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[4] or of searches on the world-wide web.
General point processes on a Euclidean space can be used for spatial 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.
[clarification needed] To define general point processes, we start with a probability space
Every instance (or event) of a point process ξ can be represented as where
denotes the Dirac measure, n is an integer-valued random variable and
of a point process N is a map from the set of all positive valued functions f on the state space of N, to
defined as follows: They play a similar role as the characteristic functions for random variable.
One important theorem says that: two point processes have the same law if their Laplace functionals are equal.
as follows : By monotone class theorem, this uniquely defines the product measure on
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
is a Poisson point process if the following two conditions hold 1)
The two conditions can be combined and written as follows : For any disjoint bounded subsets
Note that the Poisson point process is characterised by the single parameter
A Cox point process driven by the random measure
with the following two properties : It is easy to see that Poisson point process (homogeneous and inhomogeneous) follow as special cases of Cox point processes.
is called the intensity field of the Cox point process.
There have been many specific classes of Cox point processes that have been studied in detail such as: By Jensen's inequality, one can verify that Cox point processes satisfy the following inequality: for all bounded Borel subsets
stands for a Poisson point process with intensity measure
This is sometimes called clustering or attractive property of the Cox point process.
is a kernel function which expresses the positive influence of past events
is a possibly non-stationary function representing the expected, predictable, or deterministic part of the intensity, and
[29] 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,[30] 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 If the inter-event times are independent and identically distributed, the point process obtained is called a renewal process.
The intensity λ(t | Ht) of a point process on the real half-line with respect to a filtration Ht is defined as Ht can denote the history of event-point times preceding time t but can also correspond to other filtrations (for example in the case of a Cox process).
-notation, this can be written in a more compact form: The compensator of a point process, also known as the dual-predictable projection, is the integrated conditional intensity function defined by The Papangelou intensity function of a point process
The logarithmic likelihood of a parameterized simple point process conditional upon some observed data is written as The analysis of point pattern data in a compact subset S of Rn is a major object of study within spatial statistics.
In contrast, many datasets considered in classical multivariate 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 in stochastic geometry.