Hidden Markov Models. based on chapters from the book Durbin, Eddy, Krogh and Mitchison Biological Sequence Analysis via Shamir s lecture notes

Transcription

1 Hidden Markov Models based on chapters from the book Durbin, Eddy, Krogh and Mitchison Biological Sequence Analysis via Shamir s lecture notes

2 music recognition deal with variations in - actual sound - timing

3 deal with variations in - actual sound actual base - timing insertions/deletions application: gene finding

4 basic questions start with sequence of observations probabilistic model of our domain does this sequence belong to a certain family? Markov chains can we say something about the internal structure? (indirect observation) HMM: Hidden Markov Models

5 introduction discrete time discrete space 0.3 A B no state history: present state only states, transitions 1 C 0.8 P(X) P(X,Y) P(X Y) X and Y X, given Y

6 Markov model t AA A B model M = (Q,P,T) states Q initial probabilities p x transition probabilities t xy matrix / graph t AC first order: no history C observation X sequence of states probability ( observation given the model )

7 example 0.7 A B C 0.6 Q = { A, B, C } P = ( 1, 0, 0 ) T = unique starting state A P( AABBCCC M ) = A 0.4 B 0.6 C A A B B C C C = vs =

8 Markov model t AA A B initial state x 0 fixed ~ initial probabilities final state [not in this picture] t AC t 0A 0 C t 0C t 0B small values: underflow

9 comparing models M1 best explained by which model? P(X M1) vs. P(X M2) P(M1 X) vs. P(M2 X)!! M2 Bayes: P(A B) = P(B A) P(A) / P(B) P(M1 X) P(X M1) P(M1) = P(M2 X) P(X M2) P(M2)

10 bases are not random motto

11 observed frequencies A C island non island application: CpG islands + A C G T A C G T A C G T A C G T G T consecutive CG pair CG TG mostly rare, although islands occur signal (e.g.) promotor regions

12 basic questions observation: DNA sequence model: CpG islands / non-islands does this sequence belong to a certain family? Markov chains is this a CpG island (or not)? can we say something about the internal structure? HMM: Hidden Markov Models where are the CpG islands?

13 application: CpG islands + A C G T A C G T island - A C G T A C G T non island score X = ACGT = 2.82

14 application: CpG islands bits (log 2 ) LLR A C G T A C G T log-score X = ACGT log = =

15 CpG Log-Likelihood Ratio LLR A C G T A C G T LLR(ACGT) = = 1.50 ( 0.37 per base ) is a (short) sequence a CpG island? compare with observed data (normalized for length) where (in long sequence) are CpG islands? first approach: sliding window length of window?

16 empirical data is a (short) sequence a CpG island? compare with observed data (normalized for length)

17 where (in long sequence) are CpG islands? first approach: sliding window CpGplot ACCGATACGATGAGAATGAGCAATGTAGTGAATCGTTTCAGCTACT CTCTATCGTAGCATTACTATGCAGTCAGTGATGCGCGCTAGCCGCG TAGCTCGCGGTCGCATCGCTGGCCGTAGCTGCGTACGATCTGCTGT ACGCTGATCGGAGCGCTGCATCTCAACTGACTCATACTCATATGTC TACATCATCATCATTCATGTCAGTCTAGCATACTATTATCGACGAC TGATCGATCTGACTGCTAGTAGACGTACCGAGCCAGGCATACGACA TCAGTCGACT

18 CpGplot observed vs. expected percentage putative islands Islands of unusual CG composition EMBOSS_001 from 1 to 286 Observed/Expected ratio > 0.60 Percent C + Percent G > Length > 50 Length 114 ( )

19 hidden Markov model where (in long sequence) are CpG islands? second approach: hidden Markov model

20 What is a hidden Markov model? Sean R Eddy Nature Biotechnology 22, (2004) Eddy (2004)

21 weather emission probabilities P( )=0.1 P( )=0.2 P( )= H M P( )=0.3 P( )=0.4 P( )=0.3 p H = 0.4 p M = 0.2 p L = 0.4 initial probabilities transition probabilities L P( )=0.6 P( )=0.3 P( )= observed weather vs. pressure

22 weather (0.1, 0.2, 0.7) p H = 0.4 p M = 0.2 p L = H M L (0.3, 0.4, 0.3) (,, ) ( R, C, S ) 0.4 (0.6, 0.3, 0.1) P( RCCSS HHHHH ) = = 196 (x10-5 ) P( RCCSS MMMMM ) = = 432 (x10-5 ) P( RCCSS, HHHHH ) = = 1016 (x10-7 ) P( RCCSS, MMMMM ) = = 14 (x10-7 )

23 what we see x y hidden Markov model model M = (Σ,Q,T) states Q transition probabilities t AA e Ax e Ay observation observe states indirectly emission probabilities hidden A t AC C underlying process B probability observation given the model? there may be many state seq s

24 HMM main questions observation X Σ* t pq probability of this observation? most probable state sequence? how to find the model? training

25 most probable state vs. optimal path probability! probability of state * most probable state (over all state sequences) posterior decoding forward & backward probabilities * most probable path (= single state sequence) Viterbi

26 probability of observation dynamic programming: probability ending in state x i % A % B % C

27 probability ending in state probability of observation forward probability

28 weather (0.1, 0.2, 0.7) p H = 0.4 p M = 0.2 p L = H M L (0.3, 0.4, 0.3) (,, ) ( R, C, S ) (0.6, 0.3, 0.1) 0.4 P( RCCSS ) = P( RC ) 1:R 2:C H 0 41 = 4 ( )2 = 144 (x10-4 ) M 0 23 = 6 ( )4 = 576 (x10-4 ) L 0 46 = 24 ( )3 = 372 (x10-4 ) 0 1

29 posterior decoding i i-th state equals q A % B forward backward

30 HMM main questions observation X Σ* most probable state sequence t pq again: cannot try all possibilities Viterbi probability of this observation? most probable state sequence? how to find the model? training

31 Viterbi algorithm most probable state sequence for observation (1) dynamic programming: probability ending in state x i v p (i) % A % B % C

32 Viterbi algorithm (2) traceback: most probable state sequence start with final maximum x L A B C

33 CpG islands ctd estimates 8 states A + vs A - unique observation each state + A C G T A C G T A C G T A C G T

34 dishonest casino dealer

35 dishonest casino dealer

36 dishonest casino dealer Observation Viterbi LLLLLLLLLLLLFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF Forward FFLLLLLLLLLLLLFFFFFFFFLFLLLFLLFFFFFFFFFFFFFFFFFFFFLFFFFFFFFF Posterior (total) LLLLLLLLLLLLFFFFFFFFFLLLLLLFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF

37 training sequences X (i) optimize score state sequences known count transitions pq count emissions b in p divide by total transitions in p emissions in q Parameter estimation Laplace correction

38 Baum-Welch state sequences unknown Baum-Welch training based on model expected number of transitions, emissions build new (better) model & iterate sum over all training sequences X sum over all positions i sum over all training sequences X sum over all positions i with x i =b

39 concerns: guaranteed to converge target score, not Θ unstable solutions! local maximum tips: repeat for several initial Θ start with meaningful Θ Viterbi training (an alternative) determine optimal paths recompute as if paths known score may decrease! Baum-Welch training