LING124 Sequence alignment

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LING124 Sequence alignment

• September 23, 2008

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Class outline

• Sequence comparison
• String
• Vector
• Sequence alignment
• Dynamic programming (DP)
• String alignment using DP algorithm
• Exercise

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Problem

• Suppose a speaker made a mistake and said k r ?
  p
• b r ? k
• k r ? b
• k r ? b
• k r ? s p
• r ? p

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Alignment

• Identify the corresponding points

k r ? p b r ? k
k r ? p k r ? b
k r ? p k r ? s p
k r ? p r ? p
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How?

• Find the alignment such that point-wise
  comparison between the two gives you the minimal
  distance
• For strings, constituent symbols will be compared
• For speech sounds,
• each sound will be represented by a sequence of
  vectors (e.g. vector of cepstral coefficients)
• constituent vectors will be compared

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Levenshtein distance

• Minimum number of edit operations required to
  derive one string from another
• Substitution
• Insertion
• Deletion

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Euclidean distance

• X ltx1, x2, ..., xngt
• Y lty1, y2, ..., yngt

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A naive approach

• List all possible alignments
• Calculate the distance for each aligned pair
• Identify the aligned pair with the minimal
  distance
• What is wrong with this approach?

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Dynamic programming

• Break the problem into smaller overlapping
  sub-problems
• Find the optimal answer to each sub-problem
• Combine the optimal answers for the sub-problems
  to find the optimal answer for the whole problem

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String alignment

• Suppose
• Two strings
• X k r I p
• Y r I p
• Edit costs
• insdelsub1
• (n1)(m1) matrix
• n length of Y
• m length of X

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String alignment (2)

• Fill in the costs
• C(i, 0) i (0 i n)
• C(0, j) j (0 j m)
• For i in 1 to n
• For j in 1 to m
• sub0 if yixj
• sub1 if yi?xj
• C(i,j)the minimum of
• C(i-1, j-1) sub
• C(i-1, j) del
• C(i, j-1) ins

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String alignment (3)

• Find the shortest path from bottom-right to
  top-left
• Trace back
• Start from bottom-right
• Move towards top-left
• Identify adjacent cells
• Move to the one with the shortest distance
• Define trace-back priority in case of tie-score
  (e.g. left gt up gt diagonal)

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String alignment (4)

• Write the string backwards from right to left as
  you trace back
• If you reached cell(i,j) by moving diagonally
• Prefix Y with Yi
• Prefix X with Xj
• If you reached cell(i,j) by moving up
• Prefix Y with Yi
• Prefix X with _
• If you reached cell(i,j) by moving left
• Prefix Y with _
• Prefix X with Xj
• X k r I p
• Y _ r I p

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Exercise

• Suppose
• Xbrick, Ybig
• Edit costs ins del sub 1
• Trace-back priority diagonal gt up gt left
• Create the matrix
• Fill in the cells with Levenshtein distance
• Retrace to derive the alignment

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Feature-vector alignment

• Similar to string alignment except
• Point-wise comparison is typically based on
  Euclidean distance
• Various methods to reduce the time and space
  complexity
• More on this in class on Dynamic Time Warping

Figure from Salvador and Chan (2004)