Hidden Markov Models

1. Hidden Markov Models 戴玉書 L.R Rabiner, B. H. Juang, An Introduction to Hidden Markov Models Ara V. Nefian and Monson H. Hayeslll, Face detection and recognition using Hidden Markov Models

2. Outline • Markov Chain & Markov Models • Hidden Markov Models • HMM Problem -Evaluation -Decoding -Learning • Application

3. Outline • Markov Chain & Markov Models • Hidden Markov Models • HMM Problem -Evaluation -Decoding -Learning • Application

4. Markov chain property: • Probability of each subsequent state depends only on what was the previous state

5. Markov Models State State State

6. Outline • Markov Chain & Markov Models • Hidden Markov Models • HMM Problem -Evaluation -Decoding -Learning • Application

7. Hidden Markov Models • If you don’t have complete state information, but some observations at each state N - number of states : M - the number of observables: …… q1 q2 q3 q4

8. Hidden Markov Models State:{ , , } Observable:{ , } 0.1 0.3 0.9 0.7 0.8 0.2

9. Hidden Markov Models • M=(A, B, )  = initial probabilities : =(i) , i= P(si)

10. Outline • Markov Chain & Markov Models • Hidden Markov Models • HMM Problem -Evaluation -Decoding -Learning • Application

11. Evaluation • Determine the probability that a particular sequence of symbols O was generated by that model

12. Forward recursion • Initialization: • Forward recursion: • Termination:

13. Backward recursion • Initialization: • Backward recursion: • Termination:

14. Outline • Markov Chain & Markov Models • Hidden Markov Models • HMM Problem -Evaluation -Decoding -Learning • Application

15. Decoding • Given a set of symbols O determine the most likely sequence of hidden states Q that led to the observations • We want to find the state sequence Q which • maximizes P(Q|o1,o2,...,oT)

16. s1 si sN sj qt-1 qt a1j aij aNj Viterbi algorithm General idea: if best path ending in qt= sj goes through qt-1= si then it should coincide with best path ending in qt-1= si

17. Viterbi algorithm • Initialization: • Forward recursion: • Termination:

18. Viterbi algorithm

19. Outline • Markov Chain & Markov Models • Hidden Markov Models • HMM Problem -Evaluation -Decoding -Learning • Application

20. Learning problem • Given a coarse structure of the model, determine HMM parameters M=(A, B, ) that best fit training data determine these parameters

21. Baum-Welch algorithm • Define variable t(i,j) as the probability of being in state si at time t and in state sj at time t+1, given the observation sequence o1, o2, ... ,oT

22. Baum-Welch algorithm • Define variable k(i) as the probability of being in state si at time t, given the observation sequence o1,o2 ,...,oT

23. Outline • Markov Chain & Markov Models • Hidden Markov Models • HMM Problem -Evaluation problem -Decoding problem -Learning problem • Application

24. s1 s2 s3 Example 1 -character recognition • The structure of hidden states: • Observation = number of islands in the vertical slice

25. Example 1 -character recognition {1,3,2,1} • After character image segmentation the following sequence of island numbers in 4 slices was observed :

26. Example 2- face detection & recognition • The structure of hidden states:

27. Example 2- face detection • A set of face images is used in the training of one HMM model N =6 states Image:48, Training:9, Correct detection:90%,Pixels:60X90

28. Example 2- face recognition • Each individual in the database is represent by an HMM face model • A set of images representing different instances of same face are used to train each HMM N =6 states

29. Example 2- face recognition Image:400, Training :Half, Individual:40, Pixels:92X112