Poisson point processIn probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The Poisson point process is often called simply the Poisson process, but it is also called a Poisson random measure, Poisson random point field or Poisson point field.
Compound Poisson processA compound Poisson process is a continuous-time stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. To be precise, a compound Poisson process, parameterised by a rate and jump size distribution G, is a process given by where, is the counting variable of a Poisson process with rate , and are independent and identically distributed random variables, with distribution function G, which are also independent of When are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process.
Random variableA random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' can be misleading as it is not actually random nor a variable, but rather it is a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads and tails ) in a sample space (e.g., the set ) to a measurable space (e.g., in which 1 corresponding to and −1 corresponding to ), often to the real numbers.
Cox processIn probability theory, a Cox process, also known as a doubly stochastic Poisson process is a point process which is a generalization of a Poisson process where the intensity that varies across the underlying mathematical space (often space or time) is itself a stochastic process. The process is named after the statistician David Cox, who first published the model in 1955.
Point processIn 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.
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