CS 68: BIOINFORMATICS. Prof. Sara Mathieson Swarthmore College Spring 2018

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1 CS 68: BIOINFORMTICS Prof. Sara Mathieson Swarthmore College Spring 2018

2 Outline: pr 4 Lab 5 Examples HMM example in population genetics Recap Viterbi lgorithm Forward-Backward lgorithm Posterior Decoding In lab tomorrow: working in log-space Notes: Office hours TODY 1-3pm Lab 7 due tonight Lab 8: 1.5 week lab (last graded lab)

3 Lab 5 Examples

4 Lab 5: UPGM visualizations Sam & Hunter Tyler & Nathan Neeraj & Tim Ellen & Douglas

5 Lab 5: UPGM visualizations (including branch lengths) Sarah & Tommy

6 Lab 5: UPGM visualizations (including branch lengths) Genji & Eugene Sarah & Tommy

7 Lab 5: UPGM visualizations (including branch lengths) Quinn & Kelly Genji & Eugene Sarah & Tommy

8 Lab 5: dissimilarity map comparisons William

9 HMM example from population genetics Back to recombination.

10 Recombination over time Great-grandfather Great-grandmother Great-grandmother Great-grandfather Great-grandfather Grandmother Grandfather Grandmother Mother Father Child

11 Recombination over time Great-grandfather Great-grandmother Great-grandmother Great-grandfather Great-grandfather Grandmother Grandfather Grandmother Mother Father Child Zoom in on this portion of the genome

12 Recombination over time Great-grandfather Great-grandmother Great-grandmother Great-grandfather Great-grandfather Grandmother Grandfather Grandmother Mother Father Coalesce (find a common ancestor) after 2 generations Child Zoom in on this portion of the genome

13 Recombination over time Great-grandfather Great-grandmother Great-grandmother Great-grandfather Great-grandfather Grandmother Grandfather Grandmother Mother Father Child Now zoom in on *this* portion of the genome

14 Recombination over time Great-grandfather Great-grandmother Great-grandmother Great-grandfather Great-grandfather Grandmother Grandfather Grandmother Mother Father Coalesce after *3* generations Child Now zoom in on *this* portion of the genome

15 How could we encode this as an HMM? Take-home message: the tree changes across the genome! Both topology (for n > 2) and branch lengths

16 Sequence data at many sites T T C G G G

17 Tree changes along the genome! T T C G G G

18 HMM observations: sequence data T T C G G G

19 HMM hidden states: the tree T T C G G G

20 Number of possible trees grows exponentially just look at n=2 One person, two chromosomes! T G G Now the hidden state becomes the *time* of coalescence

21 PSMC: pairwise sequentially Markovian coalescent The distribution of pairwise coalescence times should be exponential with parameter 1 If this differs from the exponential distribution, there were probably population size changes If all coalescence times are very recent, small population size If all coalescence times are very ancient, large population size We can use this to reconstruct the population size change history over time! Image: wikipedia

22 PSMC: an HMM for two sequences The complete genome sequence of a Neanderthal from the ltai Mountains, Prufer et al (2014)

23 Recap Viterbi lgorithm

24 HMM definition K = num hidden states, B = num emitted states Transition probabilities: (K x K matrix) z i-1 k z i l

25 HMM definition K = num hidden states, B = num emitted states Transition probabilities: (K x K matrix) z i-1 k z i l z i Emission probabilities: (K x B matrix) k b x i

26 HMM definition K = num hidden states, B = num emitted states Transition probabilities: (K x K matrix) z i-1 k z i l z i k Emission probabilities: (K x B matrix) b way to deal with the initial state z 0 0 z 1 x i 1) Special start state with no emission 1) x 1

27 HMM definition K = num hidden states, B = num emitted states Transition probabilities: (K x K matrix) z i-1 k z i l z i k Emission probabilities: (K x B matrix) b way to deal with the initial state 1) Special start state with no emission 2) Probability distribution over initial states z 0 0 z 1 z 1 k x i π k = p(z 1 = k) x 1 1) 2) x 1

28 Viterbi lgorithm Input: observed sequence (x 1,x 2,,x L ) and transition/emission probabilities (a and e matrices) Output: most probable (i.e. most likely) hidden state sequence z *

29 Viterbi lgorithm Input: observed sequence (x 1,x 2,,x L ) and transition/emission probabilities (a and e matrices) Output: most probable (i.e. most likely) hidden state sequence z * Initialization: create a K x L matrix, this will be our dynamic programming (DP) table K L

30 Viterbi lgorithm Input: observed sequence (x 1,x 2,,x L ) and transition/emission probabilities (a and e matrices) Output: most probable (i.e. most likely) hidden state sequence z * Initialization: create a K x L matrix, this will be our dynamic programming (DP) table (Note: there are lots of ways to initialize, this avoids a special start state.) K L

31 Viterbi lgorithm Input: observed sequence (x 1,x 2,,x L ) and transition/emission probabilities (a and e matrices) Output: most probable (i.e. most likely) hidden state sequence z * Initialization: create a K x L matrix, this will be our dynamic programming (DP) table (Note: there are lots of ways to initialize, this avoids a special start state.) Recursion: K L

32 Viterbi lgorithm Input: observed sequence (x 1,x 2,,x L ) and transition/emission probabilities (a and e matrices) Output: most probable (i.e. most likely) hidden state sequence z * Initialization: create a K x L matrix, this will be our dynamic programming (DP) table (Note: there are lots of ways to initialize, this avoids a special start state.) Recursion: K keep a back pointer to the max L

33 Viterbi lgorithm Input: observed sequence (x 1,x 2,,x L ) and transition/emission probabilities (a and e matrices) Output: most probable (i.e. most likely) hidden state sequence z * Initialization: create a K x L matrix, this will be our dynamic programming (DP) table (Note: there are lots of ways to initialize, this avoids a special start state.) Recursion: K L Termination and traceback:

34 Viterbi lgorithm Input: observed sequence (x 1,x 2,,x L ) and transition/emission probabilities (a and e matrices) Output: most probable (i.e. most likely) hidden state sequence z * Initialization: create a K x L matrix, this will be our dynamic programming (DP) table (Note: there are lots of ways to initialize, this avoids a special start state.) 0 Recursion: K L Termination and traceback: z * = (1,2,2,0,3,2,1,1,2,0)

35 What is wrong with Viterbi?

36 What is wrong with Viterbi? Only one path! We can t compute the probability of being in state k at step i

37 What is wrong with Viterbi? Only one path! We can t compute the probability of being in state k at step i We don t know if there are many possible paths, all with very similar probabilities

38 What is wrong with Viterbi? Only one path! We can t compute the probability of being in state k at step i We don t know if there are many possible paths, all with very similar probabilities nd a note for later: we may not know the transition and emission probabilities

39 Forward-Backward algorithm

CISC 889 Bioinformatics (Spring 2004) Hidden Markov Models (II)

CISC 889 Bioinformatics (Spring 24) Hidden Markov Models (II) a. Likelihood: forward algorithm b. Decoding: Viterbi algorithm c. Model building: Baum-Welch algorithm Viterbi training Hidden Markov models