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Hidden Markov Models. Slides by Bill Majoros, based on Methods for Computational Gene Prediction. What is an HMM?. An HMM is a stochastic machine M =( Q , , P t , P e ) consisting of the following: a finite set of states , Q ={ q 0 , q 1 , ... , q m }

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## Hidden Markov Models

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**Hidden Markov Models**Slides by Bill Majoros, based on Methods for Computational Gene Prediction**What is an HMM?**• An HMM is astochastic machineM=(Q, , Pt, Pe) consisting of the following: • a finite set of states, Q={q0, q1, ... , qm} • a finite alphabet ={s0, s1, ... , sn} • a transition distribution Pt :Q×Q a¡ i.e.,Pt (qj |qi) • an emission distribution Pe:Q×a¡ i.e.,Pe (sj|qi) An Example 5% M1=({q0,q1,q2},{Y,R},Pt,Pe) Pt={(q0,q1,1), (q1,q1,0.8), (q1,q2,0.15), (q1,q0,0.05), (q2,q2,0.7), (q2,q1,0.3)} Pe={(q1,Y,1), (q1,R,0), (q2,Y,0), (q2,R,1)} 15% Y=0% R= 100% q2 R=0% Y= 100% q1 q0 80% 30% 70% 100%**5%**15% Y=0% R= 100% q2 R=0% Y= 100% q1 q0 80% 30% 70% 100% Probability of a Sequence P(YRYRY|M1) = a0→1b1,Ya1→2b2,Ra2→1b1,Ya1→2b2,Ra2→1b1,Ya1→0 =1 1 0.15 1 0.3 1 0.15 1 0.3 1 0.05 =0.00010125**Another Example**M2= (Q, , Pt, Pe) Q = {q0, q1, q2, q3, q4} ={A, C, G, T} q2 65% A=35% T=25% C=15% G=25% q4 q1 35% 50% A=27% T=14% C=22% G=37% A=10% T=30% C=40% G=20% 100% q0 A=11% T=17% C=43% G=29% 50% 20% 100% 80% q3**Finding the Most Probable Path**Finding the Most Probable Path q2 65% Example: CATTAATAG A=35% T=25% C=15% G=25% q4 q1 50% 35% A=27% T=14% C=22% G=37% A=10% T=30% C=40% G=20% top: 7.0×10-7 bottom: 2.8×10-9 100% q0 A=11% T=17% C=43% G=29% 20% 50% 100% 80% q3 The most probable path is: States:122222224 Sequence: CATTAATAG resulting in this parse: States: 122222224 Sequence:CATTAATAG feature 1: C feature 2: ATTAATA feature 3: G**The Viterbi Algorithm**. . . . . . sequence k k+1 k-2 k-1 states (i,k) . . .**Viterbi: Traceback**T( T( T( ... T( T(i, L-1), L-2) ..., 2), 1), 0) = 0**The Forward Algorithm : Probability of a Sequence**the single most probable path Viterbi: sum over all paths Forward: i.e., . . . . . . sequence k k+1 k-2 k-1 states (i,k) . . .**Training an HMM from Labeled Sequences**transitions emissions From the textbook, Ch. 6.3**Recall: Eukaryotic Gene Structure**complete mRNA coding segment ATG TGA exon intron exon intron exon . . . . . . . . . AG GT AG GT TGA ATG start codon donor site acceptor site donor site acceptor site stop codon**Using an HMM for Gene Prediction**Intron Donor Acceptor Exon the Markov model: Start codon Stop codon Intergenic q0 the input sequence: AGCTAGCAGTATGTCATGGCATGTTCGGAGGTAGTACGTAGAGGTAGCTAGTATAGGTCGATAGTACGCGA the most probable path: the gene prediction: exon 1 exon 2 exon 3**Higher Order Markovian Eukaryotic Recognizer(HOMER)**H17 H5 H3 H95 H27 H77**HOMER, version H3**Intron I=intron state E=exon state N=intergenic state Donor Acceptor Exon Start codon Stop codon Intergenic tested on 500 Arabidopsis genes: q0**Recall: Sensitivity and Precision**NOTE: “specificity” as defined here and throughout these slides (and the text) is really precision**HOMER, version H5**three exon states, for the three codon positions**HOMER version H17**Intron Donor Acceptor Exon Start codon Stop codon Intergenic q0 donor site acceptor site stop codon start codon**Maintaining Phase Across an Intron**01201201 2012012012 phase: + GTATGCGATAGTCAAGAGTGATCGCTAGACC sequence: | | | | | | | coordinates: 0 5 10 15 20 25 30**HOMER version H27**three separate intron models**Recall: Weight Matrices**(stop codons) T G A (start codons) T A A A T G T A G (acceptor splice sites) (donor splice sites) A G G T**HOMER version H77**positional biases near splice sites**Higher-order Markov Models**P(G) A C G C T A 0th order: P(G|C) A C G C T A 1st order: P(G|AC) A C G C T A 2nd order:**Higher-order Markov Models**0 1 2 3 4 5**Summary**• An HMM is a stochastic generative model which emits sequences • Parsing with an HMM can be accomplished using a decoding algorithm (such as Viterbi) to find the most probable (MAP) path generating the input sequence • Training of unambiguous HMM’s can be accomplished using labeled sequence training • Training of ambiguous HMM’s can be accomplished using Viterbi training or the Baum-Welch algorithm • Posterior decoding can be used to estimate the probability that a given symbol or substring was generate by a particular state (next lesson...)

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