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For example, given a homogeneous Poisson point process on the real line, the probability of finding a single point of the process in a small interval of width is approximately . In fact, such intuition is how the Poisson point process is sometimes introduced and its distribution derived.
[119] [121] The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and Lévy processes. [49] The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the ...
It describes how a Poisson point process is altered under measurable transformations. This allows construction of more complex Poisson point processes out of homogeneous Poisson point processes and can, for example, be used to simulate these more complex Poisson point processes in a similar manner to inverse transform sampling.
The simplest and most ubiquitous example of a point process is the Poisson point process, which is a spatial generalisation of the Poisson process. A Poisson (counting) process on the line can be characterised by two properties : the number of points (or events) in disjoint intervals are independent and have a Poisson distribution. A Poisson ...
Complete spatial randomness (CSR) describes a point process whereby point events occur within a given study area in a completely random fashion. It is synonymous with a homogeneous spatial Poisson process. [1] Such a process is modeled using only one parameter , i.e. the density of points within the defined area. The term complete spatial ...
Consider a collection of points {x i} in the plane ℝ 2 that form a homogeneous Poisson process Φ with constant (point) density λ.For each point of the Poisson process (i.e. x i ∈ Φ), place a disk D i with its center located at the point x i.
These properties and the definition of the homogeneous Poisson process extend to the case of the inhomogeneous (or non-homogeneous) Poisson process, which is a non-stationary stochastic process with a location-dependent density λ(x) where x is a point (usually in the plane, R 2). For more information, see the articles on the Poisson process.
Pages in category "Poisson point processes" The following 17 pages are in this category, out of 17 total. ... Homogeneous Poisson point process; M. Mapping theorem ...