Stochastic, from the Greek "stochos" or "goal", means of, relating to, or characterized by conjecture; conjectural; random. A stochastic process is one whose behavior is non-deterministic in that the next state of the environment is not fully determined by the previous state of the environment. In the mathematics of probability, a stochastic process can be thought of as a random function. ...

Mathematical theory

In mathematics, specifically in probability theory, the field of stochastic processes has for some decades been a major area of research, to which hundreds of researchers have devoted their careers. See that article for more. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Probability theory is the mathematical study of probability. ... In the mathematics of probability, a stochastic process can be thought of as 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 stochastic matrix is a square matrix whose columns are probability vectors, i. ... For the square matrix section, see square matrix. ... Please refer to Real vs. ...

Artificial intelligence

In artificial intelligence stochastic programs work by using probabilistic methods to solve problems, as in simulated annealing 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. Artificial intelligence (also known as machine intelligence and often abbreviated as AI) is intelligence exhibited by any manufactured (i. ... 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. ... 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 unpredictable (this is an example of chaos out of order). 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 can be thought of as a random function. ... Pressure (symbol: p or P) is the measure of the force that acts on a unit area. ... Gas (actually as, part of the Gnu Binutils package) is the default Gcc Back-end. ... 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 an art, entertainment, or other human activity which involves organized sound, though definitions may vary. ... Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ...

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 (May 29, 1922, Romania - February 4, 2001) was a Greek composer who spent much of his life in Paris, France. ... The word probability derives from the Latin probare (to prove, or to test). ... Game theory is a branch of applied mathematics that uses models to study interactions with formalised incentive structures (games). It has applications in a variety of fields, including economics, international relations, evolutionary biology, political science, and military strategy. ... 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. ... In mathematics and computer science, Boolean algebras, or Boolean lattices, are algebraic structures which capture the essence of the logical operations AND, OR and NOT as well as the corresponding set theoretic operations intersection, union and complement. ... The tower of a personal computer (specifically a Power Mac G5). ... John Cage John Milton Cage (September 5, 1912–August 12, 1992) was an experimental music composer and writer, possibly best known (some might say notorious) for his piece 4′ 33″, often described (somewhat erroneously) as four and a half minutes of silence. ... 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 (Simplified Chinese: 易经; Traditional Chinese: 易經, Hanyu Pinyin: Yì Jīng; Cantonese IPA: ; Cantonese Jyutping: jik6ging1; alternative romanizations include I Jing, Yi Ching, Yi King) is the oldest of the Chinese classic texts. ...

Visual arts

In the visual arts, Yoshiyuki Abe[1] (http://www.pli.jp), 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 (color separated through RGB filters). One resultant film, or plate etc. represents the cyan, magenta, yellow, and black data. Color printing is a binary system, therefore all color separations to be printed, have to 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). ... 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. Wikipedia does not yet have an article with this exact name. ... A street musician with accordion in Bremen A performance comprises an event in which generally one group of people (the performer or performers) behave in a particular way for the benefit of another group of people (the viewer or viewers, or audience). ... Parole can have different meanings depending on the context. ... In theology and philosophy, probabilism (from Latin probare, to test, approve) holds that in the absence of certainty, probability is the best criterion. ... Wikipedia does not yet have an article with this exact name. ...

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.)