In probability and statistics, point process notation comprises the range of mathematical notation used to symbolically represent random objects known as point processes, which are used in related fields such as stochastic geometry, spatial statistics and continuum percolation theory and frequently serve as mathematical models of random phenomena, representable as points, in time, space or both.
Contents
- Interpretation of point processes
- Random sequences of points
- Random set of points
- Random measures
- Dual notation
- Sums
- Expectations
- Uses in other fields
- References
The notation varies due to the histories of certain mathematical fields and the different interpretations of point processes, and borrows notation from mathematical areas of study such as measure theory and set theory.
Interpretation of point processes
The notation, as well as the terminology, of point processes depends on their setting and interpretation as mathematical objects which under certain assumptions can be interpreted as a random sequences of points, random sets of points or random counting measures.
Random sequences of points
In some mathematical frameworks, a given point process may be considered as a sequence of points with each point randomly positioned in d-dimensional Euclidean space Rd as well as some other more abstract mathematical spaces. In general, whether or not a random sequence is equivalent to the other interpretations of a point process depends on the underlying mathematical space, but this holds true for the setting of finite-dimensional Euclidean space Rd.
Random set of points
A point process is called simple if no two (or more points) coincide in location with probability one. Given that often point processes are simple and the order of the points does not matter, a collection of random points can be considered as a random set of points The theory of random sets was independently developed by David Kendall and Georges Matheron. In terms of being considered as a random set, a sequence of random points is a random closed set if the sequence has no accumulation points with probability one
A point process is often denoted by a single letter, for example
is used to denote that a random point
which highlights its interpretation as either a random sequence or random closed set of points. Furthermore, sometimes an uppercase letter denotes the point process, while a lowercase denotes a point from the process, so, for example, the point
Random measures
To denote the number of points of
where
On the other hand, the symbol:
represents the number of points of
to denote that there is the set
Dual notation
The different interpretations of point processes as random sets and counting measures is captured with the often used notation in which:
Denoting the counting measure again with
Sums
If
which has the random sequence appearance, or with set notation as:
or, equivalently, with integration notation as:
where
The dual interpretation of point processes is illustrated when writing the number of
where the indicator function
Expectations
The average or expected value of a sum of functions over a point process is written as:
where (in the random measure sense)
which is also known as the first moment measure of
Uses in other fields
Point processes are employed in other mathematical and statistical disciplines, hence the notation may be used in fields such stochastic geometry, spatial statistics or continuum percolation theory, and areas which use the methods and theory from these fields.