Poisson point process: Difference between revisions

Revision as of 14:51, 3 October 2024

Revision as of 19:25, 16 October 2024

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For the inhomogeneous case, a couple of different methods can be used depending on the nature of the intensity function <math>\textstyle \lambda(x)</math>.<ref name="ChiuStoyan2013page53to55"/> If the intensity function is sufficiently simple, then independent and random non-uniform (Cartesian or other) coordinates of the points can be generated. For example, simulating a Poisson point process on a circular window can be done for an isotropic intensity function (in polar coordinates <math>\textstyle r</math> and <math>\textstyle \theta</math>), implying it is rotationally variant or independent of <math>\textstyle \theta</math> but dependent on <math>\textstyle r</math>, by a change of variable in <math>\textstyle r</math> if the intensity function is sufficiently simple.<ref name="ChiuStoyan2013page53to55"/>

For more complicated intensity functions, one can use an [[Rejection sampling|acceptance-rejection method]], which consists of using (or 'accepting') only certain random points and not using (or 'rejecting') the other points, based on the ratio:.<ref name="Streit2010page14">{{cite book|author=Roy L. Streit|title=Poisson Point Processes: Imaging, Tracking, and Sensing|url=https://books.google.com/books?id=KAWmFYUJ5zsC|date=15 September 2010|publisher=Springer Science & Business Media|isbn=978-1-4419-6923-1|pages=14–16}}</ref>.

:<math> \frac{\lambda(x_i)}{\Lambda(W)}=\frac{\lambda(x_i)}{\int_W\lambda(x)\,\mathrm dx. } </math>

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| publisher = [[R (programming language)#CRAN|Comprehensive R Archive Network]]

| doi = 10.18637/jss.v078.i10

| doi-access = free| arxiv = 1612.01907

| s2cid = 14379617

| url = https://cran.r-project.org/web/packages/KFAS/vignettes/KFAS.pdf