Editing Stochastic

{{short description|Randomly determined process}}'''Stochastic''' ({{IPAc-en|s|t|ə|ˈ|k|æ|s|t|ɪ|k}}; {{etymology|grc|''{{wikt-lang|grc|στόχος}}'' ({{grc-transl|στόχος}})|aim, guess}})<ref name="OxfordStochastic"/> is the property of being well-described by a [[random]] [[probability distribution]].<ref name="OxfordStochastic">{{Cite dictionary |url=http://www.lexico.com/definition/Stochastic |archive-url=https://web.archive.org/web/20200102160730/https://www.lexico.com/definition/stochastic |url-status=dead |archive-date=January 2, 2020 |title=Stochastic |dictionary=[[Lexico]] UK English Dictionary |publisher=[[Oxford University Press]]}}</ref> ''Stochasticity'' and ''randomness'' are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; in everyday conversation, however, these terms are often used [[synonymous|interchangeably]]. In [[probability theory]], the formal concept of a ''[[stochastic process]]'' is also referred to as a ''random process''.<ref name="AdlerTaylor2009page7">{{cite book|author1=Robert J. Adler|author2=Jonathan E. Taylor|title=Random Fields and Geometry|url=https://books.google.com/books?id=R5BGvQ3ejloC|date=29 January 2009|publisher=Springer Science & Business Media|isbn=978-0-387-48116-6|pages=7–8}}</ref><ref name="Stirzaker2005page45">{{cite book|author=David Stirzaker|title=Stochastic Processes and Models|url=https://books.google.com/books?id=0avUelS7e7cC|year=2005|publisher=Oxford University Press|isbn=978-0-19-856814-8|page=45}}</ref><ref name="ChaumontYor2012">{{cite book|author1=Loïc Chaumont|author2=Marc Yor|title=Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, Via Conditioning|url=https://books.google.com/books?id=1dcqV9mtQloC&pg=PR4|date=19 July 2012|publisher=Cambridge University Press|isbn=978-1-107-60655-5|page=175}}</ref><ref name="Rosenblatt1962page91">{{cite book|author=Murray Rosenblatt|title=Random Processes|url=https://books.google.com/books?id=5-lQAAAAMAAJ|year=1962|publisher=Oxford University Press|page=91|isbn=9780758172174}}</ref><ref name="Kallenberg2002page24">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|date=8 January 2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|pages=24 and 25}}</ref>

Stochasticity is used in many different fields, including [[image processing]], [[signal processing]], [[computer science]], [[information theory]], [[telecommunications]],<ref name="Bressloff2014">{{cite book|author=[[Paul Bressloff|Paul C. Bressloff]]|title=Stochastic Processes in Cell Biology|url=https://books.google.com/books?id=SwZYBAAAQBAJ|date=22 August 2014|publisher=Springer|isbn=978-3-319-08488-6}}</ref> [[chemistry]],<ref name="Kampen2011">{{cite book|author=N.G. Van Kampen|title=Stochastic Processes in Physics and Chemistry|url=https://books.google.com/books?id=N6II-6HlPxEC|date=30 August 2011|publisher=Elsevier|isbn=978-0-08-047536-3}}</ref> [[ecology]],<ref name="LandeEngen2003">{{cite book|author1=Russell Lande|author2=Steinar Engen|author3=Bernt-Erik Sæther|title=Stochastic Population Dynamics in Ecology and Conservation|url=https://books.google.com/books?id=6KClauq8OekC|year=2003|publisher=Oxford University Press|isbn=978-0-19-852525-7}}</ref> [[neuroscience]],<ref name="LaingLord2010">{{cite book|author1=Carlo Laing|author2=Gabriel J Lord|title=Stochastic Methods in Neuroscience|url=https://books.google.com/books?id=RaYSDAAAQBAJ|year=2010|publisher=OUP Oxford|isbn=978-0-19-923507-0}}</ref> [[physics]],<ref name="PaulBaschnagel2013">{{cite book|author1=Wolfgang Paul|author2=Jörg Baschnagel|title=Stochastic Processes: From Physics to Finance|url=https://books.google.com/books?id=OWANAAAAQBAJ|date=11 July 2013|publisher=Springer Science & Business Media|isbn=978-3-319-00327-6}}</ref><ref name="Dougherty1999">{{cite book|author=Edward R. Dougherty|title=Random processes for image and signal processing|url=https://books.google.com/books?id=ePxDAQAAIAAJ|year=1999|publisher=SPIE Optical Engineering Press|isbn=978-0-8194-2513-3}}</ref><ref name="CoverThomas2012page71">{{cite book|author1=Thomas M. Cover|author2=Joy A. Thomas|title=Elements of Information Theory|url=https://books.google.com/books?id=VWq5GG6ycxMC&pg=PT16|date=28 November 2012|publisher=John Wiley & Sons|isbn=978-1-118-58577-1|page=71}}</ref><ref name="Baron2015">{{cite book|author=Michael Baron|title=Probability and Statistics for Computer Scientists, Second Edition|url=https://books.google.com/books?id=CwQZCwAAQBAJ|date=15 September 2015|publisher=CRC Press|isbn=978-1-4987-6060-7|page=131}}</ref> and [[cryptography]].<ref>{{cite book|author1=Jonathan Katz|author2=Yehuda Lindell|title=Introduction to Modern Cryptography: Principles and Protocols|url=https://books.google.com/books?id=ddsrGdsgN9sC&pg=PA269|date=2007-08-31|publisher=CRC Press|isbn=978-1-58488-586-3|page=26}}</ref><ref name="BaccelliBlaszczyszyn2009">{{cite book|author1=François Baccelli|author2=Bartlomiej Blaszczyszyn|title=Stochastic Geometry and Wireless Networks|url=https://books.google.com/books?id=H3ZkTN2pYS4C&pg=PA1|year=2009|publisher=Now Publishers Inc|isbn=978-1-60198-264-3|pages=200–}}</ref> It is also used in finance (e.g., [[stochastic oscillator]]), due to seemingly random changes in the different markets within the financial sector and in medicine, linguistics,  music, media, colour theory,  botany, manufacturing and geomorphology.<ref name="Steele2001">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=H06xzeRQgV4C|year=2001|publisher=Springer Science & Business Media|isbn=978-0-387-95016-7}}</ref><ref name="MusielaRutkowski2006">{{cite book|author1=Marek Musiela|author2=Marek Rutkowski|title=Martingale Methods in Financial Modelling|url=https://books.google.com/books?id=iojEts9YAxIC|date=21 January 2006|publisher=Springer Science & Business Media|isbn=978-3-540-26653-2}}</ref><ref name="Shreve2004">{{cite book|author=Steven E. Shreve|title=Stochastic Calculus for Finance II: Continuous-Time Models|url=https://books.google.com/books?id=O8kD1NwQBsQC|date=3 June 2004|publisher=Springer Science & Business Media|isbn=978-0-387-40101-0}}</ref>

==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.<ref name="OxfordStochastic"/> 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".<ref name="Sheĭnin2006page5">{{cite book|author=O. B. Sheĭnin|title=Theory of probability and statistics as exemplified in short dictums|url=https://books.google.com/books?id=XqMZAQAAIAAJ|year=2006|publisher=NG Verlag|isbn=978-3-938417-40-9|page=5}}</ref> This phrase was used, with reference to Bernoulli, by [[Ladislaus Bortkiewicz]],<ref name="SheyninStrecker2011page136">{{cite book|author1=Oscar Sheynin|author2=Heinrich Strecker|title=Alexandr A. Chuprov: Life, Work, Correspondence|url=https://books.google.com/books?id=1EJZqFIGxBIC&pg=PA9|year=2011|publisher=V&R unipress GmbH|isbn=978-3-89971-812-6|page=136}}</ref> 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 L. Doob]].<ref name="OxfordStochastic"/> 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]],<ref name="Doob1934">{{cite journal|last1=Doob|first1=Joseph|title=Stochastic Processes and Statistics|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=20|issue=6|year=1934|pages=376–379|doi=10.1073/pnas.20.6.376|pmc=1076423|pmid=16587907|bibcode=1934PNAS...20..376D|doi-access=free}}</ref><ref name="Khintchine1934">{{cite journal|last1=Khintchine|first1=A.|title=Korrelationstheorie der stationeren stochastischen Prozesse|journal=Mathematische Annalen|volume=109|issue=1|year=1934|pages=604–615|issn=0025-5831|doi=10.1007/BF01449156|s2cid=122842868}}</ref> though the German term had been used earlier in 1931 by [[Andrey Kolmogorov]].<ref name="Kolmogoroff1931page1">{{cite journal|last1=Kolmogoroff|first1=A.|title=Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung|journal=Mathematische Annalen|volume=104|issue=1|year=1931|page=1|issn=0025-5831|doi=10.1007/BF01457949|s2cid=119439925}}</ref>

==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.<ref name="Vere-Jones2006page4">{{cite book|last1=Vere-Jones|first1=David|title=Encyclopedia of Statistical Sciences|chapter=Khinchin, Aleksandr Yakovlevich|page=4|year=2006|doi=10.1002/0471667196.ess6027.pub2|isbn=0471667196}}</ref><ref name="Doob1934"/>{{efn|Doob, when citing Khinchin, uses the term 'chance variable', which used to be an alternative term for 'random variable'.<ref name="Snell2005">{{cite journal|last1=Snell|first1=J. Laurie|title=Obituary: Joseph Leonard Doob|journal=Journal of Applied Probability|volume=42|issue=1|year=2005|page=251|issn=0021-9002|doi=10.1239/jap/1110381384|doi-access=free}}</ref> }} 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 (mathematician)|Paul Lévy]], [[Wolfgang Doeblin]], and [[Harald Cramér]].<ref name="Bingham2000">{{cite journal|last1=Bingham|first1=N.|title=Studies in the history of probability and statistics XLVI. Measure into probability: from Lebesgue to Kolmogorov|journal=Biometrika|volume=87|issue=1|year=2000|pages=145–156|issn=0006-3444|doi=10.1093/biomet/87.1.145}}</ref><ref name="Cramer1976">{{cite journal|last1=Cramer|first1=Harald|title=Half a Century with Probability Theory: Some Personal Recollections|journal=The Annals of Probability|volume=4|issue=4|year=1976|pages=509–546|issn=0091-1798|doi=10.1214/aop/1176996025|doi-access=free}}</ref> Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory".<ref name="Cramer1976"/>

In mathematics, the theory of stochastic processes is an important contribution to [[probability theory]],<ref name="Applebaum2004">{{cite journal|last1=Applebaum|first1=David|title=Lévy processes: From probability to finance and quantum groups|journal=Notices of the AMS|volume=51|issue=11|year=2004|pages=1336–1347}}</ref> and continues to be an active topic of research for both theory and applications.<ref name="BlathImkeller2011">{{cite book|author1=Jochen Blath|author2=Peter Imkeller|author3=Sylvie Roelly|author3-link=Sylvie Roelly|title=Surveys in Stochastic Processes|url=https://books.google.com/books?id=CyK6KAjwdYkC&pg=PR5|year=2011|publisher=European Mathematical Society|isbn=978-3-03719-072-2|pages=5–}}</ref><ref name="Talagrand2014">{{cite book|author=Michel Talagrand|title=Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems|url=https://books.google.com/books?id=tfa5BAAAQBAJ&pg=PR4|date=12 February 2014|publisher=Springer Science & Business Media|isbn=978-3-642-54075-2|pages=4–}}</ref><ref name="Bressloff2014VII">{{cite book|author=Paul C. Bressloff|title=Stochastic Processes in Cell Biology|url=https://books.google.com/books?id=SwZYBAAAQBAJ&pg=PA1|date=22 August 2014|publisher=Springer|isbn=978-3-319-08488-6|pages=vii–ix}}</ref>

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 equation]]s and  [[integral]]s 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 (botanist, born 1773)|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]].<ref>Douglas Hubbard "How to Measure Anything: Finding the Value of Intangibles in Business" p. 46, John Wiley & Sons, 2007</ref> 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 [[simulation]]s 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 National Laboratory|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 generator]]s, which were far quicker to use than the tables of random numbers which had been previously used for statistical sampling.

==Biology==
In biological systems the technique of [[stochastic resonance]] - introducing stochastic "noise" - has been found to help improve the signal-strength of the internal feedback-loops for balance and other [[vestibular system|vestibular]] communication.<ref>{{Cite journal | doi = 10.1002/1439-7641(20020315)3:3<285::AID-CPHC285>3.0.CO;2-A | title = Stochastic Resonance in Biology How Noise Can Enhance Detection of Weak Signals and Help Improve Biological Information Processing | journal = ChemPhysChem | volume = 3 | issue = 3 | pages = 285–90 | year = 2002 | last1 = Hänggi | first1 = P. | pmid=12503175| url = https://opus.bibliothek.uni-augsburg.de/opus4/frontdoor/index/index/docId/28997 }}</ref> The technique has helped diabetic and stroke patients with balance control.<ref>{{cite journal | last1 = Priplata | first1 = A. | display-authors = etal  | year = 2006 | title = Noise-Enhanced Balance Control in Patients with Diabetes and Patients with Stroke | url = http://www.bu.edu/abl/files/fulltext.pdf | journal = Ann Neurol | volume = 59 | issue = 1 | pages = 4–12 | doi = 10.1002/ana.20670 | pmid = 16287079 | s2cid = 3140340 }}</ref>

Many biochemical events lend themselves to stochastic analysis. [[Gene expression]], for example, has a stochastic component through the molecular collisions—e.g., during binding and unbinding of [[RNA polymerase]] to a [[gene promoter]] which contributes to bursts of transcription and super-Poissonian variability in cell-to-cell RNA distributions <ref>{{Cite journal |last1=Dar |first1=Roy D. |last2=Razooky |first2=Brandon S. |last3=Singh |first3=Abhyudai |last4=Trimeloni |first4=Thomas V. |last5=McCollum |first5=James M. |last6=Cox |first6=Chris D. |last7=Simpson |first7=Michael L. |last8=Weinberger |first8=Leor S. |date=2012-10-23 |title=Transcriptional burst frequency and burst size are equally modulated across the human genome |journal=Proceedings of the National Academy of Sciences |volume=109 |issue=43 |pages=17454–17459 |doi=10.1073/pnas.1213530109 |doi-access=free |pmc=3491463 |pmid=23064634|bibcode=2012PNAS..10917454D }}</ref>—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]].<ref>{{cite journal |last=Simonton |first=Dean Keith |date=July 2003 |title=Scientific creativity as constrained stochastic behavior: the integration of product, person, and process perspectives. |url=https://psycnet.apa.org/record/2003-06077-003 |journal=Psychological Bulletin |volume=129 |issue=4 |pages=475–94 |doi=10.1037/0033-2909.129.4.475 |pmid=12848217 |access-date=March 31, 2024}}</ref>

==Computer science==
[[Stochastic ray tracing]] is the application of [[Monte Carlo simulation]] to the [[computer graphics]] [[Ray tracing (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|date=October 2013}}

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

In [[artificial intelligence]], stochastic programs work by using probabilistic methods to solve problems, as in [[simulated annealing]], [[stochastic neural network]]s, [[stochastic optimization]], [[genetic algorithm]]s, 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 [[interest rate]]s. These models are then used by financial analysts to value options on stock prices, bond prices, and on interest rates, see [[Markov chain|Markov models]]. Moreover, it is at the heart of the [[insurance industry]].

==Geomorphology==
{{main|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 [[Ferdinand de Saussure]], for example, in [[Functional theories of grammar|functionalist linguistic theory]], which argues that [[Linguistic competence|competence]] is based on [[Linguistic performance|performance]].<ref>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 (linguist)|William Croft]], [[Talmy Givon]], [[John Haiman]], [[Paul J. Hopper|Paul Hopper]], [[Marianne Mithun]] and [[Sandra Thompson (linguist)|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.
"</ref><ref>Bybee, Joan. "Usage-based phonology." p. 213 in Darnel, Mike (ed). 1999. Functionalism and Formalism in Linguistics: General papers. John Benjamins Publishing Company</ref> This distinction in functional theories of grammar should be carefully distinguished from the [[langue and parole|''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,<ref>Chomsky (1959). Review of Skinner's Verbal Behavior, Language, 35: 26–58</ref> it has also provided the foundation for modern statistical natural language processing<ref>Manning and Schütze, (1999) [https://books.google.com/books?id=YiFDxbEX3SUC Foundations of Statistical Natural Language Processing], MIT Press. Cambridge, MA</ref> and for theories of language learning and change.<ref>Bybee (2007) Frequency of use and the organization of language. Oxford: Oxford University Press</ref>

== Manufacturing ==
Manufacturing processes are assumed to be [[stochastic process]]es.  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==
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|date=October 2013}} 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 ==
{{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.{{cn|date=April 2025}} In contrast to the deterministic effect, severity is independent of dose. Only the ''probability'' of an effect increases with dose.{{cn|date=April 2025}}

==Music==
<!--[[Stochastic music]] redirects directly here.-->
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 [[Constraint (mathematics)|constraints]] in ''Achorripsis'', the [[normal distribution]] in ''ST/10'' and ''Atrées'', [[Markov chain]]s in ''Analogiques'', [[game theory]] in ''Duel'' and ''Stratégie'', [[group theory]] in ''[[Nomos Alpha]]'' (for [[Siegfried Palm]]), [[set theory]] in ''Herma'' and ''[[Eonta]]'',<ref>Ilias Chrissochoidis, Stavros Houliaras, and Christos Mitsakis, [https://www.academia.edu/249265/Set_theory_in_Xenakis_EONTA "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.</ref> and [[Brownian motion]] in ''N'Shima''.{{citation needed|date=May 2013}} Xenakis frequently used [[computer music|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 music|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 grammar]]s and [[Markov chain]]s 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|date=August 2011}}

The term [[stochastic terrorism]] has come into frequent use<ref>{{YouTube|id=2DvLUjIB3-I&|title=Anthony Scaramucci says he does not support President Trump's reelection}} published August 12, 2019 [[CNN]]</ref> 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.<ref name="Hamamoto-2002">{{cite journal |title=Empire of Death: Militarized Society and the Rise of Serial Killing and Mass Murder |author=Hamamoto, Darrell Y. |journal=New Political Science |year=2002 |volume=24 |issue=1 |pages=105–120|doi=10.1080/07393140220122662 |s2cid=145617529 }}</ref>

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

{{quote|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.{{pb}}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.<ref name="devega-neiwert-salon-2018">{{cite news |url=https://www.salon.com/2018/11/01/author-david-neiwert-on-the-outbreak-of-political-violence-expect-an-intense-period-of-terrorism/ |title=Author David Neiwert on the outbreak of political violence |work=Salon |date=1 November 2018 |access-date=13 December 2018 |author=DeVega, Chauncey}}</ref>}}

== 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. [[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 screen]]s 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.

== See also ==
* [[Jump process]]
* [[Sortition]]
* [[Stochastic process]]

==Notes==
{{notelist}}

== References ==
{{reflist}}

==Further reading==
* ''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==
* {{Wiktionary-inline|stochastic}}

{{Authority control}}

[[Category:Stochastic processes|*]]
[[Category:Mathematical terminology]]