Difference between revisions of "Stochastic process"

Latest revision as of 17:30, 20 April 2016

In probability theory, a stochastic process, or often random process, is a collection of random variables, representing the evolution of some system of random values over time.

Description

This is the probabilistic counterpart to a deterministic process (or deterministic system). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.

In the simple case of discrete time, as opposed to continuous time, a stochastic process is a sequence of random variables (for example, see Markov chain, also known as discrete-time Markov chain). The random variables corresponding to various times may be completely different, the only requirement being that these different random quantities all take values in the same space (the codomain of the function). One approach may be to model these random variables as random functions of one or several deterministic arguments (in most cases, the time parameter). Although the random values of a stochastic process at different times may be independent random variables, in most commonly considered situations they exhibit complicated statistical dependence.

Familiar examples of stochastic processes include stock market and exchange rate fluctuations, signals such as speech, audio and video, medical data such as a patient's EKG, EEG, blood pressure or temperature, and random movement such as Brownian motion or random walks.

Random field

A generalization, the random field, is defined by letting the variables be parametrized by members of a topological space instead of time. Examples of random fields include static images, random terrain (landscapes), wind waves or composition variations of a heterogeneous material.

See also

• Mathematics
• Probability theory
• Random field

External links

• Stochastic process @ Wikipedia