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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.
A visual depiction of a Poisson point process starting. In probability theory, statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
The thinning operation entails using some predefined rule to remove points from a point process to form a new point process .These thinning rules may be deterministic, that is, not random, which is the case for one of the simplest rules known as -thinning: [1] each point of is independently removed (or kept) with some probability (or ).
A point process is often denoted by a single letter, [1] [7] [8] for example , and if the point process is considered as a random set, then the corresponding notation: [1], is used to denote that a random point is an element of (or belongs to) the point process . The theory of random sets can be applied to point processes owing to this ...
Point processes have a number of interpretations, which is reflected by the various types of point process notation. [4] [9] For example, if a point belongs to or is a member of a point process, denoted by , then this can be written as: [4], and represents the point process being interpreted as a random set.
If is a generating ring of then a simple point process is uniquely determined by its values on the sets . This means that two simple point processes ξ {\displaystyle \xi } and ζ {\displaystyle \zeta } have the same distributions iff
In mathematics, a determinantal point process is a stochastic point process, the probability distribution of which is characterized as a determinant of some function. They are suited for modelling global negative correlations, and for efficient algorithms of sampling, marginalization, conditioning, and other inference tasks.
Point processes have a number of interpretations, which is reflected by the various types of point process notation. [ 3 ] [ 7 ] For example, if a point x {\displaystyle \textstyle x} belongs to or is a member of a point process, denoted by N {\displaystyle \textstyle {N}} , then this can be written as: [ 3 ]