Universitatea Tehnică a Moldovei Catedra Calculatoare Disciplina: Procese Stochastice. Raport Lucrare de laborator Nr Tema: Lanturi Markov timp discret. Transient Markov chains with stationary measures. Proc. Amer. Math. Dynamic Programming and Markov Processes. Lanturi Markov Finite si Aplicatii. ed. Editura Tehnica, Bucuresti () Iosifescu, M.: Lanturi Markov finite si aplicatii. Editura Tehnica Bucuresti () Kolmogorov, A.N.: Selected Works of A.N.

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### Lanț Markov – Wikipedia

A Markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. In probability theory and related fields, a Markov processnamed after the Russian mathematician Andrey Markovis a stochastic process that satisfies the Markov property [1] [3] [4] sometimes characterized as ” memorylessness “. Roughly 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, hence independently from such history; i.

A Markov chain is a type of Markov process that has either a discrete state space or a discrete index set often representing timebut the lajturi definition of a Markov chain varies.

Markov studied Markov processes in the early 20th century, publishing mariov first paper on the topic in Markov chains have many applications as statistical models of real-world processes, [1] [25] [26] [27] such as studying cruise control systems in motor vehiclesqueues or lines of customers arriving at an airport, exchange rates of currencies, storage systems such as damsand population growths of certain animal species.

Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlowhich are used for simulating sampling lanfuri complex probability distributions, and have found extensive application in Bayesian statistics. The adjective Markovian is used to describe something that is related to a Markov process. A Markov chain is a stochastic process with the Markov property. The system’s state space and time parameter index need to lahturi specified.

The following table gives an overview of the different instances of Markov processes for different levels of state space generality and for discrete time v. 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, i.

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: 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. The changes of state of the system are called transitions. 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 markob, 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.

Formally, the steps are the integers or natural numbersand the random process is a mapping of these to states. Since the system changes randomly, it is generally impossible markiv predict with certainty the state of a Markov chain at a given point in the future. From any position there are two possible transitions, to lantuuri 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. These probabilities are independent of whether the system was previously in 4 or 6. Another example is the dietary habits of a creature who eats only grapes, cheese, or lettuce, and whose dietary habits conform to the following rules:.

This creature’s eating habits can be modeled with a Markov chain since its choice tomorrow depends solely on what it ate today, not what it ate yesterday or any other time in the past. One statistical property that could be calculated is the expected percentage, over a long period, of the days on which the creature will eat grapes.

A series of independent events 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 step depends non-trivially on the current state.

## Markov chain

Andrey Markov studied Markov chains in the early 20th century. Other early uses of Markov chains include a diffusion model, introduced by Paul and Tatyana Ehrenfest inand a branching process, introduced by Francis Galton and Henry William Watson inpreceding the work of Markov. Andrei Kolmogorov developed in a paper a large part of the early theory of continuous-time Markov processes.

The process described here is a Markov chain on a countable state space that follows a random walk. If one pops one hundred kernels of popcorn in an oven, each kernel popping at an independent exponentially-distributed time, then this would be a continuous-time Markov process. The only thing one needs to know is the number of kernels that have popped prior to the time “t”. The process described here is an approximation of a Poisson point process – Poisson processes are also Markov processes.

To see why this is the case, suppose that in the first six draws, all five nickels and a lanturl are drawn. However, it is possible to model this scenario as a Markov process.

This new model would be represented by possible states that is, 6x6x6 states, since each of the three coin types could have zero to five coins on the table by the end of the 6 draws. After the second draw, the third draw depends on which coins have so far been drawn, but lnaturi longer only on the coins that were drawn for the first state since probabilistically important information has since been added to the scenario. A stochastic process has the Markov property if the conditional probability distribution of future states of the process depends only upon the present state, not on the sequence of events that preceded it.

A discrete-time Markov chain is a sequence of random variables X 1X 2X 3The possible values of X i form a countable set S called the state space of the chain. However, Markov chains are frequently assumed to be time-homogeneous see variations belowin which case the graph and karkov are independent of n and are thus not presented as sequences.

When time-homogeneous, the chain can be interpreted as a state machine assigning a probability of hopping from each vertex or state to an adjacent one.

## Lanț Markov

The fact that some magkov of states might have zero probability of occurring corresponds to a graph with multiple connected componentswhere we omit edges that would carry a zero transition probability.

A state diagram for a simple example is shown in the figure on the right, using a directed graph to picture the state transitions.

The states represent whether a hypothetical stock market is exhibiting mar,ov bull marketbear marketor stagnant market trend during a given week. Using the transition matrix it is possible to calculate, for example, the long-term fraction of weeks during which the market is stagnant, or the average number of weeks it will take to go from a stagnant to a bull market.

Using the transition probabilities, the steady-state probabilities indicate that A finite-state machine can be narkov as a representation of a Markov chain. The elements q markpv 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 lantuuri.

There are three equivalent definitions of the process. Define a discrete-time Markov chain Y n to describe the n th jump of the process and variables S 1S 2S 3The evolution of the process through one time step is described by. The superscript n is an indexand not an exponent. A Markov chain is said to be lanutri if it is possible lxnturi get to any state from any state. This integer is allowed to be different for each pair of states, hence the subscripts in n ij.

Allowing n to be zero means that every state is accessible from itself by definition. The accessibility relation is reflexive and transitive, but not necessarily symmetric. A communicating class is a maximal set of states C such that every pair of states in C communicates with each other.

Communication is an equivalence relationand communicating classes are the equivalence classes of this relation. The set of communicating classes forms a directed, acyclic graph by inheriting the arrows from the original state space.

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A communicating class is closed if and only if it has no outgoing arrows in this graph. A state i is marov if it is not essential. A Markov chain is said to be irreducible if its state space is a single communicating class; in other words, if it is possible to get to any state from any state.

A state i has period k if any return to state i must occur in multiples of k time steps. Formally, the period of a state is defined as. Otherwise the period is not defined. Note that even though a state has period kit may not be possible to reach the state in k steps. A Markov chain is aperiodic if every state is aperiodic.

An irreducible Markov chain only needs one aperiodic state to imply all states are aperiodic. Every state of a bipartite graph has an even period. A state i is said to be transient if, given that we start in state ithere is a non-zero probability that we will never return to i.

Formally, let the random variable T i be the first return time to state i the “hitting time”:. Therefore, state i is transient if. State i is recurrent or persistent if it is not transient. Recurrent states are guaranteed with probability 1 to have a finite hitting time. Recurrence and transience are class properties, that is, they either hold or do not hold equally for all members of a communicating class. Even if the hitting time is finite with probability 1it need not have a finite expectation.

The mean recurrence time at state i is the expected return time M i:. State i is positive recurrent or non-null persistent if M i is finite; otherwise, state i is null recurrent or null persistent. It can be shown that a state i is recurrent if and only if the expected number of visits to this state is infinite, i.

A state i is called absorbing if it is impossible to leave this state. Therefore, the state i is absorbing if and only if.

If every state can reach an absorbing state, then the Markov chain is an absorbing Markov chain. A state i is said to be ergodic if it is aperiodic and positive recurrent.

In other words, a state i is ergodic if it is recurrent, has a period of 1and has finite mean recurrence time. If all states in an irreducible Markov chain are ergodic, then the chain is said to be ergodic. It can be shown that a finite state irreducible Markov chain is ergodic if it has an aperiodic state.

More generally, a Markov chain is ergodic if there is a number N such maekov any state can be reached from any other state in any number of steps greater than or equal to a number N. 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. ,arkov, if the positive recurrent chain is both irreducible and aperiodic, it is said to have a limiting distribution; for any i and j.

There is no assumption on the starting distribution; the chain converges to the stationary distribution regardless of where it begins. However, if a state j is aperiodic, then.