Stochastic
Stochastic, from the
Greek "stochos" or "goal", means of, relating to, or characterized by
conjecture and
randomness. 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. An
antonym is
astochastic, because its being one is predicated on having an opposite.
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.
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.
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.
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 archived
http://www.wfmu.org/playlists/HPIn 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."
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.
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 one's experience with linguistic units. This way, the frequency of
usage-events determines one's knowledge of the language in question. 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.
The financial markets use stochastic models to value options on stock prices, bond prices, and on interest rates. Moreover, it is at the heart of the insurance industry.
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.)
