Inhomogeneous Poisson Point Process
A Inhomogeneous Poisson Point Process is a Poisson point process that is an inhomogeneous point process.
- AKA: Non-Homogeneous Poisson Process.
- Example(s):
- …
- Counter-Example(s):
- See: Borel Measurable, Poisson Point Process Terminology, Monotonic Function, Little-o Notation.
References
2016
- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Poisson_point_process#Inhomogeneous_Poisson_point_process Retrieved:2016-7-18.
- The inhomogeneous or nonhomogeneous Poisson point process (see Terminology) is a Poisson point process with a Poisson parameter set as some location-dependent function in the underlying space on which the Poisson process is defined. For Euclidean space [math]\displaystyle{ \textstyle R^d }[/math], this is achieved by introducing a locally integrable positive function [math]\displaystyle{ \textstyle \lambda(x) }[/math] , where [math]\displaystyle{ \textstyle x }[/math] is a [math]\displaystyle{ \textstyle d }[/math] -dimensional point located in [math]\displaystyle{ \textstyle R^d }[/math] , such that for any bounded region [math]\displaystyle{ \textstyle B }[/math] the ([math]\displaystyle{ \textstyle d }[/math] -dimensional) volume integral of [math]\displaystyle{ \textstyle \lambda (x) }[/math] over region [math]\displaystyle{ \textstyle B }[/math] is finite. In other words, if this integral, denoted by [math]\displaystyle{ \textstyle \Lambda (B) }[/math] , is: :[math]\displaystyle{ \Lambda (B)=\int_B \lambda(x) dx \lt \infty, }[/math]
where [math]\displaystyle{ \textstyle dx }[/math] is a ([math]\displaystyle{ \textstyle d }[/math] -dimensional) volume element,then for any collection of disjoint bounded Borel measurable sets [math]\displaystyle{ \textstyle B_1,\dots,B_k }[/math] , an inhomogeneous Poisson process with (intensity) function [math]\displaystyle{ \textstyle \lambda(x) }[/math] has the finite-dimensional distribution: : [math]\displaystyle{ P\{N(B_i)=n_i, i=1, \dots, k\}=\prod_{i=1}^k\frac{(\Lambda(B_i))^{n_i}}{n_i!}e^{-\Lambda(B_i)}. }[/math] Furthermore, [math]\displaystyle{ \textstyle \Lambda (B) }[/math] has the interpretation of being the expected number of points of the Poisson process located in the bounded region [math]\displaystyle{ \textstyle B }[/math] , namely : [math]\displaystyle{ \Lambda (B)= E[N(B)] . }[/math]
- The inhomogeneous or nonhomogeneous Poisson point process (see Terminology) is a Poisson point process with a Poisson parameter set as some location-dependent function in the underlying space on which the Poisson process is defined. For Euclidean space [math]\displaystyle{ \textstyle R^d }[/math], this is achieved by introducing a locally integrable positive function [math]\displaystyle{ \textstyle \lambda(x) }[/math] , where [math]\displaystyle{ \textstyle x }[/math] is a [math]\displaystyle{ \textstyle d }[/math] -dimensional point located in [math]\displaystyle{ \textstyle R^d }[/math] , such that for any bounded region [math]\displaystyle{ \textstyle B }[/math] the ([math]\displaystyle{ \textstyle d }[/math] -dimensional) volume integral of [math]\displaystyle{ \textstyle \lambda (x) }[/math] over region [math]\displaystyle{ \textstyle B }[/math] is finite. In other words, if this integral, denoted by [math]\displaystyle{ \textstyle \Lambda (B) }[/math] , is: :[math]\displaystyle{ \Lambda (B)=\int_B \lambda(x) dx \lt \infty, }[/math]