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Princess Nora University Modeling and Simulation Markov chain Arwa Ibrahim Ahmed 1 Markov chain 2 Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process usually characterized as memoryless: the next state depends only on the current state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is called the Markov property. 2 Markov chain : 3 Formally, a Markov chain is a random process with the Markov property. Often, the term "Markov chain" is used to mean a Markov process which has a discrete (finite or countable) state-space. Usually a Markov chain is defined for a discrete set of times (i.e., a discrete-time Markov chain) although some authors use the same terminology where "time" can take continuous values. 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. 3 Markov chain: 4 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. The changes of state of the system are called transitions, and the probabilities associated with various state-changes are called transition probabilities. The set of all states and transition probabilities completely characterizes a Markov chain. By convention, we assume all possible states and transitions have been included in the definition of the processes, so there is always a next state and the process goes on forever. 4 Applications of Markov chain: 5 The application and usefulness of Markov chains: Information sciences: 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 through Markov modeling of the English language. Queuing theory Markov chains are the basis for the analytical treatment of queues (queuing theory). Agner Krarup Erlang initiated the subject in 1917. This makes them critical for optimizing the performance of telecommunications networks, where messages must often compete for limited resources 5 Applications of Markov chain: 6 Internet applications The Page Rank of a webpage as used by Google is defined by a Markov chain.It is the probability to be at page j in the stationary distribution on the following Markov chain on all (known) WebPages. Economics and finance Markov chains are used in Finance and Economics to model a variety of different phenomena, including asset prices and market crashes. The first financial model to use a Markov chain was from Prasad et al. in 1974. 6 Applications of Markov chain: 7 Social sciences 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. Games 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. 7 DISCRETE-TIME FINITE-STATE MARKOV CHAINS 8 Discrete-time Markov chains : Let X be a discrete random variable, indexed by time t as X(t), that evolves in time as follows. X(t) ∈ X for all t = 0, 1, 2, . . . . State transitions can occur only at the discrete times t = 0, 1, 2, . . . . and at these times the random variable X(t) will shift from its current state x ∈ X to another state, say X’ ∈ X, with fixed probability p(x; x’) = Pr(X(t + 1) = x’ | X(t) = x) ≥ 0. 8 DISCRETE-TIME FINITE-STATE MARKOV CHAINS: 9 If | X | < 1 the stochastic process defend by X(t) is called a discrete-time, finite-state Markov chain. Without loss of generality, throughout this section we assume that the finite state space is X = {0, 1, 2, . . . , k} where k is a finite, but perhaps quite large, integer. 9 DISCRETE-TIME FINITE-STATE MARKOV CHAINS: 10 A discrete-time, finite-state Markov chain is completely characterized by the initial state at t = 0, X(0), and the function p(x; x’) defined for all (x, x’) ∈ X X X. When the stochastic process leaves the state x the transition must be either to state x’ = 0 with probability p(x, 0), or to state x’ = 1 with probability p(x, 1), . . . . , or to state x’ = k with probability p(x, k), and the sum of these probabilities must be 1. That is k x ' 0 p( x, x' ) 1 x = 0, 1, . . . . , k. Because p(x, x’) is independent of t for all (x, x’), the Markov chain is said to be homogeneous or stationary. 10 DISCRETE-TIME FINITE-STATE MARKOV CHAINS: 11 The state transition probability p(x; x’) represents the probability of a transition from state x to state x’. The corresponding (k + 1) X (k + 1) matrix P= .(0, k ) p (0,0) p (0,1)......... p (1,0) p (1, 2)......... .(1, k ) .......... .......... .......... ....... .......... .......... ....... .......... .......... .......... .......... ....... p ( k ,0) p ( k ,1).......p ( k , k ) with elements p(x, x’) is called the state transition matrix. 11 DISCRETE-TIME FINITE-STATE MARKOV CHAINS: 12 The elements of the state transition matrix p are non-negative and the elements of each row sum to 1.0. (A matrix with these properties is said to be a stochastic matrix.) 12 EXAMPLE: 13 If we know the probability that the child of a lowerclass parent becomes middle-class or upper-class, and we know similar information for the child of a middle-class or upper-class parent, what is the probability that the grandchild of a lower –class parent is middle or upper class? 13 EXAMPLE: 14 in sociology, it is convenient to classify people by income as lower-class ,middle-class and upper-class. Sociologists have found that the strongest determinate of the income class of an individual is the income class of the individual's parents. for example , if an individual in the lower-income class is said to be in state 1, an individual in the middle-income class is in state 2,and an individual in the upper -income class is in state3, then the following probabilities of change in income class from one generation to the next might apply. 14 EXAMPLE: 15 Table1 shows that if an individual is in state1 (lower income class) then there is a probability of 0.65 that any offspring will be in the lower-income class ,a probability of 0.28 that offspring will be in the middle income class ,and a probability of 0.07 that offspring will be in the upper-income class. 15 EXAMPLE: 16 state 1 2 3 1 0.65 0.28 0.07 2 0.15 0.67 0.18 3 0.12 0.36 0.52 The symbol Pij will be used for the probability of transaction from state I to state j in one generation . For example , p23 represents the probability that a person in state 2 will have offspring in state 3 , from that table above , p23 =0.18 . Also from the table ,p31= 0.12 , p22=0.67 , and so on 16 EXAMPLE: 17 The information from table can be written in other forms . Figure 1 is a transition diagram that shows the three states and probabilities of going from one state to another. 17 EXAMPLE: 18 0.28 0.65 1 2 0.15 0.67 0.12 0.18 0.07 0.52 3 0.36 18 EXAMPLE: 19 In a transition matrix, the states are indicated at the side and the top .if P represent the transition matrix for the table above, then 0.56 0.15 0.12 0.28 0.67 0.36 0.07 0.18 0.52 19 EXAMPLE: 20 A transition matrix has several features: 1. 2. 3. It is square , since all possible states must be used both as rows as columns. All entries are between 0 and 1 , inclusive ; this is because all entries represent probabilities. The sum of the entries in any row must be 1. 20 EXAMPLE: 21 The transition matrix P shows the probability of change in income class from one generation to the next . now let us investigate the probability for change in income class over two generation. For example ,if a parent in state 3(the upper income class) .what is the probability that a grandchild will be in state 2? To find out, start with a tree diagram ,as shown in fig2. The various probabilities come from transition matrix P. The arrows point to the outcomes “grandchild in state 2” grandchild in state2 is given by the sum of the probabilities indicated with arrows ,or 0.0336+0.2412+0.1872=4620 21 EXAMPLE: 22 1 1 2 O.21 3 O.36 3 1 2 O.52 3 1 2 3 (0.12) (0.65)=0.078 (0.12) (0.28)=0.0336 (0.12) (0.07)=0.0084 (0.36) (0.15)=0.054 2 (0.36) (067)=0.02412 3 (0.36) (0.18)=0.0648 (0.52) (012)=0.0624 (0.52) (0.36)=0.1872 (0.52) (0.52)=0.2704 22