Markov Processes Hamid R. Rabiee

Transcription

1 Markov Processes Hamid R. Rabiee

2 Overview Markov Property Markov Chains Definition Stationary Property Paths in Markov Chains Classification of States Steady States in MCs. 2

3 Markov Property A discrete process has the Markov property if given its value at time t, the value at time t+ is independent of values at times before t. That is: Pr X t+ = x t+ X t = x t, X t = x t,, X = x = Pr X t+ = x t+ X t = x t For all t, x t+, x t, x t, x t 2,, x. 3

4 Stationary Property A Markov Process is called stationary if Pr X t+ = u X t = v = Pr X = u X 0 = v for all t. The evolution of stationary processes don t change over time. For defining the complete joint distribution of a stationary Markov Process it is sufficient to define Pr X = u X 0 = v and Pr X 0 = v for all u and v. (why?) We will mainly consider stationary Markov processes here. 4

5 Example (Coin Tossing Game) Consider a single player game in which at every step a biased coin is tossed and according to the result, the score will be increased or decreased by one point. The game ends if either the score reaches 00 (winning) or -00 (losing). Score of the player at each step t 0 is a random variable and the sequence of scores as the game progresses forms a random process X 0, X,, X t. 5

6 Example (Coin Tossing Game) It is easy to verify that X is a stationary Markov chain: At the end of each step the score solely depends on the current score s c and the result of tossing the coin (which is independent of time and previous tosses). Stating this mathematically (for s c { 00,00}): Pr X t+ = s X t = s c, X t = s t,, X 0 = 0 = p ; s = s c + p ; s = s c 0 ; otherwise = Pr X t+ = s X t = s c = Pr X = s X 0 = s c If value of p was subject to change in time, the process would not be stationary. In the formulation we would have p t instead of p. Independent of t and s 0,, s t 6

7 Example (Tracking) Assume we want to track a target in XY plane, and so we need to model its movement. Assuming that current position and speed of the target describes everything about its future movements, we can consider the 4- dimensional state: S t = (X t, Y t, X t, Y t ) It is common to consider linear relation between S t and S t with a Gaussian additive noise: S t = A t S t + v t ; v t ~N(0, Σ) Or equivalently: f St S t s t s t = N(s t ; A t s t, Σ) There exists efficient algorithms to work with these linear- Gaussian models. 7

8 Example (Tracking) Considering the kinematics relations of a moving object we can define linear relation as: 0 Δt Δt A t = This approach can not model Acceleration the speed is changed only because of noise. Independent of t (stationary) We can model arbitrary speed change by appropriately setting the 3 rd and 4 th rows of the matrix at each time. An example of a non stationary Markov process. Another approach is to extend the states to 6- dimensions containing X t and Y t. 8

9 Example (Speech) A basic model for speech signal considers the value at time t to be a linear combination of its d previous values with a Gaussian additive noise: S t = d i= α i S t i S t is not a Markov process. + v t ; v t ~N(0, Σ) 9

10 Example (Speech) We can make a stationary Markov process by considering the d-dimensional state U t = S t, S t, S t d T : Where: A = U t = AU t + v k B α α 2 α d , B = Equivalently: f Ut U t u t u t = N u t ; A t u t, Σ I i d u t i = u t i Which (x) i is the ith dimension of vector x and I is the indicator function (used for guaranteeing consistency). Note that we have repeated X t in d consecutive states of U and there should be no inconsistency between those values. 0 0

11 Markov Process Types Two types of Markov processes based on domain of X t values: Discrete Continuous Discrete Markov processes are called Markov Chains (MC). In this session we will focus on stationary MCs.

12 Transition matrix According to the Markov property and stationary property, at each time step the process moves according to a fixed transition matrix: P X t+ = j X t = i = p ij Stochastic matrix: Rows sum up to Double stochastic matrix: Rows and columns sum up to 2

13 State Graph It is convenient to visualize a stationary Markov Chain with a transition diagram: A node represents a possible value of X t. At each time t, we say the process is in state s if X t =s. Each edge represents the probability of going from one state to another (we omit edges with zero weight). We should also specify the vector of initial probabilities π = π,, π n where π i = Pr(X 0 = i). So a stationary discrete process could be described as a person walking randomly on a graph (considering each step to depend only on the state he is currently in). The result path is called a Random Walk. 3

14 Example The transition diagram of the coin tossing game: p p p p p -p -p -p -p We modeled winning and losing by states which when we get into, we never get out. Note that if the process was not stationary we were not able to visualize it in this way: For example consider the case that p is changing in time. 4

15 Example (Modeling Weather) Example: Each day is sunny or rainy. If a day is rainy, the next day is rainy with probability α (and sunny with probability α). If the day is sunny, the next day is rainy with probability β (and sunny with probability β). S = {rainy, sunny}, P = α α β β α α β R S β 5

16 Paths If the state space is {0,} we have: pr x 2 = 0 x 0 = 0 = Pr x = x 0 = 0 p x 2 = 0 x = +Pr x = 0 x 0 = 0 p x 2 = 0 x = 0 (n) Define p ij as the probability that starting from state i, the process stops at state j after n time steps. (2) = r pik p kj p ij k= 6

17 Paths Theorem: (n) p ij = r k= (n ) pik pkj To simplify the notation we define the matrix P (n) such that (n) the value at the i th row and j th column is p ij. Corollary : P (n) can be calculated by: P (n) = P n Corollary 2: If the process starts at time 0 with distribution π on the states then after n steps the distribution is πp n. 7

18 Absorbing Markov Chain An absorbing state is one in which the probability that the process remains in that state once it enters the state is (i.e., p ii = ). A Markov chain is absorbing if it has at least one absorbing state, and if from every state it is possible to go to an absorbing state (not necessarily in one step). An absorbing Markov chain will eventually enter one of the absorbing states and never leave it. Example: The 00 and -00 states in coin tossing game Note: After playing ling enough, the player will either win or lose (with probability ). p p p p p -p -p -p -p 8

19 Absorption theorem In an absorbing Markov chain the probability that the process will be absorbed is. Proof: From each non-absorbing state s j it is possible to reach an absorbing state starting from s j. Therefore there exists p and m, such that the probability of not absorbing after m steps is at most p, in 2m steps at most p 2, etc. Since the probability of not being absorbed is monotonically decreasing, we have: lim n Pr(not absorbed after n steps) =0 9

20 Classification of States Accessibility: State j is said to be accessible from state i if starting in i it is possible that the process will ever enter state j: (P n ) ij > 0. Communication: Two states i and j that are accessible to each other are said to communicate. Every node communicates with itself: p ii 0 = Pr X 0 = i X 0 = i = Communication is an equivalence relation: It divides the state space up into a number of separate classes in which every pair of states communicate. (why?) The Markov chain is said to be irreducible if it has only one class. 20

21 Transient and Recurrent states For any state i we let f i denote the probability that, starting in state i, the process will ever reenter state i : f i = Pr n: X n = i X 0 = i) State i is said to be recurrent if f i = and transient if f i <. Theorem : State i is recurrent if and only if, starting in state i, the expected number of time periods that the process is in state i is infinite: Corollary : A transient state will only be visited a finite number of times Proof: E size n: X n = i X 0 = i = k= k Pr(size n: X n = i = k X 0 = i) = + prob size n: X n = i = X 0 = i < prob size n: X n = i = X 0 = i = 0 2

22 Transient and Recurrent states Theorem 2: State i is recurrent iff n=(p n ) ii =. Look at the text book for proof. Corollary 2: A finite state Markov chain has at least one recurrent state. If all states are transient there will be a finite number of steps that after that the process should not be in any state (which is a contradiction). 22

23 Ergodic states If state i is recurrent, then it is said to be positive recurrent if, starting in i, the expected time until the process returns to state i is finite. In a finite-state MC, all recurrent states are positive recurrent. State i is said to have period d if p n ii=0 whenever n is not divisible by d, and d is the largest integer with this property. Equivalently: d = gcd n: Pr X n = i X 0 = i > 0} A state with period is said to be aperiodic. We call an MC aperiodic if all its states are aperiodic. A state i is said to be ergodic if it is aperiodic and positive recurrent. Period, recurrence and positive recurrence are all class properties. They are shared between states of a class. 23

24 Example 0.25 Classes: {},{2,3},{4,5},{6},{7.8} Recurrent states: 6, 7, 8 => all positive Periodic states: 2, 3, 7, 8 (with period 2) Absorbing state(s): 6 Ergodic state(s): As time goes to infinity, what is the probability of being in each class? Answer: The process will be in transient classes {},{2,3},{4,5} with probability 0. Problem is symmetric for entering classes {6} and {7,8} as their only input edge is one from 5 with equal probabilities 0.25, and once it enters them, there is no way out. So at infinity probability of being in each of these two classes is

25 Example 0.25 Classes: {},{2,3},{4,5},{6},{7.8} Recurrent states: 6, 7, 8 => all positive Periodic states: 2, 3, 7, 8 (with period 2) Absorbing state(s): 6 Ergodic state(s): If the process is absorbed in {7,8} (which could be considered as an absorbing super state) what will happen after that? It will alternate between 7 and 8 to the end. So at time t probability of being in 7 (or 8) will depend on the parity of t. In general finding the exact behavior of non-ergodic states as t is not easy. (try it!) We will talk about the ergodic cases in next slides

26 Steady State Theorem: For an irreducible ergodic Markov chain lim n P n ij exists and is independent of i. Furthermore, letting π j = lim n P n ij Then π = π, π d t is unique nonnegative solution of π = π P d j= π j If the ergodicity condition is removed, lim n P n ij does not exist in general but the given equations yet have a unique solution π = π, π d t in which π j will be equal to the long run proportion of time that the Markov chain is in state j. = 26

27 Example Consider the weather model example discussed before. We want to see how will the weather be when time goes to infinity: P = α α β π 0 = απ 0 + βπ π = α π 0 + β π π 0 + π = β One of these equations is redundant. (why?) Which yields that π 0 = β and π +β α = α. +β α Exercise: In each of the following cases investigate the existence of solution and its meaning: ) α=0 and β = 2) α= and β = 0 27

28 Acknowledge Thanks to Hassan Hafez and Abbas Hosseini for preparation of slides 28

29 References Ross, Sheldon M. Introduction to probability models. 29