1. Slide 1 EE3J2 Data Mining EE3J2 Data Mining Lecture 14: Introduction to Hidden Markov Models Martin Russell

2. Slide 2 EE3J2 Data Mining Objectives  Limitations of sequence matching  Introduction to hidden Markov models (HMMs)

3. Slide 3 EE3J2 Data Mining Sequence retrieval using DP …… AAGDTDTDTDD AABBCBDAAAAAAA BABABABBCCDF GGGGDDGDGDGDGDTDTD DGDGDGDGD AABCDTAABCDTAABCDTAAB CDCDCDTGGG GGAACDTGGGGGAAA ……. Corpus of sequential data ‘query’ sequence Q …BBCCDDDGDGDGDCDTCDTTDCCC… Dynamic Programming Distance Calculation Calculate ad(S,Q) for each sequence S in corpus

4. Slide 4 EE3J2 Data Mining Limitations of ‘template matching’  This type of analysis is sometimes referred to as template matching  The ‘templates’ are the sequences in the corpus  Can think of each template as representing a ‘class’  Problem is to determine which class best fits the query  Performance will depend on precisely which template is used to represent the class

5. Slide 5 EE3J2 Data Mining Alternative path shapes  The basic units of path considered so far are: substitutioninsertiondeletion  Others are possible and may have advantages, e.g: substitutioninsertiondeletion

6. Slide 6 EE3J2 Data Mining Example

7. Slide 7 EE3J2 Data Mining Hidden Markov Models (HMMs)  One solution is to replace the individual template sequence with an ‘average’ sequence  But what is an ‘average sequence’?  One solution is to use a type of statistical model called a Hidden Markov Model

8. Slide 8 EE3J2 Data Mining HMMs  Suppose the following sequences are in same class: –ABC, YBBC, ABXC, AZ  Compute alignments: Y B B C A B C A B X C A B C A Z A B C

9. Slide 9 EE3J2 Data Mining Finite State Network Representation  The sequence consists of 3 ‘states’ –First state is ‘realised’ as A (twice) or Y (once) –Second state ‘realised’ as B (three times) or X (once) –Second state can be repeated or deleted –Third state can be ‘realised’ as C (twice) or Z (once)

10. Slide 10 EE3J2 Data Mining Network representation  Directed graph representation  Each state associated with a set of probabilities –Called the ‘state emission’ probabilities

11. Slide 11 EE3J2 Data Mining Transition probabilities  Transition probabilities control insertions and deletions of symbols 1 1 0.67 0.33 0.5 a jk =Prob(state k follows state j) Basic rule for drawing transition networks: Connect state j to state k if a jk > 0

12. Slide 12 EE3J2 Data Mining Formal Definition  A Hidden Markov Model (HMM) for the symbols  1,  2, …,  K consists of: –A number of states N –An N  N state transition probability matrix A –For each state k a set of probabilities p k (  1 ), …, p(  K ) - p(  k ) is the probability that  k occurs for state k

13. Slide 13 EE3J2 Data Mining Alignment paths for HMMs  For HMMs, alignment paths are called state sequences Y A B B B X B C A B C

14. Slide 14 EE3J2 Data Mining State-symbol trellis Y A B B B X B C A B C Rule: connect state j at symbol m with state k at symbol m+1 if a jk > 0

15. Slide 15 EE3J2 Data Mining More examples

16. Slide 16 EE3J2 Data Mining Dynamic Programming Y A B B B X B C A B C

17. Slide 17 EE3J2 Data Mining Formal Definition  A Hidden Markov Model (HMM) for the symbols  1,  2, …,  K consists of: –A number of states N –An N  N state transition probability matrix A –For each state k a set of probabilities p k (  1 ), …, p(  K ) - p(  k ) is the probability that  k occurs for state k

18. Slide 18 EE3J2 Data Mining Alignment paths for HMMs  For HMMs, alignment paths are called state sequences Y A B B B X B C A B C State sequence 

19. Slide 19 EE3J2 Data Mining The optimal state sequence  Let M be a HMM and s a sequence  Probability on previous slide depends on the state sequence  and the model, so we write:  By analogy with dynamic programming, the optimal state sequence is the sequence such that:

20. Slide 20 EE3J2 Data Mining Computing the optimal state sequence: The ‘state-symbol’ trellis Y A B B B X B C A B C Rule: connect state j at symbol m with state k at symbol m+1 if a jk > 0

21. Slide 21 EE3J2 Data Mining More examples

22. Slide 22 EE3J2 Data Mining Dynamic Programming a.k.a Viterbi Decoding Y A B B B X B C A B C k 4 K

23. Slide 23 EE3J2 Data Mining Sequence retrieval using HMMs Corpus of pre- build HMMs ‘query’ sequence Q …BBCCDDDGDGDGDCDTCDTTDCCC… Viterbi Decoding Calculate p(Q|M) for each HMM M in corpus

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28. Slide 28 EE3J2 Data Mining HMM Construction  Suppose we have a set of HMMs, each representing a different class (e.g. protein sequence)  Given an unknown sequence s: –Use Viterbi decoding to compare s with each HMM –Compute  But how do we obtain the HMM in the first place?

29. Slide 29 EE3J2 Data Mining HMM training  Given a set of example sequences S a HMM M can be built such that p(S|M) is locally maximised  Procedure is as follows: –Obtain an initial estimate of a suitable model M 0 –Apply an algorithm – the ‘Baum-Welch’ algorithm – to obtain a new model M 1 such that p(S|M 1 ) ≥ p(S|M 0 ) –Repeat to produce a sequence of HMMs M 0, M 1,…,M n with: p(S|M 0 ) ≤ p(S|M 1 ) ≤ p(S|M 2 ) ≤… ≤ p(S|M n )

30. Slide 30 EE3J2 Data Mining Local optimality M 0 M 1 …M n P(S|M) Local maximum Global maximum

31. Slide 31 EE3J2 Data Mining Summary  Hidden Markov Models  Importance of HMMs for sequence matching  Viterbi decoding  HMM training

32. Slide 32 EE3J2 Data Mining Summary  Review of template matching  Hidden Markov Models  Dynamic programming for HMMs