fbpx
Wikipedia

Stochastic

For use in technical analysis of financial instruments, see Stochastic oscillator.
This article may have too many section headers dividing up its content. Please help improve the article by merging similar sections and removing unneeded subheaders.(January 2020) ()

Stochastic (from Greek στόχος (stókhos) 'aim, guess') refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselves, these two terms are often used synonymously. Furthermore, in probability theory, the formal concept of a stochastic process is also referred to as a random process.

Stochasticity is used in many different fields, including the natural sciences such as biology, chemistry, ecology, neuroscience, and physics, as well as technology and engineering fields such as image processing, signal processing, information theory, computer science, cryptography, and telecommunications. It is also used in finance, due to seemingly random changes in financial markets as well as in medicine, linguistics, music, media, colour theory, botany, manufacturing, and geomorphology. Stochastic modeling is also used in social science.

Contents

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. In his work on probability Ars Conjectandi, originally published in Latin in 1713, Jakob Bernoulli used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics". This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz, who in 1917 wrote in German the word stochastik with a sense meaning random. The term stochastic process first appeared in English in a 1934 paper by Joseph Doob. 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, though the German term had been used earlier in 1931 by Andrey Kolmogorov.

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. Further fundamental work on probability theory and stochastic processes was done by Khinchin as well as other mathematicians such as Andrey Kolmogorov, Joseph Doob, William Feller, Maurice Fréchet, Paul Lévy, Wolfgang Doeblin, and Harald Cramér. Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory".

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

The word stochastic is used to describe other terms and objects in mathematics. Examples include a stochastic matrix, which describes a stochastic process known as 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.

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.

The Monte Carlo method is a stochastic method popularized by physics researchers Stanisław Ulam, Enrico Fermi, John von Neumann, and Nicholas Metropolis. 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 neutron. Monte Carlo methods were central to the simulations required for the Manhattan Project, though they were severely limited by the computational tools of the time. Therefore, it was only after electronic computers were first built (from 1945 on) that Monte Carlo methods began to be studied in depth. In the 1950s they were used at Los Alamos for early work relating to the development of the hydrogen bomb, and became popularized in the fields of physics, physical chemistry, and operations research. The RAND Corporation and the 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.

Stochastic resonance: In biological systems, introducing stochastic "noise" has been found to help improve the signal strength of the internal feedback loops for balance and other vestibular communication. It has been found to help diabetic and stroke patients with balance control. Many biochemical events also lend themselves to stochastic analysis. Gene expression, for example, has a stochastic component through the molecular collisions—as during binding and unbinding of RNA polymerase to a gene promoter—via the solution's Brownian motion.

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.

Stochastic ray tracing is the application of Monte Carlo simulation to the computer graphics ray tracing algorithm. "Distributed ray tracing samples the 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 processes.

In artificial intelligence, stochastic programs work by using probabilistic methods to solve problems, as in simulated annealing, stochastic neural networks, stochastic optimization, genetic algorithms, and genetic programming. A problem itself may be stochastic as well, as in planning under uncertainty.

The financial markets use stochastic models to represent the seemingly random behaviour of assets such as stocks, commodities, relative currency prices (i.e., the price of one currency compared to that of another, such as the price of US Dollar compared to that of the Euro), and interest rates. These models are then used by 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.

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

Non-deterministic approaches in language studies are largely inspired by the work of Ferdinand de Saussure, for example, in functionalist linguistic theory, which argues that competence is based on performance. 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, it has also provided the foundation for modern statistical natural language processing and for theories of language learning and change.

Manufacturing processes are assumed to be stochastic processes. This assumption is largely valid for either continuous or batch manufacturing processes. Testing and monitoring of the process is recorded using a 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.

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[citation needed] and is part of the Cinematic Contagion Systems patented by Geneva Media Holdings, and such modeling has been used in data collection from the time of the original Nielsen ratings to modern studio and television test audiences.

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.

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 Iannis Xenakis, who coined the term stochastic music. Specific examples of mathematics, statistics, and physics applied to music composition are the use of the statistical mechanics of gases in Pithoprakta, statistical distribution of points on a plane in Diamorphoses, minimal constraints in Achorripsis, the normal distribution in ST/10 and Atrées, Markov chains in Analogiques, game theory in Duel and Stratégie, group theory in Nomos Alpha (for Siegfried Palm), set theory in Herma and Eonta, and Brownian motion in N'Shima.[citation needed] Xenakis frequently used computers to produce his scores, such as the ST series including Morsima-Amorsima and Atrées, and founded CEMAMu. 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). Lejaren Hiller and 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.

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 fallen into frequent use with regard to 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.

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.

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 work-flow. Traditional line screens 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 sharper image.

  1. Doob, when citing Khinchin, uses the term 'chance variable', which used to be an alternative term for 'random variable'.
  1. "Stochastic". Lexico UK English Dictionary. Oxford University Press. n.d.
  2. Robert J. Adler; Jonathan E. Taylor (29 January 2009). Random Fields and Geometry. Springer Science & Business Media. pp. 7–8. ISBN 978-0-387-48116-6.
  3. David Stirzaker (2005). Stochastic Processes and Models. Oxford University Press. p. 45. ISBN 978-0-19-856814-8.
  4. Loïc Chaumont; Marc Yor (19 July 2012). Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, Via Conditioning. Cambridge University Press. p. 175. ISBN 978-1-107-60655-5.
  5. Murray Rosenblatt (1962). Random Processes. Oxford University Press. p. 91. ISBN 9780758172174.
  6. Olav Kallenberg (8 January 2002). Foundations of Modern Probability. Springer Science & Business Media. pp. 24 and 25. ISBN 978-0-387-95313-7.
  7. Paul C. Bressloff (22 August 2014). Stochastic Processes in Cell Biology. Springer. ISBN 978-3-319-08488-6.
  8. N.G. Van Kampen (30 August 2011). Stochastic Processes in Physics and Chemistry. Elsevier. ISBN 978-0-08-047536-3.
  9. Russell Lande; Steinar Engen; Bernt-Erik Sæther (2003). Stochastic Population Dynamics in Ecology and Conservation. Oxford University Press. ISBN 978-0-19-852525-7.
  10. Carlo Laing; Gabriel J Lord (2010). Stochastic Methods in Neuroscience. OUP Oxford. ISBN 978-0-19-923507-0.
  11. Wolfgang Paul; Jörg Baschnagel (11 July 2013). Stochastic Processes: From Physics to Finance. Springer Science & Business Media. ISBN 978-3-319-00327-6.
  12. Edward R. Dougherty (1999). Random processes for image and signal processing. SPIE Optical Engineering Press. ISBN 978-0-8194-2513-3.
  13. Thomas M. Cover; Joy A. Thomas (28 November 2012). Elements of Information Theory. John Wiley & Sons. p. 71. ISBN 978-1-118-58577-1.
  14. Michael Baron (15 September 2015). Probability and Statistics for Computer Scientists, Second Edition. CRC Press. p. 131. ISBN 978-1-4987-6060-7.
  15. Jonathan Katz; Yehuda Lindell (2007-08-31). Introduction to Modern Cryptography: Principles and Protocols. CRC Press. p. 26. ISBN 978-1-58488-586-3.
  16. François Baccelli; Bartlomiej Blaszczyszyn (2009). Stochastic Geometry and Wireless Networks. Now Publishers Inc. pp. 200–. ISBN 978-1-60198-264-3.
  17. J. Michael Steele (2001). Stochastic Calculus and Financial Applications. Springer Science & Business Media. ISBN 978-0-387-95016-7.
  18. Marek Musiela; Marek Rutkowski (21 January 2006). Martingale Methods in Financial Modelling. Springer Science & Business Media. ISBN 978-3-540-26653-2.
  19. Steven E. Shreve (3 June 2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer Science & Business Media. ISBN 978-0-387-40101-0.
  20. O. B. Sheĭnin (2006). Theory of probability and statistics as exemplified in short dictums. NG Verlag. p. 5. ISBN 978-3-938417-40-9.
  21. Oscar Sheynin; Heinrich Strecker (2011). Alexandr A. Chuprov: Life, Work, Correspondence. V&R unipress GmbH. p. 136. ISBN 978-3-89971-812-6.
  22. Doob, Joseph (1934). "Stochastic Processes and Statistics". Proceedings of the National Academy of Sciences of the United States of America. 20 (6): 376–379. Bibcode:1934PNAS...20..376D. doi:10.1073/pnas.20.6.376. PMC1076423. PMID 16587907.
  23. Khintchine, A. (1934). "Korrelationstheorie der stationeren stochastischen Prozesse". Mathematische Annalen. 109 (1): 604–615. doi:10.1007/BF01449156. ISSN 0025-5831. S2CID 122842868.
  24. Kolmogoroff, A. (1931). "Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung". Mathematische Annalen. 104 (1): 1. doi:10.1007/BF01457949. ISSN 0025-5831. S2CID 119439925.
  25. Vere-Jones, David (2006). "Khinchin, Aleksandr Yakovlevich". Encyclopedia of Statistical Sciences. p. 4. doi:10.1002/0471667196.ess6027.pub2. ISBN 0471667196.
  26. Snell, J. Laurie (2005). "Obituary: Joseph Leonard Doob". Journal of Applied Probability. 42 (1): 251. doi:10.1239/jap/1110381384. ISSN 0021-9002.
  27. Bingham, N. (2000). "Studies in the history of probability and statistics XLVI. Measure into probability: from Lebesgue to Kolmogorov". Biometrika. 87 (1): 145–156. doi:10.1093/biomet/87.1.145. ISSN 0006-3444.
  28. Cramer, Harald (1976). "Half a Century with Probability Theory: Some Personal Recollections". The Annals of Probability. 4 (4): 509–546. doi:10.1214/aop/1176996025. 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. Jochen Blath; Peter Imkeller; Sylvie Roelly (2011). Surveys in Stochastic Processes. European Mathematical Society. pp. 5–. ISBN 978-3-03719-072-2.
  31. Michel Talagrand (12 February 2014). Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems. Springer Science & Business Media. pp. 4–. ISBN 978-3-642-54075-2.
  32. Paul C. Bressloff (22 August 2014). Stochastic Processes in Cell Biology. Springer. pp. vii–ix. 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. Hänggi, P. (2002). "Stochastic Resonance in Biology How Noise Can Enhance Detection of Weak Signals and Help Improve Biological Information Processing". ChemPhysChem. 3 (3): 285–90. doi:10.1002/1439-7641(20020315)3:3<285::AID-CPHC285>3.0.CO;2-A. PMID 12503175.
  35. Priplata, A.; et al. (2006). "Noise-Enhanced Balance Control in Patients with Diabetes and Patients with Stroke"(PDF). Ann Neurol. 59 (1): 4–12. doi:10.1002/ana.20670. PMID 16287079. S2CID 3140340.
  36. Newmeyer, Frederick. 2001. "The Prague School and North American functionalist approaches to syntax" Journal of Linguistics 37, pp. 101–126. "Since most American functionalists adhere to this trend, I will refer to it and its practitioners with the initials 'USF'. Some of the more prominent USFs are Joan Bybee, William Croft, 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. "
  37. Bybee, Joan. "Usage-based phonology." p. 213 in Darnel, Mike (ed). 1999. Functionalism and Formalism in Linguistics: General papers. John Benjamins Publishing Company
  38. Chomsky (1959). Review of Skinner's Verbal Behavior, Language, 35: 26–58
  39. Manning and Schütze, (1999) Foundations of Statistical Natural Language Processing, MIT Press. Cambridge, MA
  40. Bybee (2007) Frequency of use and the organization of language. Oxford: Oxford University Press
  41. 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.
  42. Anthony Scaramucci says he does not support President Trump's reelection on YouTube published August 12, 2019 CNN
  43. Hamamoto, Darrell Y. (2002). "Empire of Death: Militarized Society and the Rise of Serial Killing and Mass Murder". New Political Science. 24 (1): 105–120. doi:10.1080/07393140220122662. S2CID 145617529.
  44. DeVega, Chauncey (1 November 2018). "Author David Neiwert on the outbreak of political violence". Salon. Retrieved13 December 2018.
  • The dictionary definition of stochastic at Wiktionary

Stochastic
Stochastic Article Talk Language Watch Edit For use in technical analysis of financial instruments see Stochastic oscillator This article may have too many section headers dividing up its content Please help improve the article by merging similar sections and removing unneeded subheaders January 2020 Learn how and when to remove this template message Stochastic from Greek stoxos stokhos aim guess 1 refers to the property of being well described by a random probability distribution 1 Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselves these two terms are often used 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 natural sciences such as biology 7 chemistry 8 ecology 9 neuroscience 10 and physics 11 as well as technology and engineering fields such as image processing signal processing 12 information theory 13 computer science 14 cryptography 15 and 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 Stochastic modeling is also used in social science Contents 1 Etymology 2 Mathematics 3 Natural science 4 Physics 5 Biology 6 Creativity 7 Computer science 8 Finance 9 Geomorphology 10 Language and linguistics 11 Manufacturing 12 Media 13 Medicine 14 Music 15 Social sciences 16 Subtractive color reproduction 17 See also 18 Notes 19 References 20 Further reading 21 External linksEtymology EditThe 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 1 In his work on probability Ars Conjectandi originally published in Latin in 1713 Jakob Bernoulli used the phrase Ars Conjectandi sive Stochastice which has been translated to the art of conjecturing or stochastics 20 This phrase was used with reference to Bernoulli by Ladislaus Bortkiewicz 21 who in 1917 wrote in German the word stochastik with a sense meaning random The term stochastic process first appeared in English in a 1934 paper by 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 EditIn 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 25 22 a Further fundamental work on probability theory and stochastic processes was done by Khinchin as well as other mathematicians such as Andrey Kolmogorov Joseph Doob William Feller Maurice Frechet Paul Levy Wolfgang Doeblin and Harald Cramer 27 28 Decades later Cramer 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 stochastic matrix which describes a stochastic process known as 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 EditOne 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 EditThe Monte Carlo method is a stochastic method popularized by physics researchers Stanislaw 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 neutron Monte Carlo methods were central to the simulations required for the Manhattan Project though they were severely limited by the computational tools of the time Therefore it was only after electronic computers were first built from 1945 on that Monte Carlo methods began to be studied in depth In the 1950s they were used at Los Alamos for early work relating to the development of the hydrogen bomb and became popularized in the fields of physics physical chemistry and operations research The RAND Corporation and the 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 EditStochastic resonance In biological systems introducing stochastic noise has been found to help improve the signal strength of the internal feedback loops for balance and other vestibular communication 34 It has been found to help diabetic and stroke patients with balance control 35 Many biochemical events also lend themselves to stochastic analysis Gene expression for example has a stochastic component through the molecular collisions as during binding and unbinding of RNA polymerase to a gene promoter via the solution s Brownian motion Creativity EditSimonton 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 Computer science EditStochastic ray tracing is the application of Monte Carlo simulation to the computer graphics ray tracing algorithm Distributed ray tracing samples the 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 processes In artificial intelligence stochastic programs work by using probabilistic methods to solve problems as in simulated annealing stochastic neural networks stochastic optimization genetic algorithms and genetic programming A problem itself may be stochastic as well as in planning under uncertainty Finance EditThe financial markets use stochastic models to represent the seemingly random behaviour of assets such as stocks commodities relative currency prices i e the price of one currency compared to that of another such as the price of US Dollar compared to that of the Euro and interest rates These models are then used by 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 EditMain article Meander Stochastic theory The formation of river meanders has been analyzed as a stochastic processLanguage and linguistics EditNon deterministic approaches in language studies are largely inspired by the work of Ferdinand de Saussure for example in functionalist linguistic theory which argues that competence is based on performance 36 37 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 38 it has also provided the foundation for modern statistical natural language processing 39 and for theories of language learning and change 40 Manufacturing EditManufacturing processes are assumed to be stochastic processes This assumption is largely valid for either continuous or batch manufacturing processes Testing and monitoring of the process is recorded using a 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 EditThe 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 citation needed and is part of the Cinematic Contagion Systems patented by Geneva Media Holdings and such modeling has been used in data collection from the time of the original Nielsen ratings to modern studio and television test audiences Medicine EditSee 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 EditIn 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 Iannis Xenakis who coined the term stochastic music Specific examples of mathematics statistics and physics applied to music composition are the use of the statistical mechanics of gases in Pithoprakta statistical distribution of points on a plane in Diamorphoses minimal constraints in Achorripsis the normal distribution in ST 10 and Atrees Markov chains in Analogiques game theory in Duel and Strategie group theory in Nomos Alpha for Siegfried Palm set theory in Herma and Eonta 41 and Brownian motion in N Shima citation needed Xenakis frequently used computers to produce his scores such as the ST series including Morsima Amorsima and Atrees and founded CEMAMu 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 Lejaren Hiller and 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 EditStochastic 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 fallen into frequent use 42 with regard to lone wolf terrorism The terms Scripted Violence and Stochastic Terrorism are linked in a cause lt gt 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 43 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 44 Subtractive color reproduction EditWhen 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 work flow Traditional line screens 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 sharper image See also EditJump process Sortition Stochastic processNotes Edit Doob when citing Khinchin uses the term chance variable which used to be an alternative term for random variable 26 References Edit a b c d Stochastic Lexico UK English Dictionary Oxford University Press n d Robert J Adler Jonathan E Taylor 29 January 2009 Random Fields and Geometry Springer Science amp Business Media pp 7 8 ISBN 978 0 387 48116 6 David Stirzaker 2005 Stochastic Processes and Models Oxford University Press p 45 ISBN 978 0 19 856814 8 Loic Chaumont Marc Yor 19 July 2012 Exercises in Probability A Guided Tour from Measure Theory to Random Processes Via Conditioning Cambridge University Press p 175 ISBN 978 1 107 60655 5 Murray Rosenblatt 1962 Random Processes Oxford University Press p 91 ISBN 9780758172174 Olav Kallenberg 8 January 2002 Foundations of Modern Probability Springer Science amp Business Media pp 24 and 25 ISBN 978 0 387 95313 7 Paul C Bressloff 22 August 2014 Stochastic Processes in Cell Biology Springer ISBN 978 3 319 08488 6 N G Van Kampen 30 August 2011 Stochastic Processes in Physics and Chemistry Elsevier ISBN 978 0 08 047536 3 Russell Lande Steinar Engen Bernt Erik Saether 2003 Stochastic Population Dynamics in Ecology and Conservation Oxford University Press ISBN 978 0 19 852525 7 Carlo Laing Gabriel J Lord 2010 Stochastic Methods in Neuroscience OUP Oxford ISBN 978 0 19 923507 0 Wolfgang Paul Jorg Baschnagel 11 July 2013 Stochastic Processes From Physics to Finance Springer Science amp Business Media ISBN 978 3 319 00327 6 Edward R Dougherty 1999 Random processes for image and signal processing SPIE Optical Engineering Press ISBN 978 0 8194 2513 3 Thomas M Cover Joy A Thomas 28 November 2012 Elements of Information Theory John Wiley amp Sons p 71 ISBN 978 1 118 58577 1 Michael Baron 15 September 2015 Probability and Statistics for Computer Scientists Second Edition CRC Press p 131 ISBN 978 1 4987 6060 7 Jonathan Katz Yehuda Lindell 2007 08 31 Introduction to Modern Cryptography Principles and Protocols CRC Press p 26 ISBN 978 1 58488 586 3 Francois Baccelli Bartlomiej Blaszczyszyn 2009 Stochastic Geometry and Wireless Networks Now Publishers Inc pp 200 ISBN 978 1 60198 264 3 J Michael Steele 2001 Stochastic Calculus and Financial Applications Springer Science amp Business Media ISBN 978 0 387 95016 7 Marek Musiela Marek Rutkowski 21 January 2006 Martingale Methods in Financial Modelling Springer Science amp Business Media ISBN 978 3 540 26653 2 Steven E Shreve 3 June 2004 Stochastic Calculus for Finance II Continuous Time Models Springer Science amp Business Media ISBN 978 0 387 40101 0 O B Sheĭnin 2006 Theory of probability and statistics as exemplified in short dictums NG Verlag p 5 ISBN 978 3 938417 40 9 Oscar Sheynin Heinrich Strecker 2011 Alexandr A Chuprov Life Work Correspondence V amp R unipress GmbH p 136 ISBN 978 3 89971 812 6 a b Doob Joseph 1934 Stochastic Processes and Statistics Proceedings of the National Academy of Sciences of the United States of America 20 6 376 379 Bibcode 1934PNAS 20 376D doi 10 1073 pnas 20 6 376 PMC 1076423 PMID 16587907 Khintchine A 1934 Korrelationstheorie der stationeren stochastischen Prozesse Mathematische Annalen 109 1 604 615 doi 10 1007 BF01449156 ISSN 0025 5831 S2CID 122842868 Kolmogoroff A 1931 Uber die analytischen Methoden in der Wahrscheinlichkeitsrechnung Mathematische Annalen 104 1 1 doi 10 1007 BF01457949 ISSN 0025 5831 S2CID 119439925 Vere Jones David 2006 Khinchin Aleksandr Yakovlevich Encyclopedia of Statistical Sciences p 4 doi 10 1002 0471667196 ess6027 pub2 ISBN 0471667196 Snell J Laurie 2005 Obituary Joseph Leonard Doob Journal of Applied Probability 42 1 251 doi 10 1239 jap 1110381384 ISSN 0021 9002 Bingham N 2000 Studies in the history of probability and statistics XLVI Measure into probability from Lebesgue to Kolmogorov Biometrika 87 1 145 156 doi 10 1093 biomet 87 1 145 ISSN 0006 3444 a b Cramer Harald 1976 Half a Century with Probability Theory Some Personal Recollections The Annals of Probability 4 4 509 546 doi 10 1214 aop 1176996025 ISSN 0091 1798 Applebaum David 2004 Levy processes From probability to finance and quantum groups Notices of the AMS 51 11 1336 1347 Jochen Blath Peter Imkeller Sylvie Roelly 2011 Surveys in Stochastic Processes European Mathematical Society pp 5 ISBN 978 3 03719 072 2 Michel Talagrand 12 February 2014 Upper and Lower Bounds for Stochastic Processes Modern Methods and Classical Problems Springer Science amp Business Media pp 4 ISBN 978 3 642 54075 2 Paul C Bressloff 22 August 2014 Stochastic Processes in Cell Biology Springer pp vii ix ISBN 978 3 319 08488 6 Douglas Hubbard How to Measure Anything Finding the Value of Intangibles in Business p 46 John Wiley amp Sons 2007 Hanggi P 2002 Stochastic Resonance in Biology How Noise Can Enhance Detection of Weak Signals and Help Improve Biological Information Processing ChemPhysChem 3 3 285 90 doi 10 1002 1439 7641 20020315 3 3 lt 285 AID CPHC285 gt 3 0 CO 2 A PMID 12503175 Priplata A et al 2006 Noise Enhanced Balance Control in Patients with Diabetes and Patients with Stroke PDF Ann Neurol 59 1 4 12 doi 10 1002 ana 20670 PMID 16287079 S2CID 3140340 Newmeyer Frederick 2001 The Prague School and North American functionalist approaches to syntax Journal of Linguistics 37 pp 101 126 Since most American functionalists adhere to this trend I will refer to it and its practitioners with the initials USF Some of the more prominent USFs are Joan Bybee William Croft 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 parole 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 Bybee Joan Usage based phonology p 213 in Darnel Mike ed 1999 Functionalism and Formalism in Linguistics General papers John Benjamins Publishing Company Chomsky 1959 Review of Skinner s Verbal Behavior Language 35 26 58 Manning and Schutze 1999 Foundations of Statistical Natural Language Processing MIT Press Cambridge MA Bybee 2007 Frequency of use and the organization of language Oxford Oxford University Press 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 Anthony Scaramucci says he does not support President Trump s reelection on YouTube published August 12 2019 CNN Hamamoto Darrell Y 2002 Empire of Death Militarized Society and the Rise of Serial Killing and Mass Murder New Political Science 24 1 105 120 doi 10 1080 07393140220122662 S2CID 145617529 DeVega Chauncey 1 November 2018 Author David Neiwert on the outbreak of political violence Salon Retrieved 13 December 2018 Further reading Edit private video See the stochastic process of an 8 foot tall 2 4 m Probability Machine comparing stock market returns to the randomness of the beans dropping through the quincunx pattern on YouTube from Index Funds Advisors IFA com 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 The Stochastic Empirical Loading and Dilution Model provides documentation and computer code for modeling stochastic processes in Visual Basic for Applications External links Edit The dictionary definition of stochastic at Wiktionary Retrieved from https en wikipedia org w index php title Stochastic amp oldid 1054403473, wikipedia, wiki, book,

books

, library,

article

, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.