Markov Models and Simulations

Yu MengDepartment of Computer Science and Engineering

Southern Methodist University

Outline

• Markov model/process/chain/property/HMM

• Matlab simulations

Markov Process/chain/model/etc.

• Markov Process

• Markov Model

• Markov Chain

• Markov Property

• Hidden Markov Model

Markov Process

• Markov process is a simple stochastic process in which the distribution of future states depends only on the present state and not on how it arrived in the present state.

Simple example-Graphic representation

Markov Property

• Many systems have the property that given present state, the past states have no influence on the future. This property is called Markov property.

• We can say Markov process is a process or simulation that satisfies Markov property.

State Space and Time Space

Time Space State Space

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Discrete Continuous

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Discrete (Markov X

Chain)

Continuous X X

Markov Chain

Let {Xt : t is in T} be a stochastic process with discrete-state space S and discrete-time space T satisfying

 

P(Xn+1 = j|Xn = i, Xn-1 = in-1, · · ·,X0 = i0)

= P(Xn+1 = j|Xn = i) 

for any set of state i0, i1, · · · , in-1, i, j in S and n ≥ 0 is called a Markov Chain.

Markov Model

• Sometimes Markov Model restricts attention to Markov chains with stationary transition probabilities. But some people tend to avoid this usage for sake of confusion.

• Markov Model is also used to refer to all Markov processes that satisfying Markov Property.

Hidden Markov Model(HMM)

• In an Hidden Markov Model(HMM), we don’t know the state sequence that the model passes through, but only some probabilistic function of it.

Elements of Hidden Markov Model

• A set of states {1, 2, ..., M} • An M-by-M transition matrix T whose i, j entry

is the probability of a transition from state i to state j.

• A set of possible outputs, or emissions, {s1, s2, ... , sN}.

• An M-by-N emission matrix E whose i,k entry gives the probability of emitting symbol sk given that the model is in state i.

HMM Example

• A weighted red coin.The probability of heads is .9 and the probability of tails is .1.

• A weighted green coin. The probability of heads is .95 and the probability of tails is .05.

• A red die, having 6 sides, labeled 1 to 6. • A green die, having 12 sides, 5 of which are

labeled 2 through 6, and the remaining 7 sides are labeled 1.

HMM Example

• Begin to toss the red die, write down the number. At each step, you flip the coin that has the same color as the die you rolled in the previous step. If the coin comes up heads, roll the same die as in the previous step. If the coin comes up tails, switch to the other die.

HMM Example

Matlab simulations

Matlab Statistics Toolbox 4.1

(Released in May 2003)• hmmdecode• hmmgenerate • hmmestimate • hmmtrain • hmmviterbi

Matlab simulations

Conclusions

• http://www-2.cs.cmu.edu/~awm/tutorials/