This chapter contains sections titled: Introduction The binomial point process The homogeneous Poisson point process Simulation of a homogeneous Poisson process Model characteristics Estimating the intensity Testing complete spatial randomness

Cited by 8

2. Point processes I - The Poisson point process

Dietrich Stoyan, Wilfrid S. Kendal, Joseph Mecke · 1987

Cited by 0

Notes on the Poisson point process

H. Paul Keeler · 2015

This work is licensed under a “CC BY-SA 3.0 ” license. The Poisson process is a type of random object known as a point process that has been the focus of much study and application. This survey aims to give an accessible but detailed account of the Poisson point process by covering its his-tory, mathematical definitions in a number of settings, and key properties as well detailing various terminology and applications of the process, with recommen-dations for further reading. 1

Cited by 0

Minimum Divergence for Poisson Point Process

Shinto Eguchi · SpringerBriefs in statistics · 2025

Cited by 0

Navigation on a Poisson point process

Bordenave, Charles · arXiv (Cornell University) · 2006

On a locally finite point set, a navigation defines a path through the point set from one point to another. The set of paths leading to a given point defines a tree known as the navigation tree. In this article, we analyze the properties of the navigation tree when the point set is a Poisson point process on $\mathbb{R}^d$. We examine the local weak convergence of the navigation tree, the asymptotic average of a functional along a path, the shape of the navigation tree and its topological ends. We illustrate our work in the small-world graphs where new results are established.

Cited by 0Open access

Another quasi-Poisson plane point process

P. A. P. Moran · Probability Theory and Related Fields · 1976

Cited by 4Open access

The two-parameter Poisson–Dirichlet point process

Kenji Handa · Bernoulli · 2009

The two-parameter Poisson–Dirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (that is, the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correlation functions. Using this, we apply the theory of point processes to reveal the mathematical structure of the two-parameter Poisson–Dirichlet distribution. Also, developing the Laplace transform approach due to Pitman and Yor, we are able to extend several results previously known for the one-parameter case. The Markov–Krein identity for the generalized Dirichlet process is discussed from the point of view of functional analysis based on the two-parameter Poisson–Dirichlet distribution.

Cited by 34Open access

The favorite point of a Poisson process

Davar Khoshnevisan, Thomas M. Lewis · Stochastic Processes and their Applications · 1995

Cited by 7

Point processes derived from the Poisson process

Eva Kielkowská · Digital Repository (National Repository of Grey Literature) · 2014

Finite point processes having a density with respect to the Poisson process are investigated. We are interested in special functionals called U-statistics. The mean value of such a functional is obtained as a product of mean values of functionals of the Poisson point process. Using difference operators and kernels of an U-statistic, we can derive a general formula for the mean value of the functional, in which the conditional intensity of a point process is used. The calculation of mean values of selected U-statistics of the Strauss process shows the application of the formula. The assumptions of the formula are verified via characteristics of the Poisson process. At the end, the results of numerical calculations of mean values based on simulations of the Strauss process are presented.

Cited by 0

On the Conway-Maxwell-Poisson point process

Ian Flint, Yan Wang, Aihua Xia · Communication in Statistics- Theory and Methods · 2023

The Poisson point process plays a pivotal role in modeling spatial point patterns. One of its key features is that the variance and the mean of the total number of points in a given region are equal, making it unsuitable for modeling point patterns that exhibit significantly different mean and variance. To tackle such point patterns, we introduce the class of Conway-Maxwell-Poisson point processes. Our model can easily be fitted with a logistic regression, its point counts in different regions are correlated and its log-likelihood in any subregion can be easily extracted. Both simulations and real data analyses have been carried out to demonstrate the performance of the proposed model.

Cited by 0Open access

On the homogeneous planar Poisson point process

R. E. Miles · Mathematical Biosciences · 1970

Cited by 283

Characterizing a random point process by a switched poisson process

Rossiter, Mark H. · Monash University · 2021

This thesis was scanned from the print manuscript for digital preservation and is copyright the author. Researchers can access this thesis by asking their local university, institution or public library to make a request on their behalf. Monash staff and postgraduate students can use the link in the References field.

Cited by 0Open access

Characterizing a random point process by a switched poisson process

Mark H. Rossiter · Figshare · 2021

Cited by 5Open access

Poisson point process modeling for polyphonic music transcription

Paul Peeling, Chung-fai Li, Simon Godsill · The Journal of the Acoustical Society of America · 2007

Peaks detected in the frequency domain spectrum of a musical chord are modeled as realizations of a nonhomogeneous Poisson point process. When several notes are superimposed to make a chord, the processes for individual notes combine to give another Poisson process, whose likelihood is easily computable. This avoids a data association step linking individual harmonics explicitly with detected peaks in the spectrum. The likelihood function is ideal for Bayesian inference about the unknown note frequencies in a chord. Here, maximum likelihood estimation of fundamental frequencies shows very promising performance on real polyphonic piano music recordings.

Cited by 14