scholarly journals Population viewpoint on Hawkes processes

2016 ◽  
Vol 48 (2) ◽  
pp. 463-480 ◽  
Author(s):  
Alexandre Boumezoued
Keyword(s):  
Shot Noise ◽  
Noise Intensity ◽  
Counting Processes ◽  
Hawkes Process ◽  
Process Definition ◽  
Hawkes Processes ◽  
Fertility Function ◽  
Age Pyramid ◽  
Intensity Process ◽  
New Distribution

AbstractIn this paper we focus on a class of linear Hawkes processes with general immigrants. These are counting processes with shot-noise intensity, including self-excited and externally excited patterns. For such processes, we introduce the concept of the age pyramid which evolves according to immigration and births. The virtue of this approach that combines an intensity process definition and a branching representation is that the population age pyramid keeps track of all past events. This is used to compute new distribution properties for a class of Hawkes processes with general immigrants which generalize the popular exponential fertility function. The pathwise construction of the Hawkes process and its underlying population is also given.

scholarly journals Multivariate Hawkes processes: an application to financial data

2011 ◽  
Vol 48 (A) ◽  
pp. 367-378 ◽  
Author(s):  
Paul Embrechts ◽  
Thomas Liniger ◽  
Lu Lin
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Statistical Estimation ◽  
Goodness Of Fit ◽  
Likelihood Estimation ◽  
Financial Data ◽  
Data Sets ◽  
Hawkes Process ◽  
Science And Engineering ◽  
Hawkes Processes

A Hawkes process is also known under the name of a self-exciting point process and has numerous applications throughout science and engineering. We derive the statistical estimation (maximum likelihood estimation) and goodness-of-fit (mainly graphical) for multivariate Hawkes processes with possibly dependent marks. As an application, we analyze two data sets from finance.

A review on Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes

2020 ◽  
pp. 1-22
Author(s):  
Jiwook Jang ◽  
Rosy Oh
Keyword(s):  
Poisson Process ◽  
Shot Noise ◽  
Value At Risk ◽  
Hawkes Process ◽  
Cox Process ◽  
Doubly Stochastic ◽  
Business Applications ◽  
Tail Conditional Expectation ◽  
Time Period ◽  
Contagion Process

Abstract The Poisson process is an essential building block to move up to complicated counting processes, such as the Cox (“doubly stochastic Poisson”) process, the Hawkes (“self-exciting”) process, exponentially decaying shot-noise Poisson (simply “shot-noise Poisson”) process and the dynamic contagion process. The Cox process provides flexibility by letting the intensity not only depending on time but also allowing it to be a stochastic process. The Hawkes process has self-exciting property and clustering effects. Shot-noise Poisson process is an extension of the Poisson process, where it is capable of displaying the frequency, magnitude and time period needed to determine the effect of points. The dynamic contagion process is a point process, where its intensity generalises the Hawkes process and Cox process with exponentially decaying shot-noise intensity. To facilitate the usage of these processes in practice, we revisit the distributional properties of the Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes. We provide simulation algorithms for these processes, which would be useful to statistical analysis, further business applications and research. As an application of the compound processes, numerical comparisons of value-at-risk and tail conditional expectation are made.

scholarly journals Hawkes process modeling of COVID-19 with mobility leading indicators and spatial covariates

2020 ◽  
Author(s):  
Wen-Hao Chiang ◽  
Xueying Liu ◽  
George Mohler
Keyword(s):  
Reproduction Number ◽  
Leading Indicators ◽  
Hawkes Process ◽  
Conditional Intensity ◽  
Short Term ◽  
Hawkes Processes ◽  
Event Clustering ◽  
Short Term Forecasting ◽  
Demographic Covariates ◽  
The U.S

AbstractHawkes processes are used in machine learning for event clustering and causal inference, while they also can be viewed as stochastic versions of popular compartmental models used in epidemiology. Here we show how to develop accurate models of COVID-19 transmission using Hawkes processes with spatial-temporal covariates. We model the conditional intensity of new COVID-19 cases and deaths in the U.S. at the county level, estimating the dynamic reproduction number of the virus within an EM algorithm through a regression on Google mobility indices and demographic covariates in the maximization step. We validate the approach on short-term forecasting tasks, showing that the Hawkes process outperforms several benchmark models currently used to track the pandemic, including an ensemble approach and a SEIR-variant. We also investigate which covariates and mobility indices are most important for building forecasts of COVID-19 in the U.S.

Rate of convergence to equilibrium of marked Hawkes processes

2002 ◽  
Vol 39 (1) ◽  
pp. 123-136 ◽  
Author(s):  
P. Brémaud ◽  
G. Nappo ◽  
G. L. Torrisi
Keyword(s):  
Rate Of Convergence ◽  
Stationary Process ◽  
Rates Of Convergence ◽  
Stopping Rules ◽  
Convergence To Equilibrium ◽  
Situation Models ◽  
Hawkes Process ◽  
Hawkes Processes ◽  
Simulation Algorithms

In this article we obtain rates of convergence to equilibrium of marked Hawkes processes in two situations. Firstly, the stationary process is the empty process, in which case we speak of the rate of extinction. Secondly, the stationary process is the unique stationary and nontrivial marked Hawkes process, in which case we speak of the rate of installation. The first situation models small epidemics, whereas the results in the second case are useful in deriving stopping rules for simulation algorithms of Hawkes processes with random marks.

scholarly journals Noise Intensity-Intensity Correlations and the Fourth Cumulant of Photo-assisted Shot Noise

2013 ◽  
Vol 3 (1) ◽  
Author(s):  
Jean-Charles Forgues ◽  
Fatou Bintou Sane ◽  
Simon Blanchard ◽  
Lafe Spietz ◽  
Christian Lupien ◽  
...  
Keyword(s):  
Shot Noise ◽  
Noise Intensity

scholarly journals Efficient Non-parametric Bayesian Hawkes Processes

2019 ◽  
Author(s):  
Rui Zhang ◽  
Christian Walder ◽  
Marian-Andrei Rizoiu ◽  
Lexing Xie
Keyword(s):  
Time Complexity ◽  
Large Scale ◽  
Goodness Of Fit ◽  
Linear Time ◽  
Synthetic Data ◽  
Hawkes Process ◽  
Hawkes Processes ◽  
Branching Structures ◽  
Current State ◽  
Non Parametric

In this paper, we develop an efficient non-parametric Bayesian estimation of the kernel function of Hawkes processes. The non-parametric Bayesian approach is important because it provides flexible Hawkes kernels and quantifies their uncertainty. Our method is based on the cluster representation of Hawkes processes. Utilizing the stationarity of the Hawkes process, we efficiently sample random branching structures and thus, we split the Hawkes process into clusters of Poisson processes. We derive two algorithms --- a block Gibbs sampler and a maximum a posteriori estimator based on expectation maximization --- and we show that our methods have a linear time complexity, both theoretically and empirically. On synthetic data, we show our methods to be able to infer flexible Hawkes triggering kernels. On two large-scale Twitter diffusion datasets, we show that our methods outperform the current state-of-the-art in goodness-of-fit and that the time complexity is linear in the size of the dataset. We also observe that on diffusions related to online videos, the learned kernels reflect the perceived longevity for different content types such as music or pets videos.

Rate of convergence to equilibrium of marked Hawkes processes

2002 ◽  
Vol 39 (01) ◽  
pp. 123-136 ◽  
Author(s):  
P. Brémaud ◽  
G. Nappo ◽  
G. L. Torrisi
Keyword(s):  
Rate Of Convergence ◽  
Stationary Process ◽  
Rates Of Convergence ◽  
Stopping Rules ◽  
Convergence To Equilibrium ◽  
Situation Models ◽  
Hawkes Process ◽  
Hawkes Processes ◽  
Simulation Algorithms

In this article we obtain rates of convergence to equilibrium of marked Hawkes processes in two situations. Firstly, the stationary process is the empty process, in which case we speak of the rate of extinction. Secondly, the stationary process is the unique stationary and nontrivial marked Hawkes process, in which case we speak of the rate of installation. The first situation models small epidemics, whereas the results in the second case are useful in deriving stopping rules for simulation algorithms of Hawkes processes with random marks.

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