Bayesian Inference for Hawkes Processes

Rasmussen, Jakob Gulddahl

doi:10.1007/s11009-011-9272-5

Cited by 110 publications

(71 citation statements)

References 16 publications

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“…The rate of decay is controlled by the term ω. The additional ω preceding the exponential term acts as a normalisation constant so that the jump factor multiplied by the response function can be viewed as the number of offspring after an event and the density of the time interval for the increase in activity [46].…”

Section: Univariate Modelmentioning

confidence: 99%

“…Second, the set of points {t i } N i=1 should be measured from time zero. However, since the Hawkes process depends on the infinite past, this assumption is not achievable in a real world setting and it may be difficult to eradicate the influence of events outside the observation period on those inside [46]. More details on how this problem was handled for this paper will be discussed in Section 4.1.…”

Section: Univariate Modelmentioning

confidence: 99%

“…Similarly the parameter s 0 represents the jump in IED attacks following a mutual-excitation, such as, retaliation against a BSF attack whilst ν controls the temporal scale over which this mutual-excitation persists. The parameter ν also acts as a normalisation constant for the response function h so that the product of the jump factor s 0 and the response function h can be interpreted as the number of offspring events and the density for the increase in activity following a mutually exciting incident [46]. In this twodimensional system the second intensity function would model the influence of past BSF and PIRA IED attacks on BSF attacks.…”

Section: Multidimensional Modelmentioning

confidence: 99%

“…As in the one-dimensional case, all parameters must be positive [23] to make the model realistic and events in the infinite past should be considered [46]. Likewise, event times must be unique [30].…”

Section: Multidimensional Modelmentioning

confidence: 99%

“…This effect is the result of events outside the observation period influencing those inside [46]. Whereas in previous studies using the Hawkes process the data analysed is in a single time series [18,29] with the segmentation of our data according to the five phases of PIRA it is possible that events in previous phases may have influenced those in future ones.…”

Section: Introductionmentioning

confidence: 97%

See 4 more Smart Citations

Spatio-temporal patterns of IED usage by the Provisional Irish Republican Army

Tench¹,

Fry²,

Gill³

2016

Eur. J. Appl. Math

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In this paper a unique dataset of improvised explosive device (IED) attacks during "The Troubles" in Northern Ireland (NI) is analysed via a Hawkes process model. It is found that this past dependent model is a good fit to IED attacks yielding key insights about the nature of terrorism in NI. We also present a novel approach to quantitatively investigate some of the sociological theory surrounding the Provisional Irish Republican Army (PIRA) which challenges previously held assumptions concerning changes seen in the organisation. Finally we extend our use of the Hawkes process model by considering a multidimensional version which permits both self and mutual-excitations. This allows us to test how the PIRA responded to past IED attacks on different geographical scales from which we find evidence for the autonomy of the organisation over the six counties of NI and Belfast. By incorporating a second dataset concerning British Security Force (BSF) interventions, the multidimensional model allows us to test counter-terrorism (CT) operations in NI where we find subsequent increases in violence.

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Section: Univariate Modelmentioning

confidence: 99%

Section: Univariate Modelmentioning

confidence: 99%

Section: Multidimensional Modelmentioning

confidence: 99%

Section: Multidimensional Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

See 3 more Smart Citations

Spatio-temporal patterns of IED usage by the Provisional Irish Republican Army

Tench¹,

Fry²,

Gill³

2016

Eur. J. Appl. Math

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A self‐exciting marked point process model for drought analysis

Genest

Jalbert

2021

Environmetrics

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A self‐exciting marked point process approach is proposed to model clustered low‐flow events. It combines a self‐exciting ground process designed to capture the temporal clustering behavior of extreme values and an extended Generalized Pareto mark distribution for the exceedances over a subasymptotic threshold. The model takes into account the dependence between the magnitude and occurrence time of exceedances and allows for closed‐form inference on tail probabilities and large quantiles. It is used to analyze daily water levels from the Rivière des Mille Îles (Québec, Canada) and to characterize drought patterns in the Montréal area. The model is useful to generate short‐term probability forecasts and to estimate the return period of major droughts. This information on the drought events is critical to water resource professionals in planning, designing, building, and managing more efficient water resource systems to hedge against the water shortage in case of extreme droughts.

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Approximation of Bayesian Hawkes process with inlabru

2023

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Hawkes process are very popular mathematical tools for modeling phenomena exhibiting a self-exciting or self-correcting behavior. Typical examples are earthquakes occurrence, wild-fires, drought, capture-recapture, crime violence, trade exchange, and social network activity. The widespread use of Hawkes process in different fields calls for fast, reproducible, reliable, easy-to-code techniques to implement such models. We offer a technique to perform approximate Bayesian inference of Hawkes process parameters based on the use of the R-package inlabru . The inlabru R-package, in turn, relies on the INLA methodology to approximate the posterior of the parameters. Our Hawkes process approximation is based on a decomposition of the log-likelihood in three parts, which are linearly approximated separately. The linear approximation is performed with respect to the mode of the parameters' posterior distribution, which is determined with an iterative gradient-based method. The approximation of the posterior parameters is therefore deterministic, ensuring full reproducibility of the results. The proposed technique only requires the user to provide the functions to calculate the different parts of the decomposed likelihood, which are internally linearly approximated by the R-package inlabru . We provide a comparison with the bayesianETAS R-package which is based on an MCMC method. The two techniques provide similar results but our approach requires two to ten times less computational time to converge, depending on the amount of data.

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Bayesian Inference for Hawkes Processes

Cited by 110 publications

References 16 publications

Spatio-temporal patterns of IED usage by the Provisional Irish Republican Army

Spatio-temporal patterns of IED usage by the Provisional Irish Republican Army

A self‐exciting marked point process model for drought analysis

Approximation of Bayesian Hawkes process with inlabru

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