Point processes

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.

Point processes are well studied objects in probability theory and a powerful tool in statistics for modeling and analyzing spatial data, which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, economics and others. Point processes on the real line form an important special case that is particularly amenable to study, 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, or of searches on the world-wide web.

Definition
Let S be locally compact second countable Hausdorff space equipped with its Borel &sigma;-algebra B. Write $$\mathfrak{N}$$ for the set of locally finite counting measures on S and $$\mathcal{N}$$ for the smallest &sigma;-algebra 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 Rn 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 of Rn, 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 a integer-valued random variable and $$X_i$$ are random elements of S.

Expectation measure
The expectation measure Eξ 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.

Point processes in spatial statistics
The analysis of point pattern data in a compact subset S of Rn is a major object of study within spatial statistics. Such data appear in a broad range of disciplines, 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 indepently generated datapoints that may be governed by one or several covariates (typically non-spatial).

Point processes on the real half-line
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, 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) inter-event times (T1, T2,...), from which the actual sequence (X1, X2,...) of event times can be obtained as


 * $$ X_k = \sum_{j=1}^{k} T_j \quad \text{for } k \geq 1. $$

If the inter-event 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 half-line is a function λ(t|Ht) defined as



\lambda(t| H_{t})=\lim_{\Delta t\to 0}\frac{1}{\Delta t}{P}(\mbox{One event occurs in the time-interval}\,[t,t+\Delta t]\,|\, H_t) ,$$

where Ht denotes the history of event times preceding time t.

Papangelou intensity function
The Papangelou intensity function of a point process $$N$$ in the $$n$$-dimensional Euclidean space $$ \mathbb{R}^n$$ is defined as: $$ \lambda_p(x)=\lim_{\delta \to 0}\frac{1}{|B_\delta (x)|}{P}\{\mbox{One event occurs in } \,B_\delta(x)\,|\, \sigma[N \setminus(B_\delta(x))] \} , $$

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 $$B_\delta(x)$$.