Multivariate Hawkes Process Research Articles

Statistical Inference for Networks of High-Dimensional Point Processes

Fueled 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.

Multivariate Hawkes process allowing for common shocks

Although the Hawkes process has been widely applied, their probability properties are difficult to obtain, depending on the model structure. This paper proposes a multivariate Hawkes process, which allows for common jumps from each marginal processes. The probability of this common jump is determined by another independent process, which represents the arrival intensity of external shocks to the system. The infinitesimal generator of the new multivariate jump process is derived. Based on that, moments and the Laplace transform are studied, which further demonstrate the advantages of this model structure.

Using the multivariate Hawkes process to study interactions between multiple species from camera trap data.

Interspecific interactions can influence species' activity and movement patterns. In particular, species may avoid or attract each other through reactive responses in space and/or time. However, data and methods to study such reactive interactions have remained scarce and were generally limited to two interacting species. At this time, the deployment of camera traps opens new opportunities but adapted statistical techniques are still required to analyze interaction patterns with such data. We present the multivariate Hawkes process (MHP) and show how it can be used to analyze interactions between several species using camera trap data. Hawkes processes use flexible pairwise interaction functions, allowing us to consider asymmetries and variations over time when depicting reactive temporal interactions. After describing the theoretical foundations of the MHP, we outline how its framework can be used to study interspecific interactions with camera trap data. We design a simulation study to evaluate the performance of the MHP and of another existing method to infer interactions from camera trap-like data. We also use the MHP to infer reactive interactions from real camera trap data for five species from South African savannas (impala Aepyceros melampus, greater kudu Tragelaphus strepsiceros, lion Panthera leo, blue wildebeest Connochaetes taurinus and Burchell's zebra Equus quagga burchelli). The simulation study shows that the MHP can be used as a tool to benchmark other methods of interspecific interaction inference and that this model can reliably infer interactions when enough data are considered. The analysis of real data highlights evidence of predator avoidance by prey and herbivore-herbivore attraction. Lastly, we present the advantages and limits of the MHP and discuss how it can be improved to infer attraction/avoidance patterns more reliably. As camera traps are increasingly used, the multivariate Hawkes process provides a promising framework to decipher the complexity of interactions structuring ecological communities.

Graph‐based mutually exciting point processes for modelling event times in docked bike‐sharing systems

AbstractThis paper introduces graph‐based mutually exciting processes (GB‐MEP) to model event times in network point processes, focusing on an application to docked bike‐sharing systems. GB‐MEP incorporates known relationships between nodes in a graph within the intensity function of a node‐based multivariate Hawkes process. This approach reduces the number of parameters to a quantity proportional to the number of nodes in the network, resulting in significant advantages for computational scalability when compared with traditional methods. The model is applied on event data observed on the Santander Cycles network in central London, demonstrating that exploiting network‐wide information related to geographical location of the stations is beneficial to improve the performance of node‐based models for applications in bike‐sharing systems. The proposed GB‐MEP framework is more generally applicable to any network point process where a distance function between nodes is available, demonstrating wider applicability.

Online score statistics for detecting clustered change in network point processes

We consider online monitoring of the network event data to detect local changes in a cluster when the affected data stream distribution shifts from one point process to another with different parameters. Specifically, we are interested in detecting a change point that causes a shift of the underlying data distribution that follows a multivariate Hawkes process with exponential decay temporal kernel, whereby the Hawkes process is considered to account for spatiotemporal correlation between observations. The proposed detection procedure is based on scan score statistics. We derive the asymptotic distribution of the statistic, which enables the self-normalizing property and facilitates the approximation of the instantaneous false alarm probability and the average run length. When detecting a change in the Hawkes process with nonvanishing self-excitation, the procedure does not require estimating the postchange network parameter while assuming the temporal decay parameter, which enjoys computational efficiency. We further present an efficient procedure to accurately determine the false discovery rate via importance sampling, as validated by numerical examples. Using simulated and real stock exchange data, we show the effectiveness of the proposed method in detecting change while enjoying computational efficiency.

On the splitting and aggregating of Hawkes processes

AbstractWe consider the random splitting and aggregating of Hawkes processes. We present the random splitting schemes using the direct approach for counting processes, as well as the immigration–birth branching representations of Hawkes processes. From the second scheme, it is shown that random split Hawkes processes are again Hawkes. We discuss functional central limit theorems (FCLTs) for the scaled split processes from the different schemes. On the other hand, aggregating multivariate Hawkes processes may not necessarily be Hawkes. We identify a necessary and sufficient condition for the aggregated process to be Hawkes. We prove an FCLT for a multivariate Hawkes process under a random splitting and then aggregating scheme (under certain conditions, transforming into a Hawkes process of a different dimension).

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

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

AntiConcealer: Reliable Detection of Adversary Concealed Behaviors in EdgeAI-Assisted IoT

Internet of Things (IoT) is one of the rapidly developing technologies today that attract huge real-world applications. However, the reality is that IoT is easily vulnerable to numerous types of cyberattacks and anomalies. Detecting them is becoming increasingly challenging day by day due to limitations with IoT devices and threat intelligence. Particularly, one of the most challenging problems is to detect the existence of malicious adversaries that continuously adapt or conceal their behaviors in IoT to hide their actions and to make the IoT security protocol ineffective. In this article, we study this problem at the IoT device level that can be a great idea to avoid potential attacks. We present <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AntiConcealer</monospace> , an edge-aided IoT framework, and propose an edge artificial intelligence-enabled approach (EdgeAI) for detecting adversary concealed behaviors in the IoT. We first develop an adversary behavior model and use this to identify mid-attack temporal patterns by learning the multivariate Hawkes process (MHP), a kind of point process as a random and finite series of events (e.g., behaviors) controlled by a probabilistic model. Naturally, learning MHP processed on EdgeAI reveals the influence of the concealed behaviors of adversaries in the IoT. These concealed behaviors are then grouped using a nonnegative weighted influence matrix. To observe the performance of the <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AntiConcealer</monospace> framework through evaluation, we employ honeypots integrated with edge servers and verify the usability and reliability of adversary behavioral identification.

Sequential Recommendation Based on Multivariate Hawkes Process Embedding With Attention.

Recommender systems are important approaches for dealing with the information overload problem in the big data era, and various kinds of auxiliary information, including time and sequential information, can help improve the performance of retrieval and recommendation tasks. However, it is still a challenging problem how to fully exploit such information to achieve high-quality recommendation results and improve users' experience. In this work, we present a novel sequential recommendation model, called multivariate Hawkes process embedding with attention (MHPE-a), which combines a temporal point process with the attention mechanism to predict the items that the target user may interact with according to her/his historical records. Specifically, the proposed approach MHPE-a can model users' sequential patterns in their temporal interaction sequences accurately with a multivariate Hawkes process. Then, we perform an accurate sequential recommendation to satisfy target users' real-time requirements based on their preferences obtained with MHPE-a from their historical records. Especially, an attention mechanism is used to leverage users' long/short-term preferences adaptively to achieve an accurate sequential recommendation. Extensive experiments are conducted on two real-world datasets (lastfm and gowalla), and the results show that MHPE-a achieves better performance than state-of-the-art baselines.

Multivariate Hawkes processes on inhomogeneous random graphs

We consider a population of N interacting neurons, represented by a multivariate Hawkes process: The firing rate of each neuron depends on the history of the connected neurons. Contrary to the mean-field framework where the interaction occurs on the complete graph, the connectivity between particles is given by a random possibly diluted and inhomogeneous graph where the probability of presence of each edge depends on the spatial position of its vertices. We address the well-posedness of this system and Law of Large Numbers results as N→∞. A crucial issue will be to understand how spatial inhomogeneity influences the large time behavior of the system.

A Metric Space for Point Process Excitations

A multivariate Hawkes process enables self- and cross-excitations through a triggering matrix that behaves like an asymmetrical covariance structure, characterizing pairwise interactions between the event types. Full-rank estimation of all interactions is often infeasible in empirical settings. Models that specialize on a spatiotemporal application alleviate this obstacle by exploiting spatial locality, allowing the dyadic relationships between events to depend only on separation in time and relative distances in real Euclidean space. Here we generalize this framework to any multivariate Hawkes process, and harness it as a vessel for embedding arbitrary event types in a hidden metric space. Specifically, we propose a Hidden Hawkes Geometry (HHG) model to uncover the hidden geometry between event excitations in a multivariate point process. The low dimensionality of the embedding regularizes the structure of the inferred interactions. We develop a number of estimators and validate the model by conducting several experiments. In particular, we investigate regional infectivity dynamics of COVID-19 in an early South Korean record and recent Los Angeles confirmed cases. By additionally performing synthetic experiments on short records as well as explorations into options markets and the Ebola epidemic, we demonstrate that learning the embedding alongside a point process uncovers salient interactions in a broad range of applications.

Mean-Variance Portfolio Selection in Contagious Markets

We consider a mean-variance portfolio selection problem in a financial market with contagion risk. The risky assets follow a jump-diffusion model, in which jumps are driven by a multivariate Hawkes process with mutual-excitation effect. The mutual-excitation feature of the Hawkes process captures the contagion risk in the sense that each price jump of an asset increases the likelihood of future jumps not only in the same asset but also in other assets. We apply the stochastic maximum principle, backward stochastic differential equation theory, and linear-quadratic control technique to solve the problem and obtain the efficient strategy and efficient frontier in semi-closed form, subject to a non-local partial differential equation. Numerical examples are provided to illustrate our results.

Neuronal network inference and membrane potential model using multivariate Hawkes processes

BackgroundIn this work, we propose to catch the complexity of the membrane potential’s dynamic of a motoneuron between its spikes, taking into account the spikes from other neurons around. Our approach relies on two types of data: extracellular recordings of multiple spikes trains and intracellular recordings of the membrane potential of a central neuron. New methodWe provide a unified framework and a complete pipeline to analyze neuronal activity from data extraction to statistical inference. To the best of our knowledge, this is the first time that a Hawkes-diffusion model is investigated on such complex data. The first step of the proposed procedure is to select a subnetwork of neurons impacting the central neuron using a multivariate Hawkes process. Then we infer a jump-diffusion dynamic in which jumps are driven from a Hawkes process, the occurrences of which correspond to the spike trains of the aforementioned subset of neurons that interact with the central neuron. ResultsFrom the Hawkes estimation step we recover a small connectivity graph which contains the central neuron, and we show that taking into account this information improves the inference of membrane potential through the proposed jump-diffusion model. A goodness of fit test is applied to validate the relevance of the Hawkes model in such context. Comparison with existing methodsWe compare an empirical inference method and two sparse estimation procedures based on the Hawkes assumption for the reconstruction of the connectivity graph using the spike-trains. Then, the Hawkes-diffusion model is competed with the simple diffusion in terms of best fit to describe the behavior of the membrane potential of a central neuron surrounded by a network. ConclusionsThe present method takes advantage of both spike trains and membrane potential to understand the behavior of a fixed neuron. The entire code has been developed and is freely available on GitHub.

Analysis of Individual High-Frequency Traders' Buy-Sell Order Strategy Based on Multivariate Hawkes Process.

Traders who instantly react to changes in the financial market and place orders in milliseconds are called high-frequency traders (HFTs). HFTs have recently become more prevalent and attracting attention in the study of market microstructures. In this study, we used data to track the order history of individual HFTs in the USD/JPY forex market to reveal how individual HFTs interact with the order book and what strategies they use to place their limit orders. Specifically, we introduced an 8-dimensional multivariate Hawkes process that included the excitations due to the occurrence of limit orders, cancel orders, and executions in the order book change, and performed maximum likelihood estimations of the limit order processes for 134 HFTs. As a result, we found that the limit order generation processes of 104 of the 134 HFTs were modeled by a multivariate Hawkes process. In this analysis of the EBS market, the HFTs whose strategies were modeled by the Hawkes process were categorized into three groups according to their excitation mechanisms: (1) those excited by executions; (2) those that were excited by the occurrences or cancellations of limit orders; and (3) those that were excited by their own orders.

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.

Corrigendum to: Multivariate Hawkes process analysis of posts on a web service about uncomfortable gender experiences

Corrigendum to: Multivariate Hawkes process analysis of posts on a web service about uncomfortable gender experiences

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

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

Multivariate Hawkes process analysis of posts on a web service about uncomfortable gender experiences

Abstract We propose a method for the modelling of the propagation of text data in web site space among some groups by using a multivariate Hawkes process with a sparse structure. For estimation, we introduced a hybrid method using a quasi-maximum likelihood estimator (QMLE) and a $L^1$-penalized QMLE. As a real example, we investigated posts on a Japanese web service about uncomfortable gender experiences, which we classified into 12 groups by age and sex, and we calculated the magnitude of the correlation between each group. In addition, we visualized the propagation structure of posts in each group by summarizing the results in a directed Hawkes graph and a heat map of time integrals of kernel functions.

Energy ETF return jump contagion: a multivariate Hawkes process approach

Compared with investing in individual stocks, ETF investment is capable of diversifying the non-systematic risk or exposure to broad market or industry sectors. The aim of this paper is to develop a jump contagion modeling framework to understand the contagion effect of market jump events of energy sector ETFs using multivariate Hawkes process modeling approach. Through analyzing intraday high-frequency market data, we find that negative index jumps lead index price discovery processes, and their influences disappear faster than the positive index jumps in both the S&P500 and the crude oil futures. And on average, the self contagion in negative jumps is stronger than the self contagion in the positive jumps across all ETF groups. However, the ETFs focused on the master limited partnership (MLP) segment show less negative self contagion and relatively stronger positive self contagion than the other energy ETFs. Overall, the influence of negative jumps on ETFs from both the equity index and the energy future index is stronger than that of the positive jumps. And the influence of the equity index (S&P500) jump on ETFs lasts longer than that of the crude oil futures index (CLC1).

Multivariate Hawkes process model of market participants behavior in the high frequency world

We propose a multivariate Hawkes process to model the interaction between the non-high frequency traders (NHFTs) behavior (Buy and sell) and high frequency traders (HFTs) behavior (Buy and sell). We apply our model to the intraday transaction data of the public sector banks stock in India, which is sampled from March 2012 to June 2012. We find that the mutually-exciting NHFT and HFT behaviors benefit the stocks, which have better average return above the average return of the public sector bank index. We further identify the granger causality relationship for mutually exciting dominating stocks that HFTs activities cause the activities of NHFTs. In other words, NHFTs are market followers in those stocks.