In statistics and probability theory, a point process is a type of random process for which any one realisation consists of a set of isolated points either in time or geographical space, or in even more general spaces. For example, the occurrence of lightning strikes might be considered as a point process in both time and geographical space if each is recorded according to its location in time and space.
Point processes are well studied objects in probability theory^{[1]}^{[2]} and the subject of powerful tools in statistics for modeling and analyzing spatial data,^{[3]}^{[4]} which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications, computational neuroscience,^{[5]} economics^{[6]} and others.
Point processes on the real line form an important special case that is particularly amenable to study,^{[7]} because the different 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 a Geiger counter, location of radio stations in a telecommunication network^{[8]} or of searches on the worldwide web.
General point process theory
In mathematics, a point process is a random element whose values are "point patterns" on a set S. While in the exact mathematical definition a point pattern is specified as a locally finite counting measure, it is sufficient for more applied purposes to think of a point pattern as a countable subset of S that has no limit points.
Definition
Let S be a locally compact second countable Hausdorff space equipped with its Borel σalgebra B(S). Write \mathfrak{N} for the set of locally finite counting measures on S and \mathcal{N} for the smallest σalgebra on \mathfrak{N} that renders all the point counts

\Phi_B : \mathfrak{N} \to \mathbb{Z}_{+}, \varrho \mapsto \varrho(B)
for relatively compact sets B in B measurable.
A point process on S is a measurable map

\xi:\Omega\to \mathfrak{N}
from a probability space (\Omega, \mathcal F, P) to the measurable space (\mathfrak{N},\mathcal{N}).
By this definition, a point process is a special case of a random measure.
The most common example for the state space S is the Euclidean space R^{n} or a subset thereof, where a particularly interesting special case is given by the real halfline [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 of R^{n}, in which case ξ is usually referred to as a particle process.
It has been noted that the term point process is not a very good one if S is not a subset of the real line, as it might suggest that ξ is a stochastic process. However, the term is well established and uncontested even in the general case.
Representation
Every point process ξ can be represented as

\xi=\sum_{i=1}^N \delta_{X_i},
where \delta denotes the Dirac measure, N is an integervalued random variable and X_i are random elements of S. If X_i's are almost surely distinct (or equivalently, almost surely \xi(x) \leq 1 for all x \in \mathbb{R}^d ), then the point process is known as simple.
Expectation measure
The expectation measure Eξ (also known as mean measure) of a point process ξ is a measure on S that assigns to every Borel subset B of S the expected number of points of ξ in B. That is,

E \xi (B) := E \bigl( \xi(B) \bigr) \quad \text{for every } B \in \mathcal{B}.
Laplace functional
The Laplace functional \Psi_{N}(f) of a point process N is a map from the set of all positive valued functions f on the state space of N, to [0,\infty) defined as follows:

\Psi_N(f)=E[\exp(N(f))]
They play a similar role as the characteristic functions for random variable. One important theorem says that: two point processes have the same law iff their Laplace functionals are equal.
Moment measure
The nth power of a point process, \xi^n, is defined on the product space S^n as follows :

\xi^n(A_1 \times \cdots \times A_n) = \prod_{i=1}^n \xi(A_i)
By monotone class theorem, this uniquely defines the product measure on (S^n,B(S^n)). The expectation E \xi^n(\cdot) is called the n th moment measure. The first moment measure is the mean measure.
Let S = \mathbb{R}^d . The joint intensities of a point process \xi w.r.t. the Lebesgue measure are functions \rho^{(k)} :(\mathbb{R}^d)^k \to [0,\infty) such that for any disjoint bounded Borel subsets B_1,\ldots,B_k

E\left(\prod_i \xi(B_i)\right) = \int_{B_1 \times \cdots \times B_k} \rho^{(k)}(x_1,\ldots,x_k) \, dx_1\cdots dx_k .
Joint intensities do not always exist for point processes. Given that moments of a random 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]}
Stationarity
A point process \xi \subset \mathbb{R}^d is said to be stationary if \xi + x := \sum_{i=1}^N \delta_{X_i + x} has the same distribution as \xi for all x \in \mathbb{R}^d. For a stationary point process, the mean measure E \xi (\cdot) = \lambda \\cdot\ for some constant \lambda \geq 0 and where \\cdot\ stands for the Lebesgue measure. This \lambda is called the intensity of the point process. A stationary point process on \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 & VereJones.^{[2]} It is to be noted that stationarity has been defined and studied for point processes in more general spaces than \mathbb{R}^d.
Examples of point processes
We shall see some examples of point processes in \mathbb{R}^d.
Poisson point process
The simplest and most ubiquitous example of a point process is the Poisson point process, which is a spatial generalisation of the Poisson 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 a Poisson distribution. A Poisson point process can also be defined using these two properties. Namely, we say that a point process \xi is a Poisson point process if the following two conditions hold
1) \xi(B_1),\ldots,\xi(B_n) are independent for disjoint subsets B_1,\ldots,B_n.
2) For any bounded subset B, \xi(B) has a Poisson distribution with parameter \lambda \B\, where \\cdot\ denotes the Lebesgue measure.
The two conditions can be combined together and written as follows : For any disjoint bounded subsets B_1,\ldots,B_n and nonnegative integers k_1,\ldots,k_n we have that

\Pr[\xi(B_i) = k_i, 1 \leq i \leq n] = \prod_i e^{\lambda \B_i\}\frac{(\lambda \B_i\)^{k_i}}{k_i!}.
The constant \lambda is called the intensity of the Poisson point process. Note that the Poisson point process is characterised by the single parameter \lambda. It is a simple, stationary point process. To be more specific one calls the above point process, an homogeneous Poisson point process. An inhomogeneous Poisson process is defined as above but by replacing \lambda \B\ with \stackrel{}{} \int_B\lambda(x) \, dx where \lambda is a nonnegative function on \mathbb{R}^d.
Cox point process
A Cox process (named after Sir David Cox) is a generalisation of the Poisson point process, in that we use random measures in place of \lambda \B\. More formally, let \Lambda be a random measure. A Cox point process driven by the random measure \Lambda is the point process \xi with the following two properties :

Given \Lambda(\cdot), \xi(B) is Poisson distributed with parameter \Lambda(B) for any bounded subset B.

For any finite collection of disjoint subsets B_1,\ldots,B_n and conditioned on \Lambda(B_1),\ldots,\Lambda(B_n), we have that \xi(B_1),\ldots,\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 E \xi(\cdot) = E \Lambda(\cdot) and thus in the special case of a Poisson point process, it is \lambda\\cdot\.
For a Cox point process, \Lambda(\cdot) is called the intensity measure. Further, if \Lambda(\cdot) has a (random) density (Radon–Nikodyn derivative) \lambda(\cdot) i.e.,

\Lambda(B) \stackrel{\text{a.s.}}{=} \int_B \lambda(x) \, dx,
then \lambda(\cdot) is called the intensity 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:^{[9]} \lambda(y) = \exp(X(y)) for a Gaussian random field X(.)

Shot noise Cox point processes:,^{[10]} \lambda(y)= \sum_{X \in \Phi} h(X,y) for a Poisson point process \Phi(\cdot) and kernel h(\cdot , \cdot)

Generalised shot noise Cox point processes:^{[11]} \lambda(y)= \sum_{X \in \Phi} h(X,y) for a point process \Phi(\cdot) and kernel h(. , .)

Lévy based Cox point processes:^{[12]} \lambda(y)= \int h(x,y)L(dx) for a Lévy basis L(\cdot) and kernel h(. , .), and

Permanental Cox point processes:^{[13]} \lambda(y) = X_1^2(y) + \cdots + X_k^2(y) for k independent Gaussian random fields X_i(\cdot)'s

Sigmoidal Gaussian Cox point processes:^{[14]} \lambda(y) = \lambda^{\star}/(1+\exp(X(y))) for a Gaussian random field X(\cdot) and random \lambda^\star > 0
By Jensen's inequality, one can verify that Cox point processes satisfy the following inequality: for all bounded Borel subsets B,

\operatorname{Var}(\xi(B)) \geq \operatorname{Var}(\xi_{\alpha}(B)) ,
where \xi_\alpha stands for a Poisson point process with intensity measure \alpha(\cdot) := E \xi(\cdot) = E \Lambda(\cdot). Thus points are distributed with greater variability in a Cox point process compared to a Poisson point process. This is sometimes called clustering or attractive property of the Cox point process.
Determinantal point processes
An important class of point processes, with applications to physics, random matrix theory, and combinatorics, is that of determinantal point processes.^{[15]}
Point processes on the real halfline
Historically the first point processes that were studied had the real half line R_{+} = [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,^{[16]} in which the points represented events in time, such as calls to a telephone exchange.
Point processes on R_{+} are typically described by giving the sequence of their (random) interevent times (T_{1}, T_{2}, ...), from which the actual sequence (X_{1}, X_{2}, ...) of event times can be obtained as

X_k = \sum_{j=1}^{k} T_j \quad \text{for } k \geq 1.
If the interevent times are independent and identically distributed, the point process obtained is called a renewal process.
Conditional intensity function
The conditional intensity function of a point process on the real halfline is a function λ(t  H_{t}) defined as

\lambda(t \mid H_t)=\lim_{\Delta t\to 0}\frac{1}{\Delta t}{P}(\text{One event occurs in the timeinterval}\,[t,t+\Delta t] \mid H_t) ,
where H_{t} denotes the history of event times preceding time t.
Papangelou intensity function
The Papangelou intensity function of a point process N in the ndimensional Euclidean space \mathbb{R}^n is defined as

\lambda_p(x)=\lim_{\delta \to 0}\frac{1}{B_\delta (x)}{P}\{\text{One event occurs in } \,B_\delta(x)\mid \sigma[N \setminus(B_\delta(x))] \} ,
where B_\delta (x) is the ball centered at x of a radius \delta, and \sigma[N \setminus(B_\delta(x))] denotes the information of the point process N outside B_\delta(x).
Point processes in spatial statistics
The analysis of point pattern data in a compact subset S of R^{n} is a major object of study within spatial statistics. Such data appear in a broad range of disciplines,^{[17]} 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 exhibit complete spatial randomness (i.e. are a realization of a spatial Poisson process) as opposed to exhibiting either spatial aggregation or spatial inhibition.
In contrast, many datasets considered in classical multivariate statistics consist of independently generated datapoints that may be governed by one or several covariates (typically nonspatial).
Apart from the applications in spatial statistics, point processes are one of the fundamental objects in stochastic geometry. Research has also focussed extensively on various models built on point processes such as Voronoi Tessellations, Random geometric graphs, Boolean model etc.
See also
References

^ Kallenberg, O. (1986). Random Measures, 4th edition. Academic Press, New York, London; AkademieVerlag, Berlin. ISBN 0123949602, MR 854102.

^ ^{a} ^{b} ^{c} Daley, D.J, VereJones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York. ISBN 0387966668, MR 950166.

^ Diggle, P. (2003). Statistical Analysis of Spatial Point Patterns, 2nd edition. Arnold, London. ISBN 0340740701.

^ Baddeley, A. (2006). Spatial point processes and their applications. In A. Baddeley, I. Bárány, R. Schneider, and W. Weil, editors, Stochastic Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13–18, 2004, Lecture Notes in Mathematics 1892, Springer. ISBN 3540381740, pp. 1–75

^ Brown, E. N., Kass, R. E., & Mitra, P. P. (2004). Multiple neural spike train data analysis: stateoftheart and future challenges. Nature Neuroscience, 7, 456–461. doi:10.1038/nn1228.

^ Robert F. Engle and Asger Lunde, 2003, "Trades and Quotes: A Bivariate Point Process". Journal of Financial Econometrics Vol. 1, No. 2, pp. 159–188

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^ Gilbert, E.N. (1961) Random plane networks. SIAM Journal, Vol. 9, No. 4.

^ Moller, J.; Syversveen, A. R.; Waagepetersen, R. P. (1998). "Log Gaussian Cox Processes". Scandinavian Journal of Statistics 25 (3): 451.

^ Moller, J. (2003) Shot noise Cox processes, Adv. Appl. Prob., 35.

^ Moller, J. and Torrisi, G.L. (2005) "Generalised Shot noise Cox processes", Adv. Appl. Prob., 37.

^ Hellmund, G., Prokesova, M. and Vedel Jensen, E.B. (2008) "Lévybased Cox point processes", Adv. Appl. Prob., 40.

^ Mccullagh,P. and Moller, J. (2006) "The permanental processes", Adv. Appl. Prob., 38.

^ Adams, R. P., Murray, I. MacKay, D. J. C. (2009) "Tractable inference in Poisson processes with Gaussian process intensities", Proceedings of the 26th International Conference on Machine Learning doi:10.1145/1553374.1553376

^ Hough, J. B., Krishnapur, M., Peres, Y., and Virág, B., Zeros of Gaussian analytic functions and determinantal point processes. University Lecture Series, 51. American Mathematical Society, Providence, RI, 2009.

^ Palm, C. (1943). Intensitätsschwankungen im Fernsprechverkehr (German). Ericsson Technics no. 44, (1943).MR 11402

^ Baddeley, A., Gregori, P., Mateu, J., Stoica, R., and Stoyan, D., editors (2006). Case Studies in Spatial Point Pattern Modelling, Lecture Notes in Statistics No. 185. Springer, New York. ISBN 0387283110.
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