Stochastic

Source: Wikipedia, the free encyclopedia.
For use in technical analysis of financial instruments, see Stochastic oscillator.

Stochastic (

synonymously. Furthermore, in probability theory, the formal concept of a stochastic process is also referred to as a random process.[2][3][4][5][6]

Stochasticity is used in many different fields, including the

telecommunications.[16] It is also used in finance, due to seemingly random changes in financial markets[17][18][19]
as well as in medicine, linguistics, music, media, colour theory, botany, manufacturing, and geomorphology.

Etymology

The word stochastic in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year 1662 as its earliest occurrence.

Joseph Doob.[1] For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term stochastischer Prozeß was used in German by Aleksandr Khinchin,[22][23] though the German term had been used earlier in 1931 by Andrey Kolmogorov.[24]

Mathematics

In the early 1930s, Aleksandr Khinchin gave the first mathematical definition of a stochastic process as a family of random variables indexed by the real line.

Maurice Fréchet, Paul Lévy, Wolfgang Doeblin, and Harald Cramér.[27][28] Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory".[28]

In mathematics, the theory of stochastic processes is an important contribution to probability theory,[29] and continues to be an active topic of research for both theory and applications.[30][31][32]

The word stochastic is used to describe other terms and objects in mathematics. Examples include a

Markov process, and stochastic calculus, which involves differential equations and integrals based on stochastic processes such as the Wiener process
, also called the Brownian motion process.

Natural science

One of the simplest continuous-time stochastic processes is Brownian motion. This was first observed by botanist Robert Brown while looking through a microscope at pollen grains in water.

Physics

The Monte Carlo method is a stochastic method popularized by physics researchers Stanisław Ulam, Enrico Fermi, John von Neumann, and Nicholas Metropolis.[33] The use of randomness and the repetitive nature of the process are analogous to the activities conducted at a casino. Methods of simulation and statistical sampling generally did the opposite: using simulation to test a previously understood deterministic problem. Though examples of an "inverted" approach do exist historically, they were not considered a general method until the popularity of the Monte Carlo method spread.

Perhaps the most famous early use was by Enrico Fermi in 1930, when he used a random method to calculate the properties of the newly discovered

U.S. Air Force
were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields.

Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of pseudorandom number generators, which were far quicker to use than the tables of random numbers which had been previously used for statistical sampling.

Biology

gene promoter—via the solution's Brownian motion
.

Creativity

Simonton (2003, Psych Bulletin) argues that creativity in science (of scientists) is a constrained stochastic behaviour such that new theories in all sciences are, at least in part, the product of a stochastic process.[36]

Computer science

integrand at many randomly chosen points and averages the results to obtain a better approximation. It is essentially an application of the Monte Carlo method to 3D computer graphics, and for this reason is also called Stochastic ray tracing."[citation needed
]

Stochastic forensics analyzes computer crime by viewing computers as stochastic steps.

In

stochastic neural networks, stochastic optimization, genetic algorithms, and genetic programming
. A problem itself may be stochastic as well, as in planning under uncertainty.

Finance

The financial markets use stochastic models to represent the seemingly random behaviour of various financial assets, including the random behavior of the price of one currency compared to that of another (such as the price of US Dollar compared to that of the Euro), and also to represent random behaviour of

quantitative analysts 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
.

Geomorphology

Main article: Meander § Stochastic theory

The formation of river meanders has been analyzed as a stochastic process.

Language and linguistics

Non-deterministic approaches in language studies are largely inspired by the work of

functionalist linguistic theory, which argues that competence is based on performance.[37][38] This distinction in functional theories of grammar should be carefully distinguished from the langue and parole distinction. To the extent that linguistic knowledge is constituted by experience with language, grammar is argued to be probabilistic and variable rather than fixed and absolute. This conception of grammar as probabilistic and variable follows from the idea that one's competence changes in accordance with one's experience with language. Though this conception has been contested,[39] it has also provided the foundation for modern statistical natural language processing[40] and for theories of language learning and change.[41]

Manufacturing

Manufacturing processes are assumed to be

process control
chart which plots a given process control parameter over time. Typically a dozen or many more parameters will be tracked simultaneously. Statistical models are used to define limit lines which define when corrective actions must be taken to bring the process back to its intended operational window.

This same approach is used in the service industry where parameters are replaced by processes related to service level agreements.

Media

The marketing and the changing movement of audience tastes and preferences, as well as the solicitation of and the scientific appeal of certain film and television debuts (i.e., their opening weekends, word-of-mouth, top-of-mind knowledge among surveyed groups, star name recognition and other elements of social media outreach and advertising), are determined in part by stochastic modeling. A recent attempt at repeat business analysis was done by Japanese scholars[

Nielsen ratings
to modern studio and television test audiences.

Medicine

See also:
Stochastic theory of hematopoiesis

Stochastic effect, or "chance effect" is one classification of radiation effects that refers to the random, statistical nature of the damage. In contrast to the deterministic effect, severity is independent of dose. Only the probability of an effect increases with dose.

Music

In music, mathematical processes based on probability can generate stochastic elements.

Stochastic processes may be used in music to compose a fixed piece or may be produced in performance. Stochastic music was pioneered by

Leonard Issacson used generative grammars and Markov chains in their 1957 Illiac Suite. Modern electronic music production techniques make these processes relatively simple to implement, and many hardware devices such as synthesizers and drum machines incorporate randomization features. Generative music
techniques are therefore readily accessible to composers, performers, and producers.

Social sciences

Stochastic social science theory is similar to systems theory in that events are interactions of systems, although with a marked emphasis on unconscious processes. The event creates its own conditions of possibility, rendering it unpredictable if simply for the number of variables involved. Stochastic social science theory can be seen as an elaboration of a kind of 'third axis' in which to situate human behavior alongside the traditional 'nature vs. nurture' opposition. See Julia Kristeva on her usage of the 'semiotic', Luce Irigaray on reverse Heideggerian epistemology, and Pierre Bourdieu on polythetic space for examples of stochastic social science theory.[citation needed]

The term stochastic terrorism has come into frequent use

lone wolf terrorism. The terms "Scripted Violence" and "Stochastic Terrorism" are linked in a "cause <> effect" relationship. "Scripted violence" rhetoric can result in an act of "stochastic terrorism". The phrase "scripted violence" has been used in social science since at least 2002.[44]

Author David Neiwert, who wrote the book Alt-America, told Salon interviewer Chauncey Devega:

Scripted violence is where a person who has a national platform describes the kind of violence that they want to be carried out. He identifies the targets and leaves it up to the listeners to carry out this violence. It is a form of terrorism. It is an act and a social phenomenon where there is an agreement to inflict massive violence on a whole segment of society. Again, this violence is led by people in high-profile positions in the media and the government. They're the ones who do the scripting, and it is ordinary people who carry it out.

Think of it like Charles Manson and his followers. Manson wrote the script; he didn't commit any of those murders. He just had his followers carry them out.[45]

Subtractive 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.

moiré but were used until stochastic screening became available. A stochastic (or frequency modulated
) dot pattern creates a sharper image.

See also

Notes

  1. ^ Doob, when citing Khinchin, uses the term 'chance variable', which used to be an alternative term for 'random variable'.[26]

References

  1. ^ a b c d "Stochastic". Lexico UK English Dictionary. Oxford University Press. Archived from the original on January 2, 2020.
  2. ISBN 978-0-387-48116-6
    .
  3. ISBN 978-0-19-856814-8
    .
  4. ISBN 978-1-107-60655-5
    .
  5. ISBN 9780758172174
    .
  6. ISBN 978-0-387-95313-7
    .
  7. ISBN 978-3-319-08488-6
    .
  8. ISBN 978-0-08-047536-3
    .
  9. ISBN 978-0-19-852525-7
    .
  10. ISBN 978-0-19-923507-0
    .
  11. ISBN 978-3-319-00327-6
    .
  12. ISBN 978-0-8194-2513-3
    .
  13. ISBN 978-1-118-58577-1
    .
  14. ISBN 978-1-4987-6060-7
    .
  15. ISBN 978-1-58488-586-3
    .
  16. ISBN 978-1-60198-264-3
    .
  17. ISBN 978-0-387-95016-7
    .
  18. ISBN 978-3-540-26653-2
    .
  19. ISBN 978-0-387-40101-0
    .
  20. ISBN 978-3-938417-40-9
    .
  21. ISBN 978-3-89971-812-6
    .
  22. ^
    PMID 16587907
    .
  23. S2CID 122842868
    .
  24. S2CID 119439925
    .
  25. ISBN 0471667196
    .
  26. ISSN 0021-9002
    .
  27. ISSN 0006-3444
    .
  28. ^
    ISSN 0091-1798
    .
  29. ^ Applebaum, David (2004). "Lévy processes: From probability to finance and quantum groups". Notices of the AMS. 51 (11): 1336–1347.
  30. ISBN 978-3-03719-072-2
    .
  31. ISBN 978-3-642-54075-2
    .
  32. ISBN 978-3-319-08488-6
    .
  33. ^ Douglas Hubbard "How to Measure Anything: Finding the Value of Intangibles in Business" p. 46, John Wiley & Sons, 2007
  34. PMID 12503175
    .
  35. S2CID 3140340
    .
  36. PMID 12848217
    . Retrieved March 31, 2024.
  37. Talmy Givon, John Haiman, Paul Hopper, Marianne Mithun and Sandra Thompson
    . In its most extreme form (Hopper 1987, 1988), USF rejects the Saussurean dichotomies such as langue vs. parôle. For early interpretivist approaches to focus, see Chomsky (1971) and Jackendoff (1972). parole and synchrony vs. diachrony. All adherents of this tendency feel that the Chomskyan advocacy of a sharp distinction between competence and performance is at best unproductive and obscurantist; at worst theoretically unmotivated. "
  38. ^ Bybee, Joan. "Usage-based phonology." p. 213 in Darnel, Mike (ed). 1999. Functionalism and Formalism in Linguistics: General papers. John Benjamins Publishing Company
  39. ^ Chomsky (1959). Review of Skinner's Verbal Behavior, Language, 35: 26–58
  40. ^ Manning and Schütze, (1999) Foundations of Statistical Natural Language Processing, MIT Press. Cambridge, MA
  41. ^ Bybee (2007) Frequency of use and the organization of language. Oxford: Oxford University Press
  42. ^ Ilias Chrissochoidis, Stavros Houliaras, and Christos Mitsakis, "Set theory in Xenakis' EONTA", in International Symposium Iannis Xenakis, ed. Anastasia Georgaki and Makis Solomos (Athens: The National and Kapodistrian University, 2005), 241–249.
  43. YouTube published August 12, 2019 CNN
  44. S2CID 145617529
    .
  45. ^ DeVega, Chauncey (1 November 2018). "Author David Neiwert on the outbreak of political violence". Salon. Retrieved 13 December 2018.

Further reading

External links

  • The dictionary definition of stochastic at Wiktionary
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