1. Introduction to Profile Hidden Markov Models
Mark Stamp
PHMM

2. Hidden Markov Models Here, we assume you know about HMMs
If not, see “A revealing introduction to hidden Markov models”
Executive summary of HMMs
HMM is a machine learning technique
Also, a discrete hill climb technique
Train model based on observation sequence
Score given sequence to see how closely it matches the model
Efficient algorithms, many useful applications
PHMM

3. HMM Notation Recall, HMM model denoted λ = (A,B,π)
Observation sequence is O
Notation:
PHMM

4. Hidden Markov Models Among the many uses for HMMs… Speech analysis
Music search engine
Malware detection
Intrusion detection systems (IDS)
Many more, and more all the time
PHMM

5. Limitations of HMMs Positional information not considered
HMM has no “memory”
Higher order models have some memory
But no explicit use of positional information
Does not handle insertions or deletions
These limitations are serious problems in some applications
In bioinformatics string comparison, sequence alignment is critical
Also, insertions and deletions occur
PHMM

6. Profile HMM
Profile HMM (PHMM) designed to overcome limitations on previous slide
In some ways, PHMM easier than HMM
In some ways, PHMM more complex
The basic idea of PHMM
Define multiple B matrices
Almost like having an HMM for each position in sequence
PHMM

7. PHMM In bioinformatics, begin by aligning multiple related sequences
Multiple sequence alignment (MSA)
This is like training phase for HMM
Generate PHMM based on given MSA
Easy, once MSA is known
Hard part is generating MSA
Then can score sequences using PHMM
Use forward algorithm, like HMM
PHMM

8. Generic View of PHMM Circles are Delete states
Diamonds are Insert states
Rectangles are Match states
Match states correspond to HMM states
Arrows are possible transitions
Each transition has associated probability
Transition probabilities are A matrix
Emission probabilities are B matrices
In PHMM, observations are emissions
Match and insert states have emissions
PHMM

9. Generic View of PHMM
Circles are Delete states, diamonds are Insert states, rectangles are Match states
Also, begin and end states
PHMM

10. PHMM Notation
Notation
PHMM

11. PHMM Match state probabilities easily determined from MSA, that is
aMi,Mi+1 transitions between match states
eMi(k) emission probability at match state
Note: other transition probabilities
For example, aMi,Ii and aMi,Di+1
Emissions at all match & insert states
Remember, emission == observation
PHMM

12. MSA First we show MSA construction Then construct PHMM from MSA
This is the difficult part
Lots of ways to do this
“Best” way depends on specific problem
Then construct PHMM from MSA
The easy part
Standard algorithm for this
How to score a sequence?
Forward algorithm, similar to HMM
PHMM

13. MSA How to construct MSA? Allow gaps to be inserted
Construct pairwise alignments
Combine pairwise alignments to obtain MSA
Allow gaps to be inserted
Makes better matches
But gaps tend to weaken scoring
So there is a tradeoff
PHMM

14. Global vs Local Alignment
In these pairwise alignment examples
“-” is gap
“|” are aligned
“*” omitted beginning and ending symbols
PHMM

15. Global vs Local Alignment
Global alignment is lossless
But gaps tend to proliferate
And gaps increase when we do MSA
More gaps implies more sequences match
So, result is less useful for scoring
We usually only consider local alignment
That is, omit ends for better alignment
For simplicity, we assume global alignment here
PHMM

16. Pairwise Alignment We allow gaps when aligning
How to score an alignment?
Based on n x n substitution matrix S
Where n is number of symbols
What algorithm(s) to align sequences?
Usually, dynamic programming
Sometimes, HMM is used
Other?
Local alignment --- more issues
PHMM

17. Pairwise Alignment Example Note gaps vs misaligned elements
Depends on S and gap penalty
PHMM

18. Substitution Matrix Masquerade detection
Detect imposter using an account
Consider 4 different operations
E == send
G == play games
C == C programming
J == Java programming
How similar are these to each other?
PHMM

19. Substitution Matrix Consider 4 different operations:
E, G, C, J
Possible substitution matrix:
Diagonal --- matches
High positive scores
Which others most similar?
J and C, so substituting C for J is a high score
Game playing/programming, very different
So substituting G for C is a negative score
PHMM

20. Substitution Matrix
Depending on problem, might be easy or very difficult to get useful S matrix
Consider masquerade detection based on UNIX commands
Sometimes difficult to say how “close” 2 commands are
Suppose aligning DNA sequences
Biological rationale for closeness of symbols
PHMM

21. Gap Penalty Generally must allow gaps to be inserted
But gaps make alignment more generic
So, less useful for scoring
Therefore, we penalize gaps
How to penalize gaps?
Linear gap penalty function
f(g) = dg (i.e., constant penalty per gap)
Affine gap penalty function
f(g) = a + e(g – 1)
Gap opening penalty a, then constant factor of e
PHMM

22. Pairwise Alignment Algorithm
We use dynamic programming
Based on S matrix, gap penalty function
Notation:
PHMM

23. Pairwise Alignment DP
Initialization:
Recursion:
PHMM

24. MSA from Pairwise Alignments
Given pairwise alignments…
…how to construct MSA?
Generic approach is “progressive alignment”
Select one pairwise alignment
Select another and combine with first
Continue to add more until all are combined
Relatively easy (good)
Gaps may proliferate, unstable (bad)
PHMM

25. MSA from Pairwise Alignments
Lots of ways to improve on generic progressive alignment
Here, we mention one such approach
Not necessarily “best” or most popular
Feng-Dolittle progressive alignment
Compute scores for all pairs of n sequences
Select n-1 alignments that a) “connect” all sequences and b) maximize pairwise scores
Then generate a minimum spanning tree
For MSA, add sequences in the order that they appear in the spanning tree
PHMM

26. MSA Construction Create pairwise alignments
Generate substitution matrix
Dynamic program for pairwise alignments
Use pairwise alignments to make MSA
Use pairwise alignments to construct spanning tree (e.g., Prim’s Algorithm)
Add sequences to MSA in spanning tree order (from highest score, insert gaps as needed)
Note: gap penalty is used
PHMM

27. MSA Example
Suppose 10 sequences, with the following pairwise alignment scores:
PHMM

28. MSA Example: Spanning Tree
Spanning tree based on scores
So process pairs in following order: (5,4), (5,8), (8,3), (3,2), (2,7), (2,1), (1,6), (6,10), (10,9)
PHMM

29. MSA Snapshot Intermediate step and final Note increase in gaps
Use “+” for neutral symbol
Then “-” for gaps in MSA
Note increase in gaps
PHMM

30. PHMM from MSA
For PHMM, must determine match and insert states & probabilities from MSA
“Conservative” columns are match states
Half or less of symbols are gaps
Other columns are insert states
Majority of symbols are gaps
Delete states are a separate issue
PHMM

31. PHMM States from MSA Consider a simpler MSA…
Columns 1,2,6 are match states 1,2,3, respectively
Since less than half gaps
Columns 3,4,5 are combined to form insert state 2
Since more than half gaps
Match states between insert
PHMM

32. PHMM Probabilities from MSA
Emission probabilities
Based on symbol distribution in match and insert states
State transition probs
Based on transitions in the MSA
PHMM

33. PHMM Probabilities from MSA
Emission probabilities:
But 0 probabilities are bad
Model “overfits” the data
So, use “add one” rule
Add one to each numerator, add total to denominators
PHMM

34. PHMM Probabilities from MSA
More emission probabilities:
But 0 probabilities are bad
Model “overfits” the data
Again, use “add one” rule
Add one to each numerator, add total to denominators
PHMM

35. PHMM Probabilities from MSA
Transition probabilities:
We look at some examples
Note that “-” is delete state
First, consider begin state:
Again, use add one rule
PHMM

36. PHMM Probabilities from MSA
Transition probabilities
When no information in MSA, set probs to uniform
For example I1 does not appear in MSA, so
PHMM

37. PHMM Probabilities from MSA
Transition probabilities, another example
What about transitions from state D1?
Can only go to M2, so
Again, use add one rule:
PHMM

38. PHMM Emission Probabilities
Emission probabilities for the given MSA
Using add-one rule
PHMM

39. PHMM Transition Probabilities
Transition probabilities for the given MSA
Using add-one rule
PHMM

40. PHMM Summary Construct pairwise alignments Use these to construct MSA
Usually, use dynamic programming
Use these to construct MSA
Lots of ways to do this
Using MSA, determine probabilities
Emission probabilities
State transition probabilities
In effect, we have trained a PHMM
Now what???
PHMM

41. PHMM Scoring
Want to score sequences to see how closely they match PHMM
How did we score sequences with HMM?
Forward algorithm
How to score sequences with PHMM?
But, algorithm is a little more complex
Due to complex state transitions
PHMM

42. Forward Algorithm Notation Indices i and j are columns in MSA
xi is ith observation symbol
qxi is distribution of xi in “random model”
Base case is
is score of x1,…,xi up to state j (note that in PHMM, i and j may not agree)
Some states undefined
Undefined states ignored in calculation
PHMM

43. Forward Algorithm Compute P(X|λ) recursively
Note that depends on , and
And corresponding state transition probs
PHMM

44. PHMM We will see examples of PHMM later In particular,
Malware detection based on opcodes
Masquerade detection based on UNIX commands
PHMM

45. References
Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids, Durbin, et al
Masquerade detection using profile hidden Markov models, L. Huang and M. Stamp, to appear in Computers and Security
Profile hidden Markov models for metamorphic virus detection, S. Attaluri, S. McGhee and M. Stamp, Journal in Computer Virology, Vol. 5, No. 2, May 2009, pp
PHMM