Сандық теория: Нүктелік процестер мен олардың қолданылуы

Probability Theory
In statistics and probability theory, a point process or point field is a collection of mathematical points randomly located on a mathematical space such as the real line or Euclidean space. Point processes can be used for spatial data analysis, which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications, computational neuroscience, economics and others. There are different mathematical interpretations of a point process, such as a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear. which corresponds to the index set in stochastic process terminology. on which it is defined, such as the real line or dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes. Point processes on the real line form an important special case that is particularly amenable to study, because the 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, location of radio stations in a telecommunication network or of searches on the world wide web.

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.

The expectation measure Eξ (also known as mean measure) 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,

An important class of point processes, with applications to physics, random matrix theory, and combinatorics, is that of determinantal point processes.

Given a sequence of non negative random variables , if they are independent and the cdf of is given by for , where is a positive constant, then is called a geometric process (GP). The geometric process has several extensions, including the α series process and the doubly geometric process.

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

If the inter event times are independent and identically distributed, the point process obtained is called a renewal process.