scholarly journals Properties of convex hull generated by inhomogeneous Poisson point process

2020 ◽  
Vol 12 (3) ◽  
pp. 81-96
Author(s):  
Isakjan Mamasalievich Khamdamov (Hamdamov)
Keyword(s):  
Convex Hull ◽  
Point Process ◽  
Poisson Point Process ◽  
Inhomogeneous Poisson Point Process

Convergence in mean of some characteristics of the convex hull

1989 ◽  
Vol 21 (3) ◽  
pp. 526-542 ◽  
Author(s):  
Henk Brozius
Keyword(s):  
Convex Hull ◽  
Point Process ◽  
Regular Variation ◽  
Poisson Point Process ◽  
Random Vectors ◽  
Area And Perimeter ◽  
Convergence In Mean ◽  
Quantities Of Interest ◽  
Independent And Identically Distributed

A sequence Xn, 1 of independent and identically distributed random vectors is considered. Under a condition of regular variation, the number of vertices of the convex hull of {X1, …, Xn} converges in distribution to the number of vertices of the convex hull of a certain Poisson point process. In this paper, it is proved without sharpening the conditions that the expectation of this number also converges; expressions are found for its limit, generalizing results of Davis et al. (1987). We also present some results concerning other quantities of interest, such as area and perimeter of the convex hull and the probability that a given point belongs to the convex hull.

scholarly journals On the peeling procedure applied to a Poisson point process

2010 ◽  
Vol 42 (3) ◽  
pp. 620-630
Author(s):  
Y. Davydov ◽  
A. Nagaev ◽  
A. Philippe
Keyword(s):  
Convex Hull ◽  
Point Process ◽  
Numerical Experiments ◽  
Poisson Point Process ◽  
Point Processes ◽  
Asymptotic Properties ◽  
Convex Hulls ◽  
Limit Shape ◽  
Empirical Point ◽  
Stable Random Vectors

In this paper we focus on the asymptotic properties of the sequence of convex hulls which arise as a result of a peeling procedure applied to the convex hull generated by a Poisson point process. Processes of the considered type are tightly connected with empirical point processes and stable random vectors. Results are given about the limit shape of the convex hulls in the case of a discrete spectral measure. We give some numerical experiments to illustrate the peeling procedure for a larger class of Poisson point processes.

scholarly journals Approximate decomposition of some modulated-Poisson Voronoi tessellations

2003 ◽  
Vol 35 (4) ◽  
pp. 847-862 ◽  
Author(s):  
Bartłomiej Błaszczyszyn ◽  
René Schott
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Voronoi Tessellation ◽  
Distribution Functions ◽  
Voronoi Tessellations ◽  
Intensity Measures ◽  
Voronoi Cells ◽  
True Value ◽  
Homogeneous Models ◽  
Inhomogeneous Poisson Point Process

We consider the Voronoi tessellation of Euclidean space that is generated by an inhomogeneous Poisson point process whose intensity takes different constant values on sets of some finite partition of the space. Considering the Voronoi cells as marks associated with points of the point process, we prove that the intensity measure (mean measure) of the marked Poisson point process admits an approximate decomposition formula. The true value is approximated by a mixture of respective intensity measures for homogeneous models, while the explicit upper bound for the remainder term can be computed numerically for a large class of practical examples. By the Campbell formula, analogous approximate decompositions are deduced for the Palm distributions of individual cells. This approach makes possible the analysis of a wide class of inhomogeneous-Poisson Voronoi tessellations, by means of formulae and estimates already established for homogeneous cases. Our analysis applies also to the Poisson process modulated by an independent stationary random partition, in which case the error of the approximation of the double-stochastic-Poisson Voronoi tessellation depends on some integrated linear contact distribution functions of the boundaries of the partition elements.

scholarly journals Inhomogeneous Poisson point process nucleation: comparison of analytical solution with cellular automata simulation

2009 ◽  
Vol 12 (2) ◽  
pp. 219-224 ◽  
Author(s):  
Paulo Rangel Rios ◽  
Douglas Jardim ◽  
Weslley Luiz da Silva Assis ◽  
Tatiana Caneda Salazar ◽  
Elena Villa
Keyword(s):  
Cellular Automata ◽  
Analytical Solution ◽  
Point Process ◽  
Poisson Point Process ◽  
Inhomogeneous Poisson Point Process

scholarly journals Approximate decomposition of some modulated-Poisson Voronoi tessellations

2003 ◽  
Vol 35 (04) ◽  
pp. 847-862
Author(s):  
Bartłomiej Błaszczyszyn ◽  
René Schott
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Voronoi Tessellation ◽  
Distribution Functions ◽  
Voronoi Tessellations ◽  
Intensity Measures ◽  
Voronoi Cells ◽  
True Value ◽  
Homogeneous Models ◽  
Inhomogeneous Poisson Point Process

We consider the Voronoi tessellation of Euclidean space that is generated by an inhomogeneous Poisson point process whose intensity takes different constant values on sets of some finite partition of the space. Considering the Voronoi cells as marks associated with points of the point process, we prove that the intensity measure (mean measure) of the marked Poisson point process admits an approximate decomposition formula. The true value is approximated by a mixture of respective intensity measures for homogeneous models, while the explicit upper bound for the remainder term can be computed numerically for a large class of practical examples. By the Campbell formula, analogous approximate decompositions are deduced for the Palm distributions of individual cells. This approach makes possible the analysis of a wide class of inhomogeneous-Poisson Voronoi tessellations, by means of formulae and estimates already established for homogeneous cases. Our analysis applies also to the Poisson process modulated by an independent stationary random partition, in which case the error of the approximation of the double-stochastic-Poisson Voronoi tessellation depends on some integrated linear contact distribution functions of the boundaries of the partition elements.

scholarly journals Cardinality estimation for random stopping sets based on Poisson point processes

2021 ◽  
Author(s):  
Nicolas Privault
Keyword(s):  
Convex Hull ◽  
Numerical Simulations ◽  
Poisson Distribution ◽  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Unbiased Estimators ◽  
Stopping Sets ◽  
Stopping Set ◽  
Standard Sampling

We construct unbiased estimators for the distribution of the number of vertices inside random stopping sets based on a Poisson point process. Our approach is based on moment identities for stopping sets, showing that the random count of points inside the complement of a stopping set S has a Poisson distribution conditionally to S. The proofs do not require the use of set-indexed martingales, and our estimators have a lower variance when compared to standard sampling. Numerical simulations are presented for examples such as the convex hull and the Voronoi flower of a Poisson point process, and their complements.

scholarly journals Testing hypotheses on population dynamics: A test for the inhomogeneous Poisson point process model

2017 ◽  
Author(s):  
Niklas Hohmann
Keyword(s):  
Population Dynamics ◽  
Point Process ◽  
Poisson Point Process ◽  
Process Model ◽  
Confidence Regions ◽  
Significance Level ◽  
Spatial Dimensions ◽  
Rate Functions ◽  
Inhomogeneous Poisson Point Process

AbstractIn this paper, a test for hypotheses on population dynamics is presented alongside an implementation of said test for R. The test is based on the assumption that the sample, consisting of points on a time axis, is a realization of a Poisson point process (PPP). There are no restrictions on the shapes of the rate functions that are regulating the PPP, type 2 errors can be calculated and the test is optimal in the sense that it is a uniform most powerful (UMP) test. So for every significance level a, the presented test has a lower type 2 error than every other test having the same significance level a. The test is applicable to all models based on PPPs, including models in spatial dimensions. It can be generalized and expanded in different ways, such as testing larger hypotheses, incorporating prior knowledge, and constructing confidence regions that can be used to obtain upper or lower bounds on rate functions.

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