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.

* 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.

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 ).

Methods for disambiguation often involve the use of corpora and Markov models.

The DCT, and in particular the DCT-II, is often used in signal and image processing, especially for lossy data compression, because it has a strong " energy compaction " property ( Ahmed, Natarajan and Rao, 1974 ; Rao and Yip, 1990 ): most of the signal information tends to be concentrated in a few low-frequency components of the DCT, approaching the Karhunen-Loève transform ( which is optimal in the decorrelation sense ) for signals based on certain limits of Markov processes.

Multi-dimensional integrals often arise in Bayesian statistics, computational physics, computational biology and computational linguistics, so Markov chain Monte Carlo methods are widely used in those fields.

The use of certain modern computational techniques for Bayesian inference, specifically the various types of Markov chain Monte Carlo techniques, have led to the need for checks, often made in graphical form, on the validity of such computations in expressing the required posterior distributions.

Secondly, methods such as Markov chain Monte Carlo or shared nearest neighbor methods often work very well on data that were considered intractable by other methods due to high dimensionality.

This property is often referred to as the Markov property of loop-erased random walk ( the relation to the usual Markov property is somewhat vague ).

In the domain of physics and probability, a Markov random field ( often abbreviated as MRF ), Markov network or undirected graphical model is a set of random variables having a Markov property described by an undirected graph.

Particle filters are the sequential ( online ) analogue of Markov chain Monte Carlo ( MCMC ) batch methods and are often similar to importance sampling methods.

This is an oversimplification, which is inaccurate and often unhelpful, particularly in probability-based machine learning techniques such as artificial neural networks and hidden Markov models.