Minimum Edit Distance. Defini'on of Minimum Edit Distance
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1 Minimum Edit Distance Defini'on of Minimum Edit Distance
2 How similar are two strings? Spell correc'on The user typed graffe Which is closest? graf grab grail giraffe Computa'onal Biology Align two sequences of nucleo'des AGGCTATCACCTGACCTCCAGGCCGATGCCC TAGCTATCACGACCGCGGTCGATTTGCCCGAC Resul'ng alignment: -AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC Also for Machine Transla'on, Informa'on Extrac'on, Speech Recogni'on
3 Edit Distance The minimum edit distance between two strings Is the minimum number of edi'ng opera'ons Inser'on Dele'on Subs'tu'on Needed to transform one into the other
4 Minimum Edit Distance Two strings and their alignment:
5 Minimum Edit Distance If each opera'on has cost of 1 Distance between these is 5 If subs'tu'ons cost 2 (Levenshtein) Distance between them is 8
6 Alignment in Computa9onal Biology Given a sequence of bases An alignment: AGGCTATCACCTGACCTCCAGGCCGATGCCC TAGCTATCACGACCGCGGTCGATTTGCCCGAC -AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC Given two sequences, align each lexer to a lexer or gap
7 Other uses of Edit Distance in NLP Evalua'ng Machine Transla'on and speech recogni'on R Spokesman confirms senior government adviser was shot! H Spokesman said the senior adviser was shot dead! S I D I! Named En'ty Extrac'on and En'ty Coreference IBM Inc. announced today IBM profits Stanford President John Hennessy announced yesterday for Stanford University President John Hennessy
8 How to find the Min Edit Distance? Searching for a path (sequence of edits) from the start string to the final string: Ini9al state: the word we re transforming Operators: insert, delete, subs'tute Goal state: the word we re trying to get to Path cost: what we want to minimize: the number of edits 8
9 Minimum Edit as Search But the space of all edit sequences is huge! We can t afford to navigate naïvely Lots of dis'nct paths wind up at the same state. We don t have to keep track of all of them Just the shortest path to each of those revisted states. 9
10 Defining Min Edit Distance For two strings X of length n Y of length m We define D(i,j) the edit distance between X[1..i] and Y[1..j] i.e., the first i characters of X and the first j characters of Y The edit distance between X and Y is thus D(n,m)
11 Minimum Edit Distance Defini'on of Minimum Edit Distance
12 Minimum Edit Distance Compu'ng Minimum Edit Distance
13 Dynamic Programming for Minimum Edit Distance Dynamic programming: A tabular computa'on of D(n,m) Solving problems by combining solu'ons to subproblems. BoXom- up We compute D(i,j) for small i,j And compute larger D(i,j) based on previously computed smaller values i.e., compute D(i,j) for all i (0 < i < n) and j (0 < j < m)
14 Ini'aliza'on D(i,0) = i! D(0,j) = j Defining Min Edit Distance (Levenshtein) Recurrence Rela'on: For each i = 1 M!! For each j = 1 N D(i-1,j) + 1! Termina'on: D(N,M) is distance! D(i,j)= min D(i,j-1) + 1! D(i-1,j-1) + 2; if X(i) Y(j)! 0; if X(i) = Y(j)!
15 The Edit Distance Table N 9 O 8 I 7 T 6 N 5 E 4 T 3 N 2 I 1 # # E X E C U T I O N
16 The Edit Distance Table N 9 O 8 I 7 T 6 N 5 E 4 T 3 N 2 I 1 # # E X E C U T I O N
17 Edit Distance N 9 O 8 I 7 T 6 N 5 E 4 T 3 N 2 I 1 # # E X E C U T I O N
18 The Edit Distance Table N O I T N E T N I # # E X E C U T I O N
19 Minimum Edit Distance Compu'ng Minimum Edit Distance
20 Minimum Edit Distance Backtrace for Compu'ng Alignments
21 Compu9ng alignments Edit distance isn t sufficient We oben need to align each character of the two strings to each other We do this by keeping a backtrace Every 'me we enter a cell, remember where we came from When we reach the end, Trace back the path from the upper right corner to read off the alignment
22 Edit Distance N 9 O 8 I 7 T 6 N 5 E 4 T 3 N 2 I 1 # # E X E C U T I O N
23 MinEdit with Backtrace
24 Adding Backtrace to Minimum Edit Distance Base condi'ons: Termina'on: D(i,0) = i D(0,j) = j D(N,M) is distance Recurrence Rela'on: For each i = 1 M!! For each j = 1 N! D(i-1,j) + 1! D(i,j)= min D(i,j-1) + 1! dele'on inser'on D(i-1,j-1) + 2; if X(i) Y(j) subs'tu'on! 0; if X(i) = Y(j)! LEFT! inser'on ptr(i,j)= DOWN! DIAG! dele'on subs'tu'on
25 The Distance Matrix Every non- decreasing path from (0,0) to (M, N) corresponds to an alignment of the two sequences Slide adapted from Serafim Batzoglou
26 Result of Backtrace Two strings and their alignment:
27 Performance Time: Space: Backtrace O(nm) O(nm) O(n+m)
28 Minimum Edit Distance Backtrace for Compu'ng Alignments
29 Minimum Edit Distance Weighted Minimum Edit Distance
30 Weighted Edit Distance Why would we add weights to the computa'on? Spell Correc'on: some lexers are more likely to be mistyped than others Biology: certain kinds of dele'ons or inser'ons are more likely than others
31 Confusion matrix for spelling errors
32
33 Ini'aliza'on: D(0,0) = 0! Weighted Min Edit Distance D(i,0) = D(i-1,0) + del[x(i)]; 1 < i N! D(0,j) = D(0,j-1) + ins[y(j)]; Recurrence Rela'on: D(i-1,j) D(i,j)= min D(i,j-1) Termina'on: D(N,M) is distance! + del[x(i)]! + ins[y(j)]! 1 < j M D(i-1,j-1) + sub[x(i),y(j)]!
34 Where did the name, dynamic programming, come from? The 1950s were not good years for mathematical research. [the] Secretary of Defense had a pathological fear and hatred of the word, research I decided therefore to use the word, programming. I wanted to get across the idea that this was dynamic, this was multistage I thought, let s take a word that has an absolutely precise meaning, namely dynamic it s impossible to use the word, dynamic, in a pejorative sense. Try thinking of some combination that will possibly give it a pejorative meaning. It s impossible. Thus, I thought dynamic programming was a good name. It was something not even a Congressman could object to. Richard Bellman, Eye of the Hurricane: an autobiography 1984.
35 Minimum Edit Distance Weighted Minimum Edit Distance
36 Minimum Edit Distance Minimum Edit Distance in Computa'onal Biology
37 Sequence Alignment AGGCTATCACCTGACCTCCAGGCCGATGCCC TAGCTATCACGACCGCGGTCGATTTGCCCGAC -AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC
38 Why sequence alignment? Comparing genes or regions from different species to find important regions determine func'on uncover evolu'onary forces Assembling fragments to sequence DNA Compare individuals to looking for muta'ons
39 Alignments in two fields In Natural Language Processing We generally talk about distance (minimized) And weights In Computa'onal Biology We generally talk about similarity (maximized) And scores
40 The Needleman- Wunsch Algorithm Ini'aliza'on: D(i,0) = -i * d! D(0,j) = -j * d Recurrence Rela'on: D(i-1,j) - d! D(i,j)= min D(i,j-1) - d! Termina'on: D(N,M) is distance!! D(i-1,j-1) + s[x(i),y(j)]!
41 The Needleman- Wunsch Matrix (Note that the origin is at the upper leb.) Slide adapted from Serafim Batzoglou
42 A variant of the basic algorithm: Maybe it is OK to have an unlimited # of gaps in the beginning and end: CTATCACCTGACCTCCAGGCCGATGCCCCTTCCGGC GCGAGTTCATCTATCAC--GACCGC--GGTCG If so, we don t want to penalize gaps at the ends Slide from Serafim Batzoglou
43 Different types of overlaps Example: 2 overlapping reads from a sequencing project Example: Search for a mouse gene within a human chromosome Slide from Serafim Batzoglou
44 The Overlap Detec9on variant Slide from Serafim Batzoglou Changes: 1. Ini'aliza'on For all i, j,!!f(i, 0) = 0!!F(0, j) = 0! 2. Termina'on! max i F(i, N)! F OPT = max!! max j F(M, j)!
45 The Local Alignment Problem Given two strings x = x 1 x M, y = y 1 y N Find substrings x, y whose similarity (op'mal global alignment value) is maximum x = aaaacccccggggxa y = Xcccgggaaccaacc Slide from Serafim Batzoglou
46 The Smith- Waterman algorithm Idea: Ignore badly aligning regions Modifica'ons to Needleman- Wunsch: Ini9aliza9on: F(0, j) = 0!!!!F(i, 0) = 0! 0!! Itera9on: F(i, j) = max F(i 1, j) d!!!!! F(i, j 1) d!!!!! F(i 1, j 1) + s(x i, y j )! Slide from Serafim Batzoglou
47 The Smith- Waterman algorithm Termina9on: 1. If we want the best local alignment Find F OPT and trace back F OPT = max i,j F(i, j) 2. If we want all local alignments scoring > t?? For all i, j find F(i, j) > t, and trace back? Complicated by overlapping local alignments Slide from Serafim Batzoglou
48 Local alignment example X = ATCAT! Y = ATTATC! Let: m = 1 (1 point for match) d = 1 (- 1 point for del/ins/sub) A! T! T! A! T! C! 0! 0! 0! 0! 0! 0! 0! A! 0! T! 0! C! 0! A! 0! T! 0!
49 Local alignment example X = ATCAT! Y = ATTATC! A! T! T! A! T! C! 0! 0! 0! 0! 0! 0! 0! A! 0! 1! 0! 0! 1! 0! 0! T! 0! 0! 2! 1! 0! 2! 0! C! 0! 0! 1! 1! 0! 1! 3! A! 0! 1! 0! 0! 2! 1! 2! T! 0! 0! 2! 0! 1! 3! 2!
50 Local alignment example X = ATCAT! Y = ATTATC! A! T! T! A! T! C! 0! 0! 0! 0! 0! 0! 0! A! 0! 1! 0! 0! 1! 0! 0! T! 0! 0! 2! 1! 0! 2! 0! C! 0! 0! 1! 1! 0! 1! 3! A! 0! 1! 0! 0! 2! 1! 2! T! 0! 0! 2! 0! 1! 3! 2!
51 Local alignment example X = ATCAT! Y = ATTATC! A! T! T! A! T! C! 0! 0! 0! 0! 0! 0! 0! A! 0! 1! 0! 0! 1! 0! 0! T! 0! 0! 2! 1! 0! 2! 0! C! 0! 0! 1! 1! 0! 1! 3! A! 0! 1! 0! 0! 2! 1! 2! T! 0! 0! 2! 0! 1! 3! 2!
52 Minimum Edit Distance Minimum Edit Distance in Computa'onal Biology
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