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ContentsContentsContents

Similarity Searches in Sequence Databases

Sang-Hyun Park

KMeD Research Group

Computer Science Department

University of California, Los Angeles

Contents

- Introduction
- Whole Sequence Searches
- Subsequence Searches
- Segment-Based Subsequence Searches
- Multi-Dimensional Subsequence Searches
- Conclusion

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What is Sequence?- A sequence is an ordered list of elements.

S = 14.3, 18.2, 22.0, 22,4, 19.5, 17.1, 15.8, 15.1

- Sequences are principal data format in many applications.

What is Similarity Search?

- Similarity search finds sequences whose changing patterns are similar to that of a query sequence.
- Example
- Detect stocks with similar growth patterns
- Find persons with similar voice clips
- Find patients whose brain tumors have similar evolution patterns
- Similarity search helps in clustering, data mining, and rule discovery.

Classification of Similarity Search

- Similarity Searches are classified as:
- Whole sequence searches
- Subsequence searches
- Example
- S = 1,2,3
- Subsequences (S) = { 1, 2, 3, 1,2, 2,3, 1,2,3 }
- In whole sequence searches, the sequence S itself is compared with a query sequence Q.
- In subsequence searches, every possible subsequence of S can be compared with a query sequence q.

Similarity Measure

- Lp Distance Metric
- L1 : Manhattan distance or city-block distance
- L2 : Euclidean distance
- L : maximum distance in any element pairs
- requires that two sequences should have the same length

Similarity Measure (2)

- Time Warping Distance
- Originally introduced in the area of speech recognition
- Allows sequences to be stretched along the time axis

3,5,6 3,3,5,6 3,3,3,5,6 3,3,3,5,5,6 …

- Each element of a sequence can be mapped to one or more neighboring elements of another sequence.
- Useful in applications where sequences may be of different lengths or different sampling rates

Q = 10, 15, 20

S = 10, 15, 16, 20

Similarity Measure (3)

- Time Warping Distance (2)
- Defined recursively
- Computed by dynamic programming technique, O(|S||Q|)

DTW (S, Q[2:-])

DTW (S[2:-], Q)

DTW (S[2:-], Q[2:-])

DTW (S, Q) = DBASE (S[1], Q[1]) + min

DBASE (S[1], Q[1]) = | S[1] – Q[1] | P

Q

Q[2:-]

Q[1]

S

S[2:-]

S[1]

16

11

12

6

13

9

10

7

10

7

8

6

6

4

5

5

3

2

3

4

1

1

2

S

3

4

3

Q

Similarity Measure (4)- Time Warping Distance (3)
- S = 4,5,6,7,6,6, Q = 3,4,3
- When using L1 as a DBASE, DTW (S, Q) = 12

| S[i]Q[j] | + min (V1,V2,V3)

S[i]

V2

V3

V1

Q[j]

False Alarm and False Dismissal

- False Alarm
- Candidates not similar to a query.
- Minimize false alarms for efficiency
- False Dismissal
- Similar sequences not retrieved by index search
- Avoid false dismissals for correctness

data sequences

candidates

candidates

false alarm

similar

seq.

similar

seq.

false dismissal

Contents

- Introduction
- Whole Sequence Searches
- Subsequence Searches
- Segment-Based Subsequence Searches
- Multi-Dimensional Subsequence Searches
- Conclusion

Problem Definition

- Input
- Set of data sequences {S}
- Query sequence Q
- Distance tolerance
- Output
- Set of data sequences whose distances to Q are within
- Similarity Measure
- Time warping distance function, DTW
- L as a distance function for each element pair
- If the distance of every element pair is within , then DTW(S,Q) .

Previous Approaches

- Naïve Scan [Ber96]
- Read every data sequence from database
- Apply dynamic programming technique
- For m data sequences with average length L, O(mL|Q|)
- FastMap-Based Technique [Yi98]
- Use FastMap technique for feature extraction
- Map features into multi-dimensional points
- Use Euclidean distance in index space for filtering
- Could not guarantee “no false dismissal”

Previous Approaches (2)

- LB-Scan [Yi98]
- Read every data sequence from database
- Apply the lower-bound distance function Dlb which satisfies the following lower-bound theorem:

Dlb (S,Q) DTW (S,Q)

- Faster than the original time warping distance function (O(|S|+|Q|) vs. O(|S||Q|))
- Guarantee no false dismissal
- Based on sequential scanning

Proposed Approach

- Goal
- No false dismissal
- High query processing performance
- Sketch
- Extract a time-warping invariant feature vector
- Build a multi-dimensional index
- Use a lower-bound distance function for filtering

Proposed Approach (2)

- Feature Extraction
- F(S) = First(S), Last(S), Max(S), Min(S)
- F(S) is invariant to time warping transformation.
- Distance Function for Feature Vectors

| First(S) First(Q) |

| Last(S) Last(Q) |

| Max(S) Max(Q) |

| Min(S) Min(Q) |

DFT (F(S), F(Q)) = max

Proposed Approach (3)

- Distance Function for Feature Vectors (2)
- Satisfies lower-bounding theorem:

DFT (F(S),F(Q)) DTW (S,Q)

- More accurate than Dlb proposed in LB-Scan
- Faster than Dlb (O(1) vs. O(|S|+|Q|))

Proposed Approach (4)

- Indexing
- Build a multi-dimensional index from a set of feature vectors
- Index entry First(S), Last(S), Max(S), Min(S), Identifier(S)
- Query Processing
- Extract a feature vector F(Q)
- Perform range queries in index space to find data points included in the following query rectangle:

[ First(Q) , First(Q) + ],[ Last(Q) , Last(Q) + ],

[ Max(Q) , Max(Q) + ], [ Min(Q) , Min(Q) + ]

- Perform post-processing to discard false alarms

Performance Evaluation

- Implementation
- Implemented with C++ on UNIX operating system
- R-tree is used as a multi-dimensional index.
- Experimental Setup
- S&P 500 stock data set (m=545, L=232)
- Random walk synthetic data set
- SunSparc Ultra-5

Performance Evaluation (2)

- Filtering Ratio
- Better-than LB-Scan

Performance Evaluation (3)

- Query Processing Time
- Faster than LB-Scan and Naïve-Scan

Contents

- Introduction
- Whole Sequence Searches
- Subsequence Searches
- Segment-Based Subsequence Searches
- Multi-Dimensional Subsequence Searches
- Conclusion

Problem Definition

- Input
- Set of data sequences {S}
- Query sequence q
- Distance tolerance
- Output
- Set of subsequences whose distances to q are within
- Similarity Measure
- Time warping distance function, DTW
- Any LP metric as a distance function for element pairs

Previous Approaches

- Naïve-Scan [Ber96]
- Read every data subsequence from database
- Apply dynamic programming technique
- For m data sequences with average length n, O(mL2|q|)

Previous Approaches (2)

- ST-Index [Fal94]
- Assume that the minimum query length (w) is known in advance.
- Locates a sliding window of size w at every possible location
- Extract a feature vector inside the window
- Map a feature vector into a point and group trails into MBR (Minimum Bounding Rectangle)
- Use Euclidean distance in index space for filtering
- Could not guarantee “no false dismissal”

Proposed Approach

- Goal
- No false dismissal
- High performance
- Support diverse similarity measure
- Sketch
- Convert into sequences of discrete symbols
- Build a sparse suffix tree
- Use a lower-bound distance function for filtering
- Apply branch-pruning to reduce the search space

Proposed Approach (2)

- Conversion
- Generate categories from the distribution of element values
- Maximum-entropy method
- Equal-interval method
- DISC method
- Convert element to the symbol of the corresponding category
- Example

A = [0, 1.0], B = [1.1, 2.0], C = [2.1, 3.0], D = [3.1, 4.0]

S = 1.3, 1.6, 2.9, 3.3, 1.5, 0.1

SC = B, B, C, D, B, A

Proposed Approach (3)

- Indexing
- Extract suffixes from sequences of discrete symbols.
- Example

From S1C= A, B, B, A,

we extract four suffixes: ABBA, BBA, BA, A

Proposed Approach (4)

- Indexing (2)
- Build a suffix tree.
- Suffix tree is originally proposed to retrieve substrings exactly matched to the query string.
- Suffix tree consists of nodes and edges.
- Each suffix is represented by the path from the root node to a leaf node.
- Labels on the path from the root to the internal node Ni represents the longest common prefix of the suffixes under Ni
- Suffix tree is built with computation and space complexity, O(mL).

Proposed Approach (4)

- Indexing (3)
- Example : suffix tree from S1C= A, B, B, A and S2C= A, B

A

B

B

B

$

A

A

B

$

$

$

$

A

$

S1C[1:-]

S2C[1:-]

S1C[4:-]

S1C[2:-]

S1C[3:-]

S2C[2:-]

Proposed Approach (5)

- Query Processing

query (q, )

Index Searching

candidates

answers

Post Processing

suffix tree

data sequences

Proposed Approach (6)

- Index Searching
- Visit each node of suffix tree by depth-first traversal.
- Build lower-bound distance table for q and edge labels.
- Inspect the last columns of newly added rows to find candidates.
- Apply branch-pruning to reduce the search space.
- Branch-pruning theorem:

If all columns of the last row of the distance table have values larger than a distance tolerance , adding more rows on this table does not yield the new values less than or equal to .

Proposed Approach (7)

- Index Searching (2)
- Example : q = 2, 2, 1, = 1.5

N1

A

1

2

2

A

2

2

1

q

…..

N2

B

D

B

1

1

1.1

D

2.1

2.1

4.1

A

1

2

2

N3

N4

A

1

2

2

2

2

1

—

q

—

2

2

1

q

…..

…..

Proposed Approach (8)

- Lower-Bound Distance Function DTW-LB

0 if v is within the range of A (A.min v) P if v is smaller than A.min (v A.max) P if v is larger than A.max

DBASE-LB (A, v) =

v

A.max

A.max

A.max

v

A.min

A.min

A.min

v

possible minimum

distance = 0

possible minimum

distance = (A.min – v)P

possible minimum

distance = (v – A.max)P

Proposed Approach (9)

- Lower-Bound Distance Function DTW-LB (2)
- satisfies the lower-bounding theorem

DTW-LB(sC, q) DTW (s,q)

- computation complexity O(|sC||q|)

DTW-LB (sC, q) = DBASE-LB(sC[1], q[1]) +

min

DTW-LB (sC, q[2:-])

DTW-LB (sC[2:-], q)

DTW-LB (sC[2:-], q[2:-])

Proposed Approach (10)

- Computation Complexity
- m is the number of data sequences.
- L is the average length of data sequences.
- The left expression is for index searching.
- The right expression is for post-processing.
- RP ( 1) is the reduction factor by branch-pruning.
- RD ( 1) is the reduction factor by sharing distance tables.
- n is the number of subsequences requiring post-processing.

Proposed Approach (11)

- Sparse Indexing
- The index size is linear to the number of suffixes stored.
- To reduce the index size, we build a sparse suffix tree (SST).
- That is, we store the suffix SC[i:-] only if SC[i] SC[i–1].
- Compaction Ratio
- Example
- SC = A, A, A, A, C, B, B
- store only three suffixes (SC[1:-], SC[5:-], and SC[6:-])
- compaction ratio C = 7/3

Proposed Approach (12)

- Sparse Indexing (2)
- When traversing the suffix tree, we need to find non-stored suffixes and compute their distances to q.
- Assume that k elements of sC have the same value.
- Then, sC[1:-] is stored but sC[i:-] (i=2,3,…,k) is not stored.
- For non-stored suffixes,

we introduce another lower-bound distance function.

DTW-LB2 (sC[i:-], q) = DTW-LB(sC, q) – (i – 1) DBASE-LB(sC[1], q[1])

- DTW-LB2 satisfies the lower-bounding theorem.
- DTW-LB2 is O(1) when DTW-LB(sC, q) is given.

Proposed Approach (13)

- Sparse Indexing (3)
- With sparse indexing, the complexity becomes:
- m is the number of data sequences.
- L is the average length of data sequences.
- C is the compaction ratio.
- n is the number of subsequences requiring post-processing.
- RP ( 1) is the reduction factor by branch-pruning.
- RD ( 1) is the reduction factor by sharing distance tables.

Performance Evaluation

- Implementation
- Implemented with C++ on UNIX operating system
- Experimental Setup
- S&P 500 stock data set (m=545, L=232)
- Random walk synthetic data set
- Maximum-Entropy (ME) categorization
- Disk-based suffix tree construction algorithm
- SunSparc Ultra-5

Performance Evaluation (2)

- Comparison with Naïve-Scan
- increasing distance-tolerances
- S&P 500 stock data set, |q|=20

Performance Evaluation (3)

- Scalability Test
- increasing average length of data sequences
- random-walk data set, |q|=20,m=200

Performance Evaluation (4)

- Scalability Test (2)
- increasing total number of data sequences
- random-walk data set, |q|=20, L=200

- Introduction
- Whole Sequence Searches
- Subsequence Searches
- Segment-Based Subsequence Searches
- Multi-Dimensional Subsequence Searches
- Conclusion

Introduction

- We extend the proposed subsequence searching method to large sequence databases.
- In the retrieval of similar subsequences with time warping distance function,
- Sequential Scanning is O(mL2|q|).
- The proposed method is O(mL2|q| / R) (R 1).
- It makes search algorithms suffer from severe performance degradation when L is very large.
- For a database with long sequences, we need a new searching scheme linear to L.

SBASS

- We propose a new searching scheme: Segment-Based Subsequence Searching scheme (SBASS)
- Sequences are divided into a series of piece-wise segments.
- When a query sequence q with k segments is submitted, q is compared with those subsequences which consist of k consecutive data segments.
- The lengths of segments may be different.
- SS represents the segmented sequence of S.

S = 4,5,8,9,11,8,4,3 |S| = 8

SS = 4,5,8,9,11, 8,4,3 |SS| = 2

SBASS (2)

- Only four subsequences of SS are compared with QS.

SS[1],SS[2], SS[2],SS[3], SS[3],SS[4], SS[4],SS[5]

S

SS[3]

SS[2]

SS[1]

SS[4]

SS[5]

SS

qS

qS[1]

qS[2]

SBASS (3)

- For SBASS scheme, we define the piece-wise time warping distance function (where k = |qS| = |sS|).
- Sequential scanning for SBASS scheme is O(mL|q|).
- We introduce an indexing technique with O(mL|q|/R) (R 1).

Sketch of Proposed Approach

- Indexing
- Convert sequences to segmented sequences.
- Extract a feature vector from each segment.
- Categorize feature vectors.
- Convert segmented sequences to sequences of symbols.
- Construct suffix tree from sequences of symbols.
- Query Processing
- Traverse the suffix tree to find candidates.
- Discard false alarms in post processing.

Segmentation

- Approach
- Divide at peak points.
- Divide further if maximum deviation from interpolation line is too large.
- Eliminate noises.
- Compaction Ratio (C) = |S| / |SS|

too large deviation

noises

Feature Extraction

- From each subsequence segment, extract a feature vector:

(V1, VL,L, +, –)

VL

+

–

V1

L

Categorization and Index Construction

- Categorization
- Group similar feature vectors together using multi-dimensional categorization methods like Multi-attribute Type Abstraction Hierarchy (MTAH).
- Assign unique symbol to each category
- Convert segmented sequences to sequences of symbols.

S = 4,5,8,8,8,8,9,11,8,4,3

SS = 4,5,8,8,8,8,9,11, 8,4,3

SF = (4,11,8,2,1), (8,3,3,0,1.5)

SC = A,B

- From sequences of symbols, construct the suffix tree.

Query Processing

- For query processing, we calculate lower-bond distances between symbols and keep them in table.
- Given the query sequence q and the distance tolerance ,
- Convert q to qS and then to qC.
- Search the suffix tree to find those subsequences whose lower-bound distances to qC are within .
- Discard false alarms in post processing.

Query Processing (2)

Index Searching

candidates

answers

q,

qS

qC

Post Processing

suffix tree

data sequences

Computation Complexity

- Sequential scanning is O(mL|q|).
- Complexity of the proposed search algorithm is :
- n is the number of subsequences contained in candidates.
- C is the compaction ratio or the average number of elements in segments.
- RD ( 1) is the reduction factor by sharing edges of suffix tree.

Performance Evaluation

- Test Set : Pseudo Periodic Synthetic Sequences
- m = 100, L = 10,000
- Achieved up to 6.5 times speed-up compared to sequential scanning.

60

50

40

SeqScan

30

time (sec)

20

Our Approach

10

0.2

0.4

0.6

0.8

1.0

distance tolerance

- Introduction
- Whole Sequence Searches
- Subsequence Searches
- Segment-Based Subsequence Searches
- Multi-Dimensional Subsequence Searches
- Conclusion

Introduction

- So far, we assumed that elements have single-dimensional numeric values.
- Now, we consider multi-dimensional sequences.
- Image Sequences
- Video Streams

Medical Image Sequence

Introduction (2)

- In multi-dimensional sequences, elements are represented by feature vectors.

S = S[1], …, S[N], S[i] = (S[i][1], …, S[i][F])

- Our proposed subsequence searching techniques are extended to the retrieval of similar multi-dimensional subsequences.

Introduction (3)

- Multi-Dimensional Time Warping Distance

DMTW (S, Q[2:-])

DMTW (S, Q) = DMBASE (S[1], Q[1]) + min DMTW (S[2:-], Q)

DMTW (S[2:-],Q[2:-])

DMBASE (S[1], Q[1]) = ( Wi | S[1][i] Q[1][i] | )

- F is the number of features in each element.
- Wi is the weight of i-th dimension.

Sketch of Our Approach

- Indexing
- Categorize multi-dimensional element values using MTAH.
- Assign unique symbols to categories.
- Convert multi-dimensional sequences into sequences of symbols.
- Construct suffix tree from a set of sequences of symbols.
- Query Processing
- Traverse suffix tree.
- Find candidates whose lower-bound distances to q are within .
- Do post processing to discard false alarms.

Application to KMeD

- In the environment of KMeD, the proposed technique is applied to the retrieval of medical image sequences having similar spatio-temporal characteristics to those of the query sequence.
- KMeD [CCT:95] has the following features:
- Query by both image and alphanumeric contents
- Model temporal, spatial and evolutionary nature of objects
- Formulate queries using conceptual and imprecise terms
- Support cooperative processing

Application to KMeD (2)

- Query
- Medical Image Sequence
- Attribute names and their relative weights
- Distance tolerance

DistFromLV (0.6)

Circularity (0.1)

Size

(0.3)

Application to KMeD (3)

Query

User Model

Query Analysis

Contour Extraction

Feature Extraction

Distance Function

matching seq.

Visual Presentation

Similarity Searches

feedback

medical image seq.

index structure

- Introduction
- Whole Sequence Searches
- Subsequence Searches
- Segment-Based Subsequence Searches
- Multi-Dimensional Subsequence Searches
- Conclusion

Summary

- Sequence is an ordered list of elements.
- Similarity search helps in clustering and data mining.
- For sequences of different lengths or different sampling rates, time warping distance is useful.
- We proposed the whole sequence searching method with spatial access method and lower-bound distance function.
- We proposed the subsequence searching method with suffix tree and lower-bound distance functions.
- We proposed the segment-based subsequence searching method for large sequence databases.
- We extended the subsequence searching method to the retrieval of similar multi-dimensional subsequences.

Contribution

- We proposed the tighter and faster lower-bound distance function for efficient whole sequence searches without false dismissal.
- We demonstrated the feasibility of using time warping similarity measure on a suffix tree.
- We introduced the branch pruning theorem and the fast lower-bound distance function for efficient subsequence searches without false dismissal.
- We applied categorization and sparse indexing for scalability.
- We applied the proposed technique to the real application (KMeD).

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