Stochastic (/stəˈkæstɪk/; from Ancient Greek στόχος (stókhos) 'aim, guess')[1] is the property of being well-described by arandomprobability distribution.[1]Stochasticity andrandomness are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; in everyday conversation these terms are often usedinterchangeably. Inprobability theory, the formal concept of astochastic process is also referred to as arandom process.[2][3][4][5][6]
Stochasticity is used in many different fields, includingactuarial science,image processing,signal processing,computer science,information theory,telecommunications,[7]chemistry,[8]ecology,[9]neuroscience,[10]physics,[11][12][13][14] andcryptography.[15][16] It is also used in finance, medicine, linguistics, music, media, colour theory, botany, manufacturing and geomorphology.[17][18][19]
The wordstochastic 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 theOxford English Dictionary gives the year 1662 as its earliest occurrence.[1] In his work on probabilityArs 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, byLadislaus Bortkiewicz,[21] who in 1917 wrote in German the wordStochastik with a sense meaning random. The termstochastic process first appeared in English in a 1934 paper byJoseph L. Doob.[1] For the term and a specific mathematical definition, Doob cited another 1934 paper, where the termstochastischer Prozeß was used in German byAleksandr Khinchin,[22][23] though the German term had been used earlier in 1931 byAndrey Kolmogorov.[24]
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.[25][22][a] Further fundamental work on probability theory and stochastic processes was done by Khinchin as well as other mathematicians such asAndrey Kolmogorov,Joseph Doob,William Feller,Maurice Fréchet,Paul Lévy,Wolfgang Doeblin, andHarald 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 toprobability theory,[29] and continues to be an active topic of research for both theory and applications.[30][31][32]
The wordstochastic is used to describe other terms and objects in mathematics. Examples include astochastic matrix, which describes a stochastic process known as aMarkov process, and stochastic calculus, which involvesdifferential equations andintegrals based on stochastic processes such as theWiener process, also called the Brownian motion process.
One of the simplest continuous-time stochastic processes isBrownian motion. This was first observed by botanistRobert Brown while looking through a microscope at pollen grains in water.
TheMonte Carlo method is a stochastic method popularized by physics researchersStanisław Ulam,Enrico Fermi,John von Neumann, andNicholas Metropolis.[33] The use ofrandomness 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 discoveredneutron. Monte Carlo methods were central to thesimulations required for theManhattan 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 atLos Alamos for early work relating to the development of thehydrogen bomb, and became popularized in the fields ofphysics,physical chemistry, andoperations research. TheRAND Corporation and theU.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 ofpseudorandom number generators, which were far quicker to use than the tables of random numbers which had been previously used for statistical sampling.
In biological systems the technique ofstochastic resonance - introducing stochastic "noise" - has been found to help improve the signal-strength of the internal feedback-loops for balance and othervestibular communication.[34] The technique has helped diabetic and stroke patients with balance control.[35]
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 ofRNA polymerase to agene promoter which contributes to bursts of transcription and super-Poissonian variability in cell-to-cell RNA distributions[36]—via the solution'sBrownian 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 astochastic process.[37]
Stochastic ray tracing is the application ofMonte Carlo simulation to thecomputer graphicsray tracing algorithm. "Distributed ray tracing samples theintegrand at many randomly chosen points and averages the results to obtain a better approximation. It is essentially an application of theMonte Carlo method to3D computer graphics, and for this reason is also calledStochastic ray tracing."[citation needed]
Stochastic forensics analyzes computer crime by viewing computers as stochastic steps.
Inartificial intelligence, stochastic programs work by using probabilistic methods to solve problems, as insimulated annealing,stochastic neural networks,stochastic optimization,genetic algorithms, andgenetic 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 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 ofinterest rates. These models are then used by financial analysts to value options on stock prices, bond prices, and on interest rates, seeMarkov models. Moreover, it is at the heart of theinsurance 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 ofFerdinand de Saussure, for example, infunctionalist linguistic theory, which argues thatcompetence is based onperformance.[38][39] This distinction in functional theories of grammar should be carefully distinguished from thelangue andparole 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,[40] it has also provided the foundation for modern statistical natural language processing[41] and for theories of language learning and change.[42]
Manufacturing processes are assumed to bestochastic processes. This assumption is largely valid for either continuous or batch manufacturing processes. Testing and monitoring of the process is recorded using aprocess 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 originalNielsen 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.[citation needed] In contrast to the deterministic effect, severity is independent of dose. Only theprobability of an effect increases with dose.[citation needed]
Inmusic,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 byIannis Xenakis, who coined the termstochastic music. Specific examples of mathematics, statistics, and physics applied to music composition are the use of thestatistical mechanics of gases inPithoprakta,statistical distribution of points on a plane inDiamorphoses, minimalconstraints inAchorripsis, thenormal distribution inST/10 andAtrées,Markov chains inAnalogiques,game theory inDuel andStratégie,group theory inNomos Alpha (forSiegfried Palm),set theory inHerma andEonta,[43] andBrownian motion inN'Shima.[citation needed] Xenakis frequently usedcomputers to produce his scores, such as theST series includingMorsima-Amorsima andAtrées, and foundedCEMAMu. Earlier,John Cage and others had composedaleatoric orindeterminate music, which is created by chance processes but does not have the strict mathematical basis (Cage'sMusic of Changes, for example, uses a system of charts based on theI-Ching).Lejaren Hiller andLeonard Issacson usedgenerative grammars andMarkov chains in their 1957Illiac 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 tosystems 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. SeeJulia Kristeva on her usage of the 'semiotic',Luce Irigaray on reverseHeideggerianepistemology, andPierre Bourdieu on polythetic space for examples of stochastic social science theory.[citation needed]
The termstochastic terrorism has come into frequent use[44] with regard tolone 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.[45]
Author David Neiwert, who wrote the bookAlt-America, toldSalon 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 likeCharles Manson and his followers. Manson wrote the script; he didn't commit any of those murders. He just had his followers carry them out.[46]
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. Traditionalline screens which areamplitude modulated had problems withmoiré but were used untilstochastic screening became available. A stochastic (orfrequency modulated) dot pattern creates a sharper image.
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The dictionary definition ofstochastic at Wiktionary