Stochastic (,
) refers to the property of being well described by a
random
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. Ind ...
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
.
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
synonymous
A synonym is a word, morpheme, or phrase that means exactly or nearly the same as another word, morpheme, or phrase in a given language. For example, in the English language, the words ''begin'', ''start'', ''commence'', and ''initiate'' are all ...
ly. Furthermore, in
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, the formal concept of a ''
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
'' is also referred to as a ''random process''.
Stochasticity is used in many different fields, including the
natural sciences
Natural science is one of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and repeatab ...
such as
biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
,
chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
,
ecology
Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overlaps wi ...
,
neuroscience
Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, development ...
,
and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
as well as
technology
Technology is the application of knowledge to reach practical goals in a specifiable and reproducible way. The word ''technology'' may also mean the product of such an endeavor. The use of technology is widely prevalent in medicine, science, ...
and
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
fields such as
image processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
,
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
,
information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...
,
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
,
cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
, and
telecommunication
Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that fe ...
s.
It is also used in finance, due to seemingly random changes in
financial market
A financial market is a market in which people trade financial securities and derivatives at low transaction costs. Some of the securities include stocks and bonds, raw materials and precious metals, which are known in the financial markets ...
s
as well as in medicine, linguistics, music, media, colour theory, botany, manufacturing, and geomorphology.
Etymology
The word ''stochastic'' in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year 1662 as its earliest occurrence.
In his work on probability ''Ars Conjectandi'', originally published in Latin in 1713,
Jakob Bernoulli
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Leib ...
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
Ladislaus Josephovich Bortkiewicz (Russian Владислав Иосифович Борткевич, German ''Ladislaus von Bortkiewicz'' or ''Ladislaus von Bortkewitsch'') (7 August 1868 – 15 July 1931) was a Russian economist and statisti ...
,
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
Joseph Leo Doob (February 27, 1910 – June 7, 2004) was an American mathematician, specializing in analysis and probability theory.
The theory of martingales was developed by Doob.
Early life and education
Doob was born in Cincinnati, Ohio, ...
.
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
Aleksandr Yakovlevich Khinchin (russian: Алекса́ндр Я́ковлевич Хи́нчин, french: Alexandre Khintchine; July 19, 1894 – November 18, 1959) was a Soviet mathematician and one of the most significant contributors to th ...
,
though the German term had been used earlier in 1931 by
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
.
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.
Further fundamental work on probability theory and stochastic processes was done by Khinchin as well as other mathematicians such as
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
,
Joseph Doob
Joseph Leo Doob (February 27, 1910 – June 7, 2004) was an American mathematician, specializing in analysis and probability theory.
The theory of martingales was developed by Doob.
Early life and education
Doob was born in Cincinnati, Ohio, ...
,
William Feller
William "Vilim" Feller (July 7, 1906 – January 14, 1970), born Vilibald Srećko Feller, was a Croatian-American mathematician specializing in probability theory.
Early life and education
Feller was born in Zagreb to Ida Oemichen-Perc, a Croa ...
,
Maurice Fréchet Maurice may refer to:
People
*Saint Maurice (died 287), Roman legionary and Christian martyr
*Maurice (emperor) or Flavius Mauricius Tiberius Augustus (539–602), Byzantine emperor
*Maurice (bishop of London) (died 1107), Lord Chancellor and Lo ...
,
Paul Lévy,
Wolfgang Doeblin
Wolfgang Doeblin, known in France as Vincent Doblin (17 March 1915 – 21 June 1940), was a French-German mathematician.
Life
A native of Berlin, Wolfgang was the son of the Jewish-German novelist and physician, Alfred Döblin, and Erna Reiss. ...
, and
Harald Cramér
Harald Cramér (; 25 September 1893 – 5 October 1985) was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. John Kingman described him as "one of the giants of statist ...
.
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
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
,
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
In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, ...
, which describes a stochastic process known as a
Markov process
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
, and stochastic calculus, which involves
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s and
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s based on stochastic processes such as the
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
, also called the Brownian motion process.
Natural science
One of the simplest continuous-time stochastic processes is
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
. This was first observed by botanist Robert Brown while looking through a microscope at pollen grains in water.
Physics
The
Monte Carlo method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi ...
is a stochastic method popularized by physics researchers
Stanisław Ulam
Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapon ...
,
Enrico Fermi
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and ...
,
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
, and
Nicholas Metropolis
Nicholas Constantine Metropolis (Greek: ; June 11, 1915 – October 17, 1999) was a Greek-American physicist.
Metropolis received his BSc (1937) and PhD in physics (1941, with Robert Mulliken) at the University of Chicago. Shortly afterwards, ...
. The use of
randomness
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual rand ...
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
The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons beh ...
. Monte Carlo methods were central to the
simulation
A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or proc ...
s required for the
Manhattan Project
The Manhattan Project was a research and development undertaking during World War II that produced the first nuclear weapons. It was led by the United States with the support of the United Kingdom and Canada. From 1942 to 1946, the project w ...
, 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
A thermonuclear weapon, fusion weapon or hydrogen bomb (H bomb) is a second-generation nuclear weapon design. Its greater sophistication affords it vastly greater destructive power than first-generation nuclear bombs, a more compact size, a lowe ...
, and became popularized in the fields of
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
physical chemistry
Physical chemistry is the study of macroscopic and microscopic phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical mecha ...
, and
operations research
Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve deci ...
. The
RAND Corporation
The RAND Corporation (from the phrase "research and development") is an American nonprofit global policy think tank created in 1948 by Douglas Aircraft Company to offer research and analysis to the United States Armed Forces. It is financed ...
and the
U.S. Air Force
The United States Air Force (USAF) is the air service branch of the United States Armed Forces, and is one of the eight uniformed services of the United States. Originally created on 1 August 1907, as a part of the United States Army Sign ...
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
A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generate ...
s, which were far quicker to use than the tables of random numbers which had been previously used for statistical sampling.
Biology
Stochastic resonance
Stochastic resonance (SR) is a phenomenon in which a signal that is normally too weak to be detected by a sensor, can be boosted by adding white noise to the signal, which contains a wide spectrum of frequencies. The frequencies in the white no ...
: 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
The Vestibular (from pt, vestíbulo, "entrance hall") is a competitive examination and is the primary and widespread entrance system used by Brazilian universities to select the students admitted.
The Vestibular usually takes place from Novem ...
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
Gene expression is the process by which information from a gene is used in the synthesis of a functional gene product that enables it to produce end products, protein or non-coding RNA, and ultimately affect a phenotype, as the final effect. The ...
, for example, has a stochastic component through the molecular collisions—as during binding and unbinding of
RNA polymerase
In molecular biology, RNA polymerase (abbreviated RNAP or RNApol), or more specifically DNA-directed/dependent RNA polymerase (DdRP), is an enzyme that synthesizes RNA from a DNA template.
Using the enzyme helicase, RNAP locally opens the ...
to a
gene promoter
In genetics, a promoter is a sequence of DNA to which proteins bind to initiate transcription of a single RNA transcript from the DNA downstream of the promoter. The RNA transcript may encode a protein (mRNA), or can have a function in and of i ...
—via the solution's
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
.
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
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
.
Computer science
Stochastic ray tracing is the application of
Monte Carlo simulation
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
to the
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
ray tracing algorithm. "
Distributed ray tracing
Distributed ray tracing, also called distribution ray tracing and stochastic ray tracing, is a refinement of ray tracing that allows for the rendering of "soft" phenomena.
Conventional ray tracing uses single rays to sample many different domain ...
samples the
integrand
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
at many randomly chosen points and averages the results to obtain a better approximation. It is essentially an application of the
Monte Carlo method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi ...
to
3D computer graphics
3D computer graphics, or “3D graphics,” sometimes called CGI, 3D-CGI or three-dimensional computer graphics are graphics that use a three-dimensional representation of geometric data (often Cartesian) that is stored in the computer for th ...
, and for this reason is also called ''Stochastic ray tracing''."
Stochastic forensics Stochastic forensics is a method to forensically reconstruct digital activity lacking artifacts, by analyzing emergent properties resulting from the stochastic nature of modern computers.Grier, Jonathan (2011)"Detecting data theft using stochasti ...
analyzes computer crime by viewing computers as stochastic processes.
In
artificial intelligence
Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech re ...
, stochastic programs work by using probabilistic methods to solve problems, as in
simulated annealing
Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. It ...
,
stochastic neural networks,
stochastic optimization
Stochastic optimization (SO) methods are optimization methods that generate and use random variables. For stochastic problems, the random variables appear in the formulation of the optimization problem itself, which involves random objective funct ...
,
genetic algorithm
In computer science and operations research, a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to gene ...
s, and
genetic programming
In artificial intelligence, genetic programming (GP) is a technique of evolving programs, starting from a population of unfit (usually random) programs, fit for a particular task by applying operations analogous to natural genetic processes to t ...
. 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 assets such as
stock
In finance, stock (also capital stock) consists of all the shares by which ownership of a corporation or company is divided.Longman Business English Dictionary: "stock - ''especially AmE'' one of the shares into which ownership of a company ...
s,
commodities
In economics, a commodity is an economic good, usually a resource, that has full or substantial fungibility: that is, the market treats instances of the good as equivalent or nearly so with no regard to who produced them.
The price of a comm ...
, relative
currency
A currency, "in circulation", from la, currens, -entis, literally meaning "running" or "traversing" is a standardization of money in any form, in use or circulation as a medium of exchange, for example banknotes and coins.
A more general def ...
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 rate
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
s. These models are then used by
quantitative analyst
Quantitative may refer to:
* Quantitative research, scientific investigation of quantitative properties
* Quantitative analysis (disambiguation)
* Quantitative verse, a metrical system in poetry
* Statistics, also known as quantitative analysis
...
s to value options on stock prices, bond prices, and on interest rates, see
Markov models
In probability theory, a Markov model is a stochastic model used to model pseudo-randomly changing systems. It is assumed that future states depend only on the current state, not on the events that occurred before it (that is, it assumes the Marko ...
. Moreover, it is at the heart of the
insurance industry
Insurance is a means of protection from financial loss in which, in exchange for a fee, a party agrees to compensate another party in the event of a certain loss, damage, or injury. It is a form of risk management, primarily used to hedge ...
.
Geomorphology
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
Ferdinand de Saussure (; ; 26 November 1857 – 22 February 1913) was a Swiss linguist, semiotician and philosopher. His ideas laid a foundation for many significant developments in both linguistics and semiotics in the 20th century. He is widel ...
, for example, in
functionalist linguistic theory, which argues that
competence is based on
performance
A performance is an act of staging or presenting a play, concert, or other form of entertainment. It is also defined as the action or process of carrying out or accomplishing an action, task, or function.
Management science
In the work place ...
. 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
Manufacturing processes are assumed to be
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
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
An industrial process control in continuous production processes is a discipline that uses industrial control systems to achieve a production level of consistency, economy and safety which could not be achieved purely by human manual control. I ...
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 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
Nielsen Media Research (NMR) is an American firm that measures media audiences, including television, radio, theatre, films (via the AMC Theatres MAP program), and newspapers. Headquartered in New York City, it is best known for the Nielsen rat ...
to modern studio and television test audiences.
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.
Music
In
music
Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...
,
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
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
Giannis Klearchou Xenakis (also spelled for professional purposes as Yannis or Iannis Xenakis; el, Γιάννης "Ιωάννης" Κλέαρχου Ξενάκης, ; 29 May 1922 – 4 February 2001) was a Romanian-born Greek-French avant-garde ...
, who coined the term ''stochastic music''. Specific examples of mathematics, statistics, and physics applied to music composition are the use of the
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
of gases in ''
Pithoprakta
''Pithoprakta'' (1955–56) is a piece by Iannis Xenakis for string orchestra (with 46 separate solo parts), two trombones, xylophone, and Woodblock (instrument), wood block, premièred by conductor Hermann Scherchen in Munich in March 1957. A typ ...
'',
statistical distribution
In statistics, an empirical distribution function (commonly also called an empirical Cumulative Distribution Function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution functio ...
of points on a plane in ''
Diamorphoses
''Diamorphoses'' ( gr, Διαμορφώσεις) is the first electroacoustic composition by Greek composer Iannis Xenakis. It was created between 1957 and 1958 and is considered a masterpiece in several academic books on history of electroacousti ...
'', minimal
constraints in ''Achorripsis'', the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
in ''ST/10'' and ''Atrées'',
Markov chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
s in ''Analogiques'',
game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
in ''Duel'' and ''Stratégie'',
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
in ''
Nomos Alpha
''Nomos Alpha'' ( el, Νόμος α΄) is a piece for solo cello composed by Iannis Xenakis in 1965, commissioned by Radio Bremen for cellist Siegfried Palm, and dedicated to mathematicians Aristoxenus of Tarentum, Évariste Galois, and Felix Kl ...
'' (for
Siegfried Palm
Siegfried Palm (25 April 1927 – 6 June 2005) was a German cellist who is known worldwide for his interpretations of contemporary music. Many 20th-century composers like Kagel, Ligeti, Xenakis, Penderecki and Zimmermann wrote music for ...
),
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
in ''Herma'' and ''
Eonta
''Eonta'' is a composition for piano, two trumpets, and three tenor trombones by Iannis Xenakis. It was written in 1963–64, and was premiered on December 16, 1964, by the Ensemble du Domaine Musical, with Yuji Takahashi on piano and Pierre Bou ...
'', and
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
in ''N'Shima''. Xenakis frequently used
computers
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These programs ...
to produce his scores, such as the ''ST'' series including ''Morsima-Amorsima'' and ''Atrées'', and founded
CEMAMu CCMIX (Center for the Composition of Music Iannis Xenakis, 2000), formerly Les Ateliers UPIC PIC workshops CEMAMu (Centre d'Etudes de Mathématique et Automatique Musicales, 1972), and EMAMu (Equipe de Mathématique et Automatique Musicales), was a ...
. Earlier,
John Cage
John Milton Cage Jr. (September 5, 1912 – August 12, 1992) was an American composer and music theorist. A pioneer of indeterminacy in music, electroacoustic music, and non-standard use of musical instruments, Cage was one of the leading fi ...
and others had composed ''
aleatoric
Aleatoricism or aleatorism, the noun associated with the adjectival aleatory and aleatoric, is a term popularised by the musical composer Pierre Boulez, but also Witold Lutosławski and Franco Evangelisti, for compositions resulting from "action ...
'' or
indeterminate music Indeterminacy is a composing approach in which some aspects of a musical work are left open to chance or to the interpreter's free choice. John Cage, a pioneer of indeterminacy, defined it as "the ability of a piece to be performed in substantially ...
, which is created by chance processes but does not have the strict mathematical basis (Cage's ''
Music of Changes
''Music of Changes'' is a piece for solo piano by John Cage. Composed in 1951 for pianist and friend David Tudor, it is a ground-breaking piece of Indeterminacy (music), indeterminate music. The process of composition involved applying decisions m ...
'', for example, uses a system of charts based on the ''
I-Ching'').
Lejaren Hiller
Lejaren Arthur Hiller Jr. (February 23, 1924, New York City – January 26, 1994, Buffalo, New York)[Lejaren Hi ...](_blank)
and
Leonard Issacson
Leonard Maxwell Isaacson (born 1925) is an American chemist and composer.
Isaacson collaborated with Lejaren Hiller on the computer-programmed acoustic composition, ''Illiac Suite'' (1957). used
generative grammar
Generative grammar, or generativism , is a linguistic theory that regards linguistics as the study of a hypothesised innate grammatical structure. It is a biological or biologistic modification of earlier structuralist theories of linguistic ...
s and
Markov chain
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
s in their 1957 ''
Illiac Suite
''Illiac Suite'' (later retitled String Quartet No. 4)Andrew Stiller, "Hiller, Lejaren (Arthur)", ''Grove Music Online'' (reviewed December 3, 2010; accessed December 14, 2014). is a 1957 composition for string quartet which is generally agreed t ...
''. 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
Generative music is a term popularized by Brian Eno to describe music that is ever-different and changing, and that is created by a system.
Historical background
In 1995 whilst working with SSEYO's Koan software (built by Tim Cole and Pete Col ...
techniques are therefore readily accessible to composers, performers, and producers.
Social sciences
Stochastic social science theory is similar to
systems theory
Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its structu ...
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
Julia Kristeva (; born Yuliya Stoyanova Krasteva, bg, Юлия Стоянова Кръстева; on 24 June 1941) is a Bulgarian-French philosopher, literary critic, semiotician, psychoanalyst, feminist, and, most recently, novelist, who has ...
on her usage of the 'semiotic',
Luce Irigaray
Luce Irigaray (born 3 May 1930) is a Belgian-born French feminist, philosopher, linguist, psycholinguist, psychoanalyst, and cultural theorist who examined the uses and misuses of language in relation to women. Irigaray's first and most well know ...
on reverse Heideggerian epistemology, and
Pierre Bourdieu
Pierre Bourdieu (; 1 August 1930 – 23 January 2002) was a French sociologist and public intellectual. Bourdieu's contributions to the sociology of education, the theory of sociology, and sociology of aesthetics have achieved wide influence i ...
on polythetic space for examples of stochastic social science theory.
The term "Stochastic Terrorism" has fallen into frequent use
[ published August 12, 2019 ]CNN
CNN (Cable News Network) is a multinational cable news channel headquartered in Atlanta, Georgia, U.S. Founded in 1980 by American media proprietor Ted Turner and Reese Schonfeld as a 24-hour cable news channel, and presently owned by ...
with regard to
lone wolf terrorism
A lone wolf attack, or lone actor attack, is a particular kind of mass murder, committed in a public setting by an individual who plans and commits the act on their own. In the United States, such attacks are usually committed with firearms. I ...
. 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:
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
Color printing or colour printing is the reproduction of an image or text in color (as opposed to simpler black and white
or monochrome printing). Any natural scene or color photograph can be optically and physiologically dissected into three ...
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
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
s which are
amplitude modulated
Amplitude modulation (AM) is a modulation technique used in electronic communication, most commonly for transmitting messages with a radio wave. In amplitude modulation, the amplitude (signal strength) of the wave is varied in proportion to t ...
had problems with
moiré but were used until
stochastic screening
Stochastic screening or FM screening is a halftone process based on pseudo-random distribution of halftone dots, using frequency modulation (FM) to change the density of dots according to the gray level desired. Traditional amplitude modulatio ...
became available. A stochastic (or
frequency modulated
Frequency modulation (FM) is the encoding of information in a carrier wave by varying the instantaneous frequency of the wave. The technology is used in telecommunications, radio broadcasting, signal processing, and computing.
In analog freque ...
) dot pattern creates a sharper image.
See also
*
Jump process
A jump process is a type of stochastic process that has discrete movements, called jumps, with random arrival times, rather than continuous movement, typically modelled as a simple or compound Poisson process.
In finance, various stochastic mod ...
*
Sortition
In governance, sortition (also known as selection by lottery, selection by lot, allotment, demarchy, stochocracy, aleatoric democracy, democratic lottery, and lottocracy) is the selection of political officials as a random sample from a larger ...
*
Stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
Notes
References
Further reading
* ''Formalized Music: Thought and Mathematics in Composition'' by
Iannis Xenakis
Giannis Klearchou Xenakis (also spelled for professional purposes as Yannis or Iannis Xenakis; el, Γιάννης "Ιωάννης" Κλέαρχου Ξενάκης, ; 29 May 1922 – 4 February 2001) was a Romanian-born Greek-French avant-garde ...
,
* ''Frequency and the Emergence of Linguistic Structure'' by Joan Bybee and Paul Hopper (eds.), / (Eur.)
* The
Stochastic Empirical Loading and Dilution Model provides documentation and computer code for modeling stochastic processes in
Visual Basic for Applications
Visual Basic for Applications (VBA) is an implementation of Microsoft's event-driven programming language Visual Basic 6.0 built into most desktop Microsoft Office applications. Although based on pre-.NET Visual Basic, which is no longer supported ...
.
External links
*
{{Authority control
*
Mathematical terminology