请输入您要查询的英文单词:

 

单词 markov process
释义

Markov process


Thesaurus
Noun1.Markov process - a simple stochastic process in which the distribution of future states depends only on the present state and not on how it arrived in the present stateMarkoff processMarkoff chain, Markov chain - a Markov process for which the parameter is discrete time valuesstochastic process - a statistical process involving a number of random variables depending on a variable parameter (which is usually time)

Markov process


Markov process

[′mär‚kȯf prä·səs] (mathematics) A stochastic process which assumes that in a series of random events the probability of an occurrence of each event depends only on the immediately preceding outcome.

Markov Process

 

an important special type of random processes, which are of great importance in applications of probability theory to various branches of natural science and technology.

An example of a Markov process is the decay of a radioactive substance. It is known that the probability of disintegration of a given atom over a short time interval dt is given by adt, where a is a constant that characterizes the rate of disintegration of the radioactive substance. This probability is independent of the fate of all the other atoms and of the age of the particular atom. Let N denote the number of atoms of a radioactive substance at some initial moment of time l = 0, and let Pn(t) denote the probability that n atoms will have disintegrated by time t. The probabilities Pn(t) satisfy the system of differential equations

where n = 1, 2, …, N. Solving this system of equations with the initial data

P0(0) = 1 Pn(0) = 0 1 ≤ nN

we obtain

Pn(t) = CNn(1 — e-αt)ne-(N — n)αt

In this example there exist 0, 1, 2, …, N disintegrated atoms at each moment of time; their number characterizes the state of the phenomenon under study.

This example may be put into the following more general scheme. Let ω1, ω2,…,ωn… be all possible states of a given system, either finite or infinite in number. At each moment the system may be in one of these states, and over the course of time it passes randomly from one state into another. This is called a Markov process if the state of a system ωi, at a given moment is determined only by the probability Pij that after time t the system will be in state >j, this probability is independent of the course of the process in the preceding period. The probabilities Pij (t) are called transition probabilities. Under very broad conditions the transition probabilities of a Markov process satisfy a finite or infinite system of linear homogeneous ordinary differential equations.

The theory of Markov processes is based on the studies of A. A. Markov (the elder), who in his works in 1907 set forth the foundations of the study of sequences of dependent tests and sums of random variables associated with them. This line of research is now known as the theory of Markov chains. The theory treats such systems that can pass from one state to another only at fully defined moments t1, t2, …, tk, …. Let pij denote the probability that the system at moment tk + 1 will be in state oωj if it is known that at tk it was in state >i. The investigation of Markov chains can be reduced to the study of the transition probability matrices ║Pij ║.

A number of physicists and engineers have also demonstrated in their research the importance of processes in which a given system undergoes random variations as a function of a given number of continuously varying parameters (time, coordinates, and so on). Research along these lines lacked a solid logical foundation. The Soviet mathematician A. N. Kolmogorov gave a general theory and classification of Markov processes in 1930. His studies provided a logically faultless mathematical foundation of the general theory of Markov processes, encompassing, in addition to processes of the type described above, diffusion-type processes, in which the state of a system is characterized by a continuously varying coordinate of the diffusing particle.

In this case, it is natural to replace the transition probabilities by the corresponding probability density f (t,x,y). Then f (t,x,y is the probability that a particle at the point x will have a coordinate between y and y + dy after time t. Kolmogorov proved that under certain general conditions the densities f (t,x,y satisfy the partial differential equation

which had previously been introduced for the physically important special case of a diffusion process by the German physicists A. Fokker and M. Planck. In this equation, the coefficient A (y) is the mean rate of variation of the coordinate y and the coefficient B (y) is the rate of the random variations about the mean. This equation lies at the foundations of many investigations into the theory of Markov processes in the USSR and abroad.

REFERENCES

Markov, A. A. Izbr. trudy: Teoriia chisel, teoriia veroiatnostei. Moscow, 1951.
Kolmogorov, A N. “Ob analiticheskikh metodakh v teorii veroiatnostei.” Uspekhi matematicheskikh nauk, 1938, fasc. 5.
Feller, W. Vvedenie v teoriiu veroiatnostei i ee prilozheniia, vols. 1-2. Moscow, 1967. (Translated from English.)
Gikhman, I. I., and A. V. Skorokhod. Vvedenie v teoriiu sluchainykh protsessov. Moscow, 1965.

B. A. SEVAST’IANOV and S. KH. SIRAZHDINOV

Markov process

(probability, simulation)A process in which the sequence ofevents can be described by a Markov chain.

Markov process


Mar·kov pro·cess

(mar'kof), a stochastic process such that the conditional probability distribution for the state at any future instant, given the present state, is unaffected by any additional knowledge of the past history of the system.

Markov process

EBM
A technique of decision analysis that contrasts to standard decision tree analysis, in which a patient moves in one direction through states—e.g., from not treated, to treated, to final outcome—which may include death. In a Markov process, a hypothetical patient could move between states, backwards and forward between continuous ambulatory peritoneal dialysis and haemodialysis, with the caveat that some states—the so-called absorbing states—once entered, cannot be left (e.g., death).
Theoretical medicine
A stochastic process in which the conditional probability distribution for a system’s state at any given instant is unaffected by the system’s previous state.

Markov process

Theoretical medicine A stochastic process in which the conditional probability distribution for a system's state at any given instant is unaffected by that system's previous state

Markov,

(Markoff), Andrei, Russian mathematician, 1865-1922. Markov chain - number of steps or events in sequence.Markov chaining - a theory used in psychiatry.Markov process - a process such that the conditional probability distribution for the state at any future instant, given the present state, is unaffected by any additional knowledge of the past history of the system.
AcronymsSeemilepost

Markov process


  • noun

Synonyms for Markov process

noun a simple stochastic process in which the distribution of future states depends only on the present state and not on how it arrived in the present state

Synonyms

  • Markoff process

Related Words

  • Markoff chain
  • Markov chain
  • stochastic process
随便看

 

英语词典包含2567994条英英释义在线翻译词条,基本涵盖了全部常用单词的英英翻译及用法,是英语学习的有利工具。

 

Copyright © 2004-2022 Newdu.com All Rights Reserved
更新时间:2025/1/9 17:54:05