A two-dimensional ‘immigration–branching' model with application to earthquake occurrence times and energies

1977 ◽  
Vol 14 (03) ◽  
pp. 464-474
Author(s):  
C. D. Lai
Keyword(s):  
Point Process ◽  
Two Dimensional ◽  
Earthquake Occurrence ◽  
Generating Functional ◽  
Cluster Centre ◽  
Branching Model ◽  
Cluster Point Process ◽  
Moment Measures ◽  
Poisson Cluster ◽  
Occurrence Times

A two-dimensional Poisson cluster point process is formulated by the use of a probability generating functional. Moment measures of both the cluster centre and member processes are discussed. An example is provided, and the magnitude frequency law is proved in this case.

A two-dimensional ‘immigration–branching' model with application to earthquake occurrence times and energies

1977 ◽  
Vol 14 (3) ◽  
pp. 464-474 ◽  
Author(s):  
C. D. Lai
Keyword(s):  
Point Process ◽  
Two Dimensional ◽  
Earthquake Occurrence ◽  
Generating Functional ◽  
Cluster Centre ◽  
Branching Model ◽  
Cluster Point Process ◽  
Moment Measures ◽  
Poisson Cluster ◽  
Occurrence Times

A two-dimensional Poisson cluster point process is formulated by the use of a probability generating functional. Moment measures of both the cluster centre and member processes are discussed. An example is provided, and the magnitude frequency law is proved in this case.

scholarly journals Markov properties of cluster processes

1996 ◽  
Vol 28 (2) ◽  
pp. 346-355 ◽  
Author(s):  
A. J. Baddeley ◽  
M. N. M. Van Lieshout ◽  
J. Møller
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Markov Property ◽  
Nearest Neighbour ◽  
Finite Range ◽  
Cluster Process ◽  
Cluster Point Process ◽  
Poisson Cluster ◽  
Markov Properties ◽  
Uniformly Bounded

We show that a Poisson cluster point process is a nearest-neighbour Markov point process [2] if the clusters have uniformly bounded diameter. It is typically not a finite-range Markov point process in the sense of Ripley and Kelly [12]. Furthermore, when the parent Poisson process is replaced by a Markov or nearest-neighbour Markov point process, the resulting cluster process is also nearest-neighbour Markov, provided all clusters are non-empty. In particular, the nearest-neighbour Markov property is preserved when points of the process are independently randomly translated, but not when they are randomly thinned.

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.

scholarly journals Markov properties of cluster processes

1996 ◽  
Vol 28 (02) ◽  
pp. 346-355 ◽  
Author(s):  
A. J. Baddeley ◽  
M. N. M. Van Lieshout ◽  
J. Møller
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Markov Property ◽  
Nearest Neighbour ◽  
Finite Range ◽  
Cluster Process ◽  
Cluster Point Process ◽  
Poisson Cluster ◽  
Markov Properties ◽  
Uniformly Bounded

We show that a Poisson cluster point process is a nearest-neighbour Markov point process [2] if the clusters have uniformly bounded diameter. It is typically not a finite-range Markov point process in the sense of Ripley and Kelly [12]. Furthermore, when the parent Poisson process is replaced by a Markov or nearest-neighbour Markov point process, the resulting cluster process is also nearest-neighbour Markov, provided all clusters are non-empty. In particular, the nearest-neighbour Markov property is preserved when points of the process are independently randomly translated, but not when they are randomly thinned.

On the generation and origin of I/f noise

1999 ◽  
Vol 4 ◽  
pp. 87-96 ◽  
Author(s):  
B. Kaulakys ◽  
T. Meškauskas
Keyword(s):  
Point Process ◽  
Solvable Model ◽  
Autoregressive Process ◽  
Wide Range ◽  
Occurrence Times

Simple analytically solvable model exhibiting 1/f spectrum in any desirably wide range of frequency is analysed. The model consists of pulses (point process) whose interevent times obey an autoregressive process with small damping. Analysis and generalizations of the model indicate to the possible origin of 1/f noise, i.e. random increments between the occurrence times of particles or pulses resulting in the clustering of the pulses.

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.

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.

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