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{{StatsPsy}}
 
{{StatsPsy}}
   
'''''Stochastic''''', from the [[Greek language|Greek]] "stochos" or "goal", means of, relating to, or characterized by conjecture; conjectural; [[random]]. The [[antonym]] is ''astochastic''.
+
'''Stochastic''' (from the [[Greek language|Greek]] ''στόχος'' for ''aim'' or ''guess'') refers to systems whose behaviour is intrinsically non-deterministic. A '''[[stochastic process]]''' is one whose behavior is non-[[deterministic system (mathematics)|deterministic]], in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac<ref>M. Kac & J. Logan, in ''Fluctuation Phenomena'', eds. E.W. Montroll & J.L. Lebowitz, North-Holland, Amsterdam, 1976</ref> and E. Nelson,<ref>E. Nelson, ''Quantum Fluctuations'', Princeton University Press, Princeton, 1985</ref> any kind of time development (be it deterministic or essentially
+
probabilistic) which is analyzable in terms of probability deserves the name of [[stochastic process]].
A '''[[stochastic process]]''' is one whose behavior is non-[[deterministic]] in that the next state of the environment is partially but not fully determined by the previous state of the environment.
 
   
 
==Mathematical theory==
 
==Mathematical theory==
In [[mathematics]], specifically in [[probability theory]], the field of [[stochastic process]]es has for some decades been a major area of research. See that article for more.
+
The use of the term ''stochastic'' to mean ''based on the theory of probability'' has been traced back to [[Ladislaus Bortkiewicz]], who meant the sense of ''making conjectures'' that the Greek term bears since ancient philosophers, and after the title of "[[Ars Conjectandi]]" that Bernoulli gave to his work on [[probability theory]].<ref>{{cite web |url=http://jeff560.tripod.com/s.html |title=Earliest Known Uses of Some of the Words of Mathematics (S) |accessdate=2009-03-10 |author=Jeff Miller et al. }}</ref>
   
A [[stochastic matrix]] is a [[matrix (mathematics)|matrix]] that has non-negative [[real number|real]] entries that sum to 1 in each column.
+
In [[mathematics]], specifically in [[probability theory]], the field of [[stochastic process]]es has been a major area of research.
  +
  +
A [[stochastic matrix]] is a [[matrix (mathematics)|matrix]] that has non-negative [[real number|real]] entries that sum to one in each column.
   
 
==Artificial intelligence==
 
==Artificial intelligence==
In [[artificial intelligence]] stochastic programs work by using probabilistic methods to solve problems, as in [[simulated annealing]], [[Stochastic neural network|neural networks]] and [[genetic algorithms]]. A problem itself may be stochastic as well, as in planning under uncertainty. A [[deterministic]] environment is much simpler for an agent to deal with.
+
In [[artificial intelligence]], stochastic programs work by using probabilistic methods to solve problems, as in [[simulated annealing]], [[stochastic neural network]]s, [[stochastic optimization]], [[genetic algorithms]], and [[genetic programming]]. A problem itself may be stochastic as well, as in planning under uncertainty. A [[Deterministic system (mathematics)|deterministic]] environment is much simpler{{Citation needed|date=October 2010}} for an agent to deal with.
   
 
==Natural science==
 
==Natural science==
An example of a [[stochastic process]] in the natural world is [[pressure]] in a [[gas]]. Even though each molecule is moving deterministically, a collection of them is computationally and practically unpredictable. A large enough set of molecules will exhibit stochastic characteristics, such as filling the container, exerting equal pressure, diffusing along concentration gradients,
 
etc. These are [[emergent property|emergent properties]] of the system.
 
   
==Music==
+
An example of a [[stochastic process]] in the natural world is [[pressure]] in a [[gas]] as modeled by the [[Wiener process]]. Even though (classically speaking) each molecule is moving in a deterministic path, the motion of a collection of them is computationally and practically unpredictable. A large enough set of molecules will exhibit stochastic characteristics, such as filling the container, exerting equal pressure, diffusing along concentration gradients,
In [[music]] '''stochastic''' elements are randomly generated elements created by strict [[mathematics|mathematical]] processes.
+
etc. These are [[emergent property|emergent properties]] of the systems.
   
Stochastic processes can be used in music either to compose a fixed piece, or produced in performance. Stochastic music was pioneered by [[Iannis Xenakis]], who used [[probability]], [[game theory]], [[group theory]], [[set theory]], and [[Boolean algebra]], and frequently used [[computer]]s to produce his scores. 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]]).
+
===Physics===
   
There is a radio show called the Stochastic Hit Parade. All shows are archieved [http://www.wfmu.org/playlists/HP http://www.wfmu.org/playlists/HP]
+
The name "Monte Carlo" for the stochastical [[Monte Carlo method]] was popularized by physics researchers [[Stanislaw Ulam]], [[Enrico Fermi]], [[John von Neumann]], and [[Nicholas Metropolis]], among others. The name is a reference to the [[Monte Carlo Casino]] in [[Monaco]] where Ulam's uncle would borrow money to gamble.<ref>Douglas Hubbard "How to Measure Anything: Finding the Value of Intangibles in Business" pg. 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.
   
==Visual arts==
+
Random methods of computation and experimentation (generally considered forms of [[stochastic simulation]]) can be arguably traced back to the earliest pioneers of probability theory (see, e.g., [[Buffon's needle]], and the work on small samples by [[William Sealy Gosset]]), but are more specifically traced to the pre-electronic computing era. The general difference usually described about a Monte Carlo form of simulation is that it systematically "inverts" the typical mode of simulation, treating deterministic problems by ''first'' finding a [[probabilistic]] [[meta-algorithm|analog]] (see [[Simulated annealing]]). Previous 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.
In the visual arts, Yoshiyuki Abe[http://www.pli.jp], has mastered the art of creation through stochastic process. His work uses geometric objects, mostly the surfaces of hyperbolic paraboloids, and the processing of stochastic elements. In his words: "No matter how you use a computer, or whichever computer you use, to create an art work is not easy. Nevertheless, I believe artists can find a new horizon in his/her creative activities by having the experience of using geometric object and/or stochastic process. For artists who want to create mathematical art through algorithm-driven parameter control, the essential element for success is artistic serendipity. This is the interesting fact of art in the perfect mathematical space."
+
  +
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 were severely limited by the computational tools at 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===
  +
  +
*[[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.<ref>Priplata A. et al. [http://www.bu.edu/abl/files/fulltext.pdf Noise-Enhanced Balance Control in Patients with Diabetes and Patients with Stroke.] Ann Neurol 2006;59:4–12. {{DOI|10.1002/ana.20670}} PMID 16287079.</ref> Many biochemical events also lend themselves to stochastic analysis. [[Gene expression]], for example, is a stochastic process due to the inherent unpredictability of molecular collisions (e.g. binding and unbinding of [[RNA polymerase]] to a [[Promoter (biology)|promoter]]) resulting from [[Brownian motion]].
  +
  +
===Medicine===
  +
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. Cancer is a stochastic effect.
  +
  +
*[[Stochastic theory of hematopoiesis]]
  +
  +
===Geomorphology===
  +
*[[meander|Stochastic theory of meander formation]]
  +
  +
===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]].
  +
  +
===Statistics is indeterministic===
  +
The results of a [[stochastic process]] (statistics) can only be known after computing it.
  +
  +
==Music==
  +
In [[music]], '''stochastic''' elements can be generated by [[mathematics|mathematical]] processes.
  +
  +
Stochastic processes can be used in music to compose a fixed piece or can be produced in performance. Stochastic music was pioneered by [[Iannis Xenakis]], who used [[probability]], [[game theory]], [[group theory]], [[set theory]], and [[Boolean algebra (logic)|Boolean algebra]], and frequently used [[computer]]s to produce his scores. 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]]).
   
 
== Color reproduction ==
 
== Color reproduction ==
When [[#color printing|color reproduction]]s are made, the image is separated into its component colors by taking multiple photographs filtered for each color. One resultant film or plate represents each of the cyan, magenta, yellow, and black data. [[Color printing]] is a binary system, where ink is either present or not present, so all color separations to be printed must be translated into dots at some stage of the workflow. Traditional [[linescreen]]s which are [[amplitude modulation|amplitude modulated]] had problems with [[moire]] but were used until stochastic screening became available. A stochastic (or frequency modulated) dot pattern creates a more photorealistic image.
+
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 modulation|amplitude modulated]] had problems with [[moiré]] but were used until [[stochastic screening]] became available. A stochastic (or [[frequency modulation|frequency modulated]]) dot pattern creates a sharper image.
   
 
==Language and linguistics==
 
==Language and linguistics==
In [[usage-based model|usage-based linguistic theories]], where it is argued that [[competence]], or langue, is based on [[performance]], or [[parole]], in the sense that linguistic knowledge is based on frequency of experience, grammar is often said to be [[probability|probabilistic]] and variable rather than fixed and absolute. This is so, because one's [[competence]] changes in accordance with ones experience with linguistic units. This way, the frequency of [[usage-events]] determines one's knowledge of the language in question.
+
Non-deterministic approaches in language studies are largely inspired by the work of [[Ferdinand de Saussure]]. In [[usage-based model|usage-based linguistic theories]], for example, where it is argued that [[Linguistic competence|competence]], or [[langue and parole|''langue'']], is based on [[performance]], or ''parole'', in the sense that linguistic knowledge is based on frequency of experience, grammar is often said to be [[probability|probabilistic]] and variable rather than fixed and absolute. This is so, because one's competence changes in accordance with one's experience with linguistic units. This way, the frequency of usage-events determines one's knowledge of the language in question.
  +
  +
==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 amount 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}}
  +
  +
==Business==
  +
===Manufacturing===
  +
  +
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 [[Control chart|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.
  +
  +
===Finance===
  +
The financial markets use stochastic models to represent the seemingly random behaviour of assets such as [[stocks]], [[commodities]] and [[interest rates]]. These models are then used by [[quantitative analyst]]s 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|insurance industry]].
  +
  +
Not to be confused with [[stochastic oscillator]]s in [[technical analysis]].
  +
  +
== References ==
  +
  +
{{reflist}}
   
 
==Further reading==
 
==Further reading==
*''Formalized Music: Thought and Mathematics in Composition'' by [[Iannis Xenakis]], ISBN 1576470792
+
* [http://www.youtube.com/watch?v=AUSKTk9ENzg See the stochastic process of an {{convert|8|ft|m|adj=mid|-tall}} Probability Machine comparing stock market returns to the randomness of the beans dropping through the quincunx pattern.] from Index Funds Advisors [http://www.ifa.com IFA.com]
*''Frequency and the Emergence of Linguistic Structure'' by Joan Bybee and Paul Hopper (eds.), ISBN 902722943/ISBN 9027229481 (Eur.)
+
*''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.)
  +
  +
==Software==
  +
* [http://www.intermorphic.com/tools/noatikl/index.html Intermorphic Noatikl], Noatikl is a stochastic / trans-generative music creativity system for [[Mac OS|Mac]] and [[Windows]] with [[Virtual Studio Technology|VST]], [[Audio Units|AU unit]] plugins, and is successor to [http://intermorphic.com/sseyo/index.html Koan].
  +
* [http://www.intermorphic.com/tools/mixtikl/index.html Intermorphic Mixtikl], Mixtikl is a 12 track generative music lab with integrated [http://www.intermorphic.com/tools/noatikl/index.html Noatikl stochastic engine] for [[iPhone]], [[iPad]], [[iPod touch]], [[Mac OS|Mac]] and [[Windows]] with [[web browser]], [[Virtual Studio Technology|VST]] and [[Audio Units|AU unit]] plugins.
   
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[[Category:Free will]]
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[[Category:Mathematical terminology]]
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[[Category:Statistical randomness]]
   
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Latest revision as of 18:19, November 12, 2011

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Stochastic (from the Greek στόχος for aim or guess) refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac[1] and E. Nelson,[2] any kind of time development (be it deterministic or essentially probabilistic) which is analyzable in terms of probability deserves the name of stochastic process.

Mathematical theoryEdit

The use of the term stochastic to mean based on the theory of probability has been traced back to Ladislaus Bortkiewicz, who meant the sense of making conjectures that the Greek term bears since ancient philosophers, and after the title of "Ars Conjectandi" that Bernoulli gave to his work on probability theory.[3]

In mathematics, specifically in probability theory, the field of stochastic processes has been a major area of research.

A stochastic matrix is a matrix that has non-negative real entries that sum to one in each column.

Artificial intelligenceEdit

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. A deterministic environment is much simpler[citation needed] for an agent to deal with.

Natural scienceEdit

An example of a stochastic process in the natural world is pressure in a gas as modeled by the Wiener process. Even though (classically speaking) each molecule is moving in a deterministic path, the motion of a collection of them is computationally and practically unpredictable. A large enough set of molecules will exhibit stochastic characteristics, such as filling the container, exerting equal pressure, diffusing along concentration gradients, etc. These are emergent properties of the systems.

PhysicsEdit

The name "Monte Carlo" for the stochastical Monte Carlo method was popularized by physics researchers Stanislaw Ulam, Enrico Fermi, John von Neumann, and Nicholas Metropolis, among others. The name is a reference to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money to gamble.[4] The use of randomness and the repetitive nature of the process are analogous to the activities conducted at a casino.

Random methods of computation and experimentation (generally considered forms of stochastic simulation) can be arguably traced back to the earliest pioneers of probability theory (see, e.g., Buffon's needle, and the work on small samples by William Sealy Gosset), but are more specifically traced to the pre-electronic computing era. The general difference usually described about a Monte Carlo form of simulation is that it systematically "inverts" the typical mode of simulation, treating deterministic problems by first finding a probabilistic analog (see Simulated annealing). Previous 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 were severely limited by the computational tools at 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.

BiologyEdit

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.[5] Many biochemical events also lend themselves to stochastic analysis. Gene expression, for example, is a stochastic process due to the inherent unpredictability of molecular collisions (e.g. binding and unbinding of RNA polymerase to a promoter) resulting from Brownian motion.

MedicineEdit

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. Cancer is a stochastic effect.

GeomorphologyEdit

CreativityEdit

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.

Statistics is indeterministicEdit

The results of a stochastic process (statistics) can only be known after computing it.

MusicEdit

In music, stochastic elements can be generated by mathematical processes.

Stochastic processes can be used in music to compose a fixed piece or can be produced in performance. Stochastic music was pioneered by Iannis Xenakis, who used probability, game theory, group theory, set theory, and Boolean algebra, and frequently used computers to produce his scores. Earlier, John Cage and others had composed aleatoric or indeterminate music, which is created by chance processes but does not have the strict mathematical basis (Cage's Music of Changes, for example, uses a system of charts based on the I-Ching).

Color reproduction Edit

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.

Language and linguisticsEdit

Non-deterministic approaches in language studies are largely inspired by the work of Ferdinand de Saussure. In usage-based linguistic theories, for example, where it is argued that competence, or langue, is based on performance, or parole, in the sense that linguistic knowledge is based on frequency of experience, grammar is often said to be probabilistic and variable rather than fixed and absolute. This is so, because one's competence changes in accordance with one's experience with linguistic units. This way, the frequency of usage-events determines one's knowledge of the language in question.

Social sciencesEdit

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 amount 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]

BusinessEdit

ManufacturingEdit

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.

FinanceEdit

The financial markets use stochastic models to represent the seemingly random behaviour of assets such as stocks, commodities 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.

Not to be confused with stochastic oscillators in technical analysis.

References Edit

  1. M. Kac & J. Logan, in Fluctuation Phenomena, eds. E.W. Montroll & J.L. Lebowitz, North-Holland, Amsterdam, 1976
  2. E. Nelson, Quantum Fluctuations, Princeton University Press, Princeton, 1985
  3. Jeff Miller et al.. Earliest Known Uses of Some of the Words of Mathematics (S). URL accessed on 2009-03-10.
  4. Douglas Hubbard "How to Measure Anything: Finding the Value of Intangibles in Business" pg. 46, John Wiley & Sons, 2007
  5. Priplata A. et al. Noise-Enhanced Balance Control in Patients with Diabetes and Patients with Stroke. Ann Neurol 2006;59:4–12.
    1. REDIRECT Template:Doi
    PMID 16287079.

Further readingEdit

SoftwareEdit

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