User:Wireless erlang/sandbox

In statistics and probability theory, a Poisson point process (or often a Poisson process or PPP) is a type of stochastic process known as a point process that consists of randomly located points on some underlying mathematical space. The process has convenient mathematical properties{kingman1992poisson}, which has led to it being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy{babu1996spatial}, biology{othmer1988models}, ecology{thompson1955spatial}, geology{connor1995three}, physics{scargle1998studies}, image processing{bertero2009image}, and telecommunications{BB2,Haenggi2009}.

The Poisson point process is often defined on the real line playing an important role in the field of queueing theory{kleinrock1975theory,kleinrock1976queueing}where it is used to model random occurrences or events happening in time including.... INCLUDE WIKI. In the plane, the point process, also known as a &rsquo;&rsquo;&rsquo;spatial Poisson process&rsquo;&rsquo;&rsquo;{baddeley2007spatial}, may represent seemingly scattered objects such as users in a wireless network{andrews2010primer,BB1,BB2,haenggi2012stochastic}, particles colliding a detector, or trees in a forest{stoyan1995stochastic}. In this setting, the process is often used as a model and serves as a cornerstone in the related fields of spatial point processes{daleyPPI2003,baddeley2007spatial}, stochastic geometry{stoyan1995stochastic}, spatial statistics{moller2003statistical,baddeley2007spatial}and continuum percolation theory{meester1996continuum}. In more abstract spaces, the Poisson point process serves as a subject of mathematical study in its own right and has a fundamental connection with random measures{kingman1992poisson}.

Despite its wide use as a stochastic model of phenomena representable as points, the inherent nature of the process implies it does not adequately describe phenomena in which there is significant interaction between the points. This has led to the sometimes overuse of the point process in mathematical models{kingman1992poisson,stoyan1995stochastic,andrews2010primer}, and has inspired other point processes, some of which are constructed via the Poisson point process, that seek to capture this interaction{stoyan1995stochastic}.

The process is named after French mathematician Siméon Denis Poisson owing to the fact that if a collection of random points in some space form a Poisson process, then the number points, loosely speaking, in a region of finite size is directly related to the Poisson distribution{daleyPPI2003}.

The process is sometimes considered the simplest of the point process{cox1980point}for it is defined with a single parameter. If this parameter is a constant, then the resulting process called a &rsquo;&rsquo;&rsquo;homogeneous&rsquo;&rsquo;&rsquo;{kingman1992poisson}or &rsquo;&rsquo;&rsquo;stationary&rsquo;&rsquo;&rsquo;{stoyan1995stochastic}) Poisson (point) process. Otherwise, the parameter depends on its location in the underlying space, which leads to the &rsquo;&rsquo;&rsquo;inhomogeneous&rsquo;&rsquo;&rsquo; or &rsquo;&rsquo;&rsquo;nonhomogeneous&rsquo;&rsquo;&rsquo; or Poisson (point) process{daleyPPI2003}.

The word &rsquo;&rsquo;point&rsquo;&rsquo; is often omitted, but the Poisson (point) process should not be confused with Poisson processes in general, which include the Poisson point process but can also, instead of points, have more complicated mathematical objects such as lines and polygons{kingman1992poisson}.

= History =

The namesake of the Poisson process stems from its relation to the Poisson distribution derived by the French mathematician Siméon Denis Poisson as a limiting case of the binomial distribution{good1986some}, which describes the probability of the sum of $$n$$ Bernoulli trials with probability $$p$$, which is often likened to the number of heads (or tails) after $$n$$ biased coin flips. For some constant $$\Lambda>0$$, as $$n$$ grows large and $$p$$ decrease towards zero such that $$np=\Lambda$$ is fixed, the Poisson distribution more closely approximates that of the binomial{grimmett2001probability}. Although Poisson derived the Poisson distribution through a limiting argument, there is no evidence to suggest that he used it in the sense of a stochastic process, and his result was not well-known during his time{daleyPPI2003}. The distribution would be studied years later in a different setting by Polish economist and statistician Ladislaus Bortkiewicz who famously used the Poisson distribution with real data to study the number of deaths from horse kicks in the Prussian army{good1986some,quine1987bortkiewicz}.

In the beginning of the 20th cenutry the Poisson process would arise almost independently in two different contexts. In 1909 the Danish mathematician and engineer Agner Krarup Erlang derived the Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang, not at the time aware of Poisson&rsquo;s earlier work, assumed that the number phone calls arriving in each subinterval of time were independent to each other, and then finding the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial. In 1910 physicists Ernest Rutherford and Hans Geiger were conducting an experiment in counting the number of alpha particles, which led to English mathematician Harry Bateman deriving the Poisson probabilities as a solution to a family of differential equations. {daleyPPI2003}.

= Overview of definitions =

The Poisson point process is one of the most studied stochastic processes in both the field of point processes and in more applied disciplines concerning random phenomena{daleyPPI2003}due to its convenient properties as a mathematical model as well as being mathematically interesting{kingman1992poisson}. Depending on the setting, the process has several equivalent definitions{tijms2003first}as well definitions of varying generality owing due to its many applications and characterizations{daleyPPI2003}. It may be defined, studied and used in one dimension (on the real line) where it is often interpreted as a counting process or part of a queueing model{tijms2003first,ross1996stochastic}; in higher dimensions such as the plane where it plays a role in stochastic geometry and spatial statistics{stoyan1995stochastic,baddeley1999crash}; or on more abstract spaces{daleyPPII2008}. Consequently, the notation, terminology and level of mathematical rigour used to define and study the Poisson point process and points processes in general vary according to the context{stoyan1995stochastic,daleyPPI2003}. Despite its several the forms and varying generality, the Poisson point process has two key properties.

First key property: Poisson distributed number of points
In all its guises, the Poisson point process is related to its namesake the Poisson distribution, which implies that the probability of a Poisson random variable $$N$$ is equal to $$n$$ is given by: $$P\{N=n\}=\frac{\Lambda^n}{m!}e^{-\Lambda} $$ where $$m!$$ denotes $$n$$ factorial and $$\Lambda$$ is the single Poisson parameter that is used to define the Poisson distribution. If a Poisson point process is defined on some underlying mathematical space, the number of points in a bounded region of the space will be a Poisson random variable with some parameter whose form will depend on the setting{kingman1992poisson}.

Second key property: complete independence
The other key property is that for a collection of disjoint (or non-overlapping) bounded subregions of the the underlying space, the number of points in each bounded subregion will be completely independent to all the others. This property, known as &rsquo;&rsquo;complete independence&rsquo;&rsquo;{daleyPPI2003}or &quot;independent scattering&quot;{stoyan1995stochastic,moller2003statistical}, is common to all Poisson point processes (and Poisson processes in general). In other words, there is a lack of interaction between different regions{feller1974introduction}, which motivates the Poisson process being sometimes called a &rsquo;&rsquo;purely&rsquo;&rsquo; or &rsquo;&rsquo;completely&rsquo;&rsquo; random process{daleyPPI2003}.

Simple point process
Another relevant property of the Poisson point process is that it is a simple process, which, in a more mathematical framework, is a direct consequence of the previous properties. The probability of a point of a simple point process existing at a single point on the underlying space is either zero or one. This implies that no two (or more) points of a Poisson point process coincide in location on the underlying space with probability one. The are many simple point processes such as the cluster, hard-core, soft-core, and determinantal point processes, {daleyPPII2008,moller2003statistical,lavancier2012statistical}.

Different definitions
To accommodate the various settings and definitions of the Poisson point process, it is often defined in the simplest case (on the real line), and then extended to a more general settings with more mathematical rigour{feller1974introduction,daleyPPI2003}. For all the instances of the Poisson point process, the Poisson distribution and complete independence play an important role {daleyPPI2003}.

= Terminology of Poisson point processes =

The terminology of Poisson point processes varies considerably as there are no standard terms{kingman1992poisson}. The homogeneous Poisson process is also called a &rsquo;&rsquo;stationary&rsquo;&rsquo; Poisson (point) process {daleyPPI2003}, and in the past it was sometimes, by William Feller and others, referred to as a &rsquo;&rsquo;Poisson ensemble&rsquo;&rsquo; of points{feller1974introduction,roberts1969nearest}.

The inhomogenous point process, as well as be being called &rsquo;&rsquo;nonhomogeneous&rsquo;&rsquo; {daleyPPI2003}or &rsquo;&rsquo;non-homogeneous&rsquo;&rsquo;{cox1980point}is sometimes referred to as the &quot;non-stationary&rsquo;&rsquo;{tijms2003first}, &rsquo;&rsquo;heterogeneous&rsquo;&rsquo;{lawson1993deviance,baudin1984multidimensional,shen2009species}or &rsquo;&rsquo;spatially dependent&rsquo;&rsquo; Poisson (point) process{krkovsek2005transmission,pan2007detection}.

The measure denoted by $$\Lambda$$ is sometimes called the &rsquo;&rsquo;parameter measure&rsquo;&rsquo; {daleyPPI2003}or &rsquo;&rsquo;intensity measure&rsquo;&rsquo;{stoyan1995stochastic}or &rsquo;&rsquo;mean measure&rsquo;&rsquo;{kingman1992poisson}. If a $$\Lambda$$ derivative or density exists, denoted by $$\lambda(x)$$, it may be called the &rsquo;&rsquo;intensity function&rsquo;&rsquo; of the general process{stoyan1995stochastic}or simply the &rsquo;&rsquo;rate&rsquo;&rsquo; or &rsquo;&rsquo;intensity&rsquo;&rsquo;{kingman1992poisson}. For the homogeneous Poisson point process, the intensity is a simply a constant, which can be referred to as the &rsquo;&rsquo;mean rate&rsquo;&rsquo; or &rsquo;&rsquo;mean density&rsquo;&rsquo;{daleyPPI2003}. For $$\lambda=1$$, the corresponding process is sometimes referred to as the &rsquo;&rsquo;&rsquo;standard Poisson&rsquo;&rsquo;&rsquo; (point) process{grandell1977point,merzbach1986characterization,moller2003statistical}.

= Homogeneous Poisson point process =

If a Poisson point process has a constant parameter, say, $$\lambda$$, then it is often called a homogeneous (or stationary) Poisson point process. The parameter, called rate or intensity, is related to the expected (or average) number of Poisson points existing in some bounded region{moller2003statistical}.

Defined on the real line
Consider two real numbers $$a$$ and $$b$$, where $$a\le b$$, and which may represent points in time. Denote by $$N(a,b]$$ the random number of points of a homogeneous Poisson point process existing with values greater than $$a$$ but less than or equal to $$b$$. If the points form or belong to a homogeneous Poisson process with parameter $$\lambda>0$$, then the probability of $$n$$ points existing that adhere to the above condition is given by: $$P\{N(a,b]=m\}=\frac{[\lambda(b-a)]^n}{n!}e^{-\lambda (b-a)} $$ In other words, $$N(a,b]$$ is a Poisson random variable with mean $$\lambda(b-a)$$. Furthermore, the number of points in any two subjoint intervals, say, $$(a_1,b_1]$$ amd $$(a_2,b_2]$$ are independent to each other, and this extends to any finite number of subjoint intervals.

For a more formal definition of a stochastic process, such as a point process, the Daniell-Kolmogorov theorem can be used. The theorem essentially says a stochastic process is characterized (or uniquely determined) by its finite-dimensional distribution, which in this context gives the joint probability of some number of points existing in each disjoint finite interval. More specifically, let $$N(a_i,b]$$ denote the number of points of (a point process) happening in the half-open interval $$(a_i,b_i]$$ where the real numbers $$a_i0$$ is defined with the finite-dimensional distribution{daleyPPI2003}: $$P\{N(a_i,b_i]=m_i, i=1, \dots, k\}=\prod_{i=1}^k\frac{[\lambda(b_i-a_i)]^{m_i}}{m_i!}e^{-\lambda(b_i-a_i)}, $$

Key properties
The above definition has two important features pertaining to the Poisson point processes in general:

– the number of points in each finite interval has a Poisson distribution.

– the number of points in disjoint intervals are independent random variables.

Furthermore, it has a third feature related to just the homogeneous Poisson process:

–the distribution of each interval only depends on the length $$b_i-a_i$$, hence they are stationary (the process is sometimes called the &rsquo;&rsquo;stationary Poisson process&rsquo;&rsquo;). In other words, for some finite $$t>0$$, the random variable $$N(a+t,b+t)$$ is independent of $$t$${daleyPPI2003}.

Law of large numbers
The quantity $$\lambda(b_i-a_i)$$ can be interpreted as the expected number of points occurring in the interval $$(a_i,b_i]$$, namely: $$E\{N(a_i,b_i] \}=\lambda(b_i-a_i) $$ where $$E$$ denotes the expectation operator. In other words, the parameter $$\lambda$$ of the Poisson process coincides with the &rsquo;&rsquo;density&rsquo;&rsquo; of points. Furthermore, the homogeneous Poisson point process adheres to its own form of the (strong) law of large numbers{kingman1992poisson}. More specifically, with probability one $$\lim_{t\rightarrow \infty}\frac{N (0,t]}{t} =\lambda. $$

Points are uniformly distributed
If the homogeneous point process is used as a mathematical model for occurrences of some phenomenon, then it will have a couple of interesting characteristics. For example, the positions of these occurrences or events on the real line (often interpreted as time) will be uniformly distributed{daleyPPI2003}. More specifically, if an event occurs (according to this process) in the interval $$(b_i-a_i]$$, then its location will be a uniform random variable defined on that interval. This property gives a natural way to stochastically simulate homogeneous Poisson processes on computers{stoyan1995stochastic}.

Memoryless property
The differences in times between two consecutive events happening will be an exponential random variable with parameter $$\lambda$$ (or mean $$1/\lambda$$). Consequently, the events or points have the memoryless property: the existence of one point existing in a finite interval does effect not the probability (distribution) of other points existing. This property is directly related to the complete independence property, however, it has no natural equivalent in higher dimensions{kingman1992poisson}.

Orderliness implies simplicity
A stochastic process with stationary increments is sometimes said to be &rsquo;&rsquo;orderly&rsquo;&rsquo;{cox1980point}(or &rsquo;&rsquo;regular&rsquo;&rsquo;{ross1996stochastic}) if $$P\{N(t,t+\delta]>1\}= o(\delta), $$

where little o notation is used. For point processes in general on the real line, this (distribution) property of orderliness implies that the process is simple{cox1980point}or has the (sample path) property of &rsquo;&rsquo;simplicity&rsquo;&rsquo;{daleyPPII2008}, which is indeed the case for the homogeneous Poisson point process.

Relationship to other processes
On the real line, the Poisson point process is a type of continuous-time Markov process known as a birth-death process (with just births and zero deaths) and is called a &rsquo;&rsquo;pure&rsquo;&rsquo;{ross1996stochastic}or &rsquo;&rsquo;simple&rsquo;&rsquo; birth process{papoulis2002probability}. The homogeneous Poisson process and its inhomogeneous counterpart defined on the real line play a fundamental role in queueing theory, which is the probability field of developing suitable stochastic models to represent the random arrival and departure of certain phenomena {feller1974introduction,ross1996stochastic,tijms2003first}. For example, customers arriving and being served or phone calls arriving at a phone interchange. In the queueing theory context, one often considers a point existing (in an interval) as an &rsquo;&rsquo;event&rsquo; {ross1996stochastic}, but this is different to the word &rsquo;&rsquo;event&rsquo;&rsquo; in the probability theory sense. It follows that $$\lambda$$ is the expected number of &rsquo;&rsquo;arrivals&rsquo;&rsquo; that occur per unit of time, and is often called the rate parameter{ross1996stochastic}.

Counting process interpretation
Often the homogeneous Poisson point process, when considered on the positive half-line, is defined as a counting process and often written as $$\{N(t), t\geq 0\}$$ {ross1996stochastic,tijms2003first}. In general, a counting process represents the total number of occurrences or events that have happened up to time and including time $$t$$. A counting process is the Poisson counting process with rate $$\lambda>0$$ if it has three the properties:

– N(0)=0

– has independent increments

– the number of events (or points) in any interval of length $$t$$ is a Poisson random variable with parameter (or mean) $$\lambda t$$.

The last condition implies $$E[N(t)]=\lambda t. $$ The Poisson counting process can also be defined by stating that the the time differences between events of the counting process are exponential variables with mean $$1/\lambda$${tijms2003first}. The time differences between the events or arrivals are known as &rsquo;&rsquo;&rsquo;interrarrival&rsquo;&rsquo;&rsquo; {ross1996stochastic,feller1974introduction}or &rsquo;&rsquo;&rsquo;interoccurence&rsquo;&rsquo;&rsquo; times{tijms2003first}. These two definitions of the Poisson counting process agree with the previous definition of the Poisson point process.

Restricted to the half-line
If the homogeneous point process is considered on just on the half-line $$[0,\infty)$$, which is often the case when $$t$$ represents time like it does for the previous counting process{ross1996stochastic,tijms2003first}, then the resulting process is not invariant under translation {kingman1992poisson}. In that case the process is no longer stationary, according to some definitions of stationarity{kingman1992poisson,stoyan1995stochastic,cox1980point}.

Defined on the plane
Instead of the real line, the Poisson point process may be defined on the plane $$R^2$$, which gives what often is called a spatial Poisson (point) process{merzbach1986characterization,lawson1993deviance}. For its definition, consider a bounded, open or closed (or more precisely, Borel) region $$B$$ of the plane. Denote by $$N(B)$$ the (random) number of points of existing in this region $$B\subset R^2$$. If the points belong to a homogeneous Poisson process with parameter $$\lambda>0$$, then the probability of $$n$$ points existing in $$B$$ is given by: $$P\{N(B)=n\}=\frac{(\lambda|B|)^n}{m!}e^{-\lambda|B|} $$ where $$|B|$$ denotes the the area of $$B$$.

More formally, for some some finite integer $$k\geq 1$$, consider a collection of disjoint, bounded Borel sets $$B_1,\dots,B_k$$. Let $$N(B_i)$$ denote the number of points of existing in $$N(B_i)$$. Then the homogeneous Poisson point process with parameter $$\lambda>0$$ has the finite-dimensional distribution{daleyPPI2003}$$P\{N(B_i)=m_i, i=1, \dots, k\}=\prod_{i=1}^k\frac{(\lambda|B_i|)^{m_i}}{m_i!}e^{-\lambda|B_i|}. $$

Models
This spatial point process features prominently in spatial statistics, stochastic geometry, and continuum percolation theory. For example, spatial Poisson models have been developed for alpha particles being detected{stoyan1995stochastic}. In recent years, it has been frequently used to model seemingly disordered or random spatial configurations of certain wireless networks.

Defined in higher dimensions
The previous homogeneous Poisson point process immediately extends to higher dimensions by replacing the notion of area with (high dimensional) volume. For some bounded region $$B$$ of Euclidean space $$R^d$$, if the points form a homogeneous Poisson process with parameter $$\lambda>0$$, then the probability of $$n$$ points existing in $$B\subset R^d$$ is given by: $$P\{N(B)=n\}=\frac{(\lambda|B|)^n}{m!}e^{-\lambda|B|} $$ where $$|B|$$ now denotes the $$n$$-dimensional volume of $$B$$. Furthermore, for a collection of disjoint, bounded Borel sets $$B_1,\dots,B_k$$. Let $$N(B_i)$$ of $$R^d$$, then the homogeneous Poisson point process with parameter $$\lambda>0$$ has the finite-dimensional distribution{daleyPPI2003}$$P\{N(B_i)=n_i, i=1, \dots, k\}=\prod_{i=1}^k\frac{(\lambda|B_i|)^{n_i}}{n_i!}e^{-\lambda|B_i|}. $$

Homogeneous Poisson point processes do not depend on the position of the underlying space through its parameter $$\lambda$$, which implies it is both a stationary process (invariant to translation) and an isotropic stochastic process{stoyan1995stochastic}. Similarly to the one-dimensional case, the homogeneous point process is restricted to some bounded subset of $$R^d$$, then depending on some definitions of stationarity, the process is no longer stationary{stoyan1995stochastic,cox1980point}.

= Inhomogeneous Poisson point process =

The Poisson parameter can be extended to a location-dependent function $$\lambda (x)$$ of some point $$x$$ in the underlying space $$R^d$$ on which the Poisson process is defined, thus introducing the &rsquo;&rsquo;&rsquo;inhomogeneous&rsquo;&rsquo;&rsquo; or &rsquo;&rsquo;&rsquo;nonhomogeneous&rsquo;&rsquo;&rsquo; Poisson point process. More specifically, let $$\lambda(x)$$ be a locally integrable positive function such that for any bounded region $$B$$ the ($$d$$-dimensional) volume integral of $$\lambda (x)$$ over region $$B$$ is finite. More succinctly, denote this volume integral, $$\Lambda (B)$$, and if: $$\Lambda (B)=\int_B \lambda(x) dV < \infty. $$ where $$dV$$ is the volume integral. Then for any collection of disjoint bounded Borel sets $$B_1,\dots,B_k$$, an inhomogeneous Poisson process with (intensity) function $$\lambda(x)$$ has the finite-dimensional distribution $$P\{N(B_i)=n_i, i=1, \dots, k\}=\prod_{i=1}^k\frac{(\Lambda(B_i))^{n_i}}{n_i!}e^{-\Lambda(B_i)}. $$ Furthermore, $$\Lambda (B)$$ has the interpretation of being the expected number of points of the Poisson process located in the bounded region $$B$$, namely $$\Lambda (B)= E[N(B)]. $$

On the real line
A useful feature of the one-dimension setting is that any inhomogeneous Poisson point process can be made homogeneous by a monotone transformation, which is achieved with the inverse of $$\Lambda $$ {kingman1992poisson}.

In higher dimensions
In the plane, $$\Lambda(B) $$ corresponds to an area integral; for example, in Cartesian coordinates $$\Lambda (B)=\int_B \lambda(x_1,x_2) dx_1 dx_2. $$ In $$R^d$$, the above integral naturally becomes a $$n$$-dimensional volume integral.

Applications
The real line, as mentioned earlier, is often interpreted as time and in this setting the inhomogeneous process is used in the fields of counting processes and in queueing theory{ross1996stochastic,tijms2003first}. On the plane, the Poisson point process is of fundamental importance in the related disciplines of stochastic geometry{stoyan1995stochastic,baddeley1999crash}, and spatial statistics{baddeley2007spatial,moller2003statistical}. This point process is not stationary owing to that fact its distribution is dependent on the location of underlying space. Hence, it can be used to model phenomena with a density that varies over some region. In other words, the phenomena have a location-dependent density. Uses for this process as a mathematical model are diverse and have appeared across various disciplines including the study of salmon and sea lice in the oceans{krkovsek2005transmission}, forestry{thompson1955spatial}, and naval search problems{lewis1979simulation}.

= Interpretation of $$\lambda(x)$$ =

The Poisson intensity function $$\lambda(x)$$ has an attractive and intuitive interpretation in the infinitesimal sense: $$\lambda(x)dV$$ is the infinitesimal probability of a point from $$\Phi$$ existing in a region of space with volume $$dV$$ located at $$x$${stoyan1995stochastic}. For example, given a homogeneous Poisson point process on the real line, loosely speaking the probability of finding a single point (of the process) in a small interval of width $$\delta$$ is approximately $$\lambda \delta x$$. In fact, such intuition is how the Poisson point process is sometimes introduced and its distribution derived {kingman1992poisson,cox1980point}.

= Notation of point processes =

The notation of the theory of point processes parallels the terminology by varying markedly, which reflects the field&rsquo;s history and the different interpretations {stoyan1995stochastic,daleyPPI2003}. The notation (and accompanying terminology used) will be depend on the setting and interpretation (as a mathematical object) of the point process.

Random sequences of points
A given point process, often denoted by a single letter {kingman1992poisson,moller2003statistical,stoyan1995stochastic}, may be considered as a sequence of points $$\{x_1, x_2,\dots \}$$ with each point randomly positioned in $$R^d$$, or some other mathematical space.{stoyan1995stochastic}.

Random set of points
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 discrete points{stoyan1995stochastic,baddeley2007spatial}. If a point process, denoted by $$\Phi$$, is considered as a random set, the corresponding notation: $$x\in \Phi, $$ is then used to denote that a random point $$x$$ belongs to a point process $$\Phi$$. The theory of random sets can be applied to point processes owing to this interpretation.

A point process $$\Phi$$ can be written as $$\{x_1, x_2,\dots \}=\{x\}_i$$ to highlight its interpretation as either a sequence or random closed set of points{stoyan1995stochastic}.

Random counting measures
To denote the number of points of $$\Phi$$ located in some set Borel $$B$$, it is sometimes written {kingman1992poisson}$$N(B) =\#( B \cap \Phi), $$ where $$N(B)$$ is a random variable and $$\#$$ is a counting measure, which gives the number of points in some set. In this expression point process is denoted by $$\Phi$$ while $$N$$ represents the number of points of $$\Phi$$ in $$B$$. In the context of random measures, one can write $$N(B)=n$$ to denote that there is the set $$B$$ that contains $$n$$ points of $$\Phi$$. In other words, $$N$$ can be considered as a random counting measure assigning some integer-valued measure to sets{stoyan1995stochastic}. The techniques of random measure theory offer another way to study point processes{stoyan1995stochastic,grandell1977point}.

Dual interpretation
The different interpretations of point processes is captured with the popular notation {stoyan1995stochastic,moller2003statistical,BB1}: $$\Phi(B) =\#( B \cap \Phi), $$ in which the collection of random points is $$\Phi$$ and the random variable $$\Phi(B)$$ is the number of points of $$\Phi$$ in $$B$$.

Sums
If $$f$$ is some (measurable) function on $$R^d$$, then the sum of $$f(x)$$ over all the points $$x$$ in $$\Phi$$ can{stoyan1995stochastic}be written as: $$f(x_1) + f(x_1) \dots $$ or more compactly as: $$\sum_{x\in \Phi}f(x) $$ or equivalently as: $$\int_N f(x) \Phi(dx) $$ where $$N$$ is the space of all random counting measures, hence putting emphasis on the interpretation of $$\Phi$$ as a random counting measure. The dual interpretation of point processes is illustrated when writing the number of $$\Phi$$ points in a set $$B$$ as: $$\Phi(B)= \sum_{x\in \Phi}1_B(x) $$ where the indicator function $$1_B(x)=1$$ if the point $$x$$ is exists in $$B$$ and zero otherwise. In this expression the random measure interpretation is on the left-hand side while the random set notation is used is on the right-hand side.

Expectations
The average or expected value of a sum of functions over a point process is written as: $$E\left[\sum_{x\in \Phi}f(x)\right] \qquad \text{or} \qquad \int_N\sum_{x\in \Phi}f(x) P(d\Phi), $$ where $$P$$ is an appropriate probability measure defined on the space of counting functions $$N$$. The mean value of $$\Phi(B)$$ can be written as: $$E[\Phi(B)]=E\left( \sum_{x\in \Phi}1_B(x)\right) \qquad \text{or} \qquad  \int_N\sum_{x\in \Phi}1_B(x) P(d\Phi). $$

= General Poisson point process =

The inhomogeneous Poisson point process can be defined more generally without the use of $$\lambda(x)$$ to introduce what is sometimes known as the &rsquo;&rsquo;&rsquo;general Poisson point process&rsquo;&rsquo;&rsquo;{stoyan1995stochastic,baddeley2007spatial}. Its definition is defined in relation to some diffuse Radon measure $$\Lambda$$, hence this measure has no atoms and is locally finite. Consequently, assuming that the underling space of the Poisson point process is $$R^d$$ (the space can be more general), then $$\Lambda(\{x\})=0$$ for any single point $$x$$ in $$R^d$$ and $$ \Lambda (B)$$ is finite for any bounded subset $$B$$ of $$R^d$${moller2003statistical}. Then a point process $$\Phi$$ is a general Poisson point process with intensity $$\Lambda$$ if it has the two following properties{stoyan1995stochastic}:

– The number of points in a bounded Borel set $$B$$ is a Poisson random variable with mean $$\Lambda(B)$$. In other words, denote the total number of points located in $$B$$ by $$\Phi(B)$$, then the probability that the random variable $$\Phi(B)$$ is equal to $$n$$ is given by: $$P \{ \Phi(B)=n\}=\frac{(\Lambda(B))^{n}}{n!}e^{-\Lambda(B)} $$ –the number of points in $$k$$ disjoint Borel sets forms $$k$$ independent random variables.

The Radon measure $$\Lambda$$ maintains its previous interpretation of being the expected number of points of $$\Phi$$ located in the bounded region $$B$$, namely $$\Lambda (B)= E[\Phi(B)]. $$ Furthermore, if $$\Lambda$$ is absolutely continuous such that it has a density (or more precisely, a Radon–Nikodym density or derivative) with respect to the Lebesgue measure, then for all Borel sets $$B$$ it can be written as: $$\Lambda (B)=\int_B \lambda(x) dV. $$

= Functions and characterizations =

In the theory of probability, operations are applied to random variables for different purposes. Sometimes these are regular expectations that give the average or variance of a random variable. Others, such as the characteristic functions (or Laplace transforms) of a random variable can be used to uniquely identify or characterize random variables and prove results like the central limit theorem{karr1993probability}. In the theory of point processes there exist analogous mathematical operations and tools{daleyPPII2008,stoyan1995stochastic}.

Moments measures
The moment measure generalizes the idea of (raw) moments of the random variable. For some integer $$n=1,2,\dots$$, the $$n\,$$th power of a point process $$\Phi$$ is defined as: $$\Phi^n(B_1,\times,\dots,\times B_n)= \Pi_{i=1}^n\Phi(B_i) $$ where $$B_1,...,B_n$$ is a set of not necessarily disjoint Borel sets which form a $$n$$-fold (Cartesian) product of sets. The $$n\,$$th moment measure is defined with respect to some Borel set $$B$$ as: $$M^n(B)=E [\Phi^n(B)]. $$ The first moment is simply: $$M^1(B)=E [\Phi(B)]. $$ For a Poisson point process with intensity measure $$\Lambda$$ this becomes $$M^1(B)=\Lambda(B), $$ which in the homogeneous case with constant intensity $$\lambda$$ means $$M^1(B)=\Lambda(B), $$ where $$|B|$$ is the area (or more generally, the Lebesgue measure) of $$B$$. The moment measure increases in complexity with $$n$$ in general with the second moment measure being given by: $$M^2(B)=\Lambda(B)^2+\text{Var}(B), $$ where $$Var(B)$$ is the variance of $$\Phi(B)$$. For the Poisson case with measure $$\Lambda$$ is $$M^2(B)=\Lambda(B)^2+\Lambda(B\times B), $$

Factorial moment measures
The definition of the $$n\,$$ moment of $$\Phi$$ allows for repetition of points since they sets in the set product are not necessarily disjoint. To account for this, the factorial moment measure is defined such that there are no points repeating in the product set. The end result is that that $$n\,$$th factorial moment measure is defined with respect to some Borel set $$B$$ as: $$M^{(n)}(A)=E [\Phi^{(n)}(A)(\Phi^{(n)}(A)-1)], $$ which for the Poisson point process with intensity measure $$\Lambda$$ gives the simple expression: $$M^{(n)}(A)=E [\Phi^{(n)}(A)], $$

Avoidance functions
Renyi proved that simple point processes are completely characterized by their avoidance function. Consequently, the only simple point process with the avoidance function is the Poisson point process.

Factorial moment measure expansions
= Operations of point processes =

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 points processes: thinning, superposition, transformation (or mapping), and random displacement. One of the reasons why the Poisson point process is often used as model is that for all four operations the Poisson process gives elegant results. In fact, under suitable conditions, all three operations when performed on a Poisson point process produce another (usually different) Poisson point process, demonstrating an aspect of mathematical closure.

Thinning
Furthermore, randomly thinning a Poisson process (with density $$\lambda(x))$$, where each point is independently removed (or kept) with some probability $$p$$ (or $$1-p$$), forms another Poisson process (with density $$(1-p)\lambda(x)$$) while the kept points also form a Poisson process (with density $$p\lambda(x)$$) that is independent to the Poisson process of removed points {kingman1992poisson,stoyan1995stochastic}.

Superposition
If there is a countable collection of independent Poisson point processes $$\Phi_1,\Phi_2\dots$$ with mean measures $$\Lambda_1,\Lambda_2,\dots$$, then their superposition $$\Phi=\cup_{i=1}^{\infty}\Phi_i, $$ also forms a point process. The superposition theorem, which stems directly from the complete independence property, says that this new point process will also be a Poisson point process with mean intensity $$\Lambda=\sum\limits_{i=1}^{\infty}\Lambda_i. $$ In other words, the union of two (or countably more) Poisson processes is another Poisson process. Another interesting result is that if a point $$x$$ is sampled from a countable $$n$$ union of Poisson processes, then the probability that the points belongs to the $$j$$th Poisson process is given by: $$P (x\in\Phi_j)=\frac{\Lambda_j}{\sum_{i=1}^{n}\Lambda_i}. $$ These result has been used in recent years to examine multi-tier cellular networks which, it is assumed, consist of &rsquo;&rsquo;tiers&rsquo;&rsquo; of wireless networks with different parameters combined into a single network. The superposition theorem of the Poisson processes says that this stochastic model of a network is also a Poisson process, thus allowing the results of Poisson processes to be used.

Homogeneous case
In the homogeneous case with constant $$\lambda_1,\lambda_2\dots$$, the two previous expressions reduce to $$\lambda=\sum\limits_{i=1}^{\infty}\lambda_i, $$ and $$P (x\in\Phi_j)=\frac{\lambda_j}{\sum_{i=1}^{n}\lambda_i}. $$

Transformation
Consider a homogeneous Poisson point process with parameter $$\lambda$$ on some underlying space (for example, the plane). If the locations of the points are mapped (ie the point process is transformed) according to some function to another underlying space (for example, a sphere), then the resulting point process is also a Poisson point process but with a different parameter $$\lambda$$.

More specifically, if a function maps a Poisson point process from one space, say, $$S$$, to the same or another space $$T$$ without placing the points all in one point, then the resulting point process is also a Poisson point process with a different intensity measure.

Random displacement
= Marked point processes =

Marking theorem
= Models =

Sea lice and salons have been modelled with Poisson process {krkovsek2005transmission}{Criticisms}

= Simulation of Poisson point processes =

= Related point processes =

Hard-core processes
= Poisson point processes on more general spaces =

For mathematical models the Poisson point process is often defined in Euclidean space, but has been generalized to more abstract spaces and plays a fundamental role in the study of random measures, which requires an understanding of certain mathematical fields such as probability theory, measure theory, topology and functional analysis.

In general, the concept of distance is of central interest in the theory of point processes, hence such processes need to be defined on mathematical spaces equipped with metrics. The necessity of convergence of sequences requires the space to be complete. Further mathematical technicalities have led to the Poisson and other point processes being defined on a complete separable metric space {daleyPPI2003}.

However, every realization of a point process in general can be regarded as a measure, which motivates point processes being considered as random measures{grandell1977point}. Using the techniques of random measures, the Poisson and other point processes has been defined in a different manner on a locally compact second countable Hausdorff space.

Textbooks on stochastic geometry and related fields
{abbrv}{PointReferences02}