Hidden Markov Models. Ron Shamir, CG 08

Transcription

1 Hidden Markov Models 1

2 Dr Richard Durbin is a graduate in mathematics from Cambridge University and one of the founder members of the Sanger Institute. He has also held carried out research at the Laboratory of Molecular Biology in Cambridge and at Harvard and Stanford Universities in the USA. He is currently head of the informatics division in the Sanger Center. Main source: Durbin et al., Biological Sequence Alignment (Cambridge, 98) 2

3 The occasionally dishonest casino A P A (1) = P A (2) = = 1/6 P A->B = P B->A = 1/2 B P B (1)= P B (5)=0.1 P B (6) = Can we tell when the loaded die is used? 3

4 CpG islands: Example - CpG islands DNA stretches (100~1000bp) with frequent CG pairs (contiguous on same strand). Rare, appear in significant genome parts. Problem (1): Given a short genome sequence, decide if it comes from a CpG island. 4

5 Preliminaries: Markov Chains Can avoid p by adding (S, A, p) 0 begin state + S: State set transition probs A 0* p: Initial state prob. vector {p(x 1 =s)} A: Transition prob. matrix a st = P(x i =t x i-1 =s) Assumption: X=x 1 x n is a random process with memory length 1, i.e.: s i S P(x i =s i x 1 =s 1,,x i-1 =s i-1 ) = P(x i =s i x i-1 =s i-1 ) = a s i-1,si Sequence probability: P(X) = p(x 1 ) i=2 L a x i-1,xi

6 Sequence probability - A C G T A C G T P(X) = p(x 1 ) i=2 L a xi-1, xi 7

7 Markov model - Example Markov model, Adding begin and end states B A T E C G 8

8 Andrei Andreyevich Markov Born: 14 June 1856 in Ryazan, Russia Died: 20 July 1922 in Petrograd (now St Petersburg), Russia Seminal contributions to central limit theorem stochastic processes random walks,. 9

9 Markov Models - Transition probs for non-cpg islands + Transition probs for CpG islands - A C G T A C G T A C G T A C G T

10 CpG islands: Fixed Window Problem (1): Given a short genome sequence X, decide if it comes from a CpG island. Solution: Model by a Markov chain. Let a + st : transition prob. in CpG islands, a - st : transition prob. outside CpG islands. Decide by log-likelihood ratio score: score( X ) P( X CpG island ) log P( X non CpG island ) 1 bits _ score( X ) n n xi 1,x log 2 i 1 ax,x a i 1 i i n log a a x i 1 i x,x 1 i 1,x i i 11

11 Discrimination of sequences via Markov Chains 48 CpG islands, tot length ~60K nt. Similar non-cpg. Durbin et. al, Fig

12 CpG islands the general case Problem(2): Detect CpG islands in a long DNA sequence. Naive Solution - Sliding windows: 1 k L-l, window: X k = (x k+1,,x k+l ) score: score(x k ) positive score potential CpG island Disadvantage: what is the length of the islands? How do we identify transitions? Idea: Use Markov chains as before, with additional (hidden) states 13

13 Hidden Markov Model (HMM) Alphabet of symbols Example: {A, C, G, T} path = 1,, n (sequence of states - simple Markov chain) Given sequence X = (x 1,,x L ): a kl = P( i =l i-1 =k), e k (b) = P(x i =b i =k) Finite set of states, capable of emitting symbols. Example: Q = {A +,C +,G +,T +,A -,C -,G -,T - } M=(, Q, ) P(X, ) = a 0, 1 i=1 L e i(x i ) a i, i+1 =(A,E) A: Transition prob. a kl k,l Q E: Emission prob. e k (b) k Q, b Joint prob. of observed sequence X and path (convention: 0 - begin, L+1 - end) Goal: Finding path * maximizing P(X, ) 14

14 Viterbi s Decoding Algorithm (finding most probable state path) Want: path maximizing P(X, ) v k (i) = prob. of most probable path ending in state k at step i. Init: v 0 (0) = 1; v k (0)=0 k>0 Step: v l (i+1)=e l (x i+1 ) max k {v k (i) a kl } End: P(X, * ) = max k {v k (L) a k0 } Time complexity: O(Ln 2 ) for n states, m symbols, L steps Can find * using back pointers. 15

15 The occasionally dishonest casino (2) A B emission probabilities 16

16 The occasionally dishonest casino (2) Ron Shamir, CG 08 17

17 HMM for CpG Islands States: A + C + G + T + A - C - G - T - Symbols: A C G T A C G T Path = 1,, n : sequence of states + A C G T A C G T A C G T A C G transition prob. T

18 HMM for CpG Islands A + T + A - T - C + G + C - G - 19

19 Posterior State Probabilities Goal: calculate P( i =k X) Our strategy: P(X, i =k) = = P(x 1,,x i, i =k) P(x i+1,,x L x 1,,x i, i =k) = P(x 1,,x i, i =k) P(x i+1,,x L i =k) P( i =k X) = P( i =k, X) / P(X) Need to compute these two terms - and P(X) 20

20 Forward Algorithm Goal: calculate P(X) = P(X, ) Approximation: take max path * from Viterbi alg. Not justified when several near maximal paths Exact alg : (a.k.a. Forward Algorithm ) f k (i) = P(x 0,,x i, i =k) Init: f 0 (0) = 1; f k (0)=0 k>0 Step: f j (i+1) = e j (x i+1 ) k f k (i) a kj End: P(X) = k f k (L) a k0 21

21 Backward Algorithm b k (i) = P(x i+1, x L i =k) init: k, b k (L) = a k0 step: b k (i) = l a kl e l (x i+1 ) b l (i+1) End: P(X) = k a 0k e k (x 1 ) b k (1) 22

22 Posterior State Probabilities (2) Goal: calculate P( i =k X) Recall: f k (i) = P(x 0,,x i, i =k) b k (i) = P(x i+1, x L i =k) Each can be used to compute P(X) P(X, i =k) = = P(x 1,,x i, i =k) P(x i+1,,x L x 1,,x i, i =k) = P(x 1,,x i, i =k) P(x i+1,,x L i =k) = f k (i) b k (i) P( i =k X) = P( i =k, X) / P(X) 23

23 Dishonest Casino (3) Durbin et al. pp

24 Posterior Decoding Now we have P( i =k X). How do we decode? 1. i* =argmax k P( i =k X) Good when interested in state at particular point path of states 1*,.., L * may not be legal 2. Define a function of interest g(i) on the states. Compute G(i X) = k P( i =k X) g(k) E.g.: g(i) =1 for states in S, 0 on the rest: G(i X) is posterior prob of symbol i coming from S e.g., CpG island 25 S={A +,C +,G +,T + }

25 Andrew Viterbi Dr. Andrew J. Viterbi is a pioneer in the field of Wireless Communications. He received his Bachelors and Masters degrees from MIT, and his Ph.D. in digital communications from the University of Southern California (USC). He taught at UCLA and consulted for the Jet Propulsion Laboratory (JPL) Immediately after obtaining his Ph.D. He was a co-founder of Linkabit in 1968, a small military contractor, and co-founded QualComm with Irwin Jacobs in He created the Viterbi Algorithm for interference suppression and efficient decoding of a digital transmission sequence, used by all four international standards for digital cellular telephony. QualComm is the recognized pioneer of the Code Division Multiple Access (CDMA) digital wireless technology, which allows many users to share the same radio frequencies, and thereby increase system capacity many times over analog system capacity. He is a Life Fellow of the IEEE, and was inducted as a member of the National Academy of Engineering in 1978 and of the National Academy of Sciences in _center/comsoc/viterbi.html