# 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 to value options on stock prices, bond prices, and on interest rates, see Markov models. Moreover, it is at the heart of the insurance industry.

Not to be confused with Stochastic Oscillators in Technical Analysis.

• 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.)
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In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic.
random is used to express lack of order, purpose, cause, or predictability in non-scientific parlance. A random process is a repeating process whose outcomes follow no describable deterministic pattern, but follow a probability distribution.
A stochastic process, or sometimes random process, is the opposite of a deterministic process (or deterministic system) in probability theory. Instead of dealing only with one possible 'reality' of how the process might evolve under time (as is the case, for example, for
Determinism is the philosophical proposition that every event, including human cognition and behavior, decision and action, is causally determined by an unbroken chain of prior occurrences.
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
Probability theory is the branch of mathematics concerned with analysis of random phenomena.[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities
A stochastic process, or sometimes random process, is the opposite of a deterministic process (or deterministic system) in probability theory. Instead of dealing only with one possible 'reality' of how the process might evolve under time (as is the case, for example, for
In mathematics, a stochastic matrix, probability matrix, or transition matrix describes the transitions of a Markov chain. It has found use in probability theory, statistics and linear algebra, as well as computer science.
matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied.
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
artificial intelligence (or AI) is "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions which maximizes its chances of success.
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. It was independently presented by S. Kirkpatrick, C. D. Gelatt and M. P.
Stochastic neural networks are a type of artificial neural networks, which is a tool of artificial intelligence. They are built by introducing random variations into the network, either by giving the network's neurons stochastic transfer functions, or by giving them stochastic
Stochastic optimization algorithms are optimization algorithms which satisfy one or both of the following properties (Spall, 2003):
1. There is random noise in the measurements of the criterion to be optimized and/or related information (such as the gradient vector of the

A genetic algorithm (GA) is a search used in computing to find exact or approximate solutions to optimization and search problems. Genetic algorithms are categorized as global search heuristics.
Determinism is the philosophical proposition that every event, including human cognition and behavior, decision and action, is causally determined by an unbroken chain of prior occurrences.
A stochastic process, or sometimes random process, is the opposite of a deterministic process (or deterministic system) in probability theory. Instead of dealing only with one possible 'reality' of how the process might evolve under time (as is the case, for example, for
Pressure (symbol: p) is the force per unit area applied on a surface in a direction perpendicular to that surface.

Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.
Gas is one of the four major states of matter, consisting of freely moving atoms or molecules without a definite shape. Compared to the solid and liquid states of matter a gas has lower density and a lower viscosity.
emergence refers to the way complex systems and patterns arise out of a multiplicity of relatively simple interactions. Like intelligence in the field of AI, or agents in distributed artificial intelligence, emergence is central to the theory of complex systems and yet very
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Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
Iannis Xenakis (Γιάννης Ξενάκης) (May 29 1922 - February 4 2001) was a Greek composer and one of the most important modernist composers of the 20th century.
Probability is the likelihood that something is the case or will happen. Probability theory is used extensively in areas such as statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of
Game theory is a branch of applied mathematics that is often used in the context of economics. It studies strategic interactions between agents. In strategic games, agents choose strategies which will maximize their return, given the strategies the other agents choose.
Group theory is the mathematical study of symmetry, as embodied in the structures known as groups. These are sets with a closed binary operation satisfying the following three properties:
1. The operation must be associative.
2. There must be an identity element.