Hidden Markov Model in Biological Sequence Analysis – Part 2
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Hidden Markov Model in Biological Sequence Analysis – Part 2 Continued discussion about estimation Using HMM in sequence analysis: Splice site recognition CpG Island Profile HMM Alignment Acknowledgement: In addition to the textbooks, a large portion of the lecture is based on: Computational Methods in Molecular Biology, edited by S. L. Salzberg, D. B.Searls and S. Kasif, pages 45-63. Elsevier, 1998.
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Hidden Markov Model in Biological Sequence Analysis – Part 2. Continued discussion about estimation Using HMM in sequence analysis: Splice site recognition CpG Island Profile HMM Alignment. - PowerPoint PPT Presentation
Transcript of Hidden Markov Model in Biological Sequence Analysis – Part 2
Hidden Markov Model in Biological Sequence Analysis – Part 2
Continued discussion about estimationUsing HMM in sequence analysis:
Splice site recognitionCpG IslandProfile HMMAlignment
Acknowledgement: In addition to the textbooks, a large portion of the lecture is based on: Computational Methods in Molecular Biology, edited by S. L. Salzberg, D. B.Searls and S. Kasif, pages 45-63. Elsevier, 1998.
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initial distributiontransition probabilitiesemission probabilities
HMM
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Initiation
1
2
b1(A2)
b2(A1)
b2(A2)
Hidden Markov Model
Common questions:
How to efficiently calculate emissions:
How to find the most likely hidden state:
How to find the most likely parameters:
)|( OP
)|(maxarg OQPQ
)|(maxarg
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)O , ,O ,(OO t21(t)
Notations:
Let denote the hidden state sequence;
Let denote the observed emission sequence;
Let denote the emission up to time t.
)()2()1( ...... TqqqQ
},......,,{ 21 TOOOO
Reminders
Calculating emissions probabilities:
itt EqOPit )()( ,,
Emissions up to step t chain at state i at step t
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Reminders
Calculating posterior probability of hidden state:
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Emissions after step t chain at state i at step t
Posterior state probabilities
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The Viterbi Algorithm
To find:
So, to maximize the conditional probability, we can simply maximize the joint probability.
Define:
)|(maxarg OQPQ
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)(),(max)|(max
OQPOPOQPOQP
OPOQPOQP
QQQ
(t)ttqqqt OEiqqqqPi
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and ,,,......,,max)( 121,...,, 121
The Viterbi Algorithm
Then our goal becomes:
Compare this to dynamic programming pairwise sequence alignment:
At every step, we aim at the maximumThe maximum depends on the last stepAt the end, we want the maximum and the trace
So, this problem is also solved by dynamic programming.
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ikitkt
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(t)ttqqqt OEiqqqqPi
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and ,,,......,,max)( 121,...,, 121
The Viterbi Algorithm
Time: 1 2 3
States: E1 0.5*0.5 0.15 0.0675
E2 0.5*0.75 0.05625 0.01125
The calculation is from left border to right border, while in the sequence alignment, it is from upper-left corner to the right and lower borders.
Again, why is this efficient calculation possible?The short memory of the Markov Chain!
The Estimation of parameters
When the topology of an HMM is known, how do we find the initial distribution, the transition probabilities and the emission probabilities, from a set of emitted sequences?
Difficulties:
Usually the number of parameters is not small. Even the simple chain in our example has 5 parameters.
Normally the landscape of the likelihood function is bumpy. Finding the global optimum is hard.
)|(argmax
OP
Basic idea
There are three pieces: λ, O, and Q. O is observed.
We can iterate between λ and Q.
When λ is fixed, we can do one of two things:
(1) find the best Q given O
or (2) integrate out Q
When Q is fixed, we can estimate λ using frequencies.
The Viterbi training
(1) Start with some initial parameters
(2) Find the most likely paths using the Viterbi algorithm.
(3) Re-estimate the parameters. On next page.
(4) Iterate until the most likely paths don’t change.
(t)ttqqqt OEiqqqqPi
t
and ,,,......,,max)( 121,...,, 121
)(max),(max iOQP TiQ
The Viterbi training
Estimating the parameters.
Transition probabilities:
Emission probabilities:
Initial probabilities: need multiple training sequences.
The Baum-Welch is a special case of the EM algorithm.The Viterbi training is equivalent to a simplified version of
the EM algorithm, replacing the expectation with the most likely classification.
General flow:
Baum-Welch algorithm
(1) Start with some initial parameters
(2) Given these parameters, P(Q|O) is known for every Q. This is done effectively by calculating the forward and the backward variables.
(3) Take expectations to obtain new parameter estimates.
(4) Iterate (2) and (3) until convergence.
Baum-Welch algorithm
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Reminder:
Estimation of emission probabilities:
Given λ
Given Q
Baum-Welch algorithm
time t t+1
State i
State j
Emission Ot Ot+1
Baum-Welch algorithm
Given λ
Baum-Welch algorithm
Estimation of transition probabilities:
Estimation of initial distribution:
Given Q
Given Q
Baum-Welch algorithmComments:
We don’t actually calculate P(Q|O). Rather, the necessary information pertaining to all the thousands of P(Q|O) is summarized in the forward- and backward- variables.
When the chain runs long, logarithm need to be used for numerical stability.
In applications, the emission is often continuous. A distribution need to be assumed, and the density used for emission probabilities.
The Baum-Welch is a special case of the EM algorithm.
The Viterbi training is equivalent to an simplified version of the EM algorithm, replacing the expectation with the most likely classification.
Common questions in HMM-based sequence analysis
HMM is very versatile in terms of coding sequence properties by model structures.
Given a sequence profile (HMM with parameters known), find the most likely hidden state path from a sequence.
A harder question:
Given a bunch of (aligned) training sequence segments,
- find the most plausible HMM structure-- train the parameters of the HMM.
Nature Biotech. 22(10): 1315
Splice site recognition E: exonI: intron5: 5’splice site
CpG Island
In the four character DNA book, the appearance of “CG” is rare, due to some issues related to the probability of mutations.
However, there are regions where “CG” is more frequent. And such regions are mostly associated with genes. (Remember, only 5% of the DNA sequence are genes. )
……GTATTTCGAAGAAT……
CpG Island
Within the “+” and “-” groups, the transition probabilities are close to shown in the table. A small probability is left for between “+” and “-” transitions.
A+ C+
G+T+
G-
A- C-
T-
CpG Island
Transition probabilities are summarized from known CpG islands.
To identify CpG island from new sequences:
Using the most likely path: run the Viterbi algorithm. When the chain passes through any of the four “+” states, call it an island
Using the posterior decoding: run the forward- and backward- algorithm. Obtain the posterior probabilities of each letter being in a CpG island.
Profile HMM --- a simple case
Motif: functional short sequence that tends to be conserved.
Profile HMM --- a simple case
For (1) accounting for sequence length, (2) computational stability, Log odds ratios are preferred.
Profile HMM --- a simple case
Profile HMM --- the general model
DeletionInsertion
Profile HMM --- the general model
Profile HMM --- the general model
Avoid over-fitting with limited number of observed sequences.Add pseudocounts proportional to the observed frequencies of the amino acids. And add pseudocounts to transitions.
Profile HMM --- searching with profile HMM
Given a profile HMM, how to find if a short sequence is such a motif?
Use the Viterbi algorithm to find the best fit and the log odds ratio:
Multiple sequence alignment(1) From a known family of highly related sequences, find the most conserved segments which have functional and/or structural implications.
Pandit et al. In Silico Biology 4, 0047
Multiple sequence alignment(2) From a known family of sequences that are related only by very small motifs (that contribute to the function), find the real motif from vast amounts of junk.
This is a harder problem that draws statisticians.Science 262, 208-214.
Heuristic methods for global alignment
Scoring: Sum of Pairs (SP score)
(1) Assume columns of the alignment is independent from each other
(2) For each column, the score is the sum of the scores between all possible pairs
(3) For the overall alignment, the score is the sum of the column scores
Heuristic methods for global alignmentMulti-dimensional dynamic programming:
2N-1 choices at each step. Impractical with more than a few sequences.
Heuristic methods for global alignmentProgressive alignment: when exhaustive search is hard, go stepwise.
Heuristic methods for global alignmentMSA: reducing multi-dimensional dynamic programming search space based on fast methods like progressive alignment.
Heuristic methods for global alignmentCLUSTALW: profile alignment rather than purely pairwise
Key improvement: Once an aligned group of sequences is established, position-specific information from the group is used to align a new sequence to it.
Heuristic methods for global alignmentAlignment by profile HMM training.We have studied the scoring system by known HMM. The difficulty here is to train the HMM from unaligned sequences.
Baum-Welch and Viterbi can be trapped by local optima. Stochastic methods like simulated annealing can help.
Heuristic methods for global alignmentAlignment by profile HMM training.
For global alignment, the number of matching states is ~mean(sequence length).
The motif discovery we will talk about also utilizes the HMM in essence.
The difference is in motif models, only a very small number of (5~20) matching states is expected. The vast amount of data is generated from a relatively simple background chain.
Modeling genes with HMM
Genes account for only 5% of the human genome.
Within each gene, there are exons (that code for protein sequence) and introns (that are skipped during transcription).
Characteristics of the exons, introns and the borders between them allow people to identify putative genes from long DNA sequences.
HMM is a model that can incorporate such characteristics.
