Hidden Markov Models Review and Applications. hidden Markov model. what we see model M = (,Q,T) states Q transition probabilities e Ax

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1 Hidden Markov Models Review and Applications 1 hidden Markov model what we see x y model M = (,Q,T) states Q transition probabilities e Ax t AA e Ay observation observe states indirectly emission probabilities hidden A B t AC C underlying process probability observation given the model Note: there may be many state sequences 2 1

2 weather P( )=0.1 P( )=0.2 P( )=0.7 H M 0.4 P( )=0.3 P( )=0.4 P( )=0.3 p H = 0.4 p M = 0.2 p L = 0.4 L P( )=0.6 P( )=0.3 P( )= Observed weather (R,S,C) vs. pressure (L,M,H). 3 CpG islands ctd. p 8 states A + vs A - unique observation each state 0.180p + A C G T A C G T (1-p)/4 A G C T 1-p 1-q - A C G T A C G T A G C T + denotes CpG island - denotes non-cpg island q 64 transitions! 4 2

3 application: CpG islands bits (log 2 ) LLR A C G T A C G T Next character C or G: positive contribution - Others: negative contribution do we measure CG content? - no answer - a propos 5 HMM main questions observation X * t pq probability of this observation? most probable state sequence? how to find the model? training 6 3

4 Three Important Questions (See also L.R. Rabiner (1989)) How likely is a given sequence? The Forward algorithm What is the most probable path for generating a given sequence? The Viterbi algorithm How can we learn the HMM parameters given a set of sequences? The Forward-Backward (Baum-Welch) algorithm 7 Decoding Problem: Viterbi algorithm (1) dynamic programming: max probability ending in state (2) traceback: most probable state sequence x L A B C 8 4

5 Posterior Decoding Problem Input: Given a Hidden Markov Model M = (Σ, Q, Θ) and a sequence X for which the generating path P is unknown. Question: For each 1 i L (the length of the path P) and k in Q compute the probability: P(π i = k X). 9 Posterior Decoding Problem P(π i = k X) gives two additional decoding possibilities: 1. Alternative path P* that follows the max probability states: argmax k { P(π i = k X) }. 2. Define a function g(k) on the states k in Q, then G(i X) = k { P(π i = k X)g(k) } We can use 2) to calculate the posterior probability os each nucleotide to be in a CpG island, using a function g(.) that equals on for all CpG-island states, and equal to 0 otherwise. 10 5

6 Posterior Decoding Forward Algorithm for given sequence X: f k (i) = P(x 1,,x i, π i = k) is equal to the probability of emitting the prefix (x 1,,x i ) and reaching state π i = k. Backward Algorithm for a given sequence X: b k (i) = P(x i+1,,x L π i = k) is equal to the probability of emitting the suffix (x i+1,,x L ) given you are in state π i = k. 11 posterior decoding i i-th state equals q A % B forward backward 12 6

7 posterior decoding i-th state equals q 0 q forward backward 13 two explanations posterior Σ best state every position path may not be allowed by model viterbi max optimal global path many paths with similar probability 14 7

8 dishonest casino dealer experiment: rolls, die reconstruction: rolls die (viterbi) 15 assignment 3, dishonest casino dealer M=6 N=2 A: B: pi:

9 assignment 3, dishonest casino dealer 1) Rolls Die FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLL Your Viterbi FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLL My Viterbi FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLL 2) Rolls Die LLLLLLFFFFFFFFFFFFLLLLLLLLLLLLLLLLFFFLLLLLLLLLLLLLLFFFFFFFFF Your Viterbi LLLLLLFFFFFFFFFFFFLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLFFFFFFFF My Viterbi FFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL 3) Rolls Die FFFFFFFFLLLLLLLLLLLLLFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL Your Viterbi FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL My Viterbi FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 4) Rolls Die LLLLLLLLFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF Your Viterbi LLLLLLLLLLLLFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF My Viterbi LLLLLLLLLLLLFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 5) Rolls Die FFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLLLLLLLLLFFFFFFFFFFF Your Viterbi FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLLLLLLFFFFFFFFFFF My Viterbi FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLLLLLLFFFFFFFFFFF 17 assignment 3, dishonest casino dealer M= 6 N= 2 A: B: pi:

10 assignment 3, dishonest casino dealer 1) Rolls Die FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLL Your Viterbi FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLL My Viterbi FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLL Second FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLL 2) Rolls Die LLLLLLFFFFFFFFFFFFLLLLLLLLLLLLLLLLFFFLLLLLLLLLLLLLLFFFFFFFFF Your Viterbi LLLLLLFFFFFFFFFFFFLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLFFFFFFFF My Viterbi FFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL Second FFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLFFFFFFFF 3) Rolls Die FFFFFFFFLLLLLLLLLLLLLFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL Your Viterbi FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFL My Viterbi FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF Second FFFFFFFFFLLLLLLLLFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 4) Rolls Die LLLLLLLLFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF Your Viterbi LLLLLLLLLLLLFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF My Viterbi LLLLLLLLLLLLFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF Second LLLLLLLLLLLLFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 5) Rolls Die FFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLLLLLLLLLFFFFFFFFFFF Your Viterbi FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLLLLLLFFFFFFFFFFF My Viterbi FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLLLLLLFFFFFFFFFFF Second FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLLLLLLFFFFFFFFFFF 19 Learning Without Hidden State Learning is simple if we know the correct path for each sequence in our training set estimate parameters by counting the number of times each parameter is used across the training set 20 10

11 Path Known: Parameter estimation 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 Laplace correction 21 Learning With Hidden State If we don t know the correct path for each sequence in our training set, consider all possible paths for the sequence Estimate parameters through a procedure that counts the expected number of times each parameter is used across the training set

12 Learning Parameters: The Baum-Welch Algorithm Also known as the Forward-Backward algorithm An Expectation Maximization (EM) algorithm EM is a family of algorithms for learning probabilistic models in problems that involve hidden states In this context, the hidden state is the path that best explains each training sequence. Note, finding the parameters of the HMM that optimally explains the given sequences is NP-Complete! 23 Learning Parameters: The Baum-Welch Algorithm Algorithm sketch: initialize parameters of model iterate until convergence calculate the expected number of times each transition or emission is used adjust the parameters to maximize the likelihood of these expected values 24 12

13 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 25 Computational Complexity of HMM Algorithms Given an HMM with S states and a sequence of length L, the complexity of the Forward, Backward and Viterbi algorithms is O( S 2 L) This assumes that the states are densely interconnected Given M sequences of length L, the complexity of Baum Welch on each iteration is O( MS 2 L) 26 13

14 Hidden Markov Models II applications 27 model structure A G C T many states & fully connected training seldom works local maxima use knowledge about the problem e.g. linear model for profile alignment: begin M 2 end 28 14

15 silent states skipping nodes [square states] silent states (no emission) quadratic vs. linear size but less modeling possibilities high/low 29 silent states: algorithm forward algorithm transition / emission as before for silent states no silent loops (!): update in topological order 30 15

16 profile alignment begin M 2 end Assume a given profile set: ungapped transition probabilities: 1 trivial alignment HMM to sequence VGAHAGEY VTGNVDEV VEADVAGH VKSNDVAD VYSTVETS FNANIPKH IAGNGAGV profile HMM P dedicated topology Let e i (b) observing symbol b at pos i 31 profile alignment (2) Given profile sequences: VGA--HAGEY VNA--NVDEV VEA--DVAGH VKG--NYDED VYS--TYETS FNA--NIPKH IAGADNGAGV I j M j M j+1 affine model open gap insert state match states emission: - background probabilities: e I (b) = p(b) - or based on alignment extension 32 16

17 profile alignment (3) Given profile Sequences: VGA--HAGEY V----NVDEV VEA--DVAGH VKG------D VYS--TYETS FNA--NIPKH IAGADNGAGV D j-1 I j M j M j+1 D j M j-1 M j M j+1 insert state match states delete state (silent) adapt Viterbi => 33 HMM for profiles / multiple alignment Deletion Insertion D I Viterbi Match begin M end M Y v j ( i) em ( x j i ). max v j 1( i 1). ty j 1 Y M, I, D I Y v j ( i) p( xi ). max v j ( i 1). ty j I Y M, I, D 1 D Y v j ( i) max v j 1( i). ty j D Y M, I, D 1 j j M same level same position j 34 17

18 profile alignment given multiple alignment Insertion / Deletion states VGA--HAGEY V----NVDEV VEA--DVAGH VKG------D VYS--TYETS FNA--NIPKH IAGADNGAGV counting transitions M 1 M / 10 M 1 I / 10 M 1 D / 10 emissions F 1+1 2/ 27 I 1+1 2/ 27 V 5+1 6/ 27 other 17x 0+1 1/ Laplace correction, i.e., adding 1 for each frequency to avoid dividing by 0 Multiple Sequence Alignment Multiple Sequence Alignment Problem: Given sequence S 1,, S n, how can they be optimally aligned? Assume a profile HMM P is known, then: - Align each sequence S i to the profile separately - Accumulate the obtained alignments to a multiple alignment - Hereby insert runs are not aligned. (Just leftjustify insert regions.) 36 18

19 Multiple Sequence Alignment Multiple Sequence Alignment Problem: Given sequence S 1,, S n, how can they be optimally aligned? Assume a profile HMM P is not known, then obtain an HMM profile P from S 1,, S n as follows: - Choose a length L for the profile HMM and intialize the transition and emission probabilities. - Train the HMM using Baum- Welch on all sequences S 1,, S n. Now obtain the multiple alignment using this HMM P as in the previous case: - Align each sequence S i to the profile separately - Accumulate the obtained alignments to a multiple alignment - Hereby insert runs are not aligned. (Just left-justify insert regions.) 37 multiple alignment with profile IAGADNGAGV VGAHAGEY FNAPNI-KH align each sequence separately accumulate alignments M and D positions align inserts leftmost I positions VGA--HAGEY FNAP-NI-KH IAGADNGAGV

20 application: gene finding 39 gene finding central dogma: DNA transcription RNA translation protein only 2%-3% coding find these regions Prokaryotes no nucleus most of genome is coding H.influenza: 70 % coding continuous genes Eukariotes nucleus part of genome is coding Human: 2% - 3% coding introns & exons 40 20

21 ORFs open reading frames 3 (or 6) start stop AUG UAA, UAG, UGA 3/64 stops (random) average protein 1000bp [much longer] search for long ORFs - miss short genes - too many found 41 codon frequencies genes are not random Leu Leucine 6 codons 6.9 Ala Alanine Trp Tryptophan 1 1 random motto coding A or T in 2nd position sometimes 90% Markov models codon triplets as states [64 states] ~ 3 rd order (but no overlap) triplet frequencies only keep 3 frames in sliding window cf. CpG 42 21

22 consensus sequence i.e. not exact n TTGAC n 18 TATAAT n 6 position weight matrix promotor regions N n pos A C G T wmm wam TATA box weight matrix model start coding cf. profile + dependencies between adjacent positions 43 Eukaryotes exons expressed introns noncoding (alternative) splicing exon intron intron donor acceptor tss transcription start site polya polyadenylation utr untranslated region 5 tss and start codon 3 stop codon and polya 44 22

23 introns: splicing consensus sequences / weight matrices intron AGGUAAGU CTGAC NCAGG < 15 bp > pyrimidine rich CT donor site branchpoint acceptor site 45 exon lengths HMM cannot model arbitrary length distributions p 1-p P(len=k) = p k (1-p) 46 23

24 generalized HMM states emit strings of symbols + length distribution parse of observation assigns subsequences to states Viterby like time consuming, hard to train GenScan: models for subsequences in genome transitions biologically consistent statistics depending on C+G content 47 phase offsets GenScan backward strand 48 24

25 however Intron 22 of the FVIII gene is unusual; it is very large (32kb) and contains a CpG island. This CpG island acts as a bidirectional promoter for two genes within the FVIII gene, F8A and F8B. F8A is transcribed in the opposite direction to factor VIII, is intronless and completely nested within intron 22. The functions of F8A and F8B are unknown although the genes are expressed ubiquitously. see (page moved!) 50 Important Papers on HMM L.R. Rabiner, A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition, Proceeding of the IEEE, Vol. 77, No. 22, February Krogh, I. Saira Mian, D. Haussler, A Hidden Markov Model that finds genes in E. coli DNA, Nucleid Acids Research, Vol. 22 (1994), pp Furthermore: R. Hassan, A combination of hidden Markov model and fuzzy model for stock market forecasting, Neurocomputing archive, Vol. 72, Issue 16-18, pp , October

26 References Lecture Craven s website: A. Baxevanis and B. F. F. Ouellette. Bioinformatics: A Practical Guide to the Analysis of Genes and Proteins (3rd ed.). John Wiley & Sons, 2004 R.Durbin, S.Eddy, A.Krogh and G.Mitchison. Biological Sequence Analysis: Probability Models of Proteins and Nucleic Acids. Cambridge University Press, 1998 N. C. Jones and P. A. Pevzner. An Introduction to Bioinformatics Algorithms. MIT Press, 2004 I. Korf, M. Yandell, and J. Bedell. BLAST. O'Reilly, 2003 L. R. Rabiner. A tutorial on hidden markov models and selected applications in speech recognition. Proc. IEEE, 77: , 1989 J. C. Setubal and J. Meidanis. Introduction to Computational Molecular Biology. PWS Pub Co., M. S. Waterman. Introduction to Computational Biology: Maps, Sequences, and Genomes. CRC Press, 1995 Krogh, I. Saira Mian, D. Haussler, A Hidden Markov Model that finds genes in E. coli DNA, Nucleid Acids Research, Vol. 22, pp , CMB2011 Assignment 5 Be sure you have done Assignment 3 before Friday March 18 th If you have any problems please contact me! The Umdhmm Package consist of implementations of the Viterbi, forward-backward and Baum-Welch algorithms in C. Download the code from Uncompress the zip-file and compile it under the operating system of your choice. In the README file it is explained how you can specify a HMM and how you can apply the standard HMM algorithms. Assignment 5: Using HMM for Biomolecular Sequences is available Tuesday March 22 nd 2011 on the website Assignment 5 due: Tuesday April 5 th