In this paper, we present a maximum likelihood method for estimating the parameters of a univariate Hawkes process with self-excitation or inhibition. Our work generalizes techniques and results that were restricted to the self-exciting scenario. The proposed estimator is implemented for the classical exponential kernel and we show that, in the inhibition context, our procedure provides more accurate estimations than current alternative approaches.
The Hawks process is a point process with a self-exciting property. It has been used to model earthquakes, social media events, infections, etc., and is getting a lot of attention. However, as a real problem, there are often situations where we can not obtain data with sufficient observation time. In such cases, it is not appropriate to approximate the error distribution of an estimator by the normal distribution. To overcome this problem, we provide a rigorous mathematical foundation of the theory for the higher-order asymptotic behavior of the one-dimensional Hawkes process with an exponenti…
The asymptotic distribution of the score test of the null hypothesis that marks do not impact the intensity of a Hawkes marked self-exciting point process is shown to be chi-squared. For local asymptotic power, the distribution against local alternatives is also established as non-central chi-squared. These asymptotic results are derived using existing asymptotic results for likelihood estimates of the unmarked Hawkes process model together with mild additional conditions on the moments and ergodicity of the marks process and an additional uniform boundedness assumption, shown to be true for t…
Multi-dimensional Hawkes process (MHP) is a class of self and mutually exciting point processes that find wide range of applications -- from prediction of earthquakes to modelling of order books in high frequency trading. This paper makes two major contributions, we first find an unbiased estimator for the log-likelihood estimator of the Hawkes process to enable efficient use of the stochastic gradient descent method for maximum likelihood estimation. The second contribution is, we propose a specific single hidden layered neural network for the non-parametric estimation of the underlying kerne…
Georgia Institute of Technology（佐治亚理工学院）；Duke University（杜克大学）
Multivariate Hawkes processes are commonly used to model streaming networked event data in a wide variety of applications. However, it remains a challenge to extract reliable inference from complex datasets with uncertainty quantification. Aiming towards this, we develop a statistical inference framework to learn causal relationships between nodes from networked data, where the underlying directed graph implies Granger causality. We provide uncertainty quantification for the maximum likelihood estimate of the network multivariate Hawkes process by providing a non-asymptotic confidence set. The…
An extension of the Hawkes model where the productivity is variable is considered. In particular, the case is considered where each point may have its own productivity and a simple analytic formula is derived for the maximum likelihood estimators of these productivities. This estimator is compared with an empirical estimator and ways are explored of stabilizing both estimators by lower truncating, smoothing, and rescaling the estimates. Properties of the estimators are explored in simulations, and the methods are applied to seismological and epidemic datasets to show and quantify substantial v…
Conjugate Bayesian Two-step Change Point Detection for Hawkes Process
Center for Applied Statistics（应用统计中心）；School of Statistics, Renmin University of China（中国人民大学统计学院）；Innovation Platform, Renmin University of China（中国人民大学创新平台）；Beijing Advanced Innovation Center for Future Blockchain and Privacy Computing（北京未来区块链与隐私计算高精尖创新中心）
We develop a quasi-likelihood analysis procedure for a general class of multivariate marked point processes. As a by-product of the general method, we establish under stability and ergodicity conditions the local asymptotic normality of the quasi-log likelihood, along with the convergence of moments of quasi-likelihood and quasi-Bayesian estimators. To illustrate the general approach, we then turn our attention to a class of multivariate marked Hawkes processes with generalized exponential kernels, comprising among others the so-called Erlang kernels. We provide explicit conditions on the kern…
Georgia Institute of Technology（佐治亚理工学院）；Duke University（杜克大学）
We present a new CUSUM procedure for sequentially detecting change-point in the self and mutual exciting processes, a.k.a. Hawkes networks using discrete events data. Hawkes networks have become a popular model for statistics and machine learning due to their capability in modeling irregularly observed data where the timing between events carries a lot of information. The problem of detecting abrupt changes in Hawkes networks arises from various applications, including neuronal imaging, sensor network, and social network monitoring. Despite this, there has not been a computationally and memory…
Multivariate point processes are widely applied to model event-type data such as natural disasters, online message exchanges, financial transactions or neuronal spike trains. One very popular point process model in which the probability of occurrences of new events depend on the past of the process is the Hawkes process. In this work we consider the nonlinear Hawkes process, which notably models excitation and inhibition phenomena between dimensions of the process. In a nonparametric Bayesian estimation framework, we obtain concentration rates of the posterior distribution on the parameters, u…
We construct a general procedure for the Quasi Likelihood Analysis applied to a multivariate point process on the real half line in an ergodic framework. More precisely, we assume that the stochastic intensity of the underlying model belongs to a family of processes indexed by a finite dimensional parameter. When a particular family of laws of large numbers applies to those processes, we establish the consistency, the asymptotic normality and the convergence of moments of both the Quasi Maximum Likelihood estimator and the Quasi Bayesian estimator. In addition, we illustrate our main results b…
We have established a sparse estimation method for the generalized exponential marked Hawkes process by the penalized method to the ordinary method (P-O) estimator. Furthermore, we evaluated the probability of correct variable selection. In order to achieve this, we established a framework for a likelihood analysis and the P-O estimation when there might be nuisance parameters and the true value of the parameter could be realized at the boundary of the parameter space. Numerical simulations are given for several important examples.
In this paper we introduce a new model named CARMA(p,q)-Hawkes process as the Hawkes model with exponential kernel implies a strictly decreasing behaviour of the autocorrelation function and empirically evidences reject the monotonicity assumption on the autocorrelation function. The proposed model is a Hawkes process where the intensity follows a Continuous Time Autoregressive Moving Average (CARMA) process and specifically is able to reproduce more realistic dependence structures. We also study the conditions of stationarity and positivity for the intensity and the strong mixing property for…
This paper proves the consistency property for the regularized maximum likelihood estimators (MLEs) of multivariate Hawkes processes (MHPs). It also develops an alternating minimization type algorithm (AA-iPALM) to compute the MLEs with guaranteed global convergence to the set of stationary points. The performance of this AA-iPALM algorithm on both synthetic and real-world data shows that AA-iPALM consistently improves over iPALM and PALM. Moreover, AA-iPALM is able to identify the causality structure in rental listings on craigslist and herd behavior in the NASDAQ ITCH dataset.
This paper presents a parametric estimation method for ill-observed linear stationary Hawkes processes. When the exact locations of points are not observed, but only counts over time intervals of fixed size, methods based on the likelihood are not feasible. We show that spectral estimation based on Whittle's method is adapted to this case and provides consistent and asymptotically normal estimators, provided a mild moment condition on the reproduction function. Simulated datasets and a case-study illustrate the performances of the estimation, notably of the reproduction function even when time…
We introduce and show the existence of a Hawkes self-exciting point process with exponentially-decreasing kernel and where parameters are time-varying. The quantity of interest is defined as the integrated parameter $T^{-1}\int_0^Tθ_t^*dt$, where $θ_t^*$ is the time-varying parameter, and we consider the high-frequency asymptotics. To estimate it naïvely, we chop the data into several blocks, compute the maximum likelihood estimator (MLE) on each block, and take the average of the local estimates. The asymptotic bias explodes asymptotically, thus we provide a non-naïve estimator which is const…
We introduce a point process regression model that is applicable to price models and limit order book models. Hawkes type autoregression in the intensity process is generalized to a stochastic regression to covariate processes. We establish the so-called quasi likelihood analysis, which gives a polynomial type large deviation estimate for the statistical random field. We derive large sample properties of the maximum likelihood type estimator and the Bayesian type estimator when the intensity processes become large under a finite time horizon. There appears non-ergodic statistics. A classical a…
In this paper we consider point processes specified on directed linear networks, i.e. linear networks with associated directions. We adapt the so-called conditional intensity function used for specifying point processes on the time line to the setting of directed linear networks. For models specified by such a conditional intensity function, we derive an explicit expression for the likelihood function, specify two simulation algorithms (the inverse method and Ogata's modified thinning algorithm), and consider methods for model checking through the use of residuals. We also extend the results a…
Global Parameter Identification and Uncertainty Quantification of reduced-order ADM1 using Maximum Likelihood Estimation and Bayesian Inference
DBFZ Deutsches Biomasseforschungszentrum gGmbH（德国生物质研究中心）；Chemnitz University of Technology（开姆尼茨工业大学）；Technische Universität Berlin（柏林工业大学）；University of Cambridge（剑桥大学）；Leipzig University（莱比锡大学）；FH Münster University of Applied Sciences（明斯特应用科学大学）
