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Details of stochastic

Stochastic, from the Greek "stochos" or "aim, guess", means of, relating to, or characterized by conjecture and randomness. A stochastic process is one whose behavior is non-deterministic in that a state does not fully determine its next state.

Mathematical theory

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

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, stochastic optimization, 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 (classically speaking) each molecule is moving in a deterministic path, the motion of 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 to compose a fixed piece or can be 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).

Colour reproduction

When colour reproductions are made, the image is separated into its component colors by taking multiple photographs filtered for each colour. One resultant film or plate represents each of the cyan, magenta, yellow, and black data. Colour 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 moiré but were used until stochastic screening became available. A stochastic (or frequency modulated) dot pattern creates a more photorealistic image.

Language and linguistics

Non-deterministic approaches in language studies are largely inspired by the work of Ferdinand de Saussure. In usage-based linguistic theories, for example, 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 one's experience with linguistic units. This way, the frequency of usage-events determines one's knowledge of the language in question. For much later work in this area, see Julia Kristeva on her usage of the 'semiotic,' Luce Irigaray on reverse Heideggerian epistomology, and Pierre Bourdieu on polythetic space for examples of stochastic social science theory.

Finance

The financial markets use stochastic models such as a Stochastic oscillator to value options on stock prices, bond prices, and on interest rates. Moreover, it is at the heart of the insurance industry.

Further reading

• Formalized Music: Thought and Mathematics in Composition by Iannis Xenakis, ISBN 1-57647-079-2
• Frequency and the Emergence of Linguistic Structure by Joan Bybee and Paul Hopper (eds.), ISBN 1-58811-028-1/ISBN 90-272-2948-1 (Eur.)