Content deleted Content added
m WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (10901) |
m Bot: Deprecating Template:Cite doi and some minor fixes |
||
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">{{
==Interpretation of point processes==
Line 11:
===Random sequences of points===
In some mathematical frameworks, a given point process may be considered as a sequence of points with each point randomly positioned in ''d''-dimensional [[Euclidean space]] '''R'''<sup>''d''</sup><ref name="stoyan1995stochastic"/> as well as some other more abstract [[mathematical space]]s. In general, whether or not a random sequence is equivalent to the other interpretations of a point process depends on the underlying mathematical space, but this holds true for the setting of finite-dimensional Euclidean space '''R'''<sup>''d''</sup>.<ref name="daleyPPII2008">{{
===Random set of points===
A point process is called ''simple'' if no two (or more points) coincide in ___location with [[Almost surely|probability one]]. Given that often point processes are simple and the order of the points does not matter, a collection of random points can be considered as a random set of points<ref name="stoyan1995stochastic"/><ref name="baddeley2007spatial">{{
A point process is often denoted by a single letter,<ref name="stoyan1995stochastic"/><ref name="kingman1992poisson">[[J. F. C. Kingman]]. ''Poisson processes'', volume 3. Oxford university press, 1992.</ref><ref name="moller2003statistical">{{
:<math> x\in {N}, </math>
Line 45:
<math> \Phi(B)=n</math>
to denote that there is the set <math> B</math> that contains <math> n</math> points of <math> {N}</math>. In other words, a point process can be considered as a [[random measure]] that assigns some non-negative integer-valued [[Measure (mathematics)|measure]] to sets.<ref name="stoyan1995stochastic"/> This interpretation has motivated a point process being considered just another name for a ''random counting measure''<ref name="molvcanov2005theory">{{
==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">{{
* <math> {N}</math> denotes a set of random points.
|