Instatistics andprobability theory, apoint process orpoint field is a set of a random number ofmathematical points randomly located on a mathematical space such as thereal line orEuclidean space.^[1][2]

Point processes on the real line form an important special case that is particularly amenable to study,^[3] because the points are ordered in a natural way, and the whole point process can be described completely by the (random) intervals between the points. These point processes are frequently used as models for random events in time, such as the arrival of customers in a queue (queueing theory), of impulses in a neuron (computational neuroscience), particles in aGeiger counter, location of radio stations in atelecommunication network^[4] or of searches on theworld-wide web.

General point processes on a Euclidean space can be used forspatial data analysis,^[5][6] which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications, computational neuroscience,^[7] economics^[8] and others.

Since point processes were historically developed by different communities, there are different mathematical interpretations of a point process, such as arandom counting measure or a random set,^[9][10] and different notations. The notations are described in detail on thepoint process notation page.

Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process,^[11][12] though it has been remarked that the difference between point processes and stochastic processes is not clear.^[12] Others consider a point process as a stochastic process, where the process is indexed by sets of the underlying space^[a] on which it is defined, such as the real line or${\displaystyle n}$-dimensional Euclidean space.^[15][16] Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.^[17][12] Sometimes the term "point process" is not preferred, as historically the word "process" denoted an evolution of some system in time, so point process is also called a random point field.^[18]

In mathematics, a point process is arandom element whose values are "point patterns" on asetS. While in the exact mathematical definition a point pattern is specified as alocally finitecounting measure, it is sufficient for more applied purposes to think of a point pattern as acountable subset ofS that has nolimit points.^{[clarification needed]}

To define general point processes, we start with a probability space${\displaystyle (\mathrm{\Omega },\mathcal{F},P)}$,and a measurable space${\displaystyle (S,\mathcal{S})}$ where${\displaystyle S}$ is alocally compactsecond countableHausdorff space and${\displaystyle \mathcal{S}}$ is itsBorel σ-algebra. Consider now an integer-valued locally finite kernel${\displaystyle \xi }$from${\displaystyle (\mathrm{\Omega },\mathcal{F})}$ into${\displaystyle (S,\mathcal{S})}$, that is, a mapping${\displaystyle \mathrm{\Omega }\times \mathcal{S}\mapsto {\mathbb{Z}}_{+}}$ such that:

1. For every${\displaystyle \omega \in \mathrm{\Omega }}$,${\displaystyle \xi (\omega ,\cdot )}$ is a (integer-valued)locally finite measure on${\displaystyle S}$.
2. For every${\displaystyle B\in \mathcal{S}}$,${\displaystyle \xi (\cdot ,B):\mathrm{\Omega }\to {\mathbb{Z}}_{+}}$ is a random variable over${\displaystyle {\mathbb{Z}}_{+}}$.

This kernel defines arandom measure in the following way. We would like to think of${\displaystyle \xi }$as defining a mapping which maps${\displaystyle \omega \in \mathrm{\Omega }}$ to a measure${\displaystyle {\xi }_{\omega }\in \mathcal{M}(\mathcal{S})}$(namely,${\displaystyle \mathrm{\Omega }\mapsto \mathcal{M}(\mathcal{S})}$),where${\displaystyle \mathcal{M}(\mathcal{S})}$ is the set of all locally finite measures on${\displaystyle S}$.Now, to make this mapping measurable, we need to define a${\displaystyle \sigma }$-field over${\displaystyle \mathcal{M}(\mathcal{S})}$.This${\displaystyle \sigma }$-field is constructed as the minimal algebra so that all evaluation maps of the form${\displaystyle {\pi }_B:\mu \mapsto \mu (B)}$, where${\displaystyle B\in \mathcal{S}}$ isrelatively compact,are measurable. Equipped with this${\displaystyle \sigma }$-field, then${\displaystyle \xi }$ is a random element, where for every${\displaystyle \omega \in \mathrm{\Omega }}$,${\displaystyle {\xi }_{\omega }}$ is a locally finite measure over${\displaystyle S}$.

Now, bya point process on${\displaystyle S}$ we simply meanan integer-valued random measure (or equivalently, integer-valuedkernel)${\displaystyle \xi }$ constructed as above.The most common example for the state spaceS is the Euclidean spaceRⁿ or a subset thereof, where a particularly interesting special case is given by the real half-line [0,∞). However, point processes are not limited to these examples and may among other things also be used if the points are themselves compact subsets ofRⁿ, in which caseξ is usually referred to as aparticle process.

Despite the namepoint process sinceS might not be a subset of the real line, as it might suggest that ξ is astochastic process.

Every instance (or event) of a point process ξ can be represented as

${\displaystyle \xi =\sum _{i=1}^n{\delta }_{X_i},}$

where${\displaystyle \delta }$ denotes theDirac measure,n is an integer-valued random variable and${\displaystyle X_i}$ are random elements ofS. If${\displaystyle X_i}$'s arealmost surely distinct (or equivalently, almost surely${\displaystyle \xi (x)\le 1}$ for all${\displaystyle x\in {\mathbb{R}}^d}$), then the point process is known assimple.

Another different but useful representation of an event (an event in the event space, i.e. a series of points) is the counting notation, where each instance is represented as an${\displaystyle N(t)}$ function, a continuous function which takes integer values:${\displaystyle N:\mathbb{R}\to {\mathbb{Z}}_0^{+}}$:

${\displaystyle N(t_1,t_2)={\int }_{t_1}^{t_2}\xi (t)dt}$

which is the number of events in the observation interval${\displaystyle (t_1,t_2]}$. It is sometimes denoted by${\displaystyle N_{t_1,t_2}}$, and${\displaystyle N_T}$ or${\displaystyle N(T)}$ mean${\displaystyle N_{0,T}}$.

Theexpectation measureEξ (also known asmean measure) of a point process ξ is a measure onS that assigns to every Borel subsetB ofS the expected number of points ofξ inB. That is,

${\displaystyle E\xi (B):=E(\xi (B))\text{for every }B\in \mathcal{B}.}$

TheLaplace functional${\displaystyle {\mathrm{\Psi }}_N(f)}$ of a point processN is amap from the set of all positive valued functionsf on the state space ofN, to${\displaystyle [0,\mathrm{\infty })}$ defined as follows:

${\displaystyle {\mathrm{\Psi }}_N(f)=E[\exp (-N(f))]}$

They play a similar role as thecharacteristic functions forrandom variable. One important theorem says that: two point processes have the same law if their Laplace functionals are equal.

The${\displaystyle n}$th power of a point process,${\displaystyle {\xi }^n,}$ is defined on the product space${\displaystyle S^n}$ as follows :

${\displaystyle {\xi }^n(A_1\times \cdots \times A_n)=\prod _{i=1}^n\xi (A_i)}$

Bymonotone class theorem, this uniquely defines the product measure on${\displaystyle (S^n,B(S^n)).}$ The expectation${\displaystyle E{\xi }^n(\cdot )}$ is calledthe${\displaystyle n}$ thmoment measure. The first moment measure is the mean measure.

Let${\displaystyle S={\mathbb{R}}^d}$ . Thejoint intensities of a point process${\displaystyle \xi }$ w.r.t. theLebesgue measure are functions${\displaystyle {\rho }^{(k)}:({\mathbb{R}}^d{)}^k\to [0,\mathrm{\infty })}$ such that for any disjoint bounded Borel subsets${\displaystyle B_1,\dots ,B_k}$

${\displaystyle E\left(\prod _i\xi (B_i)\right)={\int }_{B_1\times \cdots \times B_k}{\rho }^{(k)}(x_1,\dots ,x_k)d{x}_1\cdots d{x}_k.}$

Joint intensities do not always exist for point processes. Given thatmoments of arandom variable determine the random variable in many cases, a similar result is to be expected for joint intensities. Indeed, this has been shown in many cases.^[2]

A point process${\displaystyle \xi \subset {\mathbb{R}}^d}$ is said to bestationary if${\displaystyle \xi +x:=\sum _{i=1}^N{\delta }_{X_i+x}}$ has the same distribution as${\displaystyle \xi }$ for all${\displaystyle x\in {\mathbb{R}}^d.}$ For a stationary point process, the mean measure${\displaystyle E\xi (\cdot )=\lambda \Vert \cdot \Vert }$ for some constant${\displaystyle \lambda \ge 0}$ and where${\displaystyle \Vert \cdot \Vert }$ stands for the Lebesgue measure. This${\displaystyle \lambda }$ is called theintensity of the point process. A stationary point process on${\displaystyle {\mathbb{R}}^d}$ has almost surely either 0 or an infinite number of points in total. For more on stationary point processes and random measure, refer to Chapter 12 of Daley & Vere-Jones.^[2] Stationarity has been defined and studied for point processes in more general spaces than${\displaystyle {\mathbb{R}}^d}$.

A point process transformation is a function that maps a point process to another point process.

We shall see some examples of point processes in${\displaystyle {\mathbb{R}}^d.}$

The simplest and most ubiquitous example of a point process is thePoisson point process, which is a spatial generalisation of thePoisson process. A Poisson (counting) process on the line can be characterised by two properties : the number of points (or events) in disjoint intervals are independent and have aPoisson distribution. A Poisson point process can also be defined using these two properties. Namely, we say that a point process${\displaystyle \xi }$ is a Poisson point process if the following two conditions hold

1)${\displaystyle \xi (B_1),\dots ,\xi (B_n)}$ are independent for disjoint subsets${\displaystyle B_1,\dots ,B_n.}$

2) For any bounded subset${\displaystyle B}$,${\displaystyle \xi (B)}$ has aPoisson distribution with parameter${\displaystyle \lambda \Vert B\Vert ,}$ where${\displaystyle \Vert \cdot \Vert }$ denotes theLebesgue measure.

The two conditions can be combined and written as follows : For any disjoint bounded subsets${\displaystyle B_1,\dots ,B_n}$ and non-negative integers${\displaystyle k_1,\dots ,k_n}$ we have that

${\displaystyle Pr[\xi (B_i)=k_i,1\le i\le n]=\prod _i{e}^{-\lambda \Vert B_i\Vert }\frac{(\lambda \Vert B_i\Vert {)}^{k_i}}{k_i!}.}$

The constant${\displaystyle \lambda }$ is called the intensity of the Poisson point process. Note that the Poisson point process is characterised by the single parameter${\displaystyle \lambda .}$ It is a simple, stationary point process.To be more specific one calls the above point process a homogeneous Poisson point process. Aninhomogeneous Poisson process is defined as above but by replacing${\displaystyle \lambda \Vert B\Vert }$ with${\displaystyle {\int }_B\lambda (x)dx}$ where${\displaystyle \lambda }$ is a non-negative function on${\displaystyle {\mathbb{R}}^d.}$

ACox process (named afterSir David Cox) is a generalisation of the Poisson point process, in that we userandom measures in place of${\displaystyle \lambda \Vert B\Vert }$. More formally, let${\displaystyle \mathrm{\Lambda }}$ be arandom measure. A Cox point process driven by therandom measure${\displaystyle \mathrm{\Lambda }}$ is the point process${\displaystyle \xi }$ with the following two properties :

1. Given${\displaystyle \mathrm{\Lambda }(\cdot )}$,${\displaystyle \xi (B)}$ is Poisson distributed with parameter${\displaystyle \mathrm{\Lambda }(B)}$ for any bounded subset${\displaystyle B.}$
2. For any finite collection of disjoint subsets${\displaystyle B_1,\dots ,B_n}$ and conditioned on${\displaystyle \mathrm{\Lambda }(B_1),\dots ,\mathrm{\Lambda }(B_n),}$ we have that${\displaystyle \xi (B_1),\dots ,\xi (B_n)}$ are independent.

It is easy to see that Poisson point process (homogeneous and inhomogeneous) follow as special cases of Cox point processes. The mean measure of a Cox point process is${\displaystyle E\xi (\cdot )=E\mathrm{\Lambda }(\cdot )}$ and thus in the special case of a Poisson point process, it is${\displaystyle \lambda \Vert \cdot \Vert .}$

For a Cox point process,${\displaystyle \mathrm{\Lambda }(\cdot )}$ is called theintensity measure. Further, if${\displaystyle \mathrm{\Lambda }(\cdot )}$ has a (random) density (Radon–Nikodym derivative)${\displaystyle \lambda (\cdot )}$ i.e.,

${\displaystyle \mathrm{\Lambda }(B)\stackrel{\text{a.s.}}{=}{\int }_B\lambda (x)dx,}$

then${\displaystyle \lambda (\cdot )}$ is called theintensity field of the Cox point process. Stationarity of the intensity measures or intensity fields imply the stationarity of the corresponding Cox point processes.

There have been many specific classes of Cox point processes that have been studied in detail such as:

• Log-Gaussian Cox point processes:^[19]${\displaystyle \lambda (y)=\exp (X(y))}$ for aGaussian random field${\displaystyle X(\cdot )}$
• Shot noise Cox point processes:,^[20]${\displaystyle \lambda (y)=\sum _{X\in \mathrm{\Phi }}h(X,y)}$ for a Poisson point process${\displaystyle \mathrm{\Phi }(\cdot )}$ and kernel${\displaystyle h(\cdot ,\cdot )}$
• Generalised shot noise Cox point processes:^[21]${\displaystyle \lambda (y)=\sum _{X\in \mathrm{\Phi }}h(X,y)}$ for a point process${\displaystyle \mathrm{\Phi }(\cdot )}$ and kernel${\displaystyle h(\cdot ,\cdot )}$
• Lévy based Cox point processes:^[22]${\displaystyle \lambda (y)=\int h(x,y)L(dx)}$ for a Lévy basis${\displaystyle L(\cdot )}$ and kernel${\displaystyle h(\cdot ,\cdot )}$, and
• Permanental Cox point processes:^[23]${\displaystyle \lambda (y)=X_1^2(y)+\cdots +X_k^2(y)}$ fork independent Gaussian random fields${\displaystyle X_i(\cdot )}$'s
• Sigmoidal Gaussian Cox point processes:^[24]${\displaystyle \lambda (y)={\lambda }^{\star }/(1+\exp (-X(y)))}$ for a Gaussian random field${\displaystyle X(\cdot )}$ and random${\displaystyle {\lambda }^{\star }>0}$

By Jensen's inequality, one can verify that Cox point processes satisfy the following inequality: for all bounded Borel subsets${\displaystyle B}$,

${\displaystyle \mathrm{Var}(\xi (B))\ge \mathrm{Var}({\xi }_{\alpha }(B)),}$

where${\displaystyle {\xi }_{\alpha }}$ stands for a Poisson point process with intensity measure${\displaystyle \alpha (\cdot ):=E\xi (\cdot )=E\mathrm{\Lambda }(\cdot ).}$ Thus points are distributed with greater variability in a Cox point process compared to a Poisson point process. This is sometimes calledclustering orattractive property of the Cox point process.

An important class of point processes, with applications tophysics,random matrix theory, andcombinatorics, is that ofdeterminantal point processes.^[25]

A Hawkes process${\displaystyle N_t}$, also known as a self-exciting counting process, is a simple point process whose conditional intensity can be expressed as

where${\displaystyle \nu :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}}$ is a kernel function which expresses the positive influence of past events${\displaystyle T_i}$ on the current value of the intensity process${\displaystyle \lambda (t)}$,${\displaystyle \mu (t)}$ is a possibly non-stationary function representing the expected, predictable, or deterministic part of the intensity, and is the time of occurrence of thei-th event of the process.^[26]

Given a sequence of non-negative random variables$\{X_k,k=1,2,\dots \}$, if they are independent and the cdf of${\displaystyle X_k}$ is given by${\displaystyle F(a^{k-1}x)}$ for${\displaystyle k=1,2,\dots }$, where${\displaystyle a}$ is a positive constant, then${\displaystyle \{X_k,k=1,2,\dots \}}$ is called a geometric process (GP).^[27]

The geometric process has several extensions, including theα- series process^[28] and thedoubly geometric process.^[29]

Historically the first point processes that were studied had the real half lineR₊ = [0,∞) as their state space, which in this context is usually interpreted as time. These studies were motivated by the wish to model telecommunication systems,^[30] in which the points represented events in time, such as calls to a telephone exchange.

Point processes onR₊ are typically described by giving the sequence of their (random) inter-event times (T₁, T₂, ...), from which the actual sequence (X₁, X₂, ...) of event times can be obtained as

${\displaystyle X_k=\sum _{j=1}^k{T}_j\text{for }k\ge 1.}$

If the inter-event times are independent and identically distributed, the point process obtained is called arenewal process.

Theintensityλ(t | H_t) of a point process on the real half-line with respect to a filtrationH_t is defined as

${\displaystyle \lambda (t\mid H_t)=\underset{\mathrm{\Delta }t\to 0}{lim}\frac{1}{\mathrm{\Delta }t}Pr(\text{One event occurs in the time-interval}[t,t+\mathrm{\Delta }t]\mid H_t),}$

H_t can denote the history of event-point times preceding timet but can also correspond to other filtrations (for example in the case of a Cox process).

In the${\displaystyle N(t)}$-notation, this can be written in a more compact form:

${\displaystyle \lambda (t\mid H_t)=\underset{\mathrm{\Delta }t\to 0}{lim}\frac{1}{\mathrm{\Delta }t}Pr(N(t+\mathrm{\Delta }t)-N(t)=1\mid H_t).}$

Thecompensator of a point process, also known as thedual-predictable projection, is the integrated conditional intensity function defined by

${\displaystyle \mathrm{\Lambda }(s,u)={\int }_s^u\lambda (t\mid H_t)\mathrm{d}t}$

ThePapangelou intensity function of a point process${\displaystyle N}$ in the${\displaystyle n}$-dimensional Euclidean space${\displaystyle {\mathbb{R}}^n}$is defined as

${\displaystyle {\lambda }_p(x)=\underset{\delta \to 0}{lim}\frac{1}{|B_{\delta }(x)|}P\{\text{One event occurs in }{B}_{\delta }(x)\mid \sigma [N({\mathbb{R}}^n\setminus B_{\delta }(x))]\},}$

where${\displaystyle B_{\delta }(x)}$ is the ball centered at${\displaystyle x}$ of a radius${\displaystyle \delta }$, and${\displaystyle \sigma [N({\mathbb{R}}^n\setminus B_{\delta }(x))]}$ denotes the information of the point process${\displaystyle N}$outside${\displaystyle B_{\delta }(x)}$.

The logarithmic likelihood of a parameterized simple point process conditional upon some observed data is written as

${\displaystyle \ln \mathcal{L}(N(t{)}_{t\in [0,T]})={\int }_0^T(1-\lambda (s))ds+{\int }_0^T\ln \lambda (s)d{N}_s}$^[31]

The analysis of point pattern data in a compact subsetS ofRⁿ is a major object of study withinspatial statistics. Such data appear in a broad range of disciplines,^[32] amongst which are

• forestry and plant ecology (positions of trees or plants in general)
• epidemiology (home locations of infected patients)
• zoology (burrows or nests of animals)
• geography (positions of human settlements, towns or cities)
• seismology (epicenters of earthquakes)
• materials science (positions of defects in industrial materials)
• astronomy (locations of stars or galaxies)
• computational neuroscience (spikes of neurons).

The need to use point processes to model these kinds of data lies in their inherent spatial structure. Accordingly, a first question of interest is often whether the given data exhibitcomplete spatial randomness (i.e. are a realization of a spatialPoisson process) as opposed to exhibiting either spatial aggregation or spatial inhibition.

In contrast, many datasets considered in classicalmultivariate statistics consist of independently generated datapoints that may be governed by one or several covariates (typically non-spatial).

Apart from the applications in spatial statistics, point processes are one of the fundamental objects instochastic geometry. Research has also focussed extensively on various models built on point processes such asVoronoi tessellations,random geometric graphs, andBoolean models.

• Empirical measure
• Random measure
• Point process notation
• Point process operation
• Poisson process
• Renewal theory
• Invariant measure
• Transfer operator
• Koopman operator
• Shift operator

• Statistical data types
• Point processes
• Spatial processes

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