scholarly journals On point processes defined by angular conditions on Delaunay neighbors in the Poisson–Voronoi Tessellation

2021 ◽  
Vol 58 (4) ◽  
pp. 952-965
Author(s):  
François Baccelli ◽  
Sanket S. Kalamkar
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Voronoi Tessellation ◽  
Three Dimensions ◽  
Main Question ◽  
One Dimensional ◽  
Straight Line ◽  
Intersection Points ◽  
Set Of Points ◽  
Stationary Point Processes

AbstractConsider a homogeneous Poisson point process of the Euclidean plane and its Voronoi tessellation. The present note discusses the properties of two stationary point processes associated with the latter and depending on a parameter $\theta$ . The first is the set of points that belong to some one-dimensional facet of the Voronoi tessellation and such that the angle with which they see the two nuclei defining the facet is $\theta$ . The main question of interest on this first point process is its intensity. The second point process is that of the intersections of the said tessellation with a straight line having a random orientation. Its intensity is well known. The intersection points almost surely belong to one-dimensional facets. The main question here concerns the Palm distribution of the angle with which the points of this second point process see the two nuclei associated with the facet. We will give answers to these two questions and briefly discuss their practical motivations. We also discuss natural extensions to three dimensions.

On projected thick sections of stationary point processes and Boolean models

1996 ◽  
Vol 28 (2) ◽  
pp. 335-335
Author(s):  
Markus Kiderlen
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Linear Subspace ◽  
Isotropic Case ◽  
Thick Section ◽  
Boolean Models ◽  
Dimensional Linear Subspace ◽  
Stationary Point Process ◽  
Stationary Point Processes

For a stationary point process X of convex particles in ℝd the projected thick section process X(L) on a q-dimensional linear subspace L is considered. Formulae connecting geometric functionals, e.g. the quermass densities of X and X(L), are presented. They generalize the classical results of Miles (1976) and Davy (1976) which hold only in the isotropic case.

Bivariate stationary point processes, fundamental relations and first recurrence times

1972 ◽  
Vol 4 (02) ◽  
pp. 296-317 ◽  
Author(s):  
T. K. M. Wisniewski
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Counting Process ◽  
Derived Relations ◽  
Recurrence Times ◽  
Different Types ◽  
First Recurrence ◽  
Stationary Point Processes ◽  
Event Counting

Various types of time and event sampling of a stationary and orderly bivariate point process are considered. Fundamental relations between inter-event intervals and the event counting process are derived. Relations between first forward recurrence times and their moments for different types of sampling are obtained.

Some multivariate generalizations of results in univariate stationary point processes

1977 ◽  
Vol 14 (04) ◽  
pp. 748-757 ◽  
Author(s):  
Mark Berman
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Stationary Point Process ◽  
Stationary Point Processes

Some relationships are derived between the asynchronous and partially synchronous counting and interval processes associated with a multivariate stationary point process. A few examples are given to illustrate some of these relationships.

On the Recovery of 3D Spatial Statistics of Particles from 1D Measurements: Implications for Airborne Instruments

2014 ◽  
Vol 31 (10) ◽  
pp. 2078-2087 ◽  
Author(s):  
Michael L. Larsen ◽  
Clarissa A. Briner ◽  
Philip Boehner
Keyword(s):  
Point Processes ◽  
Pair Correlation ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional System ◽  
Cloud Droplets ◽  
Sampling Variability ◽  
One Dimensional ◽  
Natural Sampling ◽  
The One

Abstract The spatial positions of individual aerosol particles, cloud droplets, or raindrops can be modeled as a point processes in three dimensions. Characterization of three-dimensional point processes often involves the calculation or estimation of the radial distribution function (RDF) and/or the pair-correlation function (PCF) for the system. Sampling these three-dimensional systems is often impractical, however, and, consequently, these three-dimensional systems are directly measured by probing the system along a one-dimensional transect through the volume (e.g., an aircraft-mounted cloud probe measuring a thin horizontal “skewer” through a cloud). The measured RDF and PCF of these one-dimensional transects are related to (but not, in general, equal to) the RDF/PCF of the intrinsic three-dimensional systems from which the sample was taken. Previous work examined the formal mathematical relationship between the statistics of the intrinsic three-dimensional system and the one-dimensional transect; this study extends the previous work within the context of realistic sampling variability. Natural sampling variability is found to constrain substantially the usefulness of applying previous theoretical relationships. Implications for future sampling strategies are discussed.

Bivariate stationary point processes, fundamental relations and first recurrence times

1972 ◽  
Vol 4 (2) ◽  
pp. 296-317 ◽  
Author(s):  
T. K. M. Wisniewski
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Counting Process ◽  
Derived Relations ◽  
Recurrence Times ◽  
Different Types ◽  
First Recurrence ◽  
Stationary Point Processes ◽  
Event Counting

Various types of time and event sampling of a stationary and orderly bivariate point process are considered. Fundamental relations between inter-event intervals and the event counting process are derived. Relations between first forward recurrence times and their moments for different types of sampling are obtained.

Extreme values of independent stochastic processes

1977 ◽  
Vol 14 (4) ◽  
pp. 732-739 ◽  
Author(s):  
Bruce M. Brown ◽  
Sidney I. Resnick
Keyword(s):  
Stochastic Processes ◽  
Point Process ◽  
Time Scales ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extreme Values ◽  
Weak Limit ◽  
One Dimensional ◽  
Limit Process ◽  
Related Point

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.

θ-stationary point processes and their second-order analysis

1981 ◽  
Vol 18 (04) ◽  
pp. 864-878
Author(s):  
Karen Byth
Keyword(s):  
Point Process ◽  
Spatial Patterns ◽  
Stationary Point ◽  
Point Processes ◽  
Simulated Data ◽  
Second Order ◽  
Stationary Processes ◽  
Order Analysis ◽  
Stationary Point Processes ◽  
Point Of Reference

The concept of θ-stationarity for a simple second-order point process in R2 is introduced. This concept is closely related to that of isotropy. Some θ-stationary processes are defined. Techniques are given for simulating realisations of these processes. The second-order analysis of these processes which have an obvious point of reference or origin is considered. Methods are suggested for modelling spatial patterns which are realisations of such processes. These methods are illustrated using simulated data. The ideas are extended to multitype point processes.

θ-stationary point processes and their second-order analysis

1981 ◽  
Vol 18 (4) ◽  
pp. 864-878 ◽  
Author(s):  
Karen Byth
Keyword(s):  
Point Process ◽  
Spatial Patterns ◽  
Stationary Point ◽  
Point Processes ◽  
Simulated Data ◽  
Second Order ◽  
Stationary Processes ◽  
Order Analysis ◽  
Stationary Point Processes ◽  
Point Of Reference

The concept of θ-stationarity for a simple second-order point process in R2 is introduced. This concept is closely related to that of isotropy. Some θ-stationary processes are defined. Techniques are given for simulating realisations of these processes. The second-order analysis of these processes which have an obvious point of reference or origin is considered. Methods are suggested for modelling spatial patterns which are realisations of such processes. These methods are illustrated using simulated data. The ideas are extended to multitype point processes.

scholarly journals Quermass-interaction processes: conditions for stability

1999 ◽  
Vol 31 (02) ◽  
pp. 315-342 ◽  
Author(s):  
W. S. Kendall ◽  
M. N. M. van Lieshout ◽  
A. J. Baddeley
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Convex Set ◽  
Compact Convex ◽  
Main Question ◽  
Interaction Point ◽  
Interaction Processes ◽  
Area Functional ◽  
Compact Convex Set

We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let Ξ be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear combination of quermass functionals of Ξ. If only the area functional is used, then the area-interaction point process is recovered. New point processes arise if we include the perimeter length functional, or the Euler functional (number of components minus number of holes). The main question addressed by the paper is that of when the resulting point process is well-defined: geometric arguments are used to establish conditions for the point process to be stable in the sense of Ruelle.

On projected thick sections of stationary point processes and Boolean models

1996 ◽  
Vol 28 (02) ◽  
pp. 335
Author(s):  
Markus Kiderlen
Keyword(s):  
Point Process ◽  
Stationary Point ◽  
Point Processes ◽  
Linear Subspace ◽  
Isotropic Case ◽  
Thick Section ◽  
Boolean Models ◽  
Dimensional Linear Subspace ◽  
Stationary Point Process ◽  
Stationary Point Processes

For a stationary point process X of convex particles in ℝ d the projected thick section process X(L) on a q-dimensional linear subspace L is considered. Formulae connecting geometric functionals, e.g. the quermass densities of X and X(L), are presented. They generalize the classical results of Miles (1976) and Davy (1976) which hold only in the isotropic case.

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