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[[File:Markov process-example.svg|thumb|Markov process example]]
 
In [[probability theory]] and [[statistics]], a '''Markov process''' or '''Markoff process''', named after the Russian mathematician [[Andrey Markov]], is a [[stochastic process]] that satisfies the [[Markov property]]. A Markov process can be thought of as 'memoryless': loosely speaking, a process satisfies the Markov property if one can make predictions for the future of the process based solely on its present state just as well as one could knowing the process's full history. I.e., [[conditional probability|conditional]] on the present state of the system, its future and past are [[Independence (probability theory)|independent]].<ref>[http://www.britannica.com/EBchecked/topic/365797/Markov-process Markov process (mathematics)] - Britannica Online Encyclopedia</ref>
 
==Introduction==
A Markov process is a stochastic model that has the [[Markov property]]. It can be used to model a random system that changes states according to a transition rule that only depends on the current state. This article describes the Markov process in a very general sense, which is a concept that is usually specified further. Particularly, the system's [[state space]] and time parameter index needs 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 vs. continuous time.
 
{| border="1" class="wikitable" style="width: 60%;" |
! scope="col" |
! scope="col" | Countable or finite state space
! scope="col" | Continuous or general state space
|-
! scope="row" | Discrete-time
|[[Markov chain]] on a countable or finite state space || [[Harris chain]] (Markov chain on a general state space)
|-
! scope="row" style="width: 10%;" | Continuous-time
| style="width: 25%;" | [[Continuous-time Markov process]] || style="width: 25%;" |Any [[continuous stochastic process]] with the Markov property, e.g. the [[Wiener process]]
|}
 
Note that there is no definitive agreement in literature on the use of some of the terms that signify special cases of Markov processes. For example, often the term "Markov chain" is used to indicate a Markov process which has a finite or countable [[state-space]], but Markov chains on a general state space fall under the same description. Similarly, a Markov chain would usually be defined for a discrete set of times (i.e. a discrete-time Markov chain)<ref>Everitt,B.S. (2002) ''The Cambridge Dictionary of Statistics''. CUP. ISBN 0-521-81099-X</ref> although some authors use the same terminology where "time" can take continuous values.<ref>Dodge, Y. ''The Oxford Dictionary of Statistical Terms'', OUP. ISBN 0-19-920613-9</ref> 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.
 
Markov processes arise in probability and statistics in one of two ways. A [[stochastic process]], defined via a separate argument, may be shown mathematically to have the [[Markov property]], and as a consequence to have the properties that can be deduced from this for all Markov processes. Alternately, in modelling a process, one may assume the process to be Markov, and take this as the basis for a construction. In modelling terms, assuming that the Markov property holds is one of a limited number of simple ways of introducing statistical dependence into a model for a stochastic process in such a way that allows the strength of dependence at different lags to decline as the lag increases.
 
==Markov property==
{{Main|Markov property}}
 
{{Technical|section|date=December 2012}}
 
===The general case===
Let <math>(\Omega,\mathcal{F},\mathbb{P})</math> be a [[probability space]] with a [[Filtration (mathematics)#Measure theory|filtration]] <math>(\mathcal{F}_t,\ t \in T)</math>, for some ([[totally ordered]]) index set <math>T</math>; and let <math>(S,\mathcal{S})</math> be a [[measurable space]]. An ''S''-valued stochastic process <math>X=(X_t,\ t\in T)</math> adapted to the filtration is said to possess the '''Markov property''' with respect to the <math>\{\mathcal{F}_t\}</math> if, for each <math>A\in \mathcal{S}</math> and each <math>s,t\in T</math> with ''s'' < ''t'',
 
:<math>\mathbb{P}(X_t \in A |\mathcal{F}_s) = \mathbb{P}(X_t \in A| X_s).</math><ref>{{cite book |last=Durrett |first=Rick |title=Probability: Theory and Examples |edition=Fourth |location=Cambridge |publisher=Cambridge University Press |year=2010 |isbn=978-0-521-76539-8 }}</ref>
 
A '''Markov process''' is a stochastic process which satisfies the Markov property with respect to its [[Stochastic process#The natural filtration|natural filtration]].
 
===For discrete-time Markov chains===
 
In the case where <math>S</math> is a discrete set with the [[Sigma-algebra#Examples|discrete sigma algebra]] and <math>T = \mathbb{N}</math>, this can be reformulated as follows:
 
:<math>\mathbb{P}(X_n=x_n|X_{n-1}=x_{n-1}, X_{n-2}=x_{n-2}, \dots, X_0=x_0)=\mathbb{P}(X_n=x_n|X_{n-1}=x_{n-1})</math>.
 
==Examples==
 
===Gambling===
{{see also|random walk|Markov chain}}
Suppose that you start with $10 in poker chips, and you repeatedly wager $1 on a (fair) coin toss indefinitely, or until you lose all of your poker chips. If <math>X_n</math> represents the number of dollars you have in chips after ''n'' tosses, with <math>X_0 = 10</math>, then the sequence <math>\{X_n : n \in [0,\infty)\}</math> is a Markov process. If I know that you have 12 chips now, then it would be expected that with even odds, you will either have 11 or 13 chips after the next toss. This guess is not improved by the added knowledge that you started with 10 chips, then went up to 11, down to 10, up to 11, and then to 12.
 
The process described here is a Markov chain on a countable state space that follows a random walk.
 
===A birth-death process===
{{See also|birth-death process|Poisson process}}
Suppose that you are popping one hundred kernels of popcorn, and each kernel will pop at an independent, [[exponential distribution|exponentially-distributed]] time. Let <math>X_t</math> denote the number of kernels which have popped up to time ''t''. Then this is a [[continuous time Markov process]]. If after some amount of time, I want to guess how many kernels will pop in the next second, I need only to know how many kernels have popped so far. It will not help me to know ''when'' they popped, so knowing <math>X_t</math> for previous times ''t'' will not inform my guess.
 
The process described here is an approximation of a [[Poisson process]] - Poisson processes are also Markov.
 
===A non-Markov example===
Suppose that you have a coin purse containing five quarters (each worth 25c), five nickels (each worth 5c) and five dimes (each worth 10c), and one-by-one, you randomly draw coins from the purse and set them on a table. If <math>X_n</math> represents the total value of the coins set on the table after ''n'' draws, with <math>X_0 = 0</math>, then the sequence <math>\{X_n : n\in[0,\infty)\}</math> is ''not'' a Markov process.
 
To see why this is the case, suppose that in your first six draws, you draw all five nickels, and then a quarter. So <math>X_6 = \$0.50</math>. If we know not just <math>X_6</math>, 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 <math>X_7 \geq \$0.60</math> with probability 1. But if we do not know the earlier values, then based only on the value <math>X_6</math> 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 <math>X_7</math> are impacted by our knowledge of values prior to <math>X_6</math>.
 
===In analysing switching by Business Class===
{{Cleanup|section|reason=it contains mistakes in style as well as layout|date=December 2013}}
In analysing switching by Business Class customers between airlines the following data has been obtained by British Airways (BA):
 
                                Next flight by
                                    BA      Competition
    Last flight by  BA              0.85    0.15
                    Competition    0.10    0.90
 
For example if the last flight by a Business Class customer was by BA the probability that their next flight is by BA is 0.85. Business Class customers make 2 flights a year on average.
 
Currently BA have 30% of the Business Class market. What would you forecast BA's share of the Business Class market to be after two years?
 
Solution
 
We have the initial system state s1 given by s1 = [0.30, 0.70] and the transition matrix P is given by
 
    P =    |  0.85    0.15  |2  =  |  0.7375    0.2625  |
            |  0.10    0.90  |        |  0.1750    0.8250  |
 
 
where the square term arises as Business Class customers make 2 flights a year on average.
 
Hence after one year has elapsed the state of the system s2 = s1P = [0.34375, 0.65625]
 
After two years have elapsed the state of the system = s3 = s2P = [0.368, 0.632]
 
and note here that the elements of s2 and s3 add to one (as required).
 
So after two years have elapsed BA's share of the Business Class market is 36.8%
 
==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, let ''X'' be a non-Markovian process. Then define a process ''Y'', such that each state of ''Y'' represents a time-interval of states of ''X''. Mathematically, this takes the form:
 
:<math>Y(t) = \big\{ X(s): s \in [a(t), b(t)] \, \big\}.</math>
 
If ''Y'' has the Markov property, then it is a Markovian representation of ''X''.
 
An example of a non-Markovian process with a Markovian representation is an [[Autoregressive model|autoregressive]] [[time series]] of order greater than one.{{Citation needed|date=September 2010}}
 
==In popular culture==
The band [[Bad Religion]] has a song titled "The Markovian Process" on their album [[Stranger than Fiction (Bad Religion album)|Stranger Than Fiction]]. The credited writer of the song, lead singer and current [[UCLA]] professor [[Greg Graffin]], has a [[Ph.D.]] in [[zoology]] and degrees in [[anthropology]] and [[geology]].
 
==See also==
{{Div col|2}}
* [[Brownian motion]]
* [[Dynamics of Markovian particles]]
* [[Examples of Markov chains]]
* [[Interacting particle system]]
* [[Markov chain]]
* [[Markov decision process]]
* [[Markov model]]
* [[Random walk]]
* [[Semi-Markov process]]
{{Div col end}}
 
==References==
{{Reflist}}
 
==External links==
{{Refbegin}}
* {{mathworld|urlname=MarkovProcess|title=Markov process}}
{{Refend}}
 
{{Stochastic processes}}
 
{{DEFAULTSORT:Markov Process}}
[[Category:Stochastic processes]]
[[Category:Markov processes| ]]
 
[[de:Markow-Kette]]

Revision as of 08:14, 7 March 2013

Markov process example

In probability theory and statistics, a Markov process or Markoff process, named after the Russian mathematician Andrey Markov, is a stochastic process that satisfies the Markov property. A Markov process can be thought of as 'memoryless': loosely speaking, a process satisfies the Markov property if one can make predictions for the future of the process based solely on its present state just as well as one could knowing the process's full history. I.e., conditional on the present state of the system, its future and past are independent.[1]

Introduction

A Markov process is a stochastic model that has the Markov property. It can be used to model a random system that changes states according to a transition rule that only depends on the current state. This article describes the Markov process in a very general sense, which is a concept that is usually specified further. Particularly, the system's state space and time parameter index needs 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 vs. continuous time.

Countable or finite state space Continuous or general state space
Discrete-time Markov chain on a countable or finite state space Harris chain (Markov chain on a general state space)
Continuous-time Continuous-time Markov process Any continuous stochastic process with the Markov property, e.g. the Wiener process

Note that there is no definitive agreement in literature on the use of some of the terms that signify special cases of Markov processes. For example, often the term "Markov chain" is used to indicate a Markov process which has a finite or countable state-space, but Markov chains on a general state space fall under the same description. Similarly, a Markov chain would usually be defined for a discrete set of times (i.e. a discrete-time Markov chain)[2] although some authors use the same terminology where "time" can take continuous values.[3] 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.

Markov processes arise in probability and statistics in one of two ways. A stochastic process, defined via a separate argument, may be shown mathematically to have the Markov property, and as a consequence to have the properties that can be deduced from this for all Markov processes. Alternately, in modelling a process, one may assume the process to be Markov, and take this as the basis for a construction. In modelling terms, assuming that the Markov property holds is one of a limited number of simple ways of introducing statistical dependence into a model for a stochastic process in such a way that allows the strength of dependence at different lags to decline as the lag increases.

Markov property

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The general case

Let be a probability space with a filtration , for some (totally ordered) index set ; and let be a measurable space. An S-valued stochastic process adapted to the filtration is said to possess the Markov property with respect to the if, for each and each with s < t,

[4]

A Markov process is a stochastic process which satisfies the Markov property with respect to its natural filtration.

For discrete-time Markov chains

In the case where is a discrete set with the discrete sigma algebra and , this can be reformulated as follows:

.

Examples

Gambling

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In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. Suppose that you start with $10 in poker chips, and you repeatedly wager $1 on a (fair) coin toss indefinitely, or until you lose all of your poker chips. If represents the number of dollars you have in chips after n tosses, with , then the sequence is a Markov process. If I know that you have 12 chips now, then it would be expected that with even odds, you will either have 11 or 13 chips after the next toss. This guess is not improved by the added knowledge that you started with 10 chips, then went up to 11, down to 10, up to 11, and then to 12.

The process described here is a Markov chain on a countable state space that follows a random walk.

A birth-death process

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The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.

Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.

is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. Suppose that you are popping one hundred kernels of popcorn, and each kernel will pop at an independent, exponentially-distributed time. Let denote the number of kernels which have popped up to time t. Then this is a continuous time Markov process. If after some amount of time, I want to guess how many kernels will pop in the next second, I need only to know how many kernels have popped so far. It will not help me to know when they popped, so knowing for previous times t will not inform my guess.

The process described here is an approximation of a Poisson process - Poisson processes are also Markov.

A non-Markov example

Suppose that you have a coin purse containing five quarters (each worth 25c), five nickels (each worth 5c) and five dimes (each worth 10c), and one-by-one, you randomly draw coins from the purse and set them on a table. If represents the total value of the coins set on the table after n draws, with , then the sequence is not a Markov process.

To see why this is the case, suppose that in your first six draws, you draw all five nickels, and then a quarter. So . If we know not just , 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 with probability 1. But if we do not know the earlier values, then based only on the value 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 are impacted by our knowledge of values prior to .

In analysing switching by Business Class

Template:Cleanup

In analysing switching by Business Class customers between airlines the following data has been obtained by British Airways (BA): 

                                Next flight by
                                    BA      Competition
   Last flight by   BA              0.85    0.15
                    Competition     0.10    0.90

For example if the last flight by a Business Class customer was by BA the probability that their next flight is by BA is 0.85. Business Class customers make 2 flights a year on average.

Currently BA have 30% of the Business Class market. What would you forecast BA's share of the Business Class market to be after two years?

Solution

We have the initial system state s1 given by s1 = [0.30, 0.70] and the transition matrix P is given by 

   P =     |  0.85    0.15  |2   =   |  0.7375    0.2625  |
           |  0.10    0.90  |        |  0.1750    0.8250  |


where the square term arises as Business Class customers make 2 flights a year on average.

Hence after one year has elapsed the state of the system s2 = s1P = [0.34375, 0.65625]

After two years have elapsed the state of the system = s3 = s2P = [0.368, 0.632]

and note here that the elements of s2 and s3 add to one (as required).

So after two years have elapsed BA's share of the Business Class market is 36.8%

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, let X be a non-Markovian process. Then define a process Y, such that each state of Y represents a time-interval of states of X. Mathematically, this takes the form:

If Y has the Markov property, then it is a Markovian representation of X.

An example of a non-Markovian process with a Markovian representation is an autoregressive time series of order greater than one.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

In popular culture

The band Bad Religion has a song titled "The Markovian Process" on their album Stranger Than Fiction. The credited writer of the song, lead singer and current UCLA professor Greg Graffin, has a Ph.D. in zoology and degrees in anthropology and geology.

See also

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References

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External links

Template:Refbegin

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Template:Refend

Template:Stochastic processes

de:Markow-Kette

  1. ↑ Markov process (mathematics) - Britannica Online Encyclopedia
  2. ↑ Everitt,B.S. (2002) The Cambridge Dictionary of Statistics. CUP. ISBN 0-521-81099-X
  3. ↑ Dodge, Y. The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9
  4. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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