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| [[File:Markov process-example.svg|thumb|Markov process example]]
| | Are we constantly having problems with your PC? Are you always searching for ways to increase PC performance? Then this really is the article you're interested in. Here we will discuss a few of the many asked issues with regards to having we PC serve we well; how could I create my computer faster for free? How to make my computer run faster?<br><br>Before actually buying the software it is very best to check on the companies that create the software. If you will discover details on the type of reputation every firm has, perhaps the risk of malicious programs can be reduced. Software from reputed companies have aided me, plus many other users, to create my PC run faster.. If the product description refuses to look superior to we, refuses to include details about the software, does not include the scan functions, you need to go for another one which ensures you're paying for what you desire.<br><br>With RegCure to boost the start and shut down of your computer. The program shows the scan progress plus you shouldn't worry where it really is working at that time. It shows we exactly what arises. Dynamic link library section of the registry may result severe application failures. RegCure restores plus repairs the registry and keeps you from DLL. RegCure is able to create individual corrections, so it could works for the requires.<br><br>There is many factors why the computer can lose speed. Normal computer use, including surfing the Internet can get your running program in a condition where it has no choice nevertheless to slow down. The continual entering and deleting of temporary files which occur when we surf the Web leave our registries with thousands of false indicators in the running system's registry.<br><br>So to fix this, we only should be able to make all of the registry files non-corrupted again. This will dramatically speed up the loading time of your computer plus might allow you to a large number of aspects on it again. And fixing these files couldn't be easier - you only should use a tool called a [http://bestregistrycleanerfix.com/registry-reviver registry reviver].<br><br>Windows relies heavily on this database, storing everything from your newest emails to a Internet favorites in there. Because it's so important, your computer is frequently adding plus updating the files inside it. This is ok, but it will create the computer run slow, when a computer accidentally breaks its important registry files. This really is a fairly usual problem, and really makes the computer run slower every day. What happens is that because a computer is constantly utilizing 100's of registry files at once, it sometimes gets confused and create a few of them unreadable. This then makes the computer run slow, because Windows takes longer to read the files it requires.<br><br>Across the top of the scan results display page you see the tabs... Registry, Junk Files, Privacy, Bad Active X, Performance, etc. Each of these tabs may show you the results of that area. The Junk Files are primarily temporary files including internet information, photos, internet pages... And they are really taking up storage space.<br><br>There are many companies that provide the service of troubleshooting your PC every time you call them, all you need to do is sign up with them and for a tiny fee, you could have the machine constantly functioning effectively and serve we greater. |
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| 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>
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| ==Introduction==
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| 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.
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| {| border="1" class="wikitable" style="width: 60%;" |
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| ! scope="col" |
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| ! scope="col" | Countable or finite state space
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| ! scope="col" | Continuous or general state space
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| |-
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| ! scope="row" | Discrete-time
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| |[[Markov chain]] on a countable or finite state space || [[Harris chain]] (Markov chain on a general state space)
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| |-
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| ! scope="row" style="width: 10%;" | Continuous-time
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| | style="width: 25%;" | [[Continuous-time Markov process]] || style="width: 25%;" |Any [[continuous stochastic process]] with the Markov property, e.g. the [[Wiener process]]
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| |}
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| 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.
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| 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.
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| ==Markov property==
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| {{Main|Markov property}}
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| {{Technical|section|date=December 2012}}
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| ===The general case===
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| 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'',
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| :<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>
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| A '''Markov process''' is a stochastic process which satisfies the Markov property with respect to its [[Stochastic process#The natural filtration|natural filtration]].
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| ===For discrete-time Markov chains===
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| 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:
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| :<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>.
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| ==Examples==
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| ===Gambling===
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| {{see also|random walk|Markov chain}}
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| 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.
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| The process described here is a Markov chain on a countable state space that follows a random walk.
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| ===A birth-death process===
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| {{See also|birth-death process|Poisson process}}
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| 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.
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| The process described here is an approximation of a [[Poisson process]] - Poisson processes are also Markov.
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| ===A non-Markov example===
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| 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.
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| 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>.
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| ===In analysing switching by Business Class===
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| {{Cleanup|section|reason=it contains mistakes in style as well as layout|date=December 2013}}
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| In analysing switching by Business Class customers between airlines the following data has been obtained by British Airways (BA):
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| Next flight by
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| BA Competition
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| Last flight by BA 0.85 0.15
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| Competition 0.10 0.90
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| 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.
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| 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?
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| Solution
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| We have the initial system state s1 given by s1 = [0.30, 0.70] and the transition matrix P is given by
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| P = | 0.85 0.15 |2 = | 0.7375 0.2625 |
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| | 0.10 0.90 | | 0.1750 0.8250 |
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| where the square term arises as Business Class customers make 2 flights a year on average.
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| Hence after one year has elapsed the state of the system s2 = s1P = [0.34375, 0.65625]
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| After two years have elapsed the state of the system = s3 = s2P = [0.368, 0.632]
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| and note here that the elements of s2 and s3 add to one (as required).
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| So after two years have elapsed BA's share of the Business Class market is 36.8%
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| ==Markovian representations==
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| 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:
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| :<math>Y(t) = \big\{ X(s): s \in [a(t), b(t)] \, \big\}.</math>
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| If ''Y'' has the Markov property, then it is a Markovian representation of ''X''.
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| 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}}
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| ==In popular culture==
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| 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]].
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| ==See also==
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| {{Div col|2}}
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| * [[Brownian motion]]
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| * [[Dynamics of Markovian particles]]
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| * [[Examples of Markov chains]]
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| * [[Interacting particle system]]
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| * [[Markov chain]]
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| * [[Markov decision process]]
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| * [[Markov model]]
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| * [[Random walk]]
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| * [[Semi-Markov process]]
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| {{Div col end}}
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| ==References==
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| {{Reflist}}
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| ==External links==
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| {{Refbegin}}
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| * {{mathworld|urlname=MarkovProcess|title=Markov process}}
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| {{Refend}}
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| {{Stochastic processes}}
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| {{DEFAULTSORT:Markov Process}}
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| [[Category:Stochastic processes]]
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| [[Category:Markov processes| ]]
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| [[de:Markow-Kette]]
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Are we constantly having problems with your PC? Are you always searching for ways to increase PC performance? Then this really is the article you're interested in. Here we will discuss a few of the many asked issues with regards to having we PC serve we well; how could I create my computer faster for free? How to make my computer run faster?
Before actually buying the software it is very best to check on the companies that create the software. If you will discover details on the type of reputation every firm has, perhaps the risk of malicious programs can be reduced. Software from reputed companies have aided me, plus many other users, to create my PC run faster.. If the product description refuses to look superior to we, refuses to include details about the software, does not include the scan functions, you need to go for another one which ensures you're paying for what you desire.
With RegCure to boost the start and shut down of your computer. The program shows the scan progress plus you shouldn't worry where it really is working at that time. It shows we exactly what arises. Dynamic link library section of the registry may result severe application failures. RegCure restores plus repairs the registry and keeps you from DLL. RegCure is able to create individual corrections, so it could works for the requires.
There is many factors why the computer can lose speed. Normal computer use, including surfing the Internet can get your running program in a condition where it has no choice nevertheless to slow down. The continual entering and deleting of temporary files which occur when we surf the Web leave our registries with thousands of false indicators in the running system's registry.
So to fix this, we only should be able to make all of the registry files non-corrupted again. This will dramatically speed up the loading time of your computer plus might allow you to a large number of aspects on it again. And fixing these files couldn't be easier - you only should use a tool called a registry reviver.
Windows relies heavily on this database, storing everything from your newest emails to a Internet favorites in there. Because it's so important, your computer is frequently adding plus updating the files inside it. This is ok, but it will create the computer run slow, when a computer accidentally breaks its important registry files. This really is a fairly usual problem, and really makes the computer run slower every day. What happens is that because a computer is constantly utilizing 100's of registry files at once, it sometimes gets confused and create a few of them unreadable. This then makes the computer run slow, because Windows takes longer to read the files it requires.
Across the top of the scan results display page you see the tabs... Registry, Junk Files, Privacy, Bad Active X, Performance, etc. Each of these tabs may show you the results of that area. The Junk Files are primarily temporary files including internet information, photos, internet pages... And they are really taking up storage space.
There are many companies that provide the service of troubleshooting your PC every time you call them, all you need to do is sign up with them and for a tiny fee, you could have the machine constantly functioning effectively and serve we greater.