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.

Markov chains have many applications as statistical models of real-world processes.

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.

* 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.

The stationary Gauss – Markov process is a very special case because it is unique, except for some trivial exceptions.

A stationary Gauss – Markov process with variance and time constant has the following properties.

If this process is applied repeatedly the distribution converges to a stationary distribution for the Markov chain.

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.

Indeed there are further possibilities for confusion with the use of " stationary " in the context of stochastic processes ; for example a " time-homogeneous " Markov chain is sometimes said to have " stationary transition probabilities ".

On the other hand, all stationary Markov random processes are time-homogeneous.

1995 ) that the sequence of samples constitutes a Markov chain, and the stationary distribution of that Markov chain is just the sought-after joint distribution.

The reason for this is that ( 1 ) successive samples are not independent of each other but form a Markov chain with some amount of correlation ; ( 2 ) the stationary distribution of the Markov chain is the desired joint distribution over the variables, but it may take a while for that stationary distribution to be reached.

In general this gives a non-stationary Markov process, but each individual step will still be reversible, and the overall process will still have the desired stationary distribution ( as long as the chain can access all states under the fixed ordering ).

The theorem has a natural interpretation in the theory of finite Markov chains ( where it is the matrix-theoretic equivalent of the convergence of an irreducible finite Markov chain to its stationary distribution, formulated in terms of the transition matrix of the chain ; see, for example, the article on the subshift of finite type ).

After writing a series of papers on the foundations of probability and stochastic processes including martingales, Markov processes, and stationary processes, Doob realized that there was a real need for a book showing what is known about the various types of stochastic processes.

A stationary Markov Chain is reversible if the transition matrix

The detailed balance condition is stronger than that required merely for a stationary distribution ; that is, there are Markov processes with stationary distributions that do not have detailed balance.

The data is often found to contain considerable variability, or noise, and thus Hidden Markov model and change-point analysis methods are being developed to infer real copy number changes.

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.

These are discussed below, and may each be derived by means of a special case of continuous-time Markov processes known as a birth-death process.

Other equivalent classes of functions are the λ-recursive functions and the functions that can be computed by Markov algorithms.

Any version of Snakes and Ladders can be represented exactly as an absorbing Markov chain, since from any square the odds of moving to any other square are fixed and independent of any previous game history.

The analysis and processing of various types of corpora are also the subject of much work in computational linguistics, speech recognition and machine translation, where they are often used to create hidden Markov models for part of speech tagging and other purposes.

The general idea of the algorithm is to generate a series of samples that are linked in a Markov chain ( where each sample is correlated only with the directly preceding sample ).

The statistical formulation of the principle of locality is now seen to be a form of the Markov property in the broad sense ; nearest neighbors are now Markov blankets.

A Markov chain is a sequence of random variables X < sub > 1 </ sub >, X < sub > 2 </ sub >, X < sub > 3 </ sub >, ... with the Markov property, namely that, given the present state, the future and past states are independent.

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.

In a regular Markov model, the state is directly visible to the observer, and therefore the state transition probabilities are the only parameters.

Hidden Markov models are especially known for their application in temporal pattern recognition such as speech, handwriting, gesture recognition, part-of-speech tagging, musical score following, partial discharges and bioinformatics.

A hidden Markov model can be considered a generalization of a mixture model where the hidden variables ( or latent variables ), which control the mixture component to be selected for each observation, are related through a Markov process rather than independent of each other.

The process described here is an approximation of a Poisson process-Poisson processes are also Markov.

These decompositions are particularly useful for matrices that are envisioned as concatenations of particular types of row vectors or column vectors, e. g. orthogonal matrices ( whose rows and columns are unit vectors orthogonal to each other ) and Markov matrices ( whose rows or columns sum to 1 ).