Hidden Markov Model

1. Hidden Markov Model • Observation : O1,O2, . . . • States in time : q1, q2, . . . • All states : s1, s2, . . ., sN Sj Si

2. Hidden Markov Model (Cont’d) • Discrete Markov Model Degree 1 Markov Model

3. Hidden Markov Model (Cont’d) : Transition Probability from Si to Sj ,

4. Discrete Markov Model Example S1 : The weather is rainy S2 : The weather is cloudy S3 : The weather is sunny cloudy sunny rainy rainy cloudy sunny

5. Hidden Markov Model Example (Cont’d) Question 1:How much is this probability: Sunny-Sunny-Sunny-Rainy-Rainy-Sunny-Cloudy-Cloudy

6. Hidden Markov Model Example (Cont’d) The probability of being in state i in time t=1 Question 2:The probability of staying in state Si for d days if we are in state Si? d Days

7. Discrete Density HMM Components • N : Number Of States • M : Number Of Outputs • A (NxN) : State Transition Probability Matrix • B (NxM): Output Occurrence Probability in each state • (1xN): Initial State Probability : Set of HMM Parameters

8. Three Basic HMM Problems • Recognition Problem: Given an HMM and a sequence of observations O,what is the probability ? • State Decoding Problem: Given a model and a sequence of observations O, what is the most likely state sequence in the model that produced the observations? • Training Problem: Given a model and a sequence of observations O, how should we adjust model parameters in order to maximize ?

9. First Problem Solution We Know That: And

10. First Problem Solution (Cont’d) Computation Order :

11. Forward Backward Approach Computing 1) Initialization

12. Forward Backward Approach (Cont’d) 2) Induction : 3) Termination : Computation Order :

13. Backward Variable 1) Initialization 2)Induction

14. Second Problem Solution • Finding the most likely state sequence Individually most likely state :

15. Viterbi Algorithm • Define : P is the most likely state sequence with this conditions : state i , time t and observation o

16. Viterbi Algorithm (Cont’d) 1) Initialization Is the most likely state before state i at time t-1

17. Viterbi Algorithm (Cont’d) 2) Recursion

18. Viterbi Algorithm (Cont’d) 3) Termination: 4)Backtracking:

19. Third Problem Solution • Parameters Estimation using Baum-Welch Or Expectation Maximization (EM) Approach Define:

20. Third Problem Solution (Cont’d) : Expected value of the number of jumps from state i : Expected value of the number of jumps from state i to state j

21. Third Problem Solution (Cont’d)

22. Baum Auxiliary Function By this approach we will reach to a local optimum

23. Restrictions Of Reestimation Formulas

24. Continuous Observation Density • We have amounts of a PDF instead of • We have Mixture Coefficients Variance Average

25. Continuous Observation Density • Mixture in HMM M1|1 M1|2 M1|3 M2|1 M2|2 M2|3 M3|1 M3|2 M3|3 M4|1 M4|2 M4|3 S2 S3 S1 Dominant Mixture:

26. Continuous Observation Density (Cont’d) • Model Parameters: N×M×K×K 1×N N×M N×M×K N×N N : Number Of States M : Number Of Mixtures In Each State K : Dimension Of Observation Vector

27. Continuous Observation Density (Cont’d)

28. Continuous Observation Density (Cont’d) Probability of event j’th state and k’th mixture at time t

29. State Duration Modeling Sj Si Probability of staying d times in state i :

30. State Duration Modeling (Cont’d) HMM With clear duration ……. ……. Sj Si

31. State Duration Modeling (Cont’d) • HMM consideration with State Duration : • Selecting using ‘s • Selecting using • Selecting Observation Sequence using in practice we assume the following independence: • Selecting next state using transition probabilities . We also have an additional constraint:

32. Training In HMM • Maximum Likelihood (ML) • Maximum Mutual Information (MMI) • Minimum Discrimination Information (MDI)

33. Training In HMM • Maximum Likelihood (ML) . . . Observation Sequence

34. Training In HMM (Cont’d) • Maximum Mutual Information (MMI) Mutual Information

35. Training In HMM (Cont’d) • Minimum Discrimination Information (MDI) Observation : Auto correlation :