Stochastic, from the Greek "stochos" or "goal", means of, relating to, or characterized by conjecture; conjectural; random. The antonym is astochastic. In ordinary language, the word random is used to express apparent lack of purpose or cause. ... Antonyms, from the Greek anti (against) and onoma (name) are word pairs that are opposite in meaning, such as hot and cold, fat and thin, and up and down. ...

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. In the mathematics of probability, a stochastic process is a random function. ... The term deterministic may refer to: the more general notion of determinism from philosophy, see determinism a type of algorithm as discussed in computer science, see deterministic algorithm scientific determinism as used by Karl Popper and Stephen Hawking deterministic system in mathematics deterministic system in philosophy deterministic finite state machine...

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. Euclid, detail from The School of Athens by Raphael. ... Probability theory is the mathematical study of probability. ... In the mathematics of probability, a stochastic process is a random function. ...

A stochastic matrix is a matrix that has non-negative real entries that sum to 1 in each column. In mathematics, especially in probability theory and statistics, and also in linear algebra and computer science, a left stochastic matrix is a square matrix whose columns are probability vectors, i. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

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. Hondas intelligent humanoid robot AI redirects here. ... Simulated annealing (SA) is a generic probabilistic meta-algorithm for the global optimization problem, namely locating a good approximation to the global optimum of a given function in a large search space. ... Stochastic neural networks are a type of artificial neural networks, which is a tool of artificial intelligence. ... A genetic algorithm (GA) is an algorithm used to find approximate solutions to difficult-to-solve problems through application of the principles of evolutionary biology to computer science. ... The term deterministic may refer to: the more general notion of determinism from philosophy, see determinism a type of algorithm as discussed in computer science, see deterministic algorithm scientific determinism as used by Karl Popper and Stephen Hawking deterministic system in mathematics deterministic system in philosophy deterministic finite state machine...

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. In the mathematics of probability, a stochastic process is a random function. ... Pressure (symbol: p) is the force per unit area acting on a surface in a direction perpendicular to that surface. ... A gas is one of the ninety three main phases of matter (after solid and liquid,it could possibly arouse you if you inhale it and followed by plasma), that subsequently appear as a solid material is subjected to increasingly higher temperatures. ... Emergence is the process of deriving some new and coherent structures, patterns and properties in a complex system. ...

Music

In music stochastic elements are randomly generated elements created by strict mathematical processes. Music is conceptual time expressed in the structures of tones and silence. ... Euclid, detail from The School of Athens by Raphael. ...

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). Iannis Xenakis Iannis Xenakis (Ιάννης Ξενάκης) (May 29, 1922 Brăila – February 4, 2001 Paris) was a Greek composer and architect who spent much of his life in Paris. ... The word probability derives from the Latin probare (to prove, or to test). ... Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. ... Group theory is that branch of mathematics concerned with the study of groups. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ... A computer is a machine designed for manipulating data according to a list of instructions known as a program. ... John Cage For the character of John Cage from the TV show Ally McBeal see: John Cage (Character) John Milton Cage (September 5, 1912 – August 12, 1992) was an American experimental music composer, writer and visual artist. ... Aleatoric (or aleatory) music or composition, is music where some element of the composition is left to chance. ... Indeterminate music was a form of music pioneered by the late John Cage. ... The I Ching or Book of Changes is the oldest of the Chinese classic texts. ...

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. Color printing is the reproduction of an image or text in color (as opposed to simpler black and white or monochrome printing). ... Amplitude modulation (AM) is a form of modulation in which the amplitude of a carrier wave is varied in direct proportion to that of a modulating signal. ... A moiré pattern is an interference pattern created when two grids are overlaid at an angle, or when they have slightly different mesh sizes. ...

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. Look up competence, incompetence in Wiktionary, the free dictionary. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Parole can have different meanings depending on the context. ... The word probability derives from the Latin probare (to prove, or to test). ... Look up competence, incompetence in Wiktionary, the free dictionary. ...

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.)