Point process operation

The thinning operation entails using some predefined rule to remove points from a point process ${\displaystyle \textstyle {N}}$ to form a new point process ${\displaystyle \textstyle {N}_{p}}$. These thinning rules may be deterministic, that is, not random, which is the case for one of the simplest rules known as ${\displaystyle \textstyle p}$-thinning:[1] each point of ${\displaystyle \textstyle {N}}$ is independently removed (or kept) with some probability ${\displaystyle \textstyle p}$ (or ${\displaystyle \textstyle 1-p}$). This rule may be generalized by introducing a non-negative function ${\displaystyle \textstyle p(x)\leq 1}$ in order to define the located-dependent ${\displaystyle \textstyle p(x)}$-thinning where now the probability of a point being removed is ${\displaystyle \textstyle p(x)}$ and is dependent on where the point of ${\displaystyle \textstyle {N}}$ is located on the underlying space. A further generalization is to have the thinning probability ${\displaystyle \textstyle p}$ random itself.

These three operations are all types of independent thinning, which means the interaction between points has no effect on the where a point is removed (or kept). Another generalization involves dependent thinning where points of the point process are removed (or kept) depending on their location in relation to other points of the point process. Thinning can be used to create new point processes such as hard-core processes where points do not exist (due to thinning) within a certain radius of each point in the thinned point process.[1]

The superposition operation is used to combine two or more point processes together onto one underlying mathematical space or state space. If there is a countable set or collection of point processes ${\displaystyle \textstyle {N}_{1},{N}_{2}\dots }$ with mean measures ${\displaystyle \textstyle \Lambda _{1},\Lambda _{2},\dots }$, then their superposition

${\displaystyle {N}=\bigcup _{i=1}^{\infty }{N}_{i},}$

also forms a point process. 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.

In the case where each ${\displaystyle \textstyle {N}_{i}}$ is a Poisson point process, then the resulting process ${\displaystyle \textstyle {N}}$ is also a Poisson point process with mean intensity

${\displaystyle \Lambda =\sum \limits _{i=1}^{\infty }\Lambda _{i}.}$

The point operation known as clustering entails replacing every point ${\displaystyle \textstyle x}$ in a given point process ${\displaystyle \textstyle {N}}$ with a cluster of points ${\displaystyle \textstyle N^{x}}$. Each cluster is also a point process, but with a finite number of points. The union of all the clusters forms a cluster point process

${\displaystyle {N}_{c}=\bigcup _{x\in {N}}N^{x}.}$

Often is it assumed that the clusters ${\displaystyle \textstyle N^{x}}$ are all sets of finite points with each set being independent and identically distributed. Furthermore, if the original point process ${\displaystyle \textstyle {N}}$ has a constant intensity ${\displaystyle \textstyle \lambda }$, then the intensity of the cluster point process ${\displaystyle \textstyle {N}_{c}}$ will be

${\displaystyle \lambda _{c}=c\lambda ,}$

where the constant ${\displaystyle \textstyle c}$ is the mean of number of points in each ${\displaystyle \textstyle N^{x}}$.

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.[5]

The result known as the Displacement theorem[2] effectively says that the random independent displacement of points of a Poisson point process (on the same underlying space) forms another Poisson point process.

Another property that is considered useful is the ability to map a point process from one underlying space to another space. For example, a point process defined on the plane R² can be transformed from Cartesian coordinates to polar coordinates.[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.