Content deleted Content added
No edit summary |
|||
Line 3:
In [[probability]] and [[statistics]], '''point process notation''' comprises the range of [[mathematical notation]] used to symbolically represent [[random]] [[Mathematical object|objects]] known as [[point process]]es, which are used in related fields such as [[stochastic geometry]], [[spatial statistics]] and [[continuum percolation theory]] and frequently serve as [[mathematical models]] of random phenomena, representable as points, in time, space or both.
The notation varies due to the histories of certain mathematical fields and the different interpretations of point processes,<ref name="stoyan1995stochastic">D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. ''Stochastic geometry and its applications'', volume 2. Wiley Chichester, 1995.</ref><ref name="daleyPPI2003">{{Cite journal | doi = 10.1007/b97277 | first1 = D. J. | last1 = Daley | first2 = D. | last2 = Vere-Jones| title = An Introduction to the Theory of Point Processes | series = Probability and its Applications | year = 2003 | isbn = 0-387-95541-0 | pmid = | pmc = }}</ref> and borrows notation from mathematical areas of study such as [[measure theory]] and [[set theory]].<ref name="stoyan1995stochastic"/> <ref name="haenggi2012stochastic">M. Haenggi. ''Stochastic geometry for wireless networks''. Cambridge University Press, 2012.</ref>
==Interpretation of point processes==
Line 49:
==Dual notation==
The different interpretations of point processes as random sets and counting measures is captured with the often used notation <ref name="stoyan1995stochastic"/><ref name="moller2003statistical"/><ref name="BB1">{{Cite journal | last1 = Baccelli | first1 = F. O. | title = Stochastic Geometry and Wireless Networks: Volume I Theory | doi = 10.1561/1300000006 | journal = Foundations and Trends® in Networking | volume = 3 | issue = 3–4 | pages = 249–449 | year = 2009 | pmid = | pmc = }}</ref><ref name="haenggi2012stochastic"/> in which:
* <math> {N}</math> denotes a set of random points.
Line 55:
* <math> {N}(B)</math> denotes a random variable that gives the number of points of <math> {N}</math> in <math> B</math> (hence it is a random counting measure).
Denoting the counting measure again with <math> \#</math>, this dual notation implies:<ref name="haenggi2012stochastic"/>
:<math> {N}(B) =\#(B \cap {N}). </math>
|