Point process

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^[1][2] and a powerful tool 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, and others. Point processes on the real line form an important special case that is particularly amenable to study,^[5] 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.

Contents 1 General point process theory 1.1 Definition 1.2 Representation 1.3 Expectation measure 1.4 Laplace functional 2 Point processes in spatial statistics 3 Point processes on the real half-line 3.1 Conditional intensity function 3.2 Papangelou intensity function 4 See also 5 References

General point process theory

Definition

Let S be locally compact second countable Hausdorff space equipped with its Borel σ-algebra B. Write for the set of locally finite counting measures on S and for the smallest σ-algebra on that renders all the point counts

for relatively compact sets B in B measurable.
A point process on S is a measurable map

from a probability space to the measurable space .

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^{[citation needed]} that the term point process is not a very good one if S is not a subset of the real line, as it might suggests 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

where δ 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,

Laplace functional

The Laplace functional Ψ_N(f) of a point process N is a map from athe set of all positive valued functions f on the state space of N, to defined as follows:

Ψ_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^[6], 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^[7], 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.

Conditional intensity function

The conditional intensity function of a point process on the real half-line is a function λ(t|H_t) defined as

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 n-dimensional Euclidean space is defined as:

where B_δ(x) is the ball centered at x of a radius δ, and denotes the information of the point process N outside B_δ(x).

See also

• Empirical measure
• Random measure
• Poisson process
• Renewal theory

References

1. ^ Kallenberg, O. (1986). Random Measures, 4th edition. Academic Press, New York, London; Akademie-Verlag, Berlin. ISBN 0-123-94960-2, MR854102.
2. ^ Daley, D.J, Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York. ISBN 0-387-96666-8, MR950166.
3. ^ Diggle, P. (2003). Statististical Analysis of Spatial Point Patterns, 2nd edition. Arnold, London. ISBN 0-340-74070-1.
4. ^ 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 3-540-38174-0, pp. 1--75
5. ^ Last, G., Brandt, A. (1995). Marked point processes on the real line. The dynamic approach. Probability and its Applications. Springer, New York. ISBN 0-387-94547-4, MR1353912
6. ^ 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 0-387-28311-0.
7. ^ Palm, C. (1943). Intensitätsschwankungen im Fernsprechverkehr (German). Ericsson Technics no. 44, (1943).MR11402