Markov chain - Wikipedia

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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.
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In probability theory and statistics, a
Markov chain
or
Markov process
is a
stochastic process
describing a
sequence
of possible events in which the
probability
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 affairs
now
." A
countably infinite
sequence, in which the chain moves state at discrete time steps, gives a
discrete-time Markov chain
(DTMC). A
continuous-time
process is called a
continuous-time Markov chain
(CTMC). Markov processes are named in honor of the Russian mathematician
Andrey Markov
Markov chains have many applications as
statistical models
of real-world processes.
They provide the basis for general stochastic simulation methods known as
Markov chain Monte Carlo
, which are used for simulating sampling from complex
probability distributions
, and have found application in areas including
Bayesian statistics
biology
chemistry
economics
finance
information theory
physics
signal processing
, and
speech processing
The adjectives
Markovian
and
Markov
are used to describe something that is related to a Markov process.
Principles
edit
Russian mathematician
Andrey Markov
Definition
edit
A Markov process is a
stochastic process
that satisfies the
Markov 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,
conditional
on the present state of the system, its future and past states are
independent
A Markov chain is a type of Markov process that has either a discrete
state space
or 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 either
discrete or continuous time
with a countable state space (thus regardless of the nature of time),
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).
Types of Markov chains
edit
The system's
state 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 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
Any
continuous stochastic process
with the Markov property (for example, the
Wiener 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, a
discrete-time Markov chain (DTMC)
11
but a few authors use the term "Markov process" to refer to a
continuous-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 (see
Markov 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, the
state 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 or
countably 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 (see
Variations
). For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, unless mentioned otherwise.
Transitions
edit
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, a
transition 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 the
integers
or
natural numbers
, and the random process is a mapping of these to states. The Markov property states that the
conditional 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
edit
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 the
Poisson process
19
20
21
Markov was interested in studying an extension of independent random sequences, motivated by a disagreement with
Pavel Nekrasov
who claimed independence was necessary for the
weak 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 in
Eugene Onegin
, written by
Alexander Pushkin
, and proved a
central limit theorem
for such chains.
16
In 1912
Henri Poincaré
studied Markov chains on
finite groups
with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced by
Paul
and
Tatyana Ehrenfest
in 1907, and a branching process, introduced by
Francis Galton
and
Henry 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 by
Irénée-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 as
Norbert 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 the
Chapman–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 include
William Feller
, starting in 1930s, and then later
Eugene Dynkin
, starting in the 1950s.
26
Examples
edit
Main article:
Examples of Markov chains
Mark V. Shaney
is a third-order Markov chain program, and a
Markov text
generator. It ingests the sample text (the
Tao Te Ching
, or the posts of a
Usenet
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 the
gambler'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 the
Wiener process
, also known as the
Brownian motion
process, and the
Poisson 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 the
number 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
edit
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. If
{\displaystyle X_{n}}
represents the total value of the coins set on the table after
draws, with
{\displaystyle X_{0}=0}
, then the sequence
{\displaystyle \{X_{n}:n\in \mathbb {N} \}}
is
not
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. Thus
0.50
{\displaystyle X_{6}=\$0.50}
. If we know not just
{\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 that
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 value
{\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 about
{\displaystyle X_{7}}
are impacted by our knowledge of values prior to
{\displaystyle X_{6}}
However, it is possible to model this scenario as a Markov process. Instead of defining
{\displaystyle X_{n}}
to represent the
total value
of the coins on the table, we could define
{\displaystyle X_{n}}
to represent the
count
of the various coin types on the table. For instance,
{\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 by
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 state
{\displaystyle X_{1}=0,1,0}
. The probability of achieving
{\displaystyle X_{2}}
now depends on
{\displaystyle X_{1}}
; for example, the state
{\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 the
{\displaystyle X_{n}=i,j,k}
state depends exclusively on the outcome of the
{\displaystyle X_{n-1}=\ell ,m,p}
state.
Formal definition
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Discrete-time Markov chain
edit
Main article:
Discrete-time Markov chain
A discrete-time Markov chain is a sequence of
random variables
, ... with the
Markov property
, namely that the probability of moving to the next state depends only on the present state and not on the previous states:
Pr
Pr
{\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 both
conditional probabilities
are well defined, that is, if
Pr
0.
{\displaystyle \Pr(X_{1}=x_{1},\ldots ,X_{n}=x_{n})>0.}
The possible values of
form a
countable set
called the state space of the chain.
Variations
edit
Time-homogeneous Markov chains are processes where
Pr
Pr
{\displaystyle \Pr(X_{n+1}=x\mid X_{n}=y)=\Pr(X_{n}=x\mid X_{n-1}=y)}
for all
. The probability of the transition is independent of
Stationary Markov chains are processes where
Pr
Pr
{\displaystyle \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 all
and
. 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
{\displaystyle X_{0}}
is a stationary distribution of the Markov chain.
A Markov chain with memory (or a Markov chain of order
) where
is finite, is a process satisfying
Pr
Pr
for
{\displaystyle {\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 past
states. It is possible to construct a chain
{\displaystyle (Y_{n})}
from
{\displaystyle (X_{n})}
which has the 'classical' Markov property by taking as state space the ordered
-tuples of
values, i.e.,
{\displaystyle Y_{n}=\left(X_{n},X_{n-1},\ldots ,X_{n-m+1}\right)}
Finite state space
edit
If the state space is
finite
, the transition probability distribution can be represented by a
matrix
, called the transition matrix, with the (
)th
element
of
equal to
Pr
{\displaystyle p_{ij}=\Pr(X_{n+1}=j\mid X_{n}=i).}
Since each row of
sums to one and all elements are non-negative,
is a
right stochastic matrix
Stationary distribution relation to eigenvectors and simplices
edit
A stationary distribution
is a (row) vector, whose entries are non-negative and sum to 1, is unchanged by the operation of transition matrix
on it and so is defined by
{\displaystyle \pi \mathbf {P} =\pi .}
By comparing this definition with that of an
eigenvector
we see that the two concepts are related and that
{\displaystyle \pi ={\frac {e}{\sum _{i}{e_{i}}}}}
is a normalized (
{\textstyle \sum _{i}\pi _{i}=1}
) multiple of a left eigenvector
of the transition matrix
with an
eigenvalue
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 distribution
{\displaystyle \textstyle \pi _{i}}
are associated with the state space of
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
{\textstyle \sum _{i}1\cdot \pi _{i}=1}
we see that the
dot product
of π with a vector whose components are all 1 is unity and that π lies on a
simplex
Time-homogeneous Markov chain with a finite state space
edit
If the Markov chain is time-homogeneous, then the transition matrix
is the same after each step, so the
-step transition probability can be computed as the
-th power of the transition matrix,
If the Markov chain is irreducible and aperiodic, then there is a unique stationary distribution
40
Additionally, in this case
converges to a rank-one matrix in which each row is the stationary distribution
lim
{\displaystyle \lim _{k\to \infty }\mathbf {P} ^{k}=\mathbf {1} \pi }
where
is the column vector with all entries equal to 1. This is stated by the
Perron–Frobenius theorem
. If, by whatever means,
lim
{\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 matrices
, the limit
lim
{\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}}
does not exist while the stationary distribution does, as shown by this example:
{\displaystyle \mathbf {P} ={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\qquad \mathbf {P} ^{2k}=I\qquad \mathbf {P} ^{2k+1}=\mathbf {P} }
{\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. Let
be an
matrix, and define
lim
{\textstyle \mathbf {Q} =\lim _{k\to \infty }\mathbf {P} ^{k}.}
It is always true that
{\displaystyle \mathbf {QP} =\mathbf {Q} .}
Subtracting
from both sides and factoring then yields
{\displaystyle \mathbf {Q} (\mathbf {P} -\mathbf {I} _{n})=\mathbf {0} _{n,n},}
where
is the
identity matrix
of size
, and
is the
zero matrix
of size
. Multiplying together stochastic matrices always yields another stochastic matrix, so
must be a
stochastic matrix
(see the definition above). It is sometimes sufficient to use the matrix equation above and the fact that
is a stochastic matrix to solve for
. Including the fact that the sum of each the rows in
is 1, there are
n+1
equations for determining
unknowns, so it is computationally easier if on the one hand one selects one row in
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 vector
, and next left-multiplies this latter vector by the inverse of transformed former matrix to find
Here is one method for doing so: first, define the function
) to return the matrix
with its right-most column replaced with all 1's. If [
)]
−1
exists then
41
40
{\displaystyle \mathbf {Q} =f(\mathbf {0} _{n,n})[f(\mathbf {P} -\mathbf {I} _{n})]^{-1}.}
Explain: The original matrix equation is equivalent to a
system of n×n linear equations
in
variables. And there are
more linear equations from the fact that
is a right
stochastic matrix
whose each row sums to 1. So it needs any
independent linear equations of the (
) equations to solve for the
variables. In this example, the
equations from "
multiplied by the right-most column of (
)" have been replaced by the
stochastic ones.
One thing to notice is that if
has an element
on its main diagonal that is equal to 1 and the
th row or column is otherwise filled with 0's, then that row or column will remain unchanged in all of the subsequent powers
. Hence, the
th row or column of
will have the 1 and the 0's in the same positions as in
Convergence speed to the stationary distribution
edit
As stated earlier, from the equation
{\displaystyle {\boldsymbol {\pi }}={\boldsymbol {\pi }}\mathbf {P} ,}
(if exists) the stationary (or steady state) distribution
is a left eigenvector of row
stochastic matrix
. Then assuming that
is diagonalizable or equivalently that
has
linearly independent eigenvectors, speed of convergence is elaborated as follows. (For non-diagonalizable, that is,
defective matrices
, one may start with the
Jordan normal form
of
and proceed with a bit more involved set of arguments in a similar way.
42
Let
be the matrix of eigenvectors (each normalized to having an L2 norm equal to 1) where each column is a left eigenvector of
and let
be the diagonal matrix of left eigenvalues of
, that is,
= diag(
,...,
). Then by
eigendecomposition
{\displaystyle \mathbf {P} =\mathbf {U\Sigma U} ^{-1}.}
Let the eigenvalues be enumerated such that:
{\displaystyle 1=|\lambda _{1}|>|\lambda _{2}|\geq |\lambda _{3}|\geq \cdots \geq |\lambda _{n}|.}
Since
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 other
which solves the stationary distribution equation above). Let
be the
-th column of
matrix, that is,
is the left eigenvector of
corresponding to λ
. Also let
be a length
row vector that represents a valid probability distribution; since the eigenvectors
span
{\displaystyle \mathbb {R} ^{n},}
we can write
{\displaystyle \mathbf {x} ^{\mathsf {T}}=\sum _{i=1}^{n}a_{i}\mathbf {u} _{i},\qquad a_{i}\in \mathbb {R} .}
If we multiply
with
from right and continue this operation with the results, in the end we get the stationary distribution
. In other words,
xPP
...
xP
as
→ ∞. That means
for
{\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}}}
Since
is parallel to
(normalized by L2 norm) and
is a probability vector,
approaches to
as
→ ∞ with a speed in the order of
exponentially. This follows because
{\displaystyle |\lambda _{2}|\geq \cdots \geq |\lambda _{n}|,}
hence
is the dominant term. The smaller the ratio is, the faster the convergence is.
43
Random noise in the state distribution
can also speed up this convergence to the stationary distribution.
44
Continuous-time Markov chain
edit
Main article:
Continuous-time Markov chain
A continuous-time Markov chain
{\displaystyle (X_{t})_{t\geq 0}}
is defined by a finite or countable state space
, a
transition rate matrix
with dimensions equal to that of the state space and initial probability distribution defined on the state space. For
, the elements
ij
are non-negative and describe the rate of the process transitions from state
to state
. The elements
ii
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
edit
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
{\displaystyle X_{t}}
be the random variable describing the state of the process at time
, and assume the process is in a state
at time
Then, knowing
{\displaystyle X_{t}=i}
{\displaystyle X_{t+h}=j}
is independent of previous values
{\displaystyle \left(X_{s}:s, and as
→ 0 for all
and for all
Pr
{\displaystyle \Pr(X(t+h)=j\mid X(t)=i)=\delta _{ij}+q_{ij}h+o(h),}
where
{\displaystyle \delta _{ij}}
is the
Kronecker delta
, using the
little-o notation
The
{\displaystyle q_{ij}}
can be seen as measuring how quickly the transition from
to
happens.
Jump chain/holding time definition
edit
Define a discrete-time Markov chain
to describe the
th jump of the process and variables
, ... to describe holding times in each of the states where
follows the
exponential distribution
with rate parameter −
Transition probability definition
edit
For any value
= 0, 1, 2, 3, ... and times indexed up to this value of
, ... and all states recorded at these times
, ... it holds that
Pr
{\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})}
where
ij
is the solution of the
forward equation
(a
first-order differential equation
{\displaystyle P'(t)=P(t)Q}
with initial condition P(0) is the
identity matrix
Locally interacting Markov chains
edit
"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.
See
interacting particle system
and
stochastic cellular automata
(probabilistic cellular automata).
See for instance
Interaction of Markov Processes
46
or.
47
Discrete-time Markov process with general state space
edit
Main article:
Markov chains on a measurable state space
Harris chains
edit
Many results for discrete-time Markov chains with finite state space can be generalized to chains with uncountable state space through
Harris chains
The use of Markov chains in
Markov chain Monte Carlo
methods covers cases where the process follows a continuous state space.
Continuous-time Markov process with general state space
edit
The definition of Markov processes in continuous time with general state space is more technical than the above.
A continuous-time Markov process
{\displaystyle X=(X_{t})_{t\geq 0}}
is a
stochastic process
adapted to a
filtration
{\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\geq 0}}
with values in a
locally compact
Polish space
{\displaystyle (S,{\mathcal {B}}(S))}
(e.g.,
{\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}
). The latter essentially ensures that the conditional expectations of
{\displaystyle X_{t}}
are
regular
, which, in simple terms, means that they behave "nicely". Then
{\displaystyle X}
is called a
Markov process
, if it satisfies the
Markov property
, i.e., for all
{\displaystyle t\geq s\geq 0}
and
{\displaystyle A\in {\mathcal {B}}(S)}
{\displaystyle P(X_{t}\in A\mid {\mathcal {F}}_{s})=P(X_{t}\in A\mid X_{s})}
Moreover,
{\displaystyle X}
is called
time-homogeneous
, if it satisfies the weak Markov property for all
{\displaystyle t,s\geq 0}
=:
{\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
{\displaystyle (t,x,A)\mapsto P_{t}(x,A)}
is the so-called
transition function
of
{\displaystyle X}
and
{\displaystyle (P_{t})_{t\geq 0}}
the
transition 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
{\displaystyle (P_{t})_{t\geq 0}}
be a transition semigroup, i.e.,
{\displaystyle P_{t}}
is
Markov kernel
for all
{\displaystyle t\geq 0}
{\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),
{\displaystyle P_{0}(x,\cdot )=\delta _{x}}
where
{\displaystyle \delta _{x}}
is the
Dirac-measure
in
{\displaystyle x}
, and
{\displaystyle X:\Omega \times [0,\infty )\to S}
. Then
{\displaystyle X}
is a homogeneous Markov process w.r.t. the natural filtration
{\displaystyle \mathbb {F} ^{X}=(\sigma (X_{s}:0\leq s\leq t))_{t\geq 0}}
, if for all
{\displaystyle 0\leq t_{1}<...{\displaystyle A_{1},...,A_{n}\in {\mathcal {B}}(S)}
the underlying probability measure
{\displaystyle P}
satisfies
{\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 measure
{\displaystyle P}
has been specified, the above equation defines a measure
:=
{\displaystyle P^{x}:=P(\cdot \mid X_{0}=x)}
on
{\displaystyle \sigma (X_{s}:s\geq 0)}
under which the process
{\displaystyle X}
started in
{\displaystyle x}
is a Markov process by construction.
In other words, Markov processes can be defined either as stochastic processes
{\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 which
{\displaystyle X}
has the Markov property.
Properties
edit
Two states are said to
communicate
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 is
closed
if the probability of leaving the class is zero. A Markov chain is
irreducible
if there is one communicating class, the state space.
A state
has period
if
is the
greatest common divisor
of the number of transitions by which
can be reached, starting from
. That is:
gcd
Pr
{\displaystyle k=\gcd\{n>0:\Pr(X_{n}=i\mid X_{0}=i)>0\}}
The state is
periodic
if
{\displaystyle k>1}
; otherwise
{\displaystyle k=1}
and the state is
aperiodic
A state
is said to be
transient
if, starting from
, there is a non-zero probability that the chain will never return to
. It is called
recurrent
(or
persistent
) otherwise.
48
For a recurrent state
, the mean
hitting time
is defined as:
{\displaystyle M_{i}=E[T_{i}]=\sum _{n=1}^{\infty }n\cdot f_{ii}^{(n)}}
where
:=
Pr
min
{\displaystyle f_{ii}^{(n)}:=\Pr(\min\{m>0:X_{m}=i\}=n\mid X_{0}=i)}
State
is
positive recurrent
if
{\displaystyle M_{i}}
is finite and
null 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 state
is called
absorbing
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 by
{\displaystyle \pi _{i}=1/E[T_{i}]}
Ergodicity
edit
A state
is said to be
ergodic
if it is aperiodic and positive recurrent. In other words, a state
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 integer
{\displaystyle k}
such that all entries of
{\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 to
ergodic processes
, defined according to
ergodic theory
52
Some authors call a matrix
primitive
if there exists some integer
{\displaystyle k}
such that all entries of
{\displaystyle M^{k}}
are positive.
53
Some authors call it
regular
54
Index of primitivity
edit
The
index of primitivity
, or
exponent
, of a regular matrix, is the smallest
{\displaystyle k}
such that all entries of
{\displaystyle M^{k}}
are positive. The exponent is purely a graph-theoretic property, since it depends only on whether each entry of
{\displaystyle M}
is zero or positive, and therefore can be found on a directed graph with
{\displaystyle \mathrm {sign} (M)}
as its adjacency matrix.
There are several combinatorial results about the exponent when there are finitely many states. Let
{\displaystyle n}
be the number of states, then
55
The exponent is
{\displaystyle \leq (n-1)^{2}+1}
. The only case where it is an equality is when the graph of
{\displaystyle M}
goes like
and
{\displaystyle 1\to 2\to \dots \to n\to 1{\text{ and }}2}
If
{\displaystyle M}
has
{\displaystyle k\geq 1}
diagonal entries, then its exponent is
{\displaystyle \leq 2n-k-1}
If
{\displaystyle \mathrm {sign} (M)}
is symmetric, then
{\displaystyle M^{2}}
has positive diagonal entries, which by previous proposition means its exponent is
{\displaystyle \leq 2n-2}
(Dulmage-Mendelsohn theorem) The exponent is
{\displaystyle \leq n+s(n-2)}
where
{\displaystyle s}
is the
girth of the graph
. It can be improved to
{\displaystyle \leq (d+1)+s(d+1-2)}
, where
{\displaystyle d}
is the
diameter of the graph
56
Measure-preserving dynamical system
edit
If a Markov chain has a stationary distribution, then it can be converted to a
measure-preserving dynamical system
: Let the probability space be
{\displaystyle \Omega =\Sigma ^{\mathbb {N} }}
, where
{\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. Let
{\displaystyle T:\Omega \to \Omega }
be the shift operator:
{\displaystyle T(X_{0},X_{1},\dots )=(X_{1},\dots )}
. Similarly we can construct such a dynamical system with
{\displaystyle \Omega =\Sigma ^{\mathbb {Z} }}
instead.
57
Since
irreducible
Markov chains with finite state spaces have a unique stationary distribution, the above construction is unambiguous for irreducible Markov chains.
In
ergodic theory
, a measure-preserving dynamical system is called
ergodic
if any measurable subset
{\displaystyle S}
such that
{\displaystyle T^{-1}(S)=S}
implies
{\displaystyle S=\emptyset }
or
{\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 is
irreducible
if its corresponding measure-preserving dynamical system is
ergodic
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, let
be a non-Markovian process. Then define a process
, such that each state of
represents a time-interval of states of
. Mathematically, this takes the form:
{\displaystyle Y(t)={\big \{}X(s):s\in [a(t),b(t)]\,{\big \}}.}
If
has the Markov property, then it is a Markovian representation of
An example of a non-Markovian process with a Markovian representation is an
autoregressive
time series
of order greater than one.
58
Hitting times
edit
Main article:
Phase-type distribution
The
hitting 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 states
, the vector
of hitting times (where element
{\displaystyle k_{i}^{A}}
represents the
expected value
, starting in state
that the chain enters one of the states in the set
) is the minimal non-negative solution to
59
for
for
{\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 process
{\displaystyle X}
in continuous time (a CTMC or a process with general state space), the reverse process
{\displaystyle {\overleftarrow {X}}=(X_{T-t})_{t\in [0,T]}}
from a fixed time
{\displaystyle T>0}
is again a Markov process. This follows directly from the
Markov 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 time
{\displaystyle \tau }
(not necessarily a
stopping time
) the stopped process
{\displaystyle X^{\tau }=(X_{t\land \tau })_{t\geq 0}}
is a time-homogeneous Markov process, then the reverse process
{\displaystyle {\overleftarrow {X^{\tau }}}=(X_{\tau -t\land \tau }1_{\{\tau <\infty \}})_{t\geq 0}}
is again time-homogeneous.
60
If
{\displaystyle X}
is a CTMC, then by
Kelly's lemma
{\displaystyle {\overleftarrow {X}}}
has the same stationary distribution as the forward process.
A chain is said to be
reversible
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 the
stationary probability distribution
, of an
ergodic
continuous-time Markov chain,
, is by first finding its
embedded Markov chain (EMC)
. Strictly speaking, the EMC is a regular discrete-time Markov chain, sometimes referred to as a
jump process
. Each element of the one-step transition probability matrix of the EMC,
, is denoted by
ij
, and represents the
conditional probability
of transitioning from state
into state
. These conditional probabilities may be found by
if
otherwise
{\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,
may be written as
diag
{\displaystyle S=I-\left(\operatorname {diag} (Q)\right)^{-1}Q}
where
is the
identity matrix
and diag(
) is the
diagonal matrix
formed by selecting the
main diagonal
from the matrix
and setting all other elements to zero.
To find the stationary probability distribution vector, we must next find
{\displaystyle \varphi }
such that
{\displaystyle \varphi S=\varphi ,}
with
{\displaystyle \varphi }
being a row vector, such that all elements in
{\displaystyle \varphi }
are greater than 0 and
{\displaystyle \|\varphi \|_{1}}
= 1. From this,
may be found as
diag
diag
{\displaystyle \pi ={-\varphi (\operatorname {diag} (Q))^{-1} \over \left\|\varphi (\operatorname {diag} (Q))^{-1}\right\|_{1}}.}
may be periodic, even if
is not. Once
is found, it must be normalized to a
unit 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 observing
) at intervals of δ units of time. The random variables
(0),
(δ),
(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 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
edit
Main article:
Bernoulli scheme
Bernoulli 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 a
Bernoulli process
Note, however, by the
Ornstein 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 that
any
stationary 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 the
adjacency matrix
of a
finite graph
, the resulting shift is termed a
topological Markov chain
or a
subshift of finite type
61
A Markov matrix that is compatible with the adjacency matrix can then provide a
measure
on the subshift. Many chaotic
dynamical systems
are isomorphic to topological Markov chains; examples include
diffeomorphisms
of
closed manifolds
, the
Prouhet–Thue–Morse system
, the
Chacon system
sofic systems
context-free systems
and
block-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 in
thermodynamics
and
statistical 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 in
lattice QCD
simulations.
64
Chemistry
edit
Substrate
binding
Catalytic
step
{\displaystyle {\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.
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 number
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 is
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 chemicals
in 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) of
copolymers
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 to
steric 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 epitaxial
superlattice
oxide materials can be accurately described by Markov chains.
69
Biology
edit
Markov chains are used in various areas of biology. Notable examples include:
Phylogenetics
and
bioinformatics
, where most
models of DNA evolution
use continuous-time Markov chains to describe the
nucleotide
present at a given site in the
genome
Population dynamics
, where Markov chains are in particular a central tool in the theoretical study of
matrix population models
Neurobiology
, where Markov chains have been used, e.g., to simulate the mammalian neocortex.
70
Systems biology
, for instance with the modeling of viral infection of single cells.
71
Compartmental models
for disease outbreak and epidemic modeling.
Information theory
edit
Markov chains are used throughout information processing.
Claude Shannon
's famous 1948 paper
A Mathematical Theory of Communication
, which in a single step created the field of
information theory
, opens by introducing the concept of
entropy
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 effective
data compression
through
entropy encoding
techniques such as
arithmetic coding
. They also allow effective
state estimation
and
pattern recognition
. Markov chains also play an important role in
reinforcement 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 the
Viterbi algorithm
for error correction), speech recognition and
bioinformatics
(such as in rearrangements detection
73
).
The
LZMA
lossless data compression algorithm combines Markov chains with
Lempel-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, an
M/M/1 queue
is a CTMC on the non-negative integers where upward transitions from
to
+ 1 occur at rate
according to a
Poisson process
and describe job arrivals, while transitions from
to
– 1 (for
> 1) occur at rate
(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, or
{\displaystyle {\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}}
The
PageRank
of a webpage as used by
Google
is defined by a Markov chain.
76
77
78
It is the probability to be at page
{\displaystyle i}
in the stationary distribution on the following Markov chain on all (known) webpages. If
{\displaystyle N}
is the number of known webpages, and a page
{\displaystyle i}
has
{\displaystyle k_{i}}
outgoing links from it then it has transition probability
{\displaystyle {\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}}
for all pages that are linked to and
{\displaystyle {\frac {1-\alpha }{N}}}
for all pages that are not linked to. The parameter
{\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 called
Markov chain Monte Carlo
(MCMC). In recent years this has revolutionized the practicability of
Bayesian inference
methods, allowing a wide range of
posterior 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 the
efficient-market hypothesis
and random walk models were popular in the literature of the 1960s.
83
Regime-switching models of business cycles were popularized by
James 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 the
Markov switching multifractal
model of
Laurent 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 a
general 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 describing
path-dependent
arguments, where current structural configurations condition future outcomes. An example is the reformulation of the idea, originally due to
Karl Marx
's
Das Kapital
, tying
economic development
to the rise of
capitalism
. 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 the
middle class
, the ratio of urban to rural residence, the rate of
political
mobilization, etc., will generate a higher probability of transitioning from
authoritarian
to
democratic regime
89
Music
edit
Markov chains are employed in
algorithmic music composition
, particularly in
software
such as
Csound
Max
, and
SuperCollider
. In a first-order chain, the states of the system become note or pitch values, and a
probability 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 be
MIDI
note values, frequency (
Hz
), or any other desirable metric.
90
1st-order matrix
Note
0.1
0.6
0.3
0.25
0.05
0.7
0.7
0.3
2nd-order matrix
Notes
AA
0.18
0.6
0.22
AD
0.5
0.5
AG
0.15
0.75
0.1
DD
DA
0.25
0.75
DG
0.9
0.1
GG
0.4
0.4
0.2
GA
0.5
0.25
0.25
GD
A second-order Markov chain can be introduced by considering the current state
and
also the previous state, as indicated in the second table. Higher,
th-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 of
phrasal
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 games
Snakes 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 as
bunting
and
base stealing
and differences when playing on grass vs.
AstroTurf
96
Markov text generators
edit
Markov processes can also be used to
generate superficially real-looking text
given a sample document. Markov processes are used in a variety of recreational "
parody generator
" software (see
dissociated 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
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
Notes
edit
Sean Meyn; Richard L. Tweedie (2 April 2009).
Markov Chains and Stochastic Stability
. Cambridge University Press. p. 3.
ISBN
978-0-521-73182-9
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
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
"Markovian"
Oxford English Dictionary
(Online ed.). Oxford University Press.
(Subscription or
participating institution membership
required.)
Øksendal, B. K. (Bernt Karsten) (2003).
Stochastic differential equations: an introduction with applications
(6th ed.). Berlin: Springer.
ISBN
3-540-04758-1
OCLC
52203046
Søren Asmussen (15 May 2003).
Applied Probability and Queues
. Springer Science & Business Media. p. 7.
ISBN
978-0-387-00211-8
Emanuel Parzen (17 June 2015).
Stochastic Processes
. Courier Dover Publications. p. 188.
ISBN
978-0-486-79688-8
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
John Lamperti (1977).
Stochastic processes: a survey of the mathematical theory
. Springer-Verlag. pp.
106–
121.
ISBN
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A. A. Markov (1906) "Rasprostranenie zakona bol'shih chisel na velichiny, zavisyaschie drug ot druga".
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Sequential Machines and Automata Theory
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] 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.
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G. Bolch, S. Greiner, H. de Meer and K. S. Trivedi,
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External links
edit
"Markov chain"
Encyclopedia of Mathematics
EMS Press
, 2001 [1994]
Markov Chains chapter in American Mathematical Society's introductory probability book
Archived
2008-05-22 at the
Wayback Machine
Introduction to Markov Chains
on
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)
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