In probability and statistics, a point process operation or point process transformation is a type of mathematical operation performed on a random object known as a point process, which are often used as mathematical models of phenomena that can be represented as points randomly located in space.
These operations can be purely random, deterministic or both, and are used to construct new point processes, which can be then also used as mathematical models.
[2][1] Point process operations have been studied in the mathematical limit as the number of random point process operations applied approaches infinity.
This had led to convergence theorems of point process operations, which have their origins in the pioneering work of Conny Palm in 1940s and later Aleksandr Khinchin in the 1950s and 1960s who both studied point processes on the real line, in the context of studying the arrival of phone calls and queueing theory in general.
In other words, in the limit as the number of operations applied approaches infinity, the point process will converge in distribution (or weakly) to a Poisson point process or, if its measure is a random measure, to a Cox point process.
[4] Convergence results, such as the Palm-Khinchin theorem for renewal processes, are then also used to justify the use of the Poisson point process as a mathematical of various phenomena.
Point processes are mathematical objects that can be used to represent collections of points randomly scattered on some underlying mathematical space.
They have a number of interpretations, which is reflected by the various types of point process notation.
belongs to or is a member of a point process, denoted by
, then this can be written as:[1] and represents the point process as a random set.
is often written as:[1][6][7] which reflects a random measure interpretation for point processes.
A point process needs to be defined on an underlying mathematical space.
, although point processes can be defined on more abstract mathematical spaces.
[4] To develop suitable models with point processes in stochastic geometry, spatial statistics and related fields, there are number of useful transformations that can be performed on point processes including: thinning, superposition, mapping (or transformation of space), clustering, and random displacement.
This rule may be generalized by introducing a non-negative function
If there is a countable set or collection of point processes
In this expression the superposition operation is denoted by a set union), which implies the random set interpretation of point processes; see Point process notation for more information.
A mathematical model may require randomly moving points of a point process from some locations to other locations on the underlying mathematical space.
[2] This point process operation is referred to as random displacement[2] or translation.
[4] If each point in the process is displaced or translated independently to other all other points in the process, then the operation forms an independent displacement or translation.
[4] It is usually assume that all the random translations have a common probability distribution; hence the displacements form a set of independent and identically distributed random vectors in the underlying mathematical space.
Applying random displacements or translations to point processes may be used as mathematical models for mobility of objects in, for example, ecology[2] or wireless networks.
[2] Provided that the mapping (or transformation) adheres to some conditions, then a result sometimes known as the Mapping theorem[2] says that if the original process is a Poisson point process with some intensity measure, then the resulting mapped (or transformed) collection of points also forms a Poisson point process with another intensity measure.
In the theory of point processes, results have been derived to study the behaviour of the resulting point process, via convergence results, in the limit as the number of performed operations approaches infinity.
Similar convergence results have been developed for the operations of thinning and superposition (with suitable rescaling of the underlying space).