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 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.

Contact and chord length distribution of a stationary Voronoi tessellation

1998 ◽  
Vol 30 (03) ◽  
pp. 603-618 ◽  
Author(s):  
Lothar Heinrich
Keyword(s):  
Point Process ◽  
Voronoi Tessellation ◽  
Pair Correlation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Closed Form Expression ◽  
Voronoi Tessellations ◽  
Chord Length Distribution ◽  
Contact Distribution

We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝ d in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed form expression for the mean chord length in terms of the two-point Palm distribution and the pair correlation function of the generating point process. The results obtained are specified for Voronoi tessellations generated by Poisson cluster and Gibbsian processes, respectively.

Contact and chord length distribution of a stationary Voronoi tessellation

1998 ◽  
Vol 30 (3) ◽  
pp. 603-618 ◽  
Author(s):  
Lothar Heinrich
Keyword(s):  
Point Process ◽  
Voronoi Tessellation ◽  
Pair Correlation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Closed Form Expression ◽  
Voronoi Tessellations ◽  
Chord Length Distribution ◽  
Contact Distribution

We give formulae for different types of contact distribution functions for stationary (not necessarily Poisson) Voronoi tessellations in ℝd in terms of the Palm void probabilities of the generating point process. Moreover, using the well-known relationship between the linear contact distribution and the chord length distribution we derive a closed form expression for the mean chord length in terms of the two-point Palm distribution and the pair correlation function of the generating point process. The results obtained are specified for Voronoi tessellations generated by Poisson cluster and Gibbsian processes, respectively.

Bayesian analysis of deformed tessellation models

2003 ◽  
Vol 35 (1) ◽  
pp. 4-26 ◽  
Author(s):  
Paul G. Blackwell ◽  
Jesper Møller
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Cross Section ◽  
Poisson Point Process ◽  
Voronoi Tessellation ◽  
Variable Dimension ◽  
Fixed Dimension ◽  
The Individual ◽  
Inhomogeneous Poisson Point Process ◽  
Number Of Cells

We define a class of tessellation models based on perturbing or deforming standard tessellations such as the Voronoi tessellation. We show how distributions over this class of ‘deformed’ tessellations can be used to define prior distributions for models based on tessellations, and how inference for such models can be carried out using Markov chain Monte Carlo methods; stability properties of the algorithms are investigated. Our approach applies not only to fixed dimension problems, but also to variable dimension problems, in which the number of cells in the tessellation is unknown. We illustrate our methods with two real examples. The first relates to reconstructing animal territories, represented by the individual cells of a tessellation, from observation of an inhomogeneous Poisson point process. The second example involves the analysis of an image of a cross-section through a sample of metal, with the tessellation modelling the micro-crystalline structure of the metal.

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 Set Reconstruction by Voronoi Cells

2012 ◽  
Vol 44 (04) ◽  
pp. 938-953 ◽  
Author(s):  
M. Reitzner ◽  
E. Spodarev ◽  
D. Zaporozhets
Keyword(s):  
Lebesgue Measure ◽  
Point Process ◽  
Finite Volume ◽  
Poisson Point Process ◽  
Borel Set ◽  
Voronoi Cells

For a Borel set A and a homogeneous Poisson point process η in of intensity λ>0, define the Poisson–Voronoi approximation A η of A as a union of all Voronoi cells with nuclei from η lying in A. If A has a finite volume and perimeter, we find an exact asymptotic of E Vol(AΔ A η) as λ→∞, where Vol is the Lebesgue measure. Estimates for all moments of Vol(A η) and Vol(AΔ A η) together with their asymptotics for large λ are obtained as well.

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.

Bayesian analysis of deformed tessellation models

2003 ◽  
Vol 35 (01) ◽  
pp. 4-26 ◽  
Author(s):  
Paul G. Blackwell ◽  
Jesper Møller
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Cross Section ◽  
Poisson Point Process ◽  
Voronoi Tessellation ◽  
Variable Dimension ◽  
Fixed Dimension ◽  
The Individual ◽  
Inhomogeneous Poisson Point Process ◽  
Number Of Cells

We define a class of tessellation models based on perturbing or deforming standard tessellations such as the Voronoi tessellation. We show how distributions over this class of ‘deformed’ tessellations can be used to define prior distributions for models based on tessellations, and how inference for such models can be carried out using Markov chain Monte Carlo methods; stability properties of the algorithms are investigated. Our approach applies not only to fixed dimension problems, but also to variable dimension problems, in which the number of cells in the tessellation is unknown. We illustrate our methods with two real examples. The first relates to reconstructing animal territories, represented by the individual cells of a tessellation, from observation of an inhomogeneous Poisson point process. The second example involves the analysis of an image of a cross-section through a sample of metal, with the tessellation modelling the micro-crystalline structure of the metal.

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