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Markov chain

From Wikipedia, the free encyclopedia
Random process independent of past history
A diagram representing a two-state Markov process. The numbers are the probability of changing from one state to another state.
Part of a series onstatistics
Probability theory

In probability theory and statistics, aMarkov chain orMarkov process is astochastic process describing asequence of possible events in which theprobability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairsnow." Acountably infinite sequence, in which the chain moves state at discrete time steps, gives adiscrete-time Markov chain (DTMC). Acontinuous-time process is called acontinuous-time Markov chain (CTMC). Markov processes are named in honor of theRussian mathematicianAndrey Markov.

Markov chains have many applications asstatistical models of real-world processes.[1] They provide the basis for general stochastic simulation methods known asMarkov chain Monte Carlo, which are used for simulating sampling from complexprobability distributions, and have found application in areas includingBayesian statistics,biology,chemistry,economics,finance,information theory,physics,signal processing, andspeech processing.[1][2][3]

The adjectivesMarkovian andMarkov are used to describe something that is related to a Markov process.[4]

Principles

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Russian mathematicianAndrey Markov

Definition

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A Markov process is astochastic process that satisfies theMarkov property (sometimes characterized as "memorylessness"). In simpler terms, it is a process for which predictions can be made regarding future outcomes based solely on its present state and—most importantly—such predictions are just as good as the ones that could be made knowing the process's full history.[5] In other words,conditional on the present state of the system, its future and past states areindependent.

A Markov chain is a type of Markov process that has either a discretestate space or a discrete index set (often representing time), but the precise definition of a Markov chain varies.[6] For example, it is common to define a Markov chain as a Markov process in eitherdiscrete or continuous time with a countable state space (thus regardless of the nature of time),[7][8][9][10] but it is also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).[6]

Types of Markov chains

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The system'sstate space and time parameter index need to be specified. The following table gives an overview of the different instances of Markov processes for different levels of state space generality for both discrete and continuous time:

Countable state spaceContinuous or general state space
Discrete-time(discrete-time) Markov chain on a countable or finite state spaceMarkov chain on a measurable state space (for example,Harris chain)
Continuous-timeContinuous-time Markov process or Markov jump processAnycontinuous stochastic process with the Markov property (for example, theWiener process)

Note that there is no definitive agreement in the literature on the use of some of the terms that signify special cases of Markov processes. Usually the term "Markov chain" is reserved for a process with a discrete set of times, that is, adiscrete-time Markov chain (DTMC),[11] but a few authors use the term "Markov process" to refer to acontinuous-time Markov chain (CTMC) without explicit mention.[12][13][14] In addition, there are other extensions of Markov processes that are referred to as such but do not necessarily fall within any of these four categories (seeMarkov model). Moreover, the time index need not necessarily be real-valued; like with the state space, there are conceivable processes that move through index sets with other mathematical constructs. Notice that the general state space continuous-time Markov chain is general to such a degree that it has no designated term.

While the time parameter is usually discrete, thestate space of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space.[15] However, many applications of Markov chains employ finite orcountably infinite state spaces, which have a more straightforward statistical analysis. Besides time-index and state-space parameters, there are many other variations, extensions and generalizations (seeVariations). For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, unless mentioned otherwise.

Transitions

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The changes of state of the system are called transitions. The probabilities associated with various state changes are called transition probabilities. The process is characterized by a state space, atransition matrix describing the probabilities of particular transitions, and an initial state (or initial distribution) across the state space. By convention, we assume all possible states and transitions have been included in the definition of the process, so there is always a next state, and the process does not terminate.

A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps. The steps are often thought of as moments in time, but they can equally well refer to physical distance or any other discrete measurement. Formally, the steps are theintegers ornatural numbers, and the random process is a mapping of these to states. The Markov property states that theconditional probability distribution for the system at the next step (and in fact at all future steps) depends only on the current state of the system, and not additionally on the state of the system at previous steps.

Since the system changes randomly, it is generally impossible to predict with certainty the state of a Markov chain at a given point in the future. However, the statistical properties of the system's future can be predicted. In many applications, it is these statistical properties that are important.

History

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Andrey Markov studied Markov processes in the early 20th century, publishing his first paper on the topic in 1906.[16][17][18] Markov processes in continuous time were discovered long before his work in the early 20th century in the form of thePoisson process.[19][20][21] Markov was interested in studying an extension of independent random sequences, motivated by a disagreement withPavel Nekrasov who claimed independence was necessary for theweak law of large numbers to hold.[22] In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed vector of values, so proving a weak law of large numbers without the independence assumption,[16][17][18] which had been commonly regarded as a requirement for such mathematical laws to hold.[18] Markov later used Markov chains to study the distribution of vowels inEugene Onegin, written byAlexander Pushkin, and proved acentral limit theorem for such chains.[16]

In 1912Henri Poincaré studied Markov chains onfinite groups with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced byPaul andTatyana Ehrenfest in 1907, and a branching process, introduced byFrancis Galton andHenry William Watson in 1873, preceding the work of Markov.[16][17] After the work of Galton and Watson, it was later revealed that their branching process had been independently discovered and studied around three decades earlier byIrénée-Jules Bienaymé.[23] Starting in 1928,Maurice Fréchet became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains.[16][24]

Andrey Kolmogorov developed in a 1931 paper a large part of the early theory of continuous-time Markov processes.[25][26] Kolmogorov was partly inspired by Louis Bachelier's 1900 work on fluctuations in the stock market as well asNorbert Wiener's work on Einstein's model of Brownian movement.[25][27] He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes.[25][28] Independent of Kolmogorov's work,Sydney Chapman derived in a 1928 paper an equation, now called theChapman–Kolmogorov equation, in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement.[29] The differential equations are now called the Kolmogorov equations[30] or the Kolmogorov–Chapman equations.[31] Other mathematicians who contributed significantly to the foundations of Markov processes includeWilliam Feller, starting in 1930s, and then laterEugene Dynkin, starting in the 1950s.[26]

Examples

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Main article:Examples of Markov chains
  • Mark V. Shaney is a third-order Markov chain program, and aMarkov text generator. It ingests the sample text (theTao Te Ching, or the posts of aUsenet group) and creates a massive list of every sequence of three successive words (triplet) which occurs in the text. It then chooses two words at random, and looks for a word which follows those two in one of the triplets in its massive list. If there is more than one, it picks at random (identical triplets count separately, so a sequence which occurs twice is twice as likely to be picked as one which only occurs once). It then adds that word to the generated text. Then, in the same way, it picks a triplet that starts with the second and third words in the generated text, and that gives a fourth word. It adds the fourth word, then repeats with the third and fourth words, and so on.[32]
  • Random walks based on integers and thegambler's ruin problem are examples of Markov processes.[33][34] Some variations of these processes were studied hundreds of years earlier in the context of independent variables.[35][36] Two important examples of Markov processes are theWiener process, also known as theBrownian motion process, and thePoisson process,[19] which are considered the most important and central stochastic processes in the theory of stochastic processes.[37][38][39] These two processes are Markov processes in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.[33][34]
  • A famous Markov chain is the so-called "drunkard's walk", a random walk on thenumber line where, at each step, the position may change by +1 or −1 with equal probability. From any position there are two possible transitions, to the next or previous integer. The transition probabilities depend only on the current position, not on the manner in which the position was reached. For example, the transition probabilities from 5 to 4 and 5 to 6 are both 0.5, and all other transition probabilities from 5 are 0. These probabilities are independent of whether the system was previously in 4 or 6.
  • A series of independent states (for example, a series of coin flips) satisfies the formal definition of a Markov chain. However, the theory is usually applied only when the probability distribution of the next state depends on the current one.

A non-Markov example

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Suppose that there is a coin purse containing five coins worth 25¢ (quarters), five coins worth 10¢ (dimes) and five coins worth 5¢ (nickels). One by one, coins are randomly drawn from the purse and are set on a table. IfXn{\displaystyle X_{n}} represents the total value of the coins set on the table aftern draws, withX0=0{\displaystyle X_{0}=0}, then the sequence{Xn:n∈N}{\displaystyle \{X_{n}:n\in \mathbb {N} \}} isnot a Markov process.

To see why this is the case, suppose that in the first six draws, all five nickels and a quarter are drawn. ThusX6=$0.50{\displaystyle X_{6}=\$0.50}. If we know not justX6{\displaystyle X_{6}}, but the earlier values as well, then we can determine which coins have been drawn, and we know that the next coin will not be a nickel; so we can determine thatX7≥$0.60{\displaystyle X_{7}\geq \$0.60} with probability 1. But if we do not know the earlier values, then based only on the valueX6{\displaystyle X_{6}} we might guess that we had drawn four dimes and two nickels, in which case it would certainly be possible to draw another nickel next. Thus, our guesses aboutX7{\displaystyle X_{7}} are impacted by our knowledge of values prior toX6{\displaystyle X_{6}}.

However, it is possible to model this scenario as a Markov process. Instead of definingXn{\displaystyle X_{n}} to represent thetotal value of the coins on the table, we could defineXn{\displaystyle X_{n}} to represent thecount of the various coin types on the table. For instance,X6=1,0,5{\displaystyle X_{6}=1,0,5} could be defined to represent the state where there is one quarter, zero dimes, and five nickels on the table after 6 one-by-one draws. This new model could be represented by6×6×6=216{\displaystyle 6\times 6\times 6=216} possible states, where each state represents the number of coins of each type (from 0 to 5) that are on the table. (Not all of these states are reachable within 6 draws.)

Suppose that the first draw results in stateX1=0,1,0{\displaystyle X_{1}=0,1,0}. The probability of achievingX2{\displaystyle X_{2}} now depends onX1{\displaystyle X_{1}}; for example, the stateX2=1,0,1{\displaystyle X_{2}=1,0,1} is not possible. After the second draw, the third draw depends on which coins have so far been drawn, but no longer only on the coins that were drawn for the first state (since probabilistically important information has since been added to the scenario). In this way, the likelihood of theXn=i,j,k{\displaystyle X_{n}=i,j,k} state depends exclusively on the outcome of theXn−1=ℓ,m,p{\displaystyle X_{n-1}=\ell ,m,p} state.

Formal definition

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Discrete-time Markov chain

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Main article:Discrete-time Markov chain

A discrete-time Markov chain is a sequence ofrandom variablesX1,X2,X3, ... with theMarkov property, namely that the probability of moving to the next state depends only on the present state and not on the previous states:

Pr(Xn+1=x∣X1=x1,X2=x2,…,Xn=xn)=Pr(Xn+1=x∣Xn=xn),{\displaystyle \Pr(X_{n+1}=x\mid X_{1}=x_{1},X_{2}=x_{2},\ldots ,X_{n}=x_{n})=\Pr(X_{n+1}=x\mid X_{n}=x_{n}),} if bothconditional probabilities are well defined, that is, ifPr(X1=x1,…,Xn=xn)>0.{\displaystyle \Pr(X_{1}=x_{1},\ldots ,X_{n}=x_{n})>0.}

The possible values ofXi form acountable setS called the state space of the chain.

Variations

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Finite state space

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If the state space isfinite, the transition probability distribution can be represented by amatrix, called the transition matrix, with the (i,j)thelement ofP equal to

pij=Pr(Xn+1=j∣Xn=i).{\displaystyle p_{ij}=\Pr(X_{n+1}=j\mid X_{n}=i).}

Since each row ofP sums to one and all elements are non-negative,P is aright stochastic matrix.

Stationary distribution relation to eigenvectors and simplices

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A stationary distributionπ is a (row) vector, whose entries are non-negative and sum to 1, is unchanged by the operation of transition matrixP on it and so is defined by

πP=π.{\displaystyle \pi \mathbf {P} =\pi .}

By comparing this definition with that of aneigenvector we see that the two concepts are related and that

π=e∑iei{\displaystyle \pi ={\frac {e}{\sum _{i}{e_{i}}}}}

is a normalized (∑iπi=1{\textstyle \sum _{i}\pi _{i}=1}) multiple of a left eigenvectore of the transition matrixP with aneigenvalue of 1. If there is more than one unit eigenvector then a weighted sum of the corresponding stationary states is also a stationary state. But for a Markov chain one is usually more interested in a stationary state that is the limit of the sequence of distributions for some initial distribution.

The values of a stationary distributionπi{\displaystyle \textstyle \pi _{i}} are associated with the state space ofP and its eigenvectors have their relative proportions preserved. Since the components of π are positive and the constraint that their sum is unity can be rewritten as∑i1⋅πi=1{\textstyle \sum _{i}1\cdot \pi _{i}=1} we see that thedot product of π with a vector whose components are all 1 is unity and that π lies on asimplex.

Time-homogeneous Markov chain with a finite state space

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If the Markov chain is time-homogeneous, then the transition matrixP is the same after each step, so thek-step transition probability can be computed as thek-th power of the transition matrix,Pk.

If the Markov chain is irreducible and aperiodic, then there is a unique stationary distributionπ.[40] Additionally, in this casePk converges to a rank-one matrix in which each row is the stationary distributionπ:

limk→∞Pk=1π{\displaystyle \lim _{k\to \infty }\mathbf {P} ^{k}=\mathbf {1} \pi }

where1 is the column vector with all entries equal to 1. This is stated by thePerron–Frobenius theorem. If, by whatever means,limk→∞Pk{\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}} is found, then the stationary distribution of the Markov chain in question can be easily determined for any starting distribution, as will be explained below.

For some stochastic matricesP, the limitlimk→∞Pk{\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}} does not exist while the stationary distribution does, as shown by this example:

P=(0110)P2k=IP2k+1=P{\displaystyle \mathbf {P} ={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\qquad \mathbf {P} ^{2k}=I\qquad \mathbf {P} ^{2k+1}=\mathbf {P} }
(1212)(0110)=(1212){\displaystyle {\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}

(This example illustrates a periodic Markov chain.)

Because there are a number of different special cases to consider, the process of finding this limit if it exists can be a lengthy task. However, there are many techniques that can assist in finding this limit. LetP be ann×n matrix, and defineQ=limk→∞Pk.{\textstyle \mathbf {Q} =\lim _{k\to \infty }\mathbf {P} ^{k}.}

It is always true that

QP=Q.{\displaystyle \mathbf {QP} =\mathbf {Q} .}

SubtractingQ from both sides and factoring then yields

Q(P−In)=0n,n,{\displaystyle \mathbf {Q} (\mathbf {P} -\mathbf {I} _{n})=\mathbf {0} _{n,n},}

whereIn is theidentity matrix of sizen, and0n,n is thezero matrix of sizen×n. Multiplying together stochastic matrices always yields another stochastic matrix, soQ must be astochastic matrix (see the definition above). It is sometimes sufficient to use the matrix equation above and the fact thatQ is a stochastic matrix to solve forQ. Including the fact that the sum of each the rows inP is 1, there aren+1 equations for determiningn unknowns, so it is computationally easier if on the one hand one selects one row inQ and substitutes each of its elements by one, and on the other one substitutes the corresponding element (the one in the same column) in the vector0, and next left-multiplies this latter vector by the inverse of transformed former matrix to findQ.

Here is one method for doing so: first, define the functionf(A) to return the matrixA with its right-most column replaced with all 1's. If [f(P −In)]−1 exists then[41][40]

Q=f(0n,n)[f(P−In)]−1.{\displaystyle \mathbf {Q} =f(\mathbf {0} _{n,n})[f(\mathbf {P} -\mathbf {I} _{n})]^{-1}.}
Explain: The original matrix equation is equivalent to asystem of n×n linear equations inn×n variables. And there aren more linear equations from the fact thatQ is a rightstochastic matrix whose each row sums to 1. So it needs anyn×n independent linear equations of the (n×n+n) equations to solve for then×n variables. In this example, then equations from "Q multiplied by the right-most column of (P-In)" have been replaced by then stochastic ones.

One thing to notice is that ifP has an elementPi,i on its main diagonal that is equal to 1 and theith row or column is otherwise filled with 0's, then that row or column will remain unchanged in all of the subsequent powersPk. Hence, theith row or column ofQ will have the 1 and the 0's in the same positions as inP.

Convergence speed to the stationary distribution

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As stated earlier, from the equationπ=πP,{\displaystyle {\boldsymbol {\pi }}={\boldsymbol {\pi }}\mathbf {P} ,} (if exists) the stationary (or steady state) distributionπ is a left eigenvector of rowstochastic matrixP. Then assuming thatP is diagonalizable or equivalently thatP hasn linearly independent eigenvectors, speed of convergence is elaborated as follows. (For non-diagonalizable, that is,defective matrices, one may start with theJordan normal form ofP and proceed with a bit more involved set of arguments in a similar way.[42])

LetU be the matrix of eigenvectors (each normalized to having an L2 norm equal to 1) where each column is a left eigenvector ofP and letΣ be the diagonal matrix of left eigenvalues ofP, that is,Σ = diag(λ1,λ2,λ3,...,λn). Then byeigendecomposition

P=UΣU−1.{\displaystyle \mathbf {P} =\mathbf {U\Sigma U} ^{-1}.}

Let the eigenvalues be enumerated such that:

1=|λ1|>|λ2|≥|λ3|≥⋯≥|λn|.{\displaystyle 1=|\lambda _{1}|>|\lambda _{2}|\geq |\lambda _{3}|\geq \cdots \geq |\lambda _{n}|.}

SinceP is a row stochastic matrix, its largest left eigenvalue is 1. If there is a unique stationary distribution, then the largest eigenvalue and the corresponding eigenvector is unique too (because there is no otherπ which solves the stationary distribution equation above). Letui be thei-th column ofU matrix, that is,ui is the left eigenvector ofP corresponding to λi. Also letx be a lengthn row vector that represents a valid probability distribution; since the eigenvectorsui spanRn,{\displaystyle \mathbb {R} ^{n},} we can write

xT=∑i=1naiui,ai∈R.{\displaystyle \mathbf {x} ^{\mathsf {T}}=\sum _{i=1}^{n}a_{i}\mathbf {u} _{i},\qquad a_{i}\in \mathbb {R} .}

If we multiplyx withP from right and continue this operation with the results, in the end we get the stationary distributionπ. In other words,π =a1u1 ←xPP...P =xPk ask → ∞. That means

π(k)=x(UΣU−1)(UΣU−1)⋯(UΣU−1)=xUΣkU−1=(a1u1T+a2u2T+⋯+anunT)UΣkU−1=a1λ1ku1T+a2λ2ku2T+⋯+anλnkunTui⊥uj for i≠j=λ1k{a1u1T+a2(λ2λ1)ku2T+a3(λ3λ1)ku3T+⋯+an(λnλ1)kunT}{\displaystyle {\begin{aligned}{\boldsymbol {\pi }}^{(k)}&=\mathbf {x} \left(\mathbf {U\Sigma U} ^{-1}\right)\left(\mathbf {U\Sigma U} ^{-1}\right)\cdots \left(\mathbf {U\Sigma U} ^{-1}\right)\\&=\mathbf {xU\Sigma } ^{k}\mathbf {U} ^{-1}\\&=\left(a_{1}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\mathbf {u} _{2}^{\mathsf {T}}+\cdots +a_{n}\mathbf {u} _{n}^{\mathsf {T}}\right)\mathbf {U\Sigma } ^{k}\mathbf {U} ^{-1}\\&=a_{1}\lambda _{1}^{k}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\lambda _{2}^{k}\mathbf {u} _{2}^{\mathsf {T}}+\cdots +a_{n}\lambda _{n}^{k}\mathbf {u} _{n}^{\mathsf {T}}&&u_{i}\bot u_{j}{\text{ for }}i\neq j\\&=\lambda _{1}^{k}\left\{a_{1}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\left({\frac {\lambda _{2}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{2}^{\mathsf {T}}+a_{3}\left({\frac {\lambda _{3}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{3}^{\mathsf {T}}+\cdots +a_{n}\left({\frac {\lambda _{n}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{n}^{\mathsf {T}}\right\}\end{aligned}}}

Sinceπ is parallel tou1(normalized by L2 norm) andπ(k) is a probability vector,π(k) approaches toa1u1 =π ask → ∞ with a speed in the order ofλ2/λ1 exponentially. This follows because|λ2|≥⋯≥|λn|,{\displaystyle |\lambda _{2}|\geq \cdots \geq |\lambda _{n}|,} henceλ2/λ1 is the dominant term. The smaller the ratio is, the faster the convergence is.[43] Random noise in the state distributionπ can also speed up this convergence to the stationary distribution.[44]

Continuous-time Markov chain

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Main article:Continuous-time Markov chain

A continuous-time Markov chain(Xt)t≥0{\displaystyle (X_{t})_{t\geq 0}} is defined by a finite or countable state spaceS, atransition rate matrixQ with dimensions equal to that of the state space and initial probability distribution defined on the state space. Fori ≠ j, the elementsqij are non-negative and describe the rate of the process transitions from statei to statej. The elementsqii are chosen such that each row of the transition rate matrix sums to zero, while the row-sums of a probability transition matrix in a (discrete) Markov chain are all equal to one.

There are three equivalent definitions of the process.[45]

Infinitesimal definition

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The continuous time Markov chain is characterized by the transition rates, the derivatives with respect to time of the transition probabilities between states i and j.

LetXt{\displaystyle X_{t}} be the random variable describing the state of the process at timet, and assume the process is in a statei at timet. Then, knowingXt=i{\displaystyle X_{t}=i},Xt+h=j{\displaystyle X_{t+h}=j} is independent of previous values(Xs:s<t){\displaystyle \left(X_{s}:s<t\right)}, and ash → 0 for allj and for allt,Pr(X(t+h)=j∣X(t)=i)=δij+qijh+o(h),{\displaystyle \Pr(X(t+h)=j\mid X(t)=i)=\delta _{ij}+q_{ij}h+o(h),}whereδij{\displaystyle \delta _{ij}} is theKronecker delta, using thelittle-o notation.Theqij{\displaystyle q_{ij}} can be seen as measuring how quickly the transition fromi toj happens.

Jump chain/holding time definition

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Define a discrete-time Markov chainYn to describe thenth jump of the process and variablesS1,S2,S3, ... to describe holding times in each of the states whereSi follows theexponential distribution with rate parameter −qYiYi.

Transition probability definition

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For any valuen = 0, 1, 2, 3, ... and times indexed up to this value ofn:t0,t1,t2, ... and all states recorded at these timesi0,i1,i2,i3, ... it holds that

Pr(Xtn+1=in+1∣Xt0=i0,Xt1=i1,…,Xtn=in)=pinin+1(tn+1−tn){\displaystyle \Pr(X_{t_{n+1}}=i_{n+1}\mid X_{t_{0}}=i_{0},X_{t_{1}}=i_{1},\ldots ,X_{t_{n}}=i_{n})=p_{i_{n}i_{n+1}}(t_{n+1}-t_{n})}

wherepij is the solution of theforward equation (afirst-order differential equation)

P′(t)=P(t)Q{\displaystyle P'(t)=P(t)Q}

with initial condition P(0) is theidentity matrix.

Locally interacting Markov chains

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"Locally interacting Markov chains" are Markov chains with an evolution that takes into account the state of other Markov chains. This corresponds to the situation when the state space has a (Cartesian-) product form.Seeinteracting particle system andstochastic cellular automata (probabilistic cellular automata).See for instanceInteraction of Markov Processes[46]or.[47]

Discrete-time Markov process with general state space

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Main article:Markov chains on a measurable state space

Harris chains

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Many results for discrete-time Markov chains with finite state space can be generalized to chains with uncountable state space throughHarris chains.

The use of Markov chains inMarkov chain Monte Carlo methods covers cases where the process follows a continuous state space.

Continuous-time Markov process with general state space

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The definition of Markov processes in continuous time with general state space is more technical than the above.

A continuous-time Markov processX=(Xt)t≥0{\displaystyle X=(X_{t})_{t\geq 0}} is astochastic process adapted to afiltrationF=(Ft)t≥0{\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\geq 0}} with values in alocally compactPolish space(S,B(S)){\displaystyle (S,{\mathcal {B}}(S))} (e.g.,(R,B(R)){\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}). The latter essentially ensures that the conditional expectations ofXt{\displaystyle X_{t}} areregular, which, in simple terms, means that they behave "nicely". ThenX{\displaystyle X} is called aMarkov process, if it satisfies theMarkov property, i.e., for allt≥s≥0{\displaystyle t\geq s\geq 0} andA∈B(S){\displaystyle A\in {\mathcal {B}}(S)}[5]

P(Xt∈A∣Fs)=P(Xt∈A∣Xs){\displaystyle P(X_{t}\in A\mid {\mathcal {F}}_{s})=P(X_{t}\in A\mid X_{s})}.

Moreover,X{\displaystyle X} is calledtime-homogeneous, if it satisfies the weak Markov property for allt,s≥0{\displaystyle t,s\geq 0}:

P(Xt+s∈A∣Fs)=P(Xt∈A∣X0=x)|x=Xs=:Pt(Xs,A){\displaystyle P(X_{t+s}\in A\mid {\mathcal {F}}_{s})=P(X_{t}\in A\mid X_{0}=x)|_{x=X_{s}}=:P_{t}(X_{s},A)}.

The function(t,x,A)↦Pt(x,A){\displaystyle (t,x,A)\mapsto P_{t}(x,A)} is the so-calledtransition function ofX{\displaystyle X} and(Pt)t≥0{\displaystyle (P_{t})_{t\geq 0}} thetransition semigroup of the process. Transition functions are generalizations of the transition matrices used in the setting with finite state space.

In a more abstract way, Markov processes can also be defined or constructed the other way around: Let(Pt)t≥0{\displaystyle (P_{t})_{t\geq 0}} be a transition semigroup, i.e.,

  1. Pt{\displaystyle P_{t}} isMarkov kernel for allt≥0{\displaystyle t\geq 0},
  2. Pt+s(x,A)=∫SPt(y,A)Ps(x,dy)∀t,s≥0,x∈R,A∈B(S){\displaystyle P_{t+s}(x,A)=\int _{S}P_{t}(y,A)P_{s}(x,dy)\quad \forall t,s\geq 0,x\in \mathbb {R} ,A\in {\mathcal {B}}(S)} (Chapman-Kolmogorov-equation),
  3. P0(x,⋅)=δx{\displaystyle P_{0}(x,\cdot )=\delta _{x}},

whereδx{\displaystyle \delta _{x}} is theDirac-measure inx{\displaystyle x}, andX:Ω×[0,∞)→S{\displaystyle X:\Omega \times [0,\infty )\to S}. ThenX{\displaystyle X} is a homogeneous Markov process w.r.t. the natural filtrationFX=(σ(Xs:0≤s≤t))t≥0{\displaystyle \mathbb {F} ^{X}=(\sigma (X_{s}:0\leq s\leq t))_{t\geq 0}}, if for all0≤t1<...<tn{\displaystyle 0\leq t_{1}<...<t_{n}},A1,...,An∈B(S){\displaystyle A_{1},...,A_{n}\in {\mathcal {B}}(S)} the underlying probability measureP{\displaystyle P} satisfies

P(Xt1∈A1,...,Xtn∈An∣X0=x)=∫A1...∫An−1Ptn−tn−1(xn−1,An)⋯Pt1(x,dx1){\displaystyle P(X_{t_{1}}\in A_{1},...,X_{t_{n}}\in A_{n}\mid X_{0}=x)=\int _{A_{1}}...\int _{A_{n-1}}P_{t_{n}-t_{n-1}}(x_{n-1},A_{n})\cdots P_{t_{1}}(x,dx_{1})}.

Or, if no probability measureP{\displaystyle P} has been specified, the above equation defines a measurePx:=P(⋅∣X0=x){\displaystyle P^{x}:=P(\cdot \mid X_{0}=x)} onσ(Xs:s≥0){\displaystyle \sigma (X_{s}:s\geq 0)} under which the processX{\displaystyle X} started inx{\displaystyle x} is a Markov process by construction.

In other words, Markov processes can be defined either as stochastic processesX{\displaystyle X} on a filtered probability space, or indirectly in terms of a transition semigroup (i.e., the transition probabilities of the process), which induces a probability space under whichX{\displaystyle X} has the Markov property.

Properties

[edit]

Two states are said tocommunicate with each other if both are reachable from one another by a sequence of transitions that have positive probability. This is an equivalence relation which yields a set of communicating classes. A class isclosed if the probability of leaving the class is zero. A Markov chain isirreducible if there is one communicating class, the state space.

A statei has periodk ifk is thegreatest common divisor of the number of transitions by whichi can be reached, starting fromi. That is:

k=gcd{n>0:Pr(Xn=i∣X0=i)>0}{\displaystyle k=\gcd\{n>0:\Pr(X_{n}=i\mid X_{0}=i)>0\}}

The state isperiodic ifk>1{\displaystyle k>1}; otherwisek=1{\displaystyle k=1} and the state isaperiodic.

A statei is said to betransient if, starting fromi, there is a non-zero probability that the chain will never return toi. It is calledrecurrent (orpersistent) otherwise.[48] For a recurrent statei, the meanhitting time is defined as:

Mi=E[Ti]=∑n=1∞n⋅fii(n){\displaystyle M_{i}=E[T_{i}]=\sum _{n=1}^{\infty }n\cdot f_{ii}^{(n)}} wherefii(n):=Pr(min{m>0:Xm=i}=n∣X0=i){\displaystyle f_{ii}^{(n)}:=\Pr(\min\{m>0:X_{m}=i\}=n\mid X_{0}=i)}.

Statei ispositive recurrent ifMi{\displaystyle M_{i}} is finite andnull recurrent otherwise. Periodicity, transience, recurrence and positive and null recurrence are class properties — that is, if one state has the property then all states in its communicating class have the property.[49]

A statei is calledabsorbing if there are no outgoing transitions from the state.

Irreducibility

[edit]

Since periodicity is a class property, if a Markov chain is irreducible, then all its states have the same period. In particular, if one state is aperiodic, then the whole Markov chain is aperiodic.[50]

If a finite Markov chain is irreducible, then all states are positive recurrent, and it has a unique stationary distribution given byπi=1/E[Ti]{\displaystyle \pi _{i}=1/E[T_{i}]}.

Ergodicity

[edit]

A statei is said to beergodic if it is aperiodic and positive recurrent. In other words, a statei is ergodic if it is recurrent, has a period of 1, and has finite mean recurrence time.

If all states in an irreducible Markov chain are ergodic, then the chain is said to be ergodic. Equivalently, there exists some integerk{\displaystyle k} such that all entries ofMk{\displaystyle M^{k}} are positive.

It can be shown that a finite state irreducible Markov chain is ergodic if it has an aperiodic state.

A Markov chain with more than one state and just one out-going transition per state is either not irreducible or not aperiodic, hence cannot be ergodic.

Terminology

[edit]

Some authors call any irreducible, positive recurrent Markov chains ergodic, even periodic ones.[51] In fact, merely irreducible Markov chains correspond toergodic processes, defined according toergodic theory.[52]

Some authors call a matrixprimitive if there exists some integerk{\displaystyle k} such that all entries ofMk{\displaystyle M^{k}} are positive.[53] Some authors call itregular.[54]

Index of primitivity

[edit]

Theindex of primitivity, orexponent, of a regular matrix, is the smallestk{\displaystyle k} such that all entries ofMk{\displaystyle M^{k}} are positive. The exponent is purely a graph-theoretic property, since it depends only on whether each entry ofM{\displaystyle M} is zero or positive, and therefore can be found on a directed graph withsign(M){\displaystyle \mathrm {sign} (M)} as its adjacency matrix.

There are several combinatorial results about the exponent when there are finitely many states. Letn{\displaystyle n} be the number of states, then[55]

Measure-preserving dynamical system

[edit]

If a Markov chain has a stationary distribution, then it can be converted to ameasure-preserving dynamical system: Let the probability space beΩ=ΣN{\displaystyle \Omega =\Sigma ^{\mathbb {N} }}, whereΣ{\displaystyle \Sigma } is the set of all states for the Markov chain. Let the sigma-algebra on the probability space be generated by the cylinder sets. Let the probability measure be generated by the stationary distribution, and the Markov chain transition. LetT:Ω→Ω{\displaystyle T:\Omega \to \Omega } be the shift operator:T(X0,X1,…)=(X1,…){\displaystyle T(X_{0},X_{1},\dots )=(X_{1},\dots )}. Similarly we can construct such a dynamical system withΩ=ΣZ{\displaystyle \Omega =\Sigma ^{\mathbb {Z} }} instead.[57]

Sinceirreducible Markov chains with finite state spaces have a unique stationary distribution, the above construction is unambiguous for irreducible Markov chains.

Inergodic theory, a measure-preserving dynamical system is calledergodic if any measurable subsetS{\displaystyle S} such thatT−1(S)=S{\displaystyle T^{-1}(S)=S} impliesS=∅{\displaystyle S=\emptyset } orΩ{\displaystyle \Omega } (up to a null set).

The terminology is inconsistent. Given a Markov chain with a stationary distribution that is strictly positive on all states, the Markov chain isirreducible if its corresponding measure-preserving dynamical system isergodic.[52]

Markovian representations

[edit]

In some cases, apparently non-Markovian processes may still have Markovian representations, constructed by expanding the concept of the "current" and "future" states. For example, letX be a non-Markovian process. Then define a processY, such that each state ofY represents a time-interval of states ofX. Mathematically, this takes the form:

Y(t)={X(s):s∈[a(t),b(t)]}.{\displaystyle Y(t)={\big \{}X(s):s\in [a(t),b(t)]\,{\big \}}.}

IfY has the Markov property, then it is a Markovian representation ofX.

An example of a non-Markovian process with a Markovian representation is anautoregressivetime series of order greater than one.[58]

Hitting times

[edit]
Main article:Phase-type distribution

Thehitting time is the time, starting in a given set of states, until the chain arrives in a given state or set of states. The distribution of such a time period has a phase type distribution. The simplest such distribution is that of a single exponentially distributed transition.

Expected hitting times

[edit]

For a subset of statesA ⊆ S, the vectorkA of hitting times (where elementkiA{\displaystyle k_{i}^{A}} represents theexpected value, starting in statei that the chain enters one of the states in the setA) is the minimal non-negative solution to[59]

kiA=0 for i∈A−∑j∈SqijkjA=1 for i∉A.{\displaystyle {\begin{aligned}k_{i}^{A}=0&{\text{ for }}i\in A\\-\sum _{j\in S}q_{ij}k_{j}^{A}=1&{\text{ for }}i\notin A.\end{aligned}}}

Time reversal

[edit]

For a general Markov processX{\displaystyle X} in continuous time (a CTMC or a process with general state space), the reverse processX←=(XT−t)t∈[0,T]{\displaystyle {\overleftarrow {X}}=(X_{T-t})_{t\in [0,T]}} from a fixed timeT>0{\displaystyle T>0} is again a Markov process. This follows directly from theMarkov property: Informally speaking, the future and the past are independent given the present. Under time-reversal, their roles are just interchanged. However, the reverse process is not time-homogeneous in general. If for some random timeτ{\displaystyle \tau } (not necessarily astopping time) the stopped processXτ=(Xt∧τ)t≥0{\displaystyle X^{\tau }=(X_{t\land \tau })_{t\geq 0}} is a time-homogeneous Markov process, then the reverse processXτ←=(Xτ−t∧τ1{τ<∞})t≥0{\displaystyle {\overleftarrow {X^{\tau }}}=(X_{\tau -t\land \tau }1_{\{\tau <\infty \}})_{t\geq 0}} is again time-homogeneous.[60]

IfX{\displaystyle X} is a CTMC, then byKelly's lemmaX←{\displaystyle {\overleftarrow {X}}} has the same stationary distribution as the forward process.

A chain is said to bereversible if the reversed process is the same as the forward process (in distribution).Kolmogorov's criterion states that the necessary and sufficient condition for a Markov chain to be reversible is that the product of transition rates around a closed loop must be the same in both directions.

Embedded Markov chain

[edit]

One method of finding thestationary probability distribution,π, of anergodic continuous-time Markov chain,Q, is by first finding itsembedded Markov chain (EMC). Strictly speaking, the EMC is a regular discrete-time Markov chain, sometimes referred to as ajump process. Each element of the one-step transition probability matrix of the EMC,S, is denoted bysij, and represents theconditional probability of transitioning from statei into statej. These conditional probabilities may be found by

sij={qij∑k≠iqikif i≠j0otherwise.{\displaystyle s_{ij}={\begin{cases}{\frac {q_{ij}}{\sum _{k\neq i}q_{ik}}}&{\text{if }}i\neq j\\0&{\text{otherwise}}.\end{cases}}}

From this,S may be written as

S=I−(diag⁡(Q))−1Q{\displaystyle S=I-\left(\operatorname {diag} (Q)\right)^{-1}Q}

whereI is theidentity matrix and diag(Q) is thediagonal matrix formed by selecting themain diagonal from the matrixQ and setting all other elements to zero.

To find the stationary probability distribution vector, we must next findφ{\displaystyle \varphi } such that

φS=φ,{\displaystyle \varphi S=\varphi ,}

withφ{\displaystyle \varphi } being a row vector, such that all elements inφ{\displaystyle \varphi } are greater than 0 and‖φ‖1{\displaystyle \|\varphi \|_{1}} = 1. From this,π may be found as

π=−φ(diag⁡(Q))−1‖φ(diag⁡(Q))−1‖1.{\displaystyle \pi ={-\varphi (\operatorname {diag} (Q))^{-1} \over \left\|\varphi (\operatorname {diag} (Q))^{-1}\right\|_{1}}.}

(S may be periodic, even ifQ is not. Onceπ is found, it must be normalized to aunit vector.)

Another discrete-time process that may be derived from a continuous-time Markov chain is a δ-skeleton—the (discrete-time) Markov chain formed by observingX(t) at intervals of δ units of time. The random variablesX(0), X(δ), X(2δ), ... give the sequence of states visited by the δ-skeleton.

Special types of Markov chains

[edit]

Markov model

[edit]
Main article:Markov model

Markov models are used to model changing systems. There are 4 main types of models, that generalize Markov chains depending on whether every sequential state is observable or not, and whether the system is to be adjusted on the basis of observations made:

System state is fully observableSystem state is partially observable
System is autonomousMarkov chainHidden Markov model
System is controlledMarkov decision processPartially observable Markov decision process

Bernoulli scheme

[edit]
Main article:Bernoulli scheme

ABernoulli scheme is a special case of a Markov chain where the transition probability matrix has identical rows, which means that the next state is independent of even the current state (in addition to being independent of the past states). A Bernoulli scheme with only two possible states is known as aBernoulli process.

Note, however, by theOrnstein isomorphism theorem, that every aperiodic and irreducible Markov chain is isomorphic to a Bernoulli scheme;[61] thus, one might equally claim that Markov chains are a "special case" of Bernoulli schemes. The isomorphism generally requires a complicated recoding. The isomorphism theorem is even a bit stronger: it states thatanystationary stochastic process is isomorphic to a Bernoulli scheme; the Markov chain is just one such example.

Subshift of finite type

[edit]
Main article:Subshift of finite type

When the Markov matrix is replaced by theadjacency matrix of afinite graph, the resulting shift is termed atopological Markov chain or asubshift of finite type.[61] A Markov matrix that is compatible with the adjacency matrix can then provide ameasure on the subshift. Many chaoticdynamical systems are isomorphic to topological Markov chains; examples includediffeomorphisms ofclosed manifolds, theProuhet–Thue–Morse system, theChacon system,sofic systems,context-free systems andblock-coding systems.[61]

Applications

[edit]

Markov chains have been employed in a wide range of topics across the natural and social sciences, and in technological applications.

Physics

[edit]

Markovian systems appear extensively inthermodynamics andstatistical mechanics, whenever probabilities are used to represent unknown or unmodelled details of the system, if it can be assumed that the dynamics are time-invariant, and that no relevant history need be considered which is not already included in the state description.[62][63] For example, a thermodynamic state operates under a probability distribution that is difficult or expensive to acquire. Therefore, Markov Chain Monte Carlo method can be used to draw samples randomly from a black-box to approximate the probability distribution of attributes over a range of objects.[63]

Markov chains are used inlattice QCD simulations.[64]

Chemistry

[edit]
Michaelis-Menten kinetics. The enzyme (E) binds a substrate (S) and produces a product (P). Each reaction is a state transition in a Markov chain.

A reaction network is a chemical system involving multiple reactions and chemical species. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain.[65] Markov chains and continuous-time Markov processes are useful in chemistry when physical systems closely approximate the Markov property. For example, imagine a large numbern of molecules in solution in state A, each of which can undergo a chemical reaction to state B with a certain average rate. Perhaps the molecule is an enzyme, and the states refer to how it is folded. The state of any single enzyme follows a Markov chain, and since the molecules are essentially independent of each other, the number of molecules in state A or B at a time isn times the probability a given molecule is in that state.

The classical model of enzyme activity,Michaelis–Menten kinetics, can be viewed as a Markov chain, where at each time step the reaction proceeds in some direction. While Michaelis-Menten is fairly straightforward, far more complicated reaction networks can also be modeled with Markov chains.[66]

An algorithm based on a Markov chain was also used to focus the fragment-based growth of chemicalsin silico towards a desired class of compounds such as drugs or natural products.[67] As a molecule is grown, a fragment is selected from the nascent molecule as the "current" state. It is not aware of its past (that is, it is not aware of what is already bonded to it). It then transitions to the next state when a fragment is attached to it. The transition probabilities are trained on databases of authentic classes of compounds.[68]

Also, the growth (and composition) ofcopolymers may be modeled using Markov chains. Based on the reactivity ratios of the monomers that make up the growing polymer chain, the chain's composition may be calculated (for example, whether monomers tend to add in alternating fashion or in long runs of the same monomer). Due tosteric effects, second-order Markov effects may also play a role in the growth of some polymer chains.

Similarly, it has been suggested that the crystallization and growth of some epitaxialsuperlattice oxide materials can be accurately described by Markov chains.[69]

Biology

[edit]

Markov chains are used in various areas of biology. Notable examples include:

Information theory

[edit]

Markov chains are used throughout information processing.Claude Shannon's famous 1948 paperA Mathematical Theory of Communication, which in a single step created the field ofinformation theory, opens by introducing the concept ofentropy by modeling texts in a natural language (such as English) as generated by an ergodic Markov process, where each letter may depend statistically on previous letters.[72] Such idealized models can capture many of the statistical regularities of systems. Even without describing the full structure of the system perfectly, such signal models can make possible very effectivedata compression throughentropy encoding techniques such asarithmetic coding. They also allow effectivestate estimation andpattern recognition. Markov chains also play an important role inreinforcement learning.

Markov chains are also the basis for hidden Markov models, which are an important tool in such diverse fields as telephone networks (which use theViterbi algorithm for error correction), speech recognition andbioinformatics (such as in rearrangements detection[73]).

TheLZMA lossless data compression algorithm combines Markov chains withLempel-Ziv compression to achieve very high compression ratios.

Queueing theory

[edit]
Main article:Queueing theory

Markov chains are the basis for the analytical treatment of queues (queueing theory).Agner Krarup Erlang initiated the subject in 1917.[74] This makes them critical for optimizing the performance of telecommunications networks, where messages must often compete for limited resources (such as bandwidth).[75]

Numerous queueing models use continuous-time Markov chains. For example, anM/M/1 queue is a CTMC on the non-negative integers where upward transitions fromi toi + 1 occur at rateλ according to aPoisson process and describe job arrivals, while transitions fromi toi – 1 (fori > 1) occur at rateμ (job service times are exponentially distributed) and describe completed services (departures) from the queue.

Internet applications

[edit]
A state diagram that represents the PageRank algorithm with a transitional probability of M, orαki+1−αN{\displaystyle {\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}}

ThePageRank of a webpage as used byGoogle is defined by a Markov chain.[76][77][78] It is the probability to be at pagei{\displaystyle i} in the stationary distribution on the following Markov chain on all (known) webpages. IfN{\displaystyle N} is the number of known webpages, and a pagei{\displaystyle i} haski{\displaystyle k_{i}} outgoing links from it then it has transition probabilityαki+1−αN{\displaystyle {\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}} for all pages that are linked to and1−αN{\displaystyle {\frac {1-\alpha }{N}}} for all pages that are not linked to. The parameterα{\displaystyle \alpha } is taken to be about 0.85.[79]

Markov models have also been used to analyze web navigation behavior of users. A user's web link transition on a particular website can be modeled using first- or second-order Markov models and can be used to make predictions regarding future navigation and to personalize the web page for an individual user.[citation needed]

Statistics

[edit]

Markov chain methods have also become very important for generating sequences of random numbers to accurately reflect very complicated desired probability distributions, via a process calledMarkov chain Monte Carlo (MCMC). In recent years this has revolutionized the practicability ofBayesian inference methods, allowing a wide range ofposterior distributions to be simulated and their parameters found numerically.[citation needed]

Economics and finance

[edit]

Markov chains are used in finance and economics to model a variety of different phenomena, including the distribution of income, the size distribution of firms, asset prices and market crashes.D. G. Champernowne built a Markov chain model of the distribution of income in 1953.[80]Herbert A. Simon and co-author Charles Bonini used a Markov chain model to derive a stationary Yule distribution of firm sizes.[81]Louis Bachelier was the first to observe that stock prices followed a random walk.[82] The random walk was later seen as evidence in favor of theefficient-market hypothesis and random walk models were popular in the literature of the 1960s.[83] Regime-switching models of business cycles were popularized byJames D. Hamilton (1989), who used a Markov chain to model switches between periods of high and low GDP growth (or, alternatively, economic expansions and recessions).[84] A more recent example is theMarkov switching multifractal model ofLaurent E. Calvet and Adlai J. Fisher, which builds upon the convenience of earlier regime-switching models.[85][86] It uses an arbitrarily large Markov chain to drive the level of volatility of asset returns.

Dynamic macroeconomics makes heavy use of Markov chains. An example is using Markov chains to exogenously model prices of equity (stock) in ageneral equilibrium setting.[87]

Credit rating agencies produce annual tables of the transition probabilities for bonds of different credit ratings.[88]

Social sciences

[edit]

Markov chains are generally used in describingpath-dependent arguments, where current structural configurations condition future outcomes. An example is the reformulation of the idea, originally due toKarl Marx'sDas Kapital, tyingeconomic development to the rise ofcapitalism. In current research, it is common to use a Markov chain to model how once a country reaches a specific level of economic development, the configuration of structural factors, such as size of themiddle class, the ratio of urban to rural residence, the rate ofpolitical mobilization, etc., will generate a higher probability of transitioning fromauthoritarian todemocratic regime.[89]

Music

[edit]

Markov chains are employed inalgorithmic music composition, particularly insoftware such asCsound,Max, andSuperCollider. In a first-order chain, the states of the system become note or pitch values, and aprobability vector for each note is constructed, completing a transition probability matrix (see below). An algorithm is constructed to produce output note values based on the transition matrix weightings, which could beMIDI note values, frequency (Hz), or any other desirable metric.[90]

1st-order matrix
NoteAC♯E♭
A0.10.60.3
C♯0.250.050.7
E♭0.70.30
2nd-order matrix
NotesADG
AA0.180.60.22
AD0.50.50
AG0.150.750.1
DD001
DA0.2500.75
DG0.90.10
GG0.40.40.2
GA0.50.250.25
GD100

A second-order Markov chain can be introduced by considering the current stateand also the previous state, as indicated in the second table. Higher,nth-order chains tend to "group" particular notes together, while 'breaking off' into other patterns and sequences occasionally. These higher-order chains tend to generate results with a sense ofphrasal structure, rather than the 'aimless wandering' produced by a first-order system.[91]

Markov chains can be used structurally, as in Xenakis's Analogique A and B.[92] Markov chains are also used in systems which use a Markov model to react interactively to music input.[93]

Usually musical systems need to enforce specific control constraints on the finite-length sequences they generate, but control constraints are not compatible with Markov models, since they induce long-range dependencies that violate the Markov hypothesis of limited memory. In order to overcome this limitation, a new approach has been proposed.[94]

Games and sports

[edit]

Markov chains can be used to model many games of chance. The children's gamesSnakes and Ladders and "Hi Ho! Cherry-O", for example, are represented exactly by Markov chains. At each turn, the player starts in a given state (on a given square) and from there has fixed odds of moving to certain other states (squares).[citation needed]

Markov chain models have been used in advanced baseball analysis since 1960, although their use is still rare. Each half-inning of a baseball game fits the Markov chain state when the number of runners and outs are considered. During any at-bat, there are 24 possible combinations of number of outs and position of the runners. Mark Pankin shows that Markov chain models can be used to evaluate runs created for both individual players as well as a team.[95]He also discusses various kinds of strategies and play conditions: how Markov chain models have been used to analyze statistics for game situations such asbunting andbase stealing and differences when playing on grass vs.AstroTurf.[96]

Markov text generators

[edit]

Markov processes can also be used togenerate superficially real-looking text given a sample document. Markov processes are used in a variety of recreational "parody generator" software (seedissociated press, Jeff Harrison,[97]Mark V. Shaney,[98][99] and Academias Neutronium). Several open-source text generation libraries using Markov chains exist.

See also

[edit]

Notes

[edit]
  1. ^abSean Meyn; Richard L. Tweedie (2 April 2009).Markov Chains and Stochastic Stability. Cambridge University Press. p. 3.ISBN 978-0-521-73182-9.
  2. ^Reuven Y. Rubinstein; Dirk P. Kroese (20 September 2011).Simulation and the Monte Carlo Method. John Wiley & Sons. p. 225.ISBN 978-1-118-21052-9.
  3. ^Dani Gamerman; Hedibert F. Lopes (10 May 2006).Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition. CRC Press.ISBN 978-1-58488-587-0.
  4. ^"Markovian".Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription orparticipating institution membership required.)
  5. ^abØksendal, B. K. (Bernt Karsten) (2003).Stochastic differential equations: an introduction with applications (6th ed.). Berlin: Springer.ISBN 3-540-04758-1.OCLC 52203046.
  6. ^abSøren Asmussen (15 May 2003).Applied Probability and Queues. Springer Science & Business Media. p. 7.ISBN 978-0-387-00211-8.
  7. ^Emanuel Parzen (17 June 2015).Stochastic Processes. Courier Dover Publications. p. 188.ISBN 978-0-486-79688-8.
  8. ^Samuel Karlin; Howard E. Taylor (2 December 2012).A First Course in Stochastic Processes. Academic Press. pp. 29 and 30.ISBN 978-0-08-057041-9.
  9. ^John Lamperti (1977).Stochastic processes: a survey of the mathematical theory. Springer-Verlag. pp. 106–121.ISBN 978-3-540-90275-1.
  10. ^Sheldon M. Ross (1996).Stochastic processes. Wiley. pp. 174 and 231.ISBN 978-0-471-12062-9.
  11. ^Everitt, B.S. (2002)The Cambridge Dictionary of Statistics. CUP.ISBN 0-521-81099-X
  12. ^Parzen, E. (1962)Stochastic Processes, Holden-Day.ISBN 0-8162-6664-6 (Table 6.1)
  13. ^Dodge, Y. (2003)The Oxford Dictionary of Statistical Terms, OUP.ISBN 0-19-920613-9 (entry for "Markov chain")
  14. ^Dodge, Y.The Oxford Dictionary of Statistical Terms, OUP.ISBN 0-19-920613-9
  15. ^Meyn, S. Sean P., and Richard L. Tweedie. (2009)Markov chains and stochastic stability. Cambridge University Press. (Preface, p. iii)
  16. ^abcdeCharles Miller Grinstead; James Laurie Snell (1997).Introduction to Probability. American Mathematical Soc. pp. 464–466.ISBN 978-0-8218-0749-1.
  17. ^abcPierre Bremaud (9 March 2013).Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Springer Science & Business Media. p. ix.ISBN 978-1-4757-3124-8.
  18. ^abcHayes, Brian (2013). "First links in the Markov chain".American Scientist.101 (2):92–96.doi:10.1511/2013.101.92.
  19. ^abSheldon M. Ross (1996).Stochastic processes. Wiley. pp. 235 and 358.ISBN 978-0-471-12062-9.
  20. ^Jarrow, Robert; Protter, Philip (2004). "A short history of stochastic integration and mathematical finance: The early years, 1880–1970".A Festschrift for Herman Rubin. pp. 75–91.CiteSeerX 10.1.1.114.632.doi:10.1214/lnms/1196285381.ISBN 978-0-940600-61-4.
  21. ^Guttorp, Peter; Thorarinsdottir, Thordis L. (2012). "What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes".International Statistical Review.80 (2):253–268.doi:10.1111/j.1751-5823.2012.00181.x.
  22. ^Seneta, E. (1996). "Markov and the Birth of Chain Dependence Theory".International Statistical Review.64 (3):255–257.doi:10.2307/1403785.JSTOR 1403785.
  23. ^Seneta, E. (1998). "I.J. Bienaymé [1796–1878]: Criticality, Inequality, and Internationalization".International Statistical Review.66 (3):291–292.doi:10.2307/1403518.JSTOR 1403518.
  24. ^Bru B, Hertz S (2001). "Maurice Fréchet". InHeyde CC, Seneta E, Crépel P, Fienberg SE, Gani J (eds.).Statisticians of the Centuries. New York, NY: Springer. pp. 331–334.doi:10.1007/978-1-4613-0179-0_71.ISBN 978-0-387-95283-3.
  25. ^abcKendall, D. G.; Batchelor, G. K.; Bingham, N. H.; Hayman, W. K.; Hyland, J. M. E.; Lorentz, G. G.; Moffatt, H. K.; Parry, W.; Razborov, A. A.; Robinson, C. A.; Whittle, P. (1990). "Andrei Nikolaevich Kolmogorov (1903–1987)".Bulletin of the London Mathematical Society.22 (1): 33.doi:10.1112/blms/22.1.31.
  26. ^abCramér, Harald (1976)."Half a Century with Probability Theory: Some Personal Recollections".The Annals of Probability.4 (4):509–546.doi:10.1214/aop/1176996025.
  27. ^Marc Barbut; Bernard Locker; Laurent Mazliak (23 August 2016).Paul Lévy and Maurice Fréchet: 50 Years of Correspondence in 107 Letters. Springer London. p. 5.ISBN 978-1-4471-7262-8.
  28. ^Valeriy Skorokhod (5 December 2005).Basic Principles and Applications of Probability Theory. Springer Science & Business Media. p. 146.ISBN 978-3-540-26312-8.
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References

[edit]
  • A. A. Markov (1906) "Rasprostranenie zakona bol'shih chisel na velichiny, zavisyaschie drug ot druga".Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete, 2-ya seriya, tom 15, pp. 135–156.
  • A. A. Markov (1971). "Extension of the limit theorems of probability theory to a sum of variables connected in a chain". reprinted in Appendix B of: R. Howard.Dynamic Probabilistic Systems, volume 1: Markov Chains. John Wiley and Sons.
  • Classical Text in Translation:Markov, A. A. (2006). "An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains".Science in Context.19 (4). Translated by Link, David:591–600.doi:10.1017/s0269889706001074.
  • Leo Breiman (1992) [1968]Probability. Original edition published by Addison-Wesley; reprinted bySociety for Industrial and Applied MathematicsISBN 0-89871-296-3. (See Chapter 7)
  • J. L. Doob (1953)Stochastic Processes. New York: John Wiley and SonsISBN 0-471-52369-0.
  • S. P. Meyn and R. L. Tweedie (1993)Markov Chains and Stochastic Stability. London: Springer-VerlagISBN 0-387-19832-6. online:MCSS . Second edition to appear, Cambridge University Press, 2009.
  • Dynkin, Eugene Borisovich (1965).Markov Processes. Grundlehren der mathematischen Wissenschaften. Vol. I (121). Translated by Fabius, Jaap; Greenberg, Vida Lazarus; Maitra, Ashok Prasad;Majone, Giandomenico. Berlin:Springer-Verlag.doi:10.1007/978-3-662-00031-1.ISBN 978-3-662-00033-5. Title-No. 5104.;Markov Processes. Grundlehren der mathematischen Wissenschaften. Vol. II (122). 1965.doi:10.1007/978-3-662-25360-1.ISBN 978-3-662-23320-7. Title-No. 5105. (NB. This was originally published in Russian asМарковские процессы (Markovskiye protsessy) byFizmatgiz in 1963 and translated to English with the assistance of the author.)
  • S. P. Meyn.Control Techniques for Complex Networks. Cambridge University Press, 2007.ISBN 978-0-521-88441-9. Appendix contains abridged Meyn & Tweedie. online:CTCN
  • Booth, Taylor L. (1967).Sequential Machines and Automata Theory (1st ed.). New York, NY: John Wiley and Sons, Inc. Library of Congress Card Catalog Number 67-25924. ] Extensive, wide-ranging book meant for specialists, written for both theoretical computer scientists as well as electrical engineers. With detailed explanations of state minimization techniques, FSMs, Turing machines, Markov processes, and undecidability. Excellent treatment of Markov processes pp. 449ff. Discusses Z-transforms, D transforms in their context.
  • Kemeny, John G.; Hazleton Mirkil; J. Laurie Snell; Gerald L. Thompson (1959).Finite Mathematical Structures (1st ed.). Englewood Cliffs, NJ: Prentice-Hall, Inc. Library of Congress Card Catalog Number 59-12841. Classical text. cf Chapter 6Finite Markov Chains pp. 384ff.
  • John G. Kemeny &J. Laurie Snell (1960)Finite Markov Chains, D. van Nostrand CompanyISBN 0-442-04328-7
  • E. Nummelin. "General irreducible Markov chains and non-negative operators". Cambridge University Press, 1984, 2004.ISBN 0-521-60494-X
  • Seneta, E.Non-negative matrices and Markov chains. 2nd rev. ed., 1981, XVI, 288 p., Softcover Springer Series in Statistics. (Originally published by Allen & Unwin Ltd., London, 1973)ISBN 978-0-387-29765-1
  • Kishor S. Trivedi,Probability and Statistics with Reliability, Queueing, and Computer Science Applications, John Wiley & Sons, Inc. New York, 2002.ISBN 0-471-33341-7.
  • K. S. Trivedi and R.A.Sahner,SHARPE at the age of twenty-two, vol. 36, no. 4, pp. 52–57, ACM SIGMETRICS Performance Evaluation Review, 2009.
  • R. A. Sahner, K. S. Trivedi and A. Puliafito,Performance and reliability analysis of computer systems: an example-based approach using the SHARPE software package, Kluwer Academic Publishers, 1996.ISBN 0-7923-9650-2.
  • G. Bolch, S. Greiner, H. de Meer and K. S. Trivedi,Queueing Networks and Markov Chains, John Wiley, 2nd edition, 2006.ISBN 978-0-7923-9650-5.

External links

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