Markov chain

A diagram representing a two-state Markov process. The numbers are the probability of changing from one state to another state.

AMarkov chainorMarkov processis astochastic processdescribing asequenceof possible events in which theprobabilityof 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 infinitesequence, in which the chain moves state at discrete time steps, gives adiscrete-time Markov chain(DTMC). Acontinuous-timeprocess is called acontinuous-time Markov chain(CTMC). Markov processes are named in honor of theRussianmathematicianAndrey Markov.

Markov chains have many applications asstatistical modelsof real-world processes.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.

The adjectivesMarkovianandMarkovare used to describe something that is related to a Markov process.

Principles

Russian mathematicianAndrey Markov

Definition

A Markov process is astochastic processthat 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.In other words,conditionalon 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 spaceor a discrete index set (often representing time), but the precise definition of a Markov chain varies.For example, it is common to define a Markov chain as a Markov process in eitherdiscrete or continuous timewith a countable state space (thus regardless of the nature of time),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).

Types of Markov chains

The system'sstate spaceand 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 and for discrete time v. continuous time:

	Countable state space	Continuous or general state space
Discrete-time	(discrete-time) Markov chain on a countable or finite state space	Markov chain on a measurable state space(for example,Harris chain)
Continuous-time	Continuous-time Markov process or Markov jump process	Anycontinuous stochastic processwith 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),but a few authors use the term "Markov process" to refer to acontinuous-time Markov chain (CTMC)without explicit mention.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 spaceof a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space.However, many applications of Markov chains employ finite orcountably infinitestate 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

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 matrixdescribing 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 theintegersornatural numbers,and the random process is a mapping of these to states. The Markov property states that theconditional probability distributionfor 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

Andrey Markovstudied Markov processes in the early 20th century, publishing his first paper on the topic in 1906.Markov Processes in continuous time were discovered long before his work in the early 20th century in the form of thePoisson process.Markov was interested in studying an extension of independent random sequences, motivated by a disagreement withPavel Nekrasovwho claimed independence was necessary for theweak law of large numbersto hold.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,which had been commonly regarded as a requirement for such mathematical laws to hold.Markov later used Markov chains to study the distribution of vowels inEugene Onegin,written byAlexander Pushkin,and proved acentral limit theoremfor such chains.

In 1912Henri Poincaréstudied Markov chains onfinite groupswith an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced byPaulandTatyana Ehrenfestin 1907, and a branching process, introduced byFrancis GaltonandHenry William Watsonin 1873, preceding the work of Markov.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.Starting in 1928,Maurice Fréchetbecame interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains.

Andrey Kolmogorovdeveloped in a 1931 paper a large part of the early theory of continuous-time Markov processes.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.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.Independent of Kolmogorov's work,Sydney Chapmanderived in a 1928 paper an equation, now called theChapman–Kolmogorov equation,in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement.The differential equations are now called the Kolmogorov equationsor the Kolmogorov–Chapman equations.Other mathematicians who contributed significantly to the foundations of Markov processes includeWilliam Feller,starting in 1930s, and then laterEugene Dynkin,starting in the 1950s.

Examples

Main article:Examples of Markov chains

• Random walksbased on integers and thegambler's ruinproblem are examples of Markov processes.Some variations of these processes were studied hundreds of years earlier in the context of independent variables.Two important examples of Markov processes are theWiener process,also known as theBrownian motionprocess, and thePoisson process,which are considered the most important and central stochastic processes in the theory of stochastic processes.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.
• A famous Markov chain is the so-called "drunkard's walk", a random walk on thenumber linewhere, 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

Suppose that there is a coin purse containing five quarters (each worth 25¢), five dimes (each worth 10¢), and five nickels (each worth 5¢), and one by one, coins are randomly drawn from the purse and are set on a table. If$X_{n}$represents the total value of the coins set on the table afterndraws, with$X_{0}=0$,then the sequence$\{X_{n}:n\in \mathbb {N} \}$isnota Markov process.

To see why this is the case, suppose that in the first six draws, all five nickels and a quarter are drawn. Thus$X_{6}=\$0.50$.If we know not just$X_{6}$,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 that$X_{7}\geq \$0.60$with probability 1. But if we do not know the earlier values, then based only on the value$X_{6}$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 about$X_{7}$are impacted by our knowledge of values prior to$X_{6}$.

However, it is possible to model this scenario as a Markov process. Instead of defining$X_{n}$to represent thetotal valueof the coins on the table, we could define$X_{n}$to represent thecountof the various coin types on the table. For instance,$X_{6}=1,0,5$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 by$6\times 6\times 6=216$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 state$X_{1}=0,1,0$.The probability of achieving$X_{2}$now depends on$X_{1}$;for example, the state$X_{2}=1,0,1$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 the$X_{n}=i,j,k$state depends exclusively on the outcome of the$X_{n-1}=\ell,m,p$state.

Formal definition

Discrete-time Markov chain

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:

$\Pr(X_{n+1}=x\mid X_{1}=x_{1},X_{2}=x_{2},\ldots,X_{n}=x_{n})=\Pr(X_{n+1}=x\mid X_{n}=x_{n}),$if bothconditional probabilitiesare well defined, that is, if$\Pr(X_{1}=x_{1},\ldots,X_{n}=x_{n})>0.$

The possible values ofX_iform acountable setScalled the state space of the chain.

Variations

• Time-homogeneous Markov chains are processes where$$\Pr(X_{n+1}=x\mid X_{n}=y)=\Pr(X_{n}=x\mid X_{n-1}=y)$$for alln.The probability of the transition is independent ofn.
• Stationary Markov chains are processes where$$\Pr(X_{0}=x_{0},X_{1}=x_{1},\ldots,X_{k}=x_{k})=\Pr(X_{n}=x_{0},X_{n+1}=x_{1},\ldots,X_{n+k}=x_{k})$$for allnandk.Every stationary chain can be proved to be time-homogeneous by Bayes' rule.
  A necessary and sufficient condition for a time-homogeneous Markov chain to be stationary is that the distribution of$X_{0}$is a stationary distribution of the Markov chain.
• A Markov chain with memory (or a Markov chain of orderm) wheremis finite, is a process satisfying$${\begin{aligned}{}&\Pr(X_{n}=x_{n}\mid X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots,X_{1}=x_{1})\\=&\Pr(X_{n}=x_{n}\mid X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots,X_{n-m}=x_{n-m}){\text{ for }}n>m\end{aligned}}$$In other words, the future state depends on the pastmstates. It is possible to construct a chain$(Y_{n})$from$(X_{n})$which has the 'classical' Markov property by taking as state space the orderedm-tuples ofXvalues, i.e.,$Y_{n}=\left(X_{n},X_{n-1},\ldots,X_{n-m+1}\right)$.

Continuous-time Markov chain

Main article:Continuous-time Markov chain

A continuous-time Markov chain (X_t)_{t≥ 0}is defined by a finite or countable state spaceS,atransition rate matrixQwith dimensions equal to that of the state space and initial probability distribution defined on the state space. Fori≠j,the elementsq_ijare non-negative and describe the rate of the process transitions from stateito statej.The elementsq_iiare 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.

Infinitesimal definition

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.

Let$X_{t}$be the random variable describing the state of the process at timet,and assume the process is in a stateiat timet. Then, knowing$X_{t}=i$,$X_{t+h}=j$is independent of previous values$\left(X_{s}:s<t\right)$,and ash→ 0 for alljand for allt, $$\Pr(X(t+h)=j\mid X(t)=i)=\delta _{ij}+q_{ij}h+o(h),$$ where$\delta _{ij}$is theKronecker delta,using thelittle-o notation. The$q_{ij}$can be seen as measuring how quickly the transition fromitojhappens.

Jump chain/holding time definition

Define a discrete-time Markov chainY_nto describe thenth jump of the process and variablesS₁,S₂,S₃,... to describe holding times in each of the states whereS_ifollows theexponential distributionwith rate parameter −q_YiYi.

Transition probability definition

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

$\Pr(X_{t_{n+1}}=i_{n+1}\mid X_{t_{0}}=i_{0},X_{t_{1}}=i_{1},\ldots,X_{t_{n}}=i_{n})=p_{i_{n}i_{n+1}}(t_{n+1}-t_{n})$

wherep_ijis the solution of theforward equation(afirst-order differential equation)

$P'(t)=P(t)Q$

with initial condition P(0) is theidentity matrix.

Finite state space

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

$p_{ij}=\Pr(X_{n+1}=j\mid X_{n}=i).$

Since each row ofPsums to one and all elements are non-negative,Pis aright stochastic matrix.

Stationary distribution relation to eigenvectors and simplices

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

$\pi \mathbf {P} =\pi.$

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

$\pi ={\frac {e}{\sum _{i}{e_{i}}}}$

is a normalized (${\textstyle \sum _{i}\pi _{i}=1}$) multiple of a left eigenvectoreof the transition matrixPwith aneigenvalueof 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 distribution$\textstyle \pi _{i}$are associated with the state space ofPand 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${\textstyle \sum _{i}1\cdot \pi _{i}=1}$we see that thedot productof 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

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

If the Markov chain is irreducible and aperiodic, then there is a unique stationary distributionπ.Additionally, in this caseP^kconverges to a rank-one matrix in which each row is the stationary distributionπ:

$\lim _{k\to \infty }\mathbf {P} ^{k}=\mathbf {1} \pi$

where1is the column vector with all entries equal to 1. This is stated by thePerron–Frobenius theorem.If, by whatever means,${\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}}$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 limit${\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}}$does not exist while the stationary distribution does, as shown by this example:

$\mathbf {P} ={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\qquad \mathbf {P} ^{2k}=I\qquad \mathbf {P} ^{2k+1}=\mathbf {P}$

${\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}$

(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. LetPbe ann×nmatrix, and define${\textstyle \mathbf {Q} =\lim _{k\to \infty }\mathbf {P} ^{k}.}$

It is always true that

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

SubtractingQfrom both sides and factoring then yields

$\mathbf {Q} (\mathbf {P} -\mathbf {I} _{n})=\mathbf {0} _{n,n},$

whereI_nis theidentity matrixof sizen,and0_n,nis thezero matrixof sizen×n.Multiplying together stochastic matrices always yields another stochastic matrix, soQmust be astochastic matrix(see the definition above). It is sometimes sufficient to use the matrix equation above and the fact thatQis a stochastic matrix to solve forQ.Including the fact that the sum of each the rows inPis 1, there aren+1equations for determiningnunknowns, so it is computationally easier if on the one hand one selects one row inQand 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 matrixAwith its right-most column replaced with all 1's. If [f(P−I_n)]⁻¹exists then

$\mathbf {Q} =f(\mathbf {0} _{n,n})[f(\mathbf {P} -\mathbf {I} _{n})]^{-1}.$

Explain: The original matrix equation is equivalent to asystem of n×n linear equationsin n×n variables. And there are n more linear equations from the fact that Q is a rightstochastic matrixwhose each row sums to 1. So it needs any n×n independent linear equations of the (n×n+n) equations to solve for the n×n variables. In this example, the n equations from “Q multiplied by the right-most column of (P-In)” have been replaced by the n stochastic ones.

One thing to notice is that ifPhas an elementP_i,ion 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 powersP^k.Hence, theith row or column ofQwill have the 1 and the 0's in the same positions as inP.

Convergence speed to the stationary distribution

As stated earlier, from the equation${\boldsymbol {\pi }}={\boldsymbol {\pi }}\mathbf {P},$(if exists) the stationary (or steady state) distributionπis a left eigenvector of rowstochastic matrixP.Then assuming thatPis diagonalizable or equivalently thatPhasnlinearly independent eigenvectors, speed of convergence is elaborated as follows. (For non-diagonalizable, that is,defective matrices,one may start with theJordan normal formofPand proceed with a bit more involved set of arguments in a similar way.)

LetUbe the matrix of eigenvectors (each normalized to having an L2 norm equal to 1) where each column is a left eigenvector ofPand letΣbe the diagonal matrix of left eigenvalues ofP,that is,Σ= diag(λ₁,λ₂,λ₃,...,λ_n). Then byeigendecomposition

$\mathbf {P} =\mathbf {U\Sigma U} ^{-1}.$

Let the eigenvalues be enumerated such that:

$1=|\lambda _{1}|>|\lambda _{2}|\geq |\lambda _{3}|\geq \cdots \geq |\lambda _{n}|.$

SincePis 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 otherπwhich solves the stationary distribution equation above). Letu_ibe thei-th column ofUmatrix, that is,u_iis the left eigenvector ofPcorresponding to λ_i.Also letxbe a lengthnrow vector that represents a valid probability distribution; since the eigenvectorsu_ispan$\mathbb {R} ^{n},$we can write

$\mathbf {x} ^{\mathsf {T}}=\sum _{i=1}^{n}a_{i}\mathbf {u} _{i},\qquad a_{i}\in \mathbb {R}.$

If we multiplyxwithPfrom right and continue this operation with the results, in the end we get the stationary distributionπ.In other words,π=a₁u₁←xPP...P=xP^kask→ ∞. That means

${\begin{aligned}{\boldsymbol {\pi }}^{(k)}&=\mathbf {x} \left(\mathbf {U\Sigma U} ^{-1}\right)\left(\mathbf {U\Sigma U} ^{-1}\right)\cdots \left(\mathbf {U\Sigma U} ^{-1}\right)\\&=\mathbf {xU\Sigma } ^{k}\mathbf {U} ^{-1}\\&=\left(a_{1}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\mathbf {u} _{2}^{\mathsf {T}}+\cdots +a_{n}\mathbf {u} _{n}^{\mathsf {T}}\right)\mathbf {U\Sigma } ^{k}\mathbf {U} ^{-1}\\&=a_{1}\lambda _{1}^{k}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\lambda _{2}^{k}\mathbf {u} _{2}^{\mathsf {T}}+\cdots +a_{n}\lambda _{n}^{k}\mathbf {u} _{n}^{\mathsf {T}}&&u_{i}\bot u_{j}{\text{ for }}i\neq j\\&=\lambda _{1}^{k}\left\{a_{1}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\left({\frac {\lambda _{2}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{2}^{\mathsf {T}}+a_{3}\left({\frac {\lambda _{3}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{3}^{\mathsf {T}}+\cdots +a_{n}\left({\frac {\lambda _{n}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{n}^{\mathsf {T}}\right\}\end{aligned}}$

Sinceπis parallel tou₁(normalized by L2 norm) andπ^(k)is a probability vector,π^(k)approaches toa₁u₁=πask→ ∞ with a speed in the order ofλ₂/λ₁exponentially. This follows because$|\lambda _{2}|\geq \cdots \geq |\lambda _{n}|,$henceλ₂/λ₁is the dominant term. The smaller the ratio is, the faster the convergence is.Random noise in the state distributionπcan also speed up this convergence to the stationary distribution.

General state space

Main article:Markov chains on a measurable state space

Harris chains

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

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

Locally interacting Markov chains

Considering a collection of Markov chains whose evolution takes in account the state of other Markov chains, is related to the notion oflocally interacting Markov chains.This corresponds to the situation when the state space has a (Cartesian-) product form. Seeinteracting particle systemandstochastic cellular automata(probabilistic cellular automata). See for instanceInteraction of Markov Processes or.

Properties

Two states are said tocommunicatewith 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 isclosedif the probability of leaving the class is zero. A Markov chain isirreducibleif there is one communicating class, the state space.

A stateihas periodkifkis thegreatest common divisorof the number of transitions by whichican be reached, starting fromi.That is:

$k=\gcd\{n>0:\Pr(X_{n}=i\mid X_{0}=i)>0\}$

The state isperiodicif$k>1$;otherwise$k=1$and the state isaperiodic.

A stateiis said to betransientif, starting fromi,there is a non-zero probability that the chain will never return toi.It is calledrecurrent(orpersistent) otherwise.For a recurrent statei,the meanhitting timeis defined as:

$M_{i}=E[T_{i}]=\sum _{n=1}^{\infty }n\cdot f_{ii}^{(n)}.$

Stateiispositive recurrentif$M_{i}$is finite andnull recurrentotherwise. 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.

A stateiis calledabsorbingif there are no outgoing transitions from the state.

Irreducibility

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.

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

Ergodicity

A stateiis said to beergodicif it is aperiodic and positive recurrent. In other words, a stateiis 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 integer$k$such that all entries of$M^{k}$are positive.

It can be shown that a finite state irreducible Markov chain is ergodic if it has an aperiodic state. More generally, a Markov chain is ergodic if there is a numberNsuch that any state can be reached from any other state in any number of steps less or equal to a numberN.In case of a fully connected transition matrix, where all transitions have a non-zero probability, this condition is fulfilled withN= 1.

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

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

Some authors call a matrixprimitiveiff there exists some integer$k$such that all entries of$M^{k}$are positive.Some authors call itregular.

Index of primitivity

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

There are several combinatorial results about the exponent when there are finitely many states. Let$n$be the number of states, then

• The exponent is$\leq (n-1)^{2}+1$.The only case where it is an equality is when the graph of$M$goes like$1\to 2\to \dots \to n\to 1{\text{ and }}2$.
• If$M$has$k\geq 1$diagonal entries, then its exponent is$\leq 2n-k-1$.
• If$\mathrm {sign} (M)$is symmetric, then$M^{2}$has positive diagonal entries, which by previous proposition means its exponent is$\leq 2n-2$.
• (Dulmage-Mendelsohn theorem) The exponent is$\leq n+s(n-2)$where$s$is thegirth of the graph.It can be improved to$\leq (d+1)+s(d+1-2)$,where$d$is thediameter of the graph.

Measure-preserving dynamical system

If a Markov chain has a stationary distribution, then it can be converted to ameasure-preserving dynamical system:Let the probability space be$\Omega =\Sigma ^{\mathbb {N} }$,where$\Sigma$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. Let$T:\Omega \to \Omega$be the shift operator:$T(X_{0},X_{1},\dots )=(X_{1},\dots )$.Similarly we can construct such a dynamical system with$\Omega =\Sigma ^{\mathbb {Z} }$instead.

SinceirreducibleMarkov 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 called "ergodic" iff any measurable subset$S$such that$T^{-1}(S)=S$implies$S=\emptyset$or$\Omega$(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 isirreducibleiff its corresponding measure-preserving dynamical system isergodic.

Markovian representations

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, letXbe a non-Markovian process. Then define a processY,such that each state ofYrepresents a time-interval of states ofX.Mathematically, this takes the form:

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

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

An example of a non-Markovian process with a Markovian representation is anautoregressivetime seriesof order greater than one.

Hitting times

Main article:Phase-type distribution

Thehitting timeis 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

For a subset of statesA⊆S,the vectork^Aof hitting times (where element$k_{i}^{A}$represents theexpected value,starting in stateithat the chain enters one of the states in the setA) is the minimal non-negative solution to

${\begin{aligned}k_{i}^{A}=0&{\text{ for }}i\in A\\-\sum _{j\in S}q_{ij}k_{j}^{A}=1&{\text{ for }}i\notin A.\end{aligned}}$

Time reversal

For a CTMCX_t,the time-reversed process is defined to be${\hat {X}}_{t}=X_{T-t}$.ByKelly's lemmathis process has the same stationary distribution as the forward process.

A chain is said to bereversibleif the reversed process is the same as the forward process.Kolmogorov's criterionstates that the necessary and sufficient condition for a process to be reversible is that the product of transition rates around a closed loop must be the same in both directions.

Embedded Markov chain

One method of finding thestationary probability distribution,π,of anergodiccontinuous-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 bys_ij,and represents theconditional probabilityof transitioning from stateiinto statej.These conditional probabilities may be found by

$s_{ij}={\begin{cases}{\frac {q_{ij}}{\sum _{k\neq i}q_{ik}}}&{\text{if }}i\neq j\\0&{\text{otherwise}}.\end{cases}}$

From this,Smay be written as

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

whereIis theidentity matrixand diag(Q) is thediagonal matrixformed by selecting themain diagonalfrom the matrixQand setting all other elements to zero.

To find the stationary probability distribution vector, we must next find$\varphi$such that

$\varphi S=\varphi,$

with$\varphi$being a row vector, such that all elements in$\varphi$are greater than 0 and$\|\varphi \|_{1}$= 1. From this,πmay be found as

$\pi ={-\varphi (\operatorname {diag} (Q))^{-1} \over \left\|\varphi (\operatorname {diag} (Q))^{-1}\right\|_{1}}.$

(Smay be periodic, even ifQis not. Onceπis 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

Markov model

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 observable	System state is partially observable
System is autonomous	Markov chain	Hidden Markov model
System is controlled	Markov decision process	Partially observable Markov decision process

Bernoulli scheme

Main article:Bernoulli scheme

ABernoulli schemeis 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;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 processis isomorphic to a Bernoulli scheme; the Markov chain is just one such example.

Subshift of finite type

Main article:Subshift of finite type

When the Markov matrix is replaced by theadjacency matrixof afinite graph,the resulting shift is termed atopological Markov chainor asubshift of finite type.A Markov matrix that is compatible with the adjacency matrix can then provide ameasureon the subshift. Many chaoticdynamical systemsare isomorphic to topological Markov chains; examples includediffeomorphismsofclosed manifolds,theProuhet–Thue–Morse system,theChacon system,sofic systems,context-free systemsandblock-coding systems.

Applications

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

Physics

Markovian systems appear extensively inthermodynamicsandstatistical 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.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.

Markov chains are used inlattice QCDsimulations.

Chemistry

${\ce {{E}+{\underset {Substrate \atop binding}{S<=>E}}{\overset {Catalytic \atop step}{S->E}}+P}}$

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.Markov chains and continuous-time Markov processes are useful in chemistry when physical systems closely approximate the Markov property. For example, imagine a large numbernof 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 isntimes 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.

An algorithm based on a Markov chain was also used to focus the fragment-based growth of chemicalsin silicotowards a desired class of compounds such as drugs or natural products.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.

Also, the growth (and composition) ofcopolymersmay 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 epitaxialsuperlatticeoxide materials can be accurately described by Markov chains.

Biology

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

• Phylogeneticsandbioinformatics,where mostmodels of DNA evolutionuse continuous-time Markov chains to describe thenucleotidepresent at a given site in thegenome.
• Population dynamics,where Markov chains are in particular a central tool in the theoretical study ofmatrix population models.
• Neurobiology,where Markov chains have been used, e.g., to simulate the mammalian neocortex.
• Systems biology,for instance with the modeling of viral infection of single cells.
• Compartmental modelsfor disease outbreak and epidemic modeling.

Testing

Several theorists have proposed the idea of the Markov chain statistical test (MCST), a method of conjoining Markov chains to form a "Markov blanket",arranging these chains in several recursive layers (" wafering ") and producing more efficient test sets—samples—as a replacement for exhaustive testing.

Solar irradiance variability

Solar irradiancevariability assessments are useful forsolar powerapplications. Solar irradiance variability at any location over time is mainly a consequence of the deterministic variability of the sun's path across the sky dome and the variability in cloudiness. The variability of accessible solar irradiance on Earth's surface has been modeled using Markov chains,also including modeling the two states of clear and cloudiness as a two-state Markov chain.

Speech recognition

Hidden Markov modelshave been used inautomatic speech recognitionsystems.

Information theory

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 ofentropyby 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.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 compressionthroughentropy encodingtechniques such asarithmetic coding.They also allow effectivestate estimationandpattern 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 algorithmfor error correction), speech recognition andbioinformatics(such as in rearrangements detection).

TheLZMAlossless data compression algorithm combines Markov chains withLempel-Ziv compressionto achieve very high compression ratios.

Queueing theory

Main article:Queueing theory

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

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

Internet applications

A state diagram that represents the PageRank algorithm with a transitional probability of M, or${\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}$.

ThePageRankof a webpage as used byGoogleis defined by a Markov chain.It is the probability to be at page$i$in the stationary distribution on the following Markov chain on all (known) webpages. If$N$is the number of known webpages, and a page$i$has$k_{i}$links to it then it has transition probability${\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}$for all pages that are linked to and${\frac {1-\alpha }{N}}$for all pages that are not linked to. The parameter$\alpha$is taken to be about 0.15.

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.

Statistics

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 inferencemethods, allowing a wide range ofposterior distributionsto be simulated and their parameters found numerically.

Economics and finance

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. Champernownebuilt a Markov chain model of the distribution of income in 1953.Herbert A. Simonand co-author Charles Bonini used a Markov chain model to derive a stationary Yule distribution of firm sizes.Louis Bachelierwas the first to observe that stock prices followed a random walk.The random walk was later seen as evidence in favor of theefficient-market hypothesisand random walk models were popular in the literature of the 1960s.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).A more recent example is theMarkov switching multifractalmodel ofLaurent E. Calvetand Adlai J. Fisher, which builds upon the convenience of earlier regime-switching models.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 equilibriumsetting.

Credit rating agenciesproduce annual tables of the transition probabilities for bonds of different credit ratings.

Social sciences

Markov chains are generally used in describingpath-dependentarguments, where current structural configurations condition future outcomes. An example is the reformulation of the idea, originally due toKarl Marx'sDas Kapital,tyingeconomic developmentto 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 ofpoliticalmobilization, etc., will generate a higher probability of transitioning fromauthoritariantodemocratic regime.

Games

Markov chains can be used to model many games of chance. The children's gamesSnakes and Laddersand "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).

Music

Markov chains are employed inalgorithmic music composition,particularly insoftwaresuch asCsound,Max,andSuperCollider.In a first-order chain, the states of the system become note or pitch values, and aprobability vectorfor 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 beMIDInote values, frequency (Hz), or any other desirable metric.

1st-order matrix

Note	A	C♯	E♭
A	0.1	0.6	0.3
C♯	0.25	0.05	0.7
E♭	0.7	0.3	0

2nd-order matrix

Notes	A	D	G
AA	0.18	0.6	0.22
AD	0.5	0.5	0
AG	0.15	0.75	0.1
DD	0	0	1
DA	0.25	0	0.75
DG	0.9	0.1	0
GG	0.4	0.4	0.2
GA	0.5	0.25	0.25
GD	1	0	0

A second-order Markov chain can be introduced by considering the current stateandalso 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 ofphrasalstructure, rather than the 'aimless wandering' produced by a first-order system.

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

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.

Baseball

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. He also discusses various kinds of strategies and play conditions: how Markov chain models have been used to analyze statistics for game situations such asbuntingandbase stealingand differences when playing on grass vs.AstroTurf.

Markov text generators

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

Probabilistic forecasting

Markov chains have been used for forecasting in several areas: for example, price trends,wind power,stochastic terrorism,andsolar irradiance.The Markov chain forecasting models utilize a variety of settings, from discretizing the time series,to hidden Markov models combined with wavelets,and the Markov chain mixture distribution model (MCM).

See also

• Dynamics of Markovian particles
• Gauss–Markov process
• Markov chain approximation method
• Markov chain geostatistics
• Markov chain mixing time
• Markov chain tree theorem
• Markov decision process
• Markov information source
• Markov odometer
• Markov operator
• Markov random field
• Master equation
• Quantum Markov chain
• Semi-Markov process
• Stochastic cellular automaton
• Telescoping Markov chain
• Variable-order Markov model

References

• 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 MathematicsISBN0-89871-296-3.(See Chapter 7)
• J. L. Doob(1953)Stochastic Processes.New York: John Wiley and SonsISBN0-471-52369-0.
• S. P. Meyn and R. L. Tweedie (1993)Markov Chains and Stochastic Stability.London: Springer-VerlagISBN0-387-19832-6.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.ISBN978-3-662-00033-5.Title-No. 5104.;Markov Processes.Vol. II (122). 1965.doi:10.1007/978-3-662-25360-1.ISBN978-3-662-23320-7.Title-No. 5105.(NB. This was originally published in Russian asМарковские процессы(Markovskiye protsessy) byFizmatgizin 1963 and translated to English with the assistance of the author.)
• S. P. Meyn.Control Techniques for Complex Networks.Cambridge University Press, 2007.ISBN978-0-521-88441-9.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 Chainspp. 384ff.
• John G. Kemeny&J. Laurie Snell(1960)Finite Markov Chains,D. van Nostrand CompanyISBN0-442-04328-7
• E. Nummelin. "General irreducible Markov chains and non-negative operators". Cambridge University Press, 1984, 2004.ISBN0-521-60494-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)ISBN978-0-387-29765-1
• Kishor S. Trivedi,Probability and Statistics with Reliability, Queueing, and Computer Science Applications,John Wiley & Sons, Inc. New York, 2002.ISBN0-471-33341-7.
• 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.ISBN0-7923-9650-2.
• G. Bolch, S. Greiner, H. de Meer and K. S. Trivedi,Queueing Networks and Markov Chains,John Wiley, 2nd edition, 2006.ISBN978-0-7923-9650-5.

External links

• "Markov chain",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Markov Chains chapter in American Mathematical Society's introductory probability bookArchived2008-05-22 at theWayback Machine
• Introduction to Markov ChainsonYouTube
• A visual explanation of Markov Chains
• Original paper by A.A Markov (1913): An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains (translated from Russian)