Poisson point process: Difference between revisions

Revision as of 17:58, 12 September 2022

Revision as of 18:01, 12 September 2022

Line 149:

==Inhomogeneous Poisson point process==

[[File:Inhomogeneouspoissonprocess.svg|thumb|Graph of an inhomogeneous Poisson point process on the real line. The events are marked with black crosses, the time-dependent rate <math> \lambda(t) </math> is given by the function marked red.]]

The '''inhomogeneous''' or '''nonhomogeneous''' '''Poisson point process''' (see [[#Terminology|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>\textstyle \textbf{R}^d</math>, this is achieved by introducing a locally integrable positive function <math>\lambda\colon\mathbb{R}^d\to[0,\infty)</math>, such that for anyevery bounded region <math>\textstyle B</math> the (<math>\textstyle d</math>-dimensional) volume integral of <math>\textstyle \lambda (x)</math> over region <math>\textstyle B</math> is finite. In other words, if this integral, denoted by <math>\textstyle \Lambda (B)</math>, is:<ref name="MollerWaagepetersen2003page14"/>

:<math> \Lambda (B)=\int_B \lambda(x)\,\mathrm dx < \infty, </math>

where <math>\textstyle{\mathrm dx}</math> is a (<math>\textstyle d</math>-dimensional) volume element,{{efn|Instead of <math>\textstyle \lambda(x)</math> and <math>\textstyle{\mathrm d}x</math>, one could write, for example, in (two-dimensional) polar coordinates <math>\textstyle \lambda(r,\theta)</math> and <math display="inline"> r\,dr\,d\theta</math> , where <math>\textstyle r</math> and <math>\textstyle \theta</math> denote the radial and angular coordinates respectively, and so <math>\textstyle{\mathrm d}x</math> would be an area element in this example.}} then for anyevery collection of disjoint bounded [[Borel measurable]] sets <math>\textstyle B_1,\dots,B_k</math>, an inhomogeneous Poisson process with (intensity) function <math>\textstyle \lambda(x)</math> has the finite-dimensional distribution:<ref name="DaleyVere-Jones2007IIpage31"/>

:<math> \Pr \{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>

Line 370:

:<math> G(v)=\operatorname E \left[\prod_{x\in N} v(x) \right] </math>

where the product is performed for all the points in <math display=inline>\textstyle {N} </math>. If the intensity measure <math>\textstyle \Lambda</math> of <math>\textstyle {N}</math> is locally finite, then the <math display=inline>\textstyle G</math> is well-defined for any measurable function <math>\textstyle u</math> on <math>\textstyle \textbf{R}^d</math>. For a Poisson point process with intensity measure <math>\textstyle \Lambda</math> the generating functional is given by:

:<math> G(v)=e^{-\int_{\textbf{R}^d} [1-v(x)]\,\Lambda(\mathrm dx)}, </math>