, and was shown using simulations to be powerful at discriminating between a process with causal triggering and a process where the clustering is merely due to spatial-temporal inhomogeneity. Here, the test is applied to data on measles, Chlamydia, and Lyme disease in the United States, to see if the test can discern between diseases that are highly contagious, moderately contagious, and not directly contagious from human to human. Measles is a highly contagious disease that spreads rapidly through populations, so it can potentially be modeled accurately using a Hawkes model. Chlamydia is a sexually transmitted disease that is not as highly contagious as measles since the level of contact needed for exposure is much higher than for measles. Lyme disease is non-contagious from human to human but cases tend to be highly clustered, as the disease is primarily spread through ticks, and this exposure is much more likely to happen during warmer weather. Further, the test is applied to data on adolescent suicides in the United States, in order to investigate the hypothesis that such suicides are an epidemic spread by social contagion. The results show that the test is able to measure the degree of contagion of a disease, and the results suggest that there is indeed a small but statistically significant element of contagion to youth suicides.

Hawkes Models and Their Applications

The Hawkes process is a model for counting the number of arrivals to a system that exhibits the self-exciting property—that one arrival creates a heightened chance of further arrivals in the near future. The model and its generalizations have been applied in a plethora of disparate domains, though two particularly developed applications are in seismology and in finance. As the original model is elegantly simple, generalizations have been proposed that track marks for each arrival, are multivariate, have a spatial component, are driven by renewal processes, treat time as discrete, and so on. This article creates a cohesive review of the traditional Hawkes model and the modern generalizations, providing details on their construction and simulation algorithms, and giving key references to the appropriate literature for a detailed treatment.

Investigation of Valve Seat Cone Angle on Small Opening Direct-Acting Relief Valve Cavitation Noise

Direct-acting relief valves are important pressure-control components in hydraulic systems; however, noise problems are now common. This study aimed to reduce and numerically analyze the valve cavitation and noise using the Zwart–Gerber–Belamri (ZBG) model with the Ffowcs Williams and Hawkings (FW–H) model to optimize the design based on the sound field perspective. First, a direct-acting relief valve flow field model was established to determine the relationship between the seat structure and the degree of cavitation through a CFD (Computational Fluid Dynamics) simulation. Second, sound field analysis was conducted based on the cavitation and non-cavitation flow fields, respectively, and the resulting noise levels were compared. Finally, prototypes of the relief valve were manufactured, and noise levels between the original and optimized valves were compared. The results revealed that cavitation within the relief valve generated noise while optimizing the valve seat cone angle suppressed this phenomenon, thereby reducing the noise emitted by the optimized valve by 18.2 dB compared to the original valve. These findings can serve as a guide for designing and optimizing direct-acting relief valves.

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

In customer support contact centers, every service interaction involves a messaging dialogue between a customer and an agent; together, they exchange information, solve problems, and collectively co-produce the service. Because the service progression is shaped by the history of conversation thus far, we propose a bivariate marked Hawkes process cluster model of the customer-agent interaction. To evaluate our stochastic model of service, we apply it to an industry contact center data set containing nearly 5 million messages. Through both a novel residual analysis comparison and several Monte Carlo goodness-of-fit tests, we show that the Hawkes cluster model indeed captures dynamics at the heart of the service and surpasses classic models that do not incorporate the service history. Furthermore, in an entirely data-driven simulation, we demonstrate how this history-dependent model can be leveraged operationally to inform a prediction-based routing policy. We show that widely used and well-studied customer routing policies can be outperformed with simple modifications according to the Hawkes model. Through analysis of a stylized model proposed in the contact center literature, we prove that service heterogeneity can cause this underperformance and, moreover, that such heterogeneity will occur if service closures are not carefully managed. This paper was accepted by Elena Katok, operations management. Funding: The authors are grateful for the generous support of this work by the National Science Foundation Division of Graduate Education [Grant DGE-1650441] (A. Daw), the Israel Science Foundation [Grant 336/19] (G. B. Yom-Tov), and the United States-Israel Binational Science Foundation [Grant 2022095] (A. Daw, G. B. Yom-Tov). Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2021.04060 .

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.

Numerical simulation of sound pressure characteristic in steam injection process

This numerical simulation study has been performed to investigate the dynamic characteristics, oscillation characteristics, and sound pressure of direct contact condensation at low steam mass flux. The volume of fluid model, Ffowcs Williams and Hawkings model, and condensation model are used to simulate the acoustic signal, which is produced during the steam injection process. The simulation results are verified by the experimental results to keep the applicability and accuracy of the model. The results show that the acoustic signal is related to the steam flow patterns. The vibration of the bubble volume, caused by the separation of bubbles, is the important factor for the generation of the acoustic signal. When the subcooled temperature is large, the chugging regime appears during the injection process. The sound pressure amplitude of the steam injection process increases significantly, with the increase in subcooled temperature. In the low-frequency region, which is below 10 Hz, the larger the subcooled temperature increases, the greater the sound pressure amplitude becomes.

A Hawkes model with CARMA(p,q) intensity

In this paper we introduce a new model, named CARMA(p,q)-Hawkes, as the Hawkes model with exponential kernel implies a strictly decreasing behavior of the autocorrelation function while empirical evidences reject its monotonicity. The proposed model is a Hawkes process where the intensity follows a Continuous Time Autoregressive Moving Average (CARMA) process. We also study the conditions for the stationarity and the positivity of the intensity and the strong mixing property for the increments. Furthermore, we present two estimation case studies based respectively on the likelihood and on the autocorrelation function.

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.

A vocalization in male Japanese bush warblers in response to both predators and conspecific females

AbstractMale Japanese bush warblers (Cettia diphone) produce not only typical songs but also long, conspicuous vocalizations (continuous songs) during the breeding season. In my preliminary observations, male warblers produced continuous songs in response to both the appearance of a predator and the calls of conspecific females. Although predator‐elicited vocalizations by males have been studied in some species, there are no known cases in which an acoustically identical vocalization is also triggered in response to conspecific females. Here, I examined whether these continuous songs triggered in response to predators and females were acoustically similar. In field experiments, I presented dummies of a Eurasian sparrowhawk and a female warbler before (April) and after (June) female arrival at the breeding ground. Before female arrival, 11.1% of males responded with continuous songs to each of the hawk and female presentations; after female arrival, the respective values were 27.6% and 48.3%. Thus, more males responded to the model presentations with continuous songs after female arrival. There was no difference in the acoustic characteristics of the continuous songs between the two models. The males approached the female model more intensively than the hawk model. These findings do not support the hypothesis that the song functions as an alarm or predator deterrent. The function of these continuous songs remains unclear, but this study is the first to show an example where males produce the same vocalization in response to both predators and conspecific females. The continuous song contains a trill‐like structure, which can be considered a performance‐related song trait. Therefore, the continuous song may possibly be an honest signal of male quality for females. Further studies are required to understand the function of this novel vocalization, particularly the effects of female breeding stage on male singing activity and the female response to the songs.

Study on the flow mechanism and frequency characteristics of rales in lower respiratory tract.

The frequency characteristics of lung sounds have great significance for noninvasive diagnosis of respiratory diseases. The rales in the lower respiratory tract region that can provide rich information about symptoms of respiratory diseases are not clear. In this paper, a three-dimensional idealized bifurcated lower respiratory tract geometric model, which contains 3rd to 13th generation (G3-G13) bronchi is constructed, where , and then the large eddy simulation and volume of fluid are used to study the fluid flow characteristics. Ffowcs Williams and Hawkings model are subsequently used to study the frequency characteristics of rale of different generations of bronchi. The results showed that bronchial blockage and sputum movement will enhance the turbulence intensity and vortex shedding intensity of flow. The dominant frequency and highest value of sound pressure level (SPL) of rhonchi/moist crackles decrease with the increase of bronchial generation. The change rates of dominant frequency of rhonchi/moist crackles in adjacent generations were 5.0 ± 0.1~9.1 ± 0.2% and 3.1 ± 0.1~11.9 ± 0.3%, respectively, which is concentrated in 290~420Hz and 200~300Hz, respectively. The change rates of SPL of rhonchi/moist crackles were 8.8 ± 0.1~15.7 ± 0.1% and 7.1 ± 0.1~19.5 ± 0.2%, respectively, which is concentrated in 28~50dB and 16~32dB, respectively. In the same generation of bronchus (e.g., G8, G9) with the same degree of initial blockage, the dominant frequency and SPL of moist crackles can be 3.7 ± 0.2% and 4.5 ± 0.3% slightly higher than that of rhonchi, respectively. This research is conducive to the establishment of a rapid and accurate noninvasive diagnosis system for respiratory diseases.

An objective approach to identifying individual atrial fibrillation triggers: A simulation study

Background and Objective:Growing evidence shows that certain acute exposures, especially alcohol, may trigger episodes of paroxysmal atrial fibrillation (AF). However, there is a lack of methods for assessing the relation between triggers and AF episodes in individual patients. The present paper proposes an approach to identifying AF triggers based on the assumption that the post-trigger AF burden is larger than the pre-trigger AF burden during the analysis time interval. Method:For the purpose of identification, a measure of relational strength between pre- and post-trigger burden is introduced, accounting for the cumulative effect of the triggers contained in the observation interval. The proposed approach is explored for different types of AF episode pattern, generated using the alternating, bivariate Hawkes model, whose conditional intensity function is designed to account for the effect of alcohol. In total, 7200 different AF patterns were generated for different numbers of AF triggers and alcohol units. Results:The simulation study demonstrates that, depending on the pattern type, the relational strength increases 3–6 times with alcohol consumption in comparison with no consumption. Conclusions:The proposed approach to identifying triggers in individual patients with paroxysmal AF should facilitate the implementation of longitudinal studies for the objective assessment of trigger effect on AF occurrence.

Asymptotic results for a class of Markovian self-exciting processes

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.

Recurrent neural network based parameter estimation of Hawkes model on high-frequency financial data

This study examines the use of a recurrent neural network for estimating the parameters of a Hawkes model based on high-frequency financial data, and subsequently, for computing volatility. Neural networks have shown promising results in various fields, and interest in finance is also growing. Our approach demonstrates significantly faster computational performance compared to traditional maximum likelihood estimation methods while yielding comparable accuracy in both simulation and empirical studies. Furthermore, we demonstrate the application of this method for real-time volatility measurement, enabling the continuous estimation of financial volatility as new price data keeps coming from the market.

A single queue reactive Hawkes model for the order flow

A single queue reactive Hawkes model for the order flow

Large Deviations for Hawkes Processes with Randomized Baseline Intensity

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.

Numerical Modeling of Cavitation Rates and Noise Acoustics of Marine Propellers

The study numerically investigated the noise dissipation, cavitation, output power, and energy produced by marine propellers. A Ffowcs Williams–Hawkings (FW–H) model was used to determine the effects of three different marine propellers with three to five blades and a fixed advancing ratio. The large-eddy Simulations model best predicted the turbulent structures’ spatial and temporal variation, which would better illustrate the flow physics. It was found that a high angle of incidence between the blade’s leading edge and the water flow direction typically causes the hub vortex to cavitate. The roll-up of the cavitating tip vortex was closely related to propeller noise. The five-blade propeller was quieter under the same dynamic conditions, such as the advancing ratio, compared to three- or four-blade propellers.

Multi-kernel property in high-frequency price dynamics under Hawkes model

Abstract This study investigates and uses multi-kernel Hawkes models to describe a high-frequency mid-price process. Each kernel represents a different responsive speed of market participants. Using the conditional Hessian, we examine whether the numerical optimizer effectively finds the global maximum of the log-likelihood function under complicated modeling. Empirical studies that use stock prices in the US equity market show the existence of multi-kernels classified as ultra-high-frequency (UHF), very-high-frequency (VHF), and high-frequency (HF). We estimate the conditional expectations of arrival times and the degree of contribution to the high-frequency activities for each kernel.

A Hawkes Model Approach to Modeling Price Spikes in the Japanese Electricity Market

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.

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.