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.

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.

A new graph related to the directions of nearest neighbours in a point process

2003 ◽  
Vol 35 (1) ◽  
pp. 47-55 ◽  
Author(s):  
S. N. Chiu ◽  
I. S. Molchanov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Point Process ◽  
Process Simulation ◽  
Point Processes ◽  
Nearest Neighbour ◽  
Simulation Studies ◽  
Nearest Neighbours ◽  
Typical Point

This paper introduces a new graph constructed from a point process. The idea is to connect a point with its nearest neighbour, then to the second nearest and continue this process until the point belongs to the interior of the convex hull of these nearest neighbours. The number of such neighbours is called the degree of a point. We derive the distribution of the degree of the typical point in a Poisson process, prove a central limit theorem for the sum of degrees, and propose an edge-corrected estimator of the distribution of the degree that is unbiased for a stationary Poisson process. Simulation studies show that this degree is a useful concept that allows the separation of clustering and repulsive behaviour of point processes.

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.

scholarly journals Prediction in a Poisson cluster model

2010 ◽  
Vol 47 (02) ◽  
pp. 350-366 ◽  
Author(s):  
Muneya Matsui ◽  
Thomas Mikosch
Keyword(s):  
Poisson Process ◽  
Cluster Model ◽  
Arrival Time ◽  
Finite Interval ◽  
Compound Poisson Process ◽  
Time Interval ◽  
Future Time ◽  
Homogeneous Poisson Process ◽  
Cluster Process ◽  
Poisson Cluster

We consider a Poisson cluster model, motivated by insurance applications. At each claim arrival time, modeled by the point of a homogeneous Poisson process, we start a cluster process which represents the number or amount of payments triggered by the arrival of a claim in a portfolio. The cluster process is a Lévy or truncated compound Poisson process. Given the observations of the process over a finite interval, we consider the expected value of the number and amount of payments in a future time interval. We also give bounds for the error encountered in this prediction procedure.

scholarly journals Prediction in a Poisson cluster model

2010 ◽  
Vol 47 (2) ◽  
pp. 350-366 ◽  
Author(s):  
Muneya Matsui ◽  
Thomas Mikosch
Keyword(s):  
Poisson Process ◽  
Cluster Model ◽  
Arrival Time ◽  
Finite Interval ◽  
Compound Poisson Process ◽  
Time Interval ◽  
Future Time ◽  
Homogeneous Poisson Process ◽  
Cluster Process ◽  
Poisson Cluster

We consider a Poisson cluster model, motivated by insurance applications. At each claim arrival time, modeled by the point of a homogeneous Poisson process, we start a cluster process which represents the number or amount of payments triggered by the arrival of a claim in a portfolio. The cluster process is a Lévy or truncated compound Poisson process. Given the observations of the process over a finite interval, we consider the expected value of the number and amount of payments in a future time interval. We also give bounds for the error encountered in this prediction procedure.

Brownian Survival in a Clusterized Trapping Medium

1998 ◽  
Vol 10 (02) ◽  
pp. 147-189 ◽  
Author(s):  
Sergio Albeverio ◽  
Leonid V. Bogachev
Keyword(s):  
Brownian Particle ◽  
Cluster Process ◽  
Poisson Cluster Process ◽  
Cluster Point Process ◽  
Long Time ◽  
Fkg Inequalities ◽  
Poisson Cluster ◽  
Random Traps ◽  
Physical Literature ◽  
Trapping Process

The survival problem for a Brownian particle moving among random traps is considered in the case where the traps are correlated in a particular fashion being gathered in clusters. It is assumed that the clusters are statistically identical and independent of each other and are distributed in space according to a Poisson law. Mathematically, such a trapping medium is described via a Poisson cluster point process. We prove that the particle survival probability is increased at all times as compared to the case of noncorrelated (Poissonian) traps, which implies the slowdown of the trapping process. It is shown that this effect may be interpreted as the manifestation of the trap "attraction", thus supporting assertions on the qualitative influence of the trap "interaction" on the trapping rate claimed earlier in the physical literature. The long-time survival asymptotics (of Donsker–Varadhan type) is also derived. By way of appendix, FKG inequalities for certain functionals are proven and the limiting distribution for a Poisson cluster process, under clusters' scaling, is determined.

Focusing of the scan statistic and geometric clique number

2002 ◽  
Vol 34 (4) ◽  
pp. 739-753 ◽  
Author(s):  
Mathew D. Penrose
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Poisson Point Process ◽  
Dimensional Space ◽  
Clique Number ◽  
Geometric Graph ◽  
Finite Range ◽  
Scan Statistic ◽  
Constant Intensity ◽  
Binomial Point Process

Given sets C and R in d-dimensional space, take a constant intensity Poisson point process on R; the associated scan statistic S is the maximum number of Poisson points in any translate of C. As R becomes large with C fixed, bounded and open but otherwise arbitrary, the distribution of S becomes concentrated on at most two adjacent integers. A similar result holds when the underlying Poisson process is replaced by a binomial point process, and these results can be extended to a large class of nonuniform distributions. Also, similar results hold for other finite-range scanning schemes such as the clique number of a geometric graph.

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.

Focusing of the scan statistic and geometric clique number

2002 ◽  
Vol 34 (04) ◽  
pp. 739-753 ◽  
Author(s):  
Mathew D. Penrose
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Poisson Point Process ◽  
Dimensional Space ◽  
Clique Number ◽  
Geometric Graph ◽  
Finite Range ◽  
Scan Statistic ◽  
Constant Intensity ◽  
Binomial Point Process

Given sets C and R in d-dimensional space, take a constant intensity Poisson point process on R; the associated scan statistic S is the maximum number of Poisson points in any translate of C. As R becomes large with C fixed, bounded and open but otherwise arbitrary, the distribution of S becomes concentrated on at most two adjacent integers. A similar result holds when the underlying Poisson process is replaced by a binomial point process, and these results can be extended to a large class of nonuniform distributions. Also, similar results hold for other finite-range scanning schemes such as the clique number of a geometric graph.

A new graph related to the directions of nearest neighbours in a point process

2003 ◽  
Vol 35 (01) ◽  
pp. 47-55
Author(s):  
S. N. Chiu ◽  
I. S. Molchanov
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Point Process ◽  
Process Simulation ◽  
Point Processes ◽  
Nearest Neighbour ◽  
Simulation Studies ◽  
Nearest Neighbours ◽  
Typical Point

This paper introduces a new graph constructed from a point process. The idea is to connect a point with its nearest neighbour, then to the second nearest and continue this process until the point belongs to the interior of the convex hull of these nearest neighbours. The number of such neighbours is called the degree of a point. We derive the distribution of the degree of the typical point in a Poisson process, prove a central limit theorem for the sum of degrees, and propose an edge-corrected estimator of the distribution of the degree that is unbiased for a stationary Poisson process. Simulation studies show that this degree is a useful concept that allows the separation of clustering and repulsive behaviour of point processes.

Sign in / Sign up

Export Citation Format

Share Document