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.

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.

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 On Laslett's transform for the Boolean model

2001 ◽  
Vol 33 (1) ◽  
pp. 1-5 ◽  
Author(s):  
A. D. Barbour ◽  
V. Schmidt
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Simple Proof ◽  
Poisson Point Process ◽  
Boolean Model ◽  
Random Sets ◽  
Convex Compact ◽  
Homogeneous Poisson Process ◽  
Vertical Strip ◽  
Cumulative Process

Consider the Boolean model in ℝ2, where the germs form a homogeneous Poisson point process with intensity λ and the grains are convex compact random sets. It is known (see, e.g., Cressie (1993, Section 9.5.3)) that Laslett's rule transforms the exposed tangent points of the Boolean model into a homogeneous Poisson process with the same intensity. In the present paper, we give a simple proof of this result, which is based on a martingale argument. We also consider the cumulative process of uncovered area in a vertical strip and show that a (linear) Poisson process with intensity λ can be embedded in it.

scholarly journals Estimation of the local intensity of a cyclic Poisson process by means of nearest neighbor distances

2002 ◽  
Vol 1 (1) ◽  
pp. 1
Author(s):  
I W. MANGKU
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Poisson Point Process ◽  
Nearest Neighbor ◽  
Single Realization ◽  
Local Intensity ◽  
Strongly Consistent ◽  
Nearest Neighbor Distances

We consider the problem of estimating the local intensity of a cyclic Poisson point process, when we know the period. We suppose that only a single realization of the cyclic Poisson point process is observed within a bounded 'window', and our aim is to estimate consistently the local intensity at a given point. A nearest neighbor estimator of the local intensity is proposed, and we show that our estimator is weakly and strongly consistent, as the window expands.

Properties of the bivariate delayed Poisson process

1975 ◽  
Vol 12 (02) ◽  
pp. 257-268 ◽  
Author(s):  
A. J. Lawrance ◽  
P. A. W. Lewis
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Joint Distribution ◽  
Poisson Point Process ◽  
Initial Conditions ◽  
Arbitrary Time ◽  
Delay Distribution ◽  
Counting Distribution

The bivariate Poisson point process introduced in Cox and Lewis (1972), and there called the bivariate delayed Poisson process, is studied further; the process arises from pairs of delays on the events of a Poisson process. In particular, results are obtained for the stationary initial conditions, the joint distribution of the number of operative delays at an arbitrary time, the asynchronous counting distribution, and two semi-synchronous interval distributions. The joint delay distribution employed allows for dependence and two-sided delays, but the model retains the independence between different pairs of delays.

scholarly journals On Limit Theorem for the Number of Vertices of the Convex Hulls in a Unit Disk

2020 ◽  
pp. 275-284
Author(s):  
Isakjan M. Khamdamov
Keyword(s):  
Limit Theorem ◽  
Unit Disk ◽  
Point Process ◽  
Poisson Point Process ◽  
Convex Hulls ◽  
Strong Mixing ◽  
The Central Limit Theorem ◽  
Asymptotic Expressions ◽  
Independent Observations ◽  
Binomial Point Process

This paper is devoted to further investigation of the property of a number of vertices of convex hulls generated by independent observations of a two-dimensional random vector with regular distributions near the boundary of support when it is a unit disk. Following P. Groeneboom [4], the Binomial point process is approximated by the Poisson point process near the boundary of support and vertex processes of convex hulls are constructed. The properties of strong mixing and martingality of vertex processes are investigated. Using these properties, asymptotic expressions are obtained for the expectations and variance of the vertex processes that correspond to the results previously obtained by H. Carnal [2]. Further, using the properties of strong mixing of vertex processes, the central limit theorem for a number of vertices of a convex hull is proved

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.

Properties of the bivariate delayed Poisson process

1975 ◽  
Vol 12 (2) ◽  
pp. 257-268 ◽  
Author(s):  
A. J. Lawrance ◽  
P. A. W. Lewis
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Joint Distribution ◽  
Poisson Point Process ◽  
Initial Conditions ◽  
Arbitrary Time ◽  
Delay Distribution ◽  
Counting Distribution

The bivariate Poisson point process introduced in Cox and Lewis (1972), and there called the bivariate delayed Poisson process, is studied further; the process arises from pairs of delays on the events of a Poisson process. In particular, results are obtained for the stationary initial conditions, the joint distribution of the number of operative delays at an arbitrary time, the asynchronous counting distribution, and two semi-synchronous interval distributions. The joint delay distribution employed allows for dependence and two-sided delays, but the model retains the independence between different pairs of delays.

On the rate of Poisson process approximation to a Bernoulli process

1997 ◽  
Vol 34 (4) ◽  
pp. 898-907 ◽  
Author(s):  
Aihua Xia
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Discrete Time ◽  
Wasserstein Distance ◽  
Stein’S Method ◽  
Upper Bounds ◽  
Stein's Method ◽  
Bernoulli Process ◽  
Logarithmic Factor ◽  
Process Approximation

This note gives the rate for a Wasserstein distance between the distribution of a Bernoulli process on discrete time and that of a Poisson process, using Stein's method and Palm theory. The result here highlights the possibility that the logarithmic factor involved in the upper bounds established by Barbour and Brown (1992) and Barbour et al. (1995) may be superfluous in the true Wasserstein distance between the distributions of a point process and a Poisson process.

Convergence to equilibrium in a traffic model with restricted passing

1979 ◽  
Vol 16 (4) ◽  
pp. 881-889 ◽  
Author(s):  
Hans Dieter Unkelbach
Keyword(s):  
Poisson Process ◽  
Limit Theorems ◽  
Point Processes ◽  
Road Traffic ◽  
Poisson Processes ◽  
Traffic Model ◽  
Cluster Point ◽  
Convergence To Equilibrium ◽  
Constant Intensity

A road traffic model with restricted passing, formulated by Newell (1966), is described by conditional cluster point processes and analytically handled by generating functionals of point processes.The traffic distributions in either space or time are in equilibrium, if the fast cars form a Poisson process with constant intensity combined with Poisson-distributed queues behind the slow cars (Brill (1971)). It is shown that this state of equilibrium is stable, which means that this state will be reached asymptotically for general initial traffic distributions. Furthermore the queues behind the slow cars dissolve asymptotically like independent Poisson processes with diminishing rate, also independent of the process of non-queuing cars. To get these results limit theorems for conditional cluster point processes are formulated.

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