Machine Improvisation builds upon a long musical tradition of statistical modeling that began with Hiller and Isaacson's Illiac Suite for String Quartet ( 1957 ) and Xenakis ' uses of Markov chains and stochastic processes.

Ordinary differential equations appear in the movement of heavenly bodies ( planets, stars and galaxies ); optimization occurs in portfolio management ; numerical linear algebra is important for data analysis ; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

Andrey Markov introduced the notion of Markov chains ( 1906 ), which played an important role in stochastic processes theory and its applications.

Pseudorandom number generators are widely used in such applications as computer modeling ( e. g., Markov chains ), statistics, experimental design, etc.

There have also been other algorithms based on Markov chains.

Many other examples of Markov chains exist.

Markov chains are often described by a directed graph, where the edges are labeled by the probabilities of going from one state to the other states.

* Time-homogeneous Markov chains ( or stationary Markov chains ) are processes where

* Examples of Markov chains

Specific examples of mathematics, statistics, and physics applied to music composition are the use of the statistical mechanics of gases in Pithoprakta, statistical distribution of points on a plane in Diamorphoses, minimal constraints in Achorripsis, the normal distribution in ST / 10 and Atrées, Markov chains in Analogiques, game theory in Duel and Stratégie, group theory in Nomos Alpha ( for Siegfried Palm ), set theory in Herma and Eonta, and Brownian motion in N ' Shima.

This page contains examples of Markov chains in action.

Markov chains

Often, random walks are assumed to be Markov chains or Markov processes, but other, more complicated walks are also of interest.

Markov chains form a common context for applications in probability theory.

Specific examples of mathematics, statistics, and physics applied to music composition are the use of the statistical mechanics of gases in Pithoprakta, statistical distribution of points on a plane in Diamorphoses, minimal constraints in Achorripsis, the normal distribution in ST / 10 and Atrées, Markov chains in Analogiques, game theory in Duel and Stratégie, group theory in Nomos Alpha ( for Siegfried Palm ), set theory in Herma and Eonta, and Brownian motion in N ' Shima.

* Elementary probability theory and Markov chains

** Examples of Markov chains

* Nearly completely decomposable, a property of some Markov chains

Prominent examples of stochastic algorithms are Markov chains and various uses of Gaussian distributions.

have used Markov Chain Monte Carlo methods to investigate the algorithm used by the UNSW group to determine from the quasar spectra, and have found that the algorithm appears to produce correct uncertainties and maximum likelihood estimates for for particular models.

* Continuous-time Markov processes have a continuous index.

In his study of stochastic processes ( random processes ), especially Markov processes, Kolmogorov and the British mathematician Sydney Chapman independently developed the pivotal set of equations in the field, which have been give the name of the Chapman – Kolmogorov equations.

A stochastic process, defined via a separate argument, may be shown mathematically to have the Markov property, and as a consequence to have the properties that can be deduced from this for all Markov processes.

If represents the number of dollars you have in chips after n tosses, with, then the sequence is a Markov process.

Let denote the number of kernels which have popped up to time t. Then this is a continuous time, non-homogenous Markov process.

Hidden Markov models have been used to produce probability scores for a family of possible multiple sequence alignments for a given query set ; although early HMM-based methods produced underwhelming performance, later applications have found them especially effective in detecting remotely related sequences because they are less susceptible to noise created by conservative or semiconservative substitutions.

In statistics, the Gauss – Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear regression model in which the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimator ( BLUE ) of the coefficients is given by the ordinary least squares estimator.

The underlying model is a Bayesian model similar to a hidden Markov model but where the state space of the latent variables is continuous and where all latent and observed variables have Gaussian distributions.

Markov algorithms have been shown to be Turing-complete, which means that they are suitable as a general model of computation and can represent any mathematical expression from its simple notation.

A good chain will have rapid mixing — the stationary distribution is reached quickly starting from an arbitrary position — described further under Markov chain mixing time.

A Markov chain is constructed in such a way as to have the integrand as its equilibrium distribution.

Markov chain Monte Carlo methods that change dimensionality have also long been used in statistical physics applications, where for some problems a distribution that is a grand canonical ensemble is used ( e. g., when the number of molecules in a box is variable ).

Yet, though Bulgarian émigré dissident Georgi Markov wrote that " served the Soviet Union more ardently than the Soviet leaders themselves did ," in many ways he can be said to have exploited the USSR for political purposes, with Bulgaria serving a buffer between the USSR and NATO.

Both the terms " Markov property " and " strong Markov property " have been used in connection with a particular " memoryless " property of the exponential distribution.

In the 1980s, there was a dramatic growth in research and applications of Bayesian methods, mostly attributed to the discovery of Markov chain Monte Carlo methods, which removed many of the computational problems, and an increasing interest in nonstandard, complex applications.

Constraints on many cosmological parameters can be obtained from their effects on the power spectrum, and results are often calculated using Markov Chain Monte Carlo sampling techniques.

In the 1980s, there was a dramatic growth in research and applications of Bayesian methods, mostly attributed to the discovery of Markov chain Monte Carlo methods, which removed many of the computational problems, and an increasing interest in nonstandard, complex applications.

The Markov chain is started from an arbitrary initial value and the algorithm is run for many iterations until this initial state is " forgotten ".

In machine learning, the environment is typically formulated as a Markov decision process ( MDP ), and many reinforcement learning algorithms for this context are highly related to dynamic programming techniques.

It is named after the Russian mathematician Andrey Markov, although it appeared earlier in the work of Pafnuty Chebyshev ( Markov's teacher ), and many sources, especially in analysis, refer to it as Chebyshev's inequality or Bienaymé's inequality.

It is still used widely in cellular phones for error correcting codes, as well as for speech recognition, DNA analysis, and many other applications of Hidden Markov models.

However, this problem has been largely overcome by the advent of simulation-based Bayesian inference, especially using Markov chain Monte Carlo methods, which suffices for many practical problems.

Very roughly, it is a stochastic conformally invariant partial differential equation which allows to catch the Markov property of loop-erased random walk ( and many other probabilistic processes ).

When viewed in a more general setting, the canonical ensemble is known as the Gibbs measure, where, because it has the Markov property of statistical independence, it occurs in many settings outside of the field of physics.