Often this space is " d "-dimensional Euclidean space denoted here by \ textstyle \ textbf { R } ^ { d }, although point processes can be defined on more mathematical spaces.
22.
But the Poisson point process can be defined on the n-dimensional Euclidean space or other mathematical spaces, where it is often interpreted as a random set or a random counting measure, instead of a stochastic process.
23.
In statistics and probability theory, a "'point process "'is a collection of mathematical points randomly located on some underlying mathematical space such as the real line, the Cartesian plane, or more abstract spaces.
24.
A Poisson point process is defined on some underlying mathematical space, called a "'carrier space "', or "'state space "', though the latter term has a different meaning in the context of stochastic processes.
25.
Since these processes are often used to represent collections of points randomly scattered in space, time or both, the underlying space is usually " d "-dimensional Euclidean space denoted here by "'R "'d, but they can be defined on more abstract mathematical spaces.
26.
Since these processes are often used to represent collections of points randomly scattered in space, time or both, the underlying space is usually " d "-dimensional Euclidean space denoted here by \ textstyle \ textbf { R } ^ { d }, but they can be defined on more abstract mathematical spaces.
27.
Since these processes are often used to represent collections of points randomly scattered in physical space, time or both, the underlying space is usually " d "-dimensional Euclidean space denoted here by \ textstyle \ textbf { R } ^ { d }, but they can be defined on more abstract mathematical spaces.
28.
The Poisson point process can be defined, studied and used in one dimension, for example, on the real line, where it can be interpreted as a counting process or part of a queueing model; in higher dimensions such as the plane where it plays a role in stochastic geometry or on more general mathematical spaces.
29.
In other words, for each point of the original Poisson process, there is an independent and identically distributed non-negative random variable, and then the compound Poisson process is then formed from the sum of all the random variables corresponding to points of the Poisson process located in a some region of the underlying mathematical space.
30.
In physics and mathematics, the "'dimension "'of a mathematical space ( or object ) is informally defined as the minimum number of coordinates needed to specify any dimension of two because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere.
