Point process notation: Difference between revisions

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{{Short description|Mathematical notation used in probability and statistics}}

{{ProbabilityTopics}}

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==Interpretation of point processes==

The notation, as well as the terminology, of point processes depends on their setting and interpretation as mathematical objects which under certain assumptions can be interpreted as random [[sequences]] of points, random [[Set (mathematics)|sets]] of points or random [[random counting measure]]s.<ref name="stoyan1995stochastic"/>

===Random sequences of points===

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where <math> \Phi(B)</math> is a [[random variable]] and <math> \#</math> is a [[counting measure]], which gives the number of points in some set. In this [[mathematical expression]] the point process is denoted by:

:<math> {N}</math>.

On the other hand, the symbol:

:<math> \Phi </math>

represents the number of points of <math> {N}</math> in <math> B</math>. In the context of random measures, one can write:

:<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">{{Cite book | doi = 10.1007/1-84628-150-4 | title = Theory of Random Sets | url = https://archive.org/details/probabilityitsap0000unse_i5l1 | url-access = registration | first = Ilya | last = Molchanov| series = Probability and Its Applications | year = 2005 | isbn = 978-1-85233-892-3 }}</ref>{{rp|106}} and the techniques of random measure theory offering another way to study point processes,<ref name="stoyan1995stochastic"/><ref name="grandell1977point">{{cite journal | last1 = Grandell | first1 = Jan | year = 1977 | title = Point Processes and Random Measures | journal = Advances in Applied Probability | volume = 9 | issue = 3 | pages = 502–526 | jstor = 1426111 | doi = 10.2307/1426111 | s2cid = 124650005 }}</ref> which also induces the use of the various notations used in [[Integral#Terminology and notation|integration]] and measure theory. {{efn|As discussed in Chapter 1 of Stoyan, Kendall and Mechke,<ref name="stoyan1995stochastic"/> varying [[integral]] notation in general applies to all integrals here and elsewhere.}}

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:<math> \int_{\textbf{R}^d} f(x) {N}(dx) </math>

where which puts an emphasis on the interpretation of <math> {N}</math> being a random counting measure. An alternative integration notation may be used to write this integral as:

:<math> \int_{\textbf{R}^d} f \, d{N} </math>

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{{DEFAULTSORT:Mathematical Notation}}

[[Category:Mathematical notation|*]]

[[Category:Point processes|N]]