Stochastic

Encyclopedia

**Stochastic**refers to systems whose behaviour is intrinsically non-deterministic. A

**stochastic process**

is one whose behavior is non-deterministic

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

Deterministic system (mathematics)

In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.-Examples:...

, 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

Mark Kac

Mark Kac was a Polish mathematician. His main interest was probability theory. His question, "Can one hear the shape of a drum?" set off research into spectral theory, with the idea of understanding the extent to which the spectrum allows one to read back the geometry. Kac completed his Ph.D...

and E. Nelson, any kind of time development (be it deterministic or essentially

probabilistic) which is analyzable in terms of probability deserves the name of stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

.

## Mathematical theory

The use of the term*stochastic*to mean

*based on the theory of probability*has been traced back to Ladislaus Bortkiewicz

Ladislaus Bortkiewicz

Ladislaus Josephovich Bortkiewicz , August 7, 1868 – July 15, 1931) was a Russian economist and statistician of Polish descent, who lived most of his professional life in Germany, where he taught at Strassburg University and Berlin University...

, who meant the sense of

*making conjectures*that the Greek term bears since ancient philosophers, and after the title of "Ars Conjectandi

Ars Conjectandi

Ars Conjectandi is a combinatorial mathematical paper written by Jakob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The seminal work consolidated, most notably among other combinatorial topics, probability theory: indeed, it is widely regarded as...

" that Bernoulli gave to his work on probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

.

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, specifically in probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, the field of stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

es has been a major area of research.

A stochastic matrix

Stochastic matrix

In mathematics, a stochastic matrix is a matrix used to describe the transitions of a Markov chain. It has found use in probability theory, statistics and linear algebra, as well as computer science...

is a matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

that has non-negative real

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

entries that sum to one in each column.

## Artificial intelligence

In artificial intelligenceArtificial intelligence

Artificial intelligence is the intelligence of machines and the branch of computer science that aims to create it. AI textbooks define the field as "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions that maximize its...

, stochastic programs work by using probabilistic methods to solve problems, as in simulated annealing

Simulated annealing

Simulated annealing is a generic probabilistic metaheuristic for the global optimization problem of locating a good approximation to the global optimum of a given function in a large search space. It is often used when the search space is discrete...

, stochastic neural network

Stochastic neural network

Stochastic neural networks are a type of artificial neural networks, which is a tool of artificial intelligence. They are built by introducing random variations into the network, either by giving the network's neurons stochastic transfer functions, or by giving them stochastic weights...

s, stochastic optimization

Stochastic optimization

Stochastic optimization 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 involve random objective functions or random constraints, for example. Stochastic...

, genetic algorithms, and genetic programming

Genetic programming

In artificial intelligence, genetic programming is an evolutionary algorithm-based methodology inspired by biological evolution to find computer programs that perform a user-defined task. It is a specialization of genetic algorithms where each individual is a computer program...

. A problem itself may be stochastic as well, as in planning under uncertainty. A deterministic

Deterministic system (mathematics)

In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.-Examples:...

environment is much simpler for an agent to deal with.

## Natural science

An example of a stochastic processStochastic process

in the natural world is pressure

Pressure

Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

in a gas

Gas

Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...

as modeled by the Wiener process

Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

. 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.

### Physics

The name "Monte Carlo" for the stochastical Monte Carlo methodMonte Carlo method

Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

was popularized by physics researchers Stanislaw Ulam, Enrico Fermi

Enrico Fermi

Enrico Fermi was an Italian-born, naturalized American physicist particularly known for his work on the development of the first nuclear reactor, Chicago Pile-1, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics...

, John von Neumann

John von Neumann

John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

, and Nicholas Metropolis

Nicholas Metropolis

Nicholas Constantine Metropolis was a Greek American physicist.-Work:Metropolis received his B.Sc. and Ph.D. degrees in physics at the University of Chicago...

, among others. The name is a reference to the Monte Carlo Casino

Monte Carlo Casino

The Monte Carlo Casino is a gambling and entertainment complex located in Monte Carlo, Monaco. It includes a casino, the Grand Théâtre de Monte Carlo, and the office of Les Ballets de Monte Carlo....

in Monaco

Monaco

Monaco , officially the Principality of Monaco , is a sovereign city state on the French Riviera. It is bordered on three sides by its neighbour, France, and its centre is about from Italy. Its area is with a population of 35,986 as of 2011 and is the most densely populated country in the...

where Ulam's uncle would borrow money to gamble. The use of randomness

Randomness

Randomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability of events....

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

Stochastic simulation

Stochastic simulation algorithms and methods were initially developed to analyse chemical reactions involving large numbers of species with complex reaction kinetics. The first algorithm, the Gillespie algorithm was proposed by Dan Gillespie in 1977...

) can be arguably traced back to the earliest pioneers of probability theory (see, e.g., Buffon's needle

Buffon's needle

In mathematics, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon:Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry...

, and the work on small samples by William Sealy Gosset

William Sealy Gosset

William Sealy Gosset is famous as a statistician, best known by his pen name Student and for his work on Student's t-distribution....

), 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

Simulated annealing

Simulated annealing is a generic probabilistic metaheuristic for the global optimization problem of locating a good approximation to the global optimum of a given function in a large search space. It is often used when the search space is discrete...

). 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

Neutron

The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...

. Monte Carlo methods were central to the simulation

Simulation

Simulation is the imitation of some real thing available, state of affairs, or process. The act of simulating something generally entails representing certain key characteristics or behaviours of a selected physical or abstract system....

s required for the Manhattan Project

Manhattan Project

The Manhattan Project was a research and development program, led by the United States with participation from the United Kingdom and Canada, that produced the first atomic bomb during World War II. From 1942 to 1946, the project was under the direction of Major General Leslie Groves of the US Army...

, 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

Los Alamos National Laboratory

Los Alamos National Laboratory is a United States Department of Energy national laboratory, managed and operated by Los Alamos National Security , located in Los Alamos, New Mexico...

for early work relating to the development of the hydrogen bomb, and became popularized in the fields of physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, physical chemistry

Physical chemistry

Physical chemistry is the study of macroscopic, atomic, subatomic, and particulate phenomena in chemical systems in terms of physical laws and concepts...

, and operations research

Operations research

Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...

. 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

Pseudorandom number generator

A pseudorandom number generator , also known as a deterministic random bit generator , is an algorithm for generating a sequence of numbers that approximates the properties of random numbers...

s, which were far quicker to use than the tables of random numbers which had been previously used for statistical sampling.

### Biology

- Stochastic resonanceStochastic resonanceStochastic resonance is a phenomenon that occurs in a threshold measurement system when an appropriate measure of information transfer is maximized in the presence of a non-zero level of stochastic input noise thereby lowering the response...

In biological systems, introducing stochastic 'noise' has been found to help improve the signal strength of the internal feedback loops for balance and other vestibular communication. It has been found to help diabetic and stroke patients with balance control. Many biochemical events also lend themselves to stochastic analysis. Gene expression

Gene expression

Gene expression is the process by which information from a gene is used in the synthesis of a functional gene product. These products are often proteins, but in non-protein coding genes such as ribosomal RNA , transfer RNA or small nuclear RNA genes, the product is a functional RNA...

, for example, is a stochastic process due to the inherent unpredictability of molecular collisions (e.g. binding and unbinding of RNA polymerase

RNA polymerase

RNA polymerase is an enzyme that produces RNA. In cells, RNAP is needed for constructing RNA chains from DNA genes as templates, a process called transcription. RNA polymerase enzymes are essential to life and are found in all organisms and many viruses...

to a promoter) resulting from Brownian motion

Brownian motion

Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

.

### 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

### 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 processStochastic process

.

### Statistics is indeterministic

The results of a stochastic processStochastic process

(statistics) can only be known after computing it.

## Music

In musicMusic

Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

,

**stochastic**elements can be generated by mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

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

Iannis Xenakis

Iannis Xenakis was a Romanian-born Greek ethnic, naturalized French composer, music theorist, and architect-engineer. He is commonly recognized as one of the most important post-war avant-garde composers...

, who used probability

Probability

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, game theory

Game theory

Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

, group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, set theory

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, and Boolean algebra, and frequently used computer

Computer

A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...

s to produce his scores. Earlier, John Cage

John Cage

John Milton Cage Jr. was an American composer, music theorist, writer, philosopher and artist. A pioneer of indeterminacy in music, electroacoustic music, and non-standard use of musical instruments, Cage was one of the leading figures of the post-war avant-garde...

and others had composed

*aleatoric*

or indeterminate music, which is created by chance processes but does not have the strict mathematical basis (Cage's

Aleatoric music

Aleatoric music is music in which some element of the composition is left to chance, and/or some primary element of a composed work's realization is left to the determination of its performer...

*Music of Changes*

, for example, uses a system of charts based on the I-Ching).

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 Cage's earliest fully indeterminate instrumental work. The process of composition involved applying decisions made using the I Ching, a Chinese classic text that is commonly used as a...

## 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 printingColor printing

Color printing or Colour printing is the reproduction of an image or text in color...

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

Amplitude modulation

Amplitude modulation is a technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. AM works by varying the strength of the transmitted signal in relation to the information being sent...

had problems with moiré but were used until stochastic screening

Stochastic screening

Stochastic screening or FM screening is a halftone process based on pseudo-random distribution of halftone dots, using frequency modulation to change the density of dots according to the gray level desired...

became available. A stochastic (or frequency modulated

Frequency modulation

In telecommunications and signal processing, frequency modulation conveys information over a carrier wave by varying its instantaneous frequency. This contrasts with amplitude modulation, in which the amplitude of the carrier is varied while its frequency remains constant...

) dot pattern creates a sharper image.

## Language and linguistics

Non-deterministic approaches in language studies are largely inspired by the work of Ferdinand de SaussureFerdinand de Saussure

Ferdinand de Saussure was a Swiss linguist whose ideas laid a foundation for many significant developments in linguistics in the 20th century. He is widely considered one of the fathers of 20th-century linguistics...

. In usage-based linguistic theories, for example, where it is argued that competence

Linguistic competence

Linguistic competence is the system of linguistic knowledge possessed by native speakers of a language, it is in contrast to the concept of Linguistic performance, the way the language system is used in communication...

, or

*langue*, is based on performance

Performance

A performance, in performing arts, generally comprises an event in which a performer or group of performers behave in a particular way for another group of people, the audience. Choral music and ballet are examples. Usually the performers participate in rehearsals beforehand. Afterwards audience...

, or

*parole*, in the sense that linguistic knowledge is based on frequency of experience, grammar is often said to be probabilistic

Probability

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

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 theorySystems theory

Systems theory is the transdisciplinary study of systems in general, with the goal of elucidating principles that can be applied to all types of systems at all nesting levels in all fields of research...

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

Julia Kristeva

Julia Kristeva is a Bulgarian-French philosopher, literary critic, psychoanalyst, sociologist, feminist, and, most recently, novelist, who has lived in France since the mid-1960s. She is now a Professor at the University Paris Diderot...

on her usage of the 'semiotic', Luce Irigaray

Luce Irigaray

Luce Irigaray is a Belgian feminist, philosopher, linguist, psychoanalyst, sociologist and cultural theorist. She is best known for her works Speculum of the Other Woman and This Sex Which Is Not One .-Biography:...

on reverse Heideggerian epistemology, and Pierre Bourdieu

Pierre Bourdieu

Pierre Bourdieu was a French sociologist, anthropologist, and philosopher.Starting from the role of economic capital for social positioning, Bourdieu pioneered investigative frameworks and terminologies such as cultural, social, and symbolic capital, and the concepts of habitus, field or location,...

on polythetic space for examples of stochastic social science theory.

### 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 process control chartControl chart

Control charts, also known as Shewhart charts or process-behaviour charts, in statistical process control are tools used to determine whether or not a manufacturing or business process is in a state of statistical control.- Overview :...

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 stocksStocks

Stocks are devices used in the medieval and colonial American times as a form of physical punishment involving public humiliation. The stocks partially immobilized its victims and they were often exposed in a public place such as the site of a market to the scorn of those who passed by...

, commodities and interest rates. These models are then used by quantitative analyst

Quantitative analyst

A quantitative analyst is a person who works in finance using numerical or quantitative techniques. Similar work is done in most other modern industries, but the work is not always called quantitative analysis...

s to value options on stock prices, bond prices, and on interest rates, see Markov models

Markov chain

A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...

. Moreover, it is at the heart of the insurance industry

Insurance

In law and economics, insurance is a form of risk management primarily used to hedge against the risk of a contingent, uncertain loss. Insurance is defined as the equitable transfer of the risk of a loss, from one entity to another, in exchange for payment. An insurer is a company selling the...

.

Not to be confused with stochastic oscillator

Stochastic oscillator

In technical analysis of securities trading, the stochastic oscillator is a momentum indicator that uses support and resistance levels. Dr. George Lane promoted this indicator in the 1950s. The term stochastic refers to the location of a current price in relation to its price range over a period...

s in technical analysis

Technical analysis

In finance, technical analysis is security analysis discipline for forecasting the direction of prices through the study of past market data, primarily price and volume. Behavioral economics and quantitative analysis incorporate technical analysis, which being an aspect of active management stands...

.

## Further reading

- See the stochastic process of an 8 feet (2.4 m) Probability Machine comparing stock market returns to the randomness of the beans dropping through the quincunx pattern. from Index Funds Advisors IFA.com
*Formalized Music: Thought and Mathematics in Composition*by Iannis XenakisIannis XenakisIannis Xenakis was a Romanian-born Greek ethnic, naturalized French composer, music theorist, and architect-engineer. He is commonly recognized as one of the most important post-war avant-garde composers...

, 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

- Intermorphic Noatikl, Noatikl is a stochastic / trans-generative music creativity system for MacMac OSMac OS is a series of graphical user interface-based operating systems developed by Apple Inc. for their Macintosh line of computer systems. The Macintosh user experience is credited with popularizing the graphical user interface...

and Windows with VSTVirtual Studio TechnologySteinberg's Virtual Studio Technology is an interface for integrating software audio synthesizer and effect plugins with audio editors and hard-disk recording systems. VST and similar technologies use digital signal processing to simulate traditional recording studio hardware with software...

, AU unitAudio UnitsAudio Units are a system-level plug-in architecture provided by Core Audio in Mac OS X developed by Apple Computer. Audio Units are a set of application programming interface services provided by the operating system to generate, process, receive, or otherwise manipulate streams of audio in...

plugins, and is successor to Koan. - Intermorphic Mixtikl, Mixtikl is a 12 track generative music lab with integrated Noatikl stochastic engine for iPhoneIPhoneThe iPhone is a line of Internet and multimedia-enabled smartphones marketed by Apple Inc. The first iPhone was unveiled by Steve Jobs, then CEO of Apple, on January 9, 2007, and released on June 29, 2007...

, iPadIPadThe iPad is a line of tablet computers designed, developed and marketed by Apple Inc., primarily as a platform for audio-visual media including books, periodicals, movies, music, games, and web content. The iPad was introduced on January 27, 2010 by Apple's then-CEO Steve Jobs. Its size and...

, iPod touchIPod touchThe iPod Touch is a portable media player, personal digital assistant, handheld game console, and Wi-Fi mobile device designed and marketed by Apple Inc. The iPod Touch adds the multi-touch graphical user interface to the iPod line...

, MacMac OSMac OS is a series of graphical user interface-based operating systems developed by Apple Inc. for their Macintosh line of computer systems. The Macintosh user experience is credited with popularizing the graphical user interface...

and Windows with web browserWeb browserA web browser is a software application for retrieving, presenting, and traversing information resources on the World Wide Web. An information resource is identified by a Uniform Resource Identifier and may be a web page, image, video, or other piece of content...

, VSTVirtual Studio TechnologySteinberg's Virtual Studio Technology is an interface for integrating software audio synthesizer and effect plugins with audio editors and hard-disk recording systems. VST and similar technologies use digital signal processing to simulate traditional recording studio hardware with software...

and AU unitAudio UnitsAudio Units are a system-level plug-in architecture provided by Core Audio in Mac OS X developed by Apple Computer. Audio Units are a set of application programming interface services provided by the operating system to generate, process, receive, or otherwise manipulate streams of audio in...

plugins.