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Stochastic, from the Greek "stochos" or "goal", means of, relating to, or characterized by conjecture; conjectural; random. The antonym is astochastic.

A stochastic process is one whose behavior is non-deterministic in that the next state of the environment is partially but not fully determined by the previous state of the environment.

Mathematical theory

In mathematics, specifically in probability theory, the field of stochastic processes has for some decades been a major area of research. See that article for more.

A stochastic matrix is a matrix that has non-negative real entries that sum to 1 in each column.

Artificial intelligence

In artificial intelligence stochastic programs work by using probabilistic methods to solve problems, as in simulated annealing, neural networks and genetic algorithms. A problem itself may be stochastic as well, as in planning under uncertainty. A deterministic environment is much simpler for an agent to deal with.

Natural science

An example of a stochastic process in the natural world is pressure in a gas. Even though each molecule is moving deterministically, a collection of them is computationally and practically unpredictable. A large enough set of molecules will exhibit stochastic characteristics, such as filling the container, exerting equal pressure, diffusing along concentration gradients, etc. These are emergent properties of the system.

Music

In music stochastic elements are randomly generated elements created by strict mathematical processes.

Stochastic processes can be used in music either to compose a fixed piece, or produced in performance. Stochastic music was pioneered by Iannis Xenakis, who used probability, game theory, group theory, set theory, and Boolean algebra, and frequently used computers to produce his scores. Earlier, John Cage and others had composed aleatoric or indeterminate music, which is created by chance processes but does not have the strict mathematical basis (Cage's Music of Changes, for example, uses a system of charts based on the I-Ching).

There is a radio show called the Stochastic Hit Parade. All shows are archieved http://www.wfmu.org/playlists/HP

Visual arts

In the visual arts, Yoshiyuki Abe[1], has mastered the art of creation through stochastic process. His work uses geometric objects, mostly the surfaces of hyperbolic paraboloids, and the processing of stochastic elements. In his words: "No matter how you use a computer, or whichever computer you use, to create an art work is not easy. Nevertheless, I believe artists can find a new horizon in his/her creative activities by having the experience of using geometric object and/or stochastic process. For artists who want to create mathematical art through algorithm-driven parameter control, the essential element for success is artistic serendipity. This is the interesting fact of art in the perfect mathematical space."

Color reproduction

When color reproductions are made, the image is separated into its component colors by taking multiple photographs filtered for each color. One resultant film or plate represents each of the cyan, magenta, yellow, and black data. Color printing is a binary system, where ink is either present or not present, so all color separations to be printed must be translated into dots at some stage of the workflow. Traditional linescreens which are amplitude modulated had problems with moire but were used until stochastic screening became available. A stochastic (or frequency modulated) dot pattern creates a more photorealistic image.

Language and linguistics

In usage-based linguistic theories, where it is argued that competence, or langue, is based on performance, or parole, in the sense that linguistic knowledge is based on frequency of experience, grammar is often said to be probabilistic and variable rather than fixed and absolute. This is so, because one's competence changes in accordance with ones experience with linguistic units. This way, the frequency of usage-events determines one's knowledge of the language in question.

Further reading

  • Formalized Music: Thought and Mathematics in Composition by Iannis Xenakis, ISBN 1576470792
  • Frequency and the Emergence of Linguistic Structure by Joan Bybee and Paul Hopper (eds.), ISBN 902722943/ISBN 9027229481 (Eur.)

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