EE3J2 Data MiningSlide 1

EE3J2 Data Mining

Lecture 12: Sequence Analysis (2)

Martin Russell

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Objectives

Revise dynamic programming Examples

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Alignment pathA C X C C D

A

B

C

D

d(C,X)

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Accumulated Distance

The accumulated distance along the path p is the sum of distances along its length

Large accumulative distance = poor matches between symbols = poor path

Small accumulative distance = good matches between symbols = good path

The path with the smallest accumulated distance is called the optimal path

Computed using Dynamic Programming

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Dynamic Programming

A C X C C D A

B

C

D

Accumulated distance to this point…

…is minimum of accumulated distances to possible previous points

Plus local, incremental cost

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Formally…

nmdKnmad

nmdnmad

nmdKnmad

nmad

INS

DEL

,1,

,1,1

,,1

min,

Accumulated distance up to the point (m,n)

Deletion penalty

‘Local’ distance between mth symbol in sequence 1 and nth symbol in sequence 2

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Example application: sequence retrieval

……

AAGDTDTDTDD

AABBCBDAAAAAAA

BABABABBCCDF

GGGGDDGDGDGDGDTDTD

DGDGDGDGD

AABCDTAABCDTAABCDTAAB

CDCDCDTGGG

GGAACDTGGGGGAAA

…….

…….

Corpus of sequential data

‘query’ sequence Q

…BBCCDDDGDGDGDCDTCDTTDCCC…

Dynamic Programming

Distance Calculation Calculate ad(S,Q)

for each sequence S in corpus

QSadSS

,minargˆ

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Example: Edit DistanceS1 = AABCD KDEL=0

S2 = ABCCD KINS = 0

Distance matrix

A B C C DA 0 1 1 1 1

A 0 1 1 1 1

B 1 0 1 1 1

C 1 1 0 0 1

D 1 1 1 1 0

Accumulated distance matrix

A B C C DA 0 1 2 3 4

A 0 1 2 3 4

B 1 0 1 2 3

C 2 1 0 0 1

D 2 1 1 1 0

Forward path matrix

A B C C D

A \ _ _ _ _

A | _ _ _ _

B | \ _ _ _

C | | \ _ _

D | | | | \

A B C C D

A \ _ _ _ _

A | _ _ _ _

B | \ _ _ _

C | | \ _ _

D | | | | \

AABCCDAABCCD

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Example 2: Edit DistanceS1 = AABCD KDEL=2

S2 = ABCCD KINS = 2

Distance matrix

A B C C DA 0 1 1 1 1

A 0 1 1 1 1

B 1 0 1 1 1

C 1 1 0 0 1

D 1 1 1 1 0

Accumulated distance matrix

A B C C DA 0 3 6 9 12

A 2 1 4 7 10

B 5 2 2 5 8

C 8 5 2 2 5

D 11 8 5 3 2

Forward path matrix

A B C C D

A \ _ _ _ _

A | \ _ _ _

B | \ \ _ _

C | | \ \ _

D | | | \ \

A B C C D

A \ _ _ _ _

A | \ _ _ _

B | \ \ _ _

C | | \ \ _

D | | | | \

ABCCDABCCD

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edit-dist.c

New C program on course website Computes the edit distance between two sequences Prints out:

– Distance matrix

– Forward accumulated distance matrix

– Forward path matrix

– Optimal path

– Optimal alignment

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edit-dist.c

Format:edit-dist seq1 seq2 <Kdel> <Kins>

Seq1 and seq2 are the sequences <Kdel> and <Kins> optional, default 0

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Matching partial sequences

In some applications the interest is in whether one sequence matches a subsequence of another sequence

Example: Bioinformatics– Look for examples of a simple DNA sequence within a

more complex sequence

– Infer evolutionary relationship between two organisms

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Partial alignment

Simple intuitive solution is to allow Dynamic Programming to:– Start at any point in the first row

– End at any point in the final row

Then proceed as before Unfortunately this has limitations…

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Finding matching sub-sequencesStart DP

from here

Best scoring end point

Lower cost path

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Backwards Pass DP

nmdKnmad

nmdnmad

nmdKnmad

nmad

INS

DEL

,1,

,1,1

,,1

min,

Forward pass

nmdKnmad

nmdnmad

nmdKnmad

nmad

INS

DEL

,1,

,1,1

,,1

min,

Backward pass

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Backwards Pass DP

Starts in bottom row, works right-to-left and bottom-to-top

Otherwise, backwards accumulated distance matrix and backwards path matrix calculations analogous with forward-pass DP

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Forward-backward DP

Suppose that we have done a complete forward DP and a complete backward DP

We will have two path matrices:

– Forward path matrix

– Backward path matrix For any point in bottom row can trace-back through forward

path matrix and recover path ending in top row For any point in top row can trace-back through backward

path matrix and recover path ending in bottom row

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Matching sub-sequences

Choose a point in the bottom row. Traceback though forward path matrix

Identify start of path. Then traceback through backward path matrix

Are paths the same? If so, then we have a matching

subsequence

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Matching subsequences

If a path occurs as a consequence of tracing-back through the forward path matrix and tracing-back through the backward path matrix, then the corresponding section of the horizontal sequence is called a matching subsequence

The matching subsequences are those which achieve a good match with the vertical pattern

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Matching subsequences

A

B

B

C

X Z A B C C Y Z

matching subsequence

We say that this subsequence most closely resembles the original sequence ABBC

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Summary

Revision of Dynamic Programming Examples: Edit distance Motivation for interest in optimal subsequences Forward and backward dynamic programming Matching subsequences, subsequences which most

closely resemble a given sequence