A cluster process representation of a self-exciting process

1974 ◽  
Vol 11 (3) ◽  
pp. 493-503 ◽  
Author(s):  
Alan G. Hawkes ◽  
David Oakes
Keyword(s):  
Point Processes ◽  
Generating Functional ◽  
Cluster Process ◽  
Process Representation ◽  
Age Dependent ◽  
Birth Processes ◽  
Poisson Cluster ◽  
Cluster Processes

It is shown that all stationary self-exciting point processes with finite intensity may be represented as Poisson cluster processes which are age-dependent immigration-birth processes, and their existence is established. This result is used to derive some counting and interval properties of these processes using the probability generating functional.

A cluster process representation of a self-exciting process

1974 ◽  
Vol 11 (03) ◽  
pp. 493-503 ◽  
Author(s):  
Alan G. Hawkes ◽  
David Oakes
Keyword(s):  
Point Processes ◽  
Generating Functional ◽  
Cluster Process ◽  
Process Representation ◽  
Age Dependent ◽  
Birth Processes ◽  
Poisson Cluster ◽  
Cluster Processes

It is shown that all stationary self-exciting point processes with finite intensity may be represented as Poisson cluster processes which are age-dependent immigration-birth processes, and their existence is established. This result is used to derive some counting and interval properties of these processes using the probability generating functional.

Count distributions, orderliness and invariance of Poisson cluster processes

1979 ◽  
Vol 16 (02) ◽  
pp. 261-273 ◽  
Author(s):  
Larry P. Ammann ◽  
Peter F. Thall
Keyword(s):  
Present Article ◽  
Generating Functional ◽  
Cluster Process ◽  
Multiple Events ◽  
Poisson Cluster Process ◽  
Finite Dimensional ◽  
The Family ◽  
Moment Measures ◽  
Poisson Cluster ◽  
Cluster Processes

The probability generating functional (p.g.fl.) of a non-homogeneous Poisson cluster process is characterized in Ammann and Thall (1977) via a decomposition of the KLM measure of the process. This p.g.fl. representation is utilized in the present article to show that the family 𝒟 of Poisson cluster processes with a.s. finite clusters is invariant under a class of cluster transformations. Explicit expressions for the finite-dimensional count distributions, product moment measures, and the distribution of clusters are derived in terms of the KLM measure. It is also shown that an element of 𝒟 has no multiple events iff the points of each cluster are a.s. distinct.

Count distributions, orderliness and invariance of Poisson cluster processes

1979 ◽  
Vol 16 (2) ◽  
pp. 261-273 ◽  
Author(s):  
Larry P. Ammann ◽  
Peter F. Thall
Keyword(s):  
Present Article ◽  
Generating Functional ◽  
Cluster Process ◽  
Multiple Events ◽  
Poisson Cluster Process ◽  
Finite Dimensional ◽  
The Family ◽  
Moment Measures ◽  
Poisson Cluster ◽  
Cluster Processes

The probability generating functional (p.g.fl.) of a non-homogeneous Poisson cluster process is characterized in Ammann and Thall (1977) via a decomposition of the KLM measure of the process. This p.g.fl. representation is utilized in the present article to show that the family 𝒟 of Poisson cluster processes with a.s. finite clusters is invariant under a class of cluster transformations. Explicit expressions for the finite-dimensional count distributions, product moment measures, and the distribution of clusters are derived in terms of the KLM measure. It is also shown that an element of 𝒟 has no multiple events iff the points of each cluster are a.s. distinct.

Note on the determination of cluster centers from a realization of a multidimensional Poisson cluster process

1983 ◽  
Vol 20 (01) ◽  
pp. 136-143
Author(s):  
Michel Baudin
Keyword(s):  
Point Processes ◽  
Linear Method ◽  
Joint Probability ◽  
Observation Window ◽  
Generating Functional ◽  
Cluster Process ◽  
Poisson Cluster Process ◽  
Actual Use ◽  
Poisson Cluster

This is the sequel to a previous paper (Baudin (1981)). The joint probability generating functional of two point processes is introduced as a tool to compute the conditional intensity of the process of cluster centers of a multidimensional Poisson cluster process when a realization is given in a bounded observation window. An explicit formula is derived but it is too complicated for actual use; a linear method for practical estimation is discussed.

Note on the determination of cluster centers from a realization of a multidimensional Poisson cluster process

1983 ◽  
Vol 20 (1) ◽  
pp. 136-143 ◽  
Author(s):  
Michel Baudin
Keyword(s):  
Point Processes ◽  
Linear Method ◽  
Joint Probability ◽  
Observation Window ◽  
Generating Functional ◽  
Cluster Process ◽  
Poisson Cluster Process ◽  
Actual Use ◽  
Poisson Cluster

This is the sequel to a previous paper (Baudin (1981)). The joint probability generating functional of two point processes is introduced as a tool to compute the conditional intensity of the process of cluster centers of a multidimensional Poisson cluster process when a realization is given in a bounded observation window. An explicit formula is derived but it is too complicated for actual use; a linear method for practical estimation is discussed.

scholarly journals Small and large scale behavior of moments of Poisson cluster processes

2017 ◽  
Vol 21 ◽  
pp. 369-393
Author(s):  
Nelson Antunes ◽  
Vladas Pipiras ◽  
Patrice Abry ◽  
Darryl Veitch
Keyword(s):  
Large Scale ◽  
Point Processes ◽  
Real Life ◽  
Real Data ◽  
Internet Traffic ◽  
Model Parameters ◽  
Scale Free ◽  
Life Phenomena ◽  
Poisson Cluster ◽  
Cluster Processes

Poisson cluster processes are special point processes that find use in modeling Internet traffic, neural spike trains, computer failure times and other real-life phenomena. The focus of this work is on the various moments and cumulants of Poisson cluster processes, and specifically on their behavior at small and large scales. Under suitable assumptions motivated by the multiscale behavior of Internet traffic, it is shown that all these various quantities satisfy scale free (scaling) relations at both small and large scales. Only some of these relations turn out to carry information about salient model parameters of interest, and consequently can be used in the inference of the scaling behavior of Poisson cluster processes. At large scales, the derived results complement those available in the literature on the distributional convergence of normalized Poisson cluster processes, and also bring forward a more practical interpretation of the so-called slow and fast growth regimes. Finally, the results are applied to a real data trace from Internet traffic.

Results in the asymptotic and equilibrium theory of Poisson cluster processes

1973 ◽  
Vol 10 (4) ◽  
pp. 807-823 ◽  
Author(s):  
M. Westcott
Keyword(s):  
Asymptotic Distribution ◽  
General Formula ◽  
Real Line ◽  
Probability Generating Function ◽  
Cluster Process ◽  
Poisson Cluster Process ◽  
Asymptotic Formulae ◽  
Equilibrium Process ◽  
Poisson Cluster ◽  
Cluster Processes

This paper contains a detailed study of the Poisson cluster process on the real line, concentrating on two aspects; first, the asymptotic distribution of the number of points in [0,t) as t→ ∞ for both transient and equilibrium cluster processes and, secondly, a general formula for the probability generating function of the equilibrium process. Asymptotic formulae for cumulants of the process are also derived. The results obtained generalize those of previous writers. The approach is analytical, in contrast to the probabilistic treatment of P. A. W. Lewis.

Results in the asymptotic and equilibrium theory of Poisson cluster processes

1973 ◽  
Vol 10 (04) ◽  
pp. 807-823 ◽  
Author(s):  
M. Westcott
Keyword(s):  
Asymptotic Distribution ◽  
General Formula ◽  
Real Line ◽  
Probability Generating Function ◽  
Cluster Process ◽  
Poisson Cluster Process ◽  
Asymptotic Formulae ◽  
Equilibrium Process ◽  
Poisson Cluster ◽  
Cluster Processes

This paper contains a detailed study of the Poisson cluster process on the real line, concentrating on two aspects; first, the asymptotic distribution of the number of points in [0,t) as t→ ∞ for both transient and equilibrium cluster processes and, secondly, a general formula for the probability generating function of the equilibrium process. Asymptotic formulae for cumulants of the process are also derived. The results obtained generalize those of previous writers. The approach is analytical, in contrast to the probabilistic treatment of P. A. W. Lewis.

An analysis of the Pólya point process

1983 ◽  
Vol 15 (01) ◽  
pp. 39-53 ◽  
Author(s):  
Ed Waymire ◽  
Vijay K. Gupta
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Critical Value ◽  
General Representation ◽  
Infinitely Divisible ◽  
Generating Functional ◽  
Representation Problem ◽  
Polya Process ◽  
Pólya Process ◽  
Stochastic Point Processes

The Pólya process is employed to illustrate certain features of the structure of infinitely divisible stochastic point processes in connection with the representation for the probability generating functional introduced by Milne and Westcott in 1972. The Pólya process is used to provide a counterexample to the result of Ammann and Thall which states that the class of stochastic point processes with the Milne and Westcott representation is the class of regular infinitely divisble point processes. So the general representation problem is still unsolved. By carrying the analysis of the Pólya process further it is possible to see the extent to which the general representation is valid. In fact it is shown in the case of the Pólya process that there is a critical value of a parameter above which the representation breaks down. This leads to a proper version of the representation in the case of regular infinitely divisible point processes.

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