FTW: Fast Similarity Search under the Time Warping Distance Yasushi - - PowerPoint PPT Presentation

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FTW: Fast Similarity Search under the Time Warping Distance Yasushi - - PowerPoint PPT Presentation

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FTW: Fast Similarity Search under the Time Warping Distance Yasushi Sakurai (NTT Cyber Space Labs) Masatoshi Yoshikawa (Nagoya Univ.) Christos Faloutsos (Carnegie Mellon Univ.) Motivation n Time-series data q many applications n computational

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FTW: Fast Similarity Search under the Time Warping Distance

Yasushi Sakurai (NTT Cyber Space Labs) Masatoshi Yoshikawa (Nagoya Univ.) Christos Faloutsos (Carnegie Mellon Univ.)

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Motivation

n Time-series data

q many applications n computational biology, astrophysics, geology,

meteorology, multimedia, economics

n Similarity search

q Euclidean distance q DTW (Dynamic Time Warping) n Useful for different sequence lengths n Different sampling rates n scaling along the time axis

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Mini-introduction to DTW

n DTW allows sequences to be stretched along the

time axis

q Minimize the distance of sequences q Insert ‘stutters’ into a sequence q THEN compute the (Euclidean) distance

‘stutters’:

  • riginal
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Mini-introduction to DTW

n DTW is computed by dynamic programming

q Warping path: set of grid cells in the time warping

matrix

data sequence P of length N query sequence Q of length M pN qM pi qj q1 p1 P Q p1 pi pN q1 qj qM p-stutters q-stutters Optimum warping path (the best alignment)

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Mini-introduction to DTW

ï î ï í ì

  • +
  • =

= ) 1 , 1 ( ) , 1 ( ) 1 , ( min ) , ( ) , ( ) , ( j i f j i f j i f q p j i f M N f Q P D

j i dtw

q-stutter no stutter p-stutter

n DTW is computed by dynamic programming

p1, p2, …, pi,; q1, q2, …, qj

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Mini-introduction to DTW

n Global constraints limit the warping scope

q Warping scope: area that the warping path is allowed to

visit

P Q p1 pi pN q1 qj qM P Q p1 pi pN q1 qj qM Itakura Parallelogram Sakoe-Chiba Band

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Mini-introduction to DTW

n Width of the warping scope W is user-defined

P Q p1 pi pN q1 qj qM Sakoe-Chiba Band

W1

P Q p1 pi pN q1 qj qM

W2

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Motivation

n Similarity search for time-series data

q DTW (Dynamic Time Warping)

n scaling along the time axis

But…

n High search cost O(NM) n prohibitive for long sequences

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Our Solution, FTW

n Requirements:

  • 1. Fast
  • 2. No false dismissals
  • 3. No restriction on the sequence length

n It should handle data sequences of different lengths

  • 4. Support for any, as well as for no restriction on

“warping scope”

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Problem Definition

n Given

q S time-series data sequences of unequal lengths

{P1, P2, …, PS},

q a query sequence Q, q an integer k, q (optionally) a warping scope W,

n Find the k-nearest neighbors of Q from the

data sequence set by using DTW with W

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Overview

n Introduction n Related work n Main ideas n Experimental results n Conclusions

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Related Work

n Sequence indexing

q Agrawal et al. (FODO 1998) q Keogh et al. (SIGMOD 2001) q …

n Subsequence matching

q Faloutsos et al. (SIGMOD 1994) q Moon et al. (SIGMOD 2002) q …

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Related Work

n Fast sequence matching for DTW

q Yi et al. (ICDE 1998) q Kim et al. (ICDE 2001) q Chu et al. (SDM 2002) q Keogh (VLDB 2002) q Zhu et al. (SIGMOD 2003) q …

n None of the existing methods for DTW fulfills all

the requirements

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Overview

n Introduction n Related work n Main ideas n Experimental results n Conclusions

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Main Idea (1) - LBS

n LBS (Lower Bounding distance measure with

Segmentation)

n P

A : Approximate sequences

q

: segment range

q

: upper value

q

: lower value

q t: length of time intervals*

) : (

U i L i R i

p p p =

U i

p

R i

p

L i

p

A

P

R

p1

R

p4

R

p3

t t t t

R

p2

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Main Idea (1) - LBS

R j

q

R i

p

n Compute lower bounding distance

q Distance of the two ranges and :

distance of their two closest points

R j

q

R i

p

Time Value Lower bound Time Value Lower bound =0

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Main Idea (1) - LBS

n Compute lower bounding distance

q Distance of the two ranges and :

distance of their two closest points

R j

q

ï ï î ï ï í ì >

  • >
  • =

) ( ) ( ) ( ) , (

  • therwise

p q p q q p q p q p D

U i L j U i L j U j L i U j L i R j R i seg R i

p

details

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Main Idea (1) - LBS

P

Q

P

Q

n Exact DTW distance

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Main Idea (1) - LBS

n Compute lower bounding distance from P

A and QA

n Use a dynamic programming approach

A

P

A

Q

A

P

A

Q

) , ( ) , ( Q P D Q P D

dtw A A lbs

£

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Main Idea (1) - LBS

n Compute lower bounding distance from P

A and QA

n Use a dynamic programming approach

A

P

A

Q

) , ( ) , ( Q P D Q P D

dtw A A lbs

£

P

Q

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Main Idea (2) - EarlyStopping

n Exploit the fact that we have found k-near neighbors

at distance dcb

q dcb: k-nearest neighbor distance (the Current Best)

the exact distance of the best k candidates so far

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Main Idea (2) - EarlyStopping

n Exclude useless warping paths by using

q Omit g(1,3) if q Omit g(4,1) if

A

P

A

Q

g(1,2) g(3,1)

A

P

A

Q

cb

d g > ) 2 , 1 (

cb

d

cb

d g > ) 1 , 3 (

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Main Idea (3) - Refinement

n Q: How to choose t (length of time intervals)?

A

P

A

Q

g(1,2) g(3,1)

A

P

A

Q

t t

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Main Idea (3) - Refinement

n Q: How to choose t (length of intervals)? n A: Use multiple granularities, as follows:

A

P

A

Q

g(1,2) g(3,1)

A

P

A

Q

t t

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Main Idea (3) - Refinement

n Compute the lower bounding distance from the

coarsest sequences as the first refinement step

n Ignore P if , otherwise:

A

P

A

Q

g(1,2) g(3,1)

A

P

A

Q

cb A A lbs

d Q P D > ) , (

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Main Idea (3) - Refinement

n … compute the distance from more accurate

sequences as the second refinement step

n … repeat

A

P

A

Q

A

Q

A

P

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Main Idea (3) - Refinement

n … until the finest granularity n Update the list of k-nearest neighbors if

P

Q

P

Q

cb dtw

d Q P D £ ) , (

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Overview

n Introduction n Related work n Main ideas n Experimental results n Conclusions

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Experimental results

n Setup

q Intel Xeon 2.8GHz, 1GB memory, Linux q Datasets:

Temperature, Fintime, RandomWalk

q Four different time intervals (for n=2048)

t1=2, t2=8, t3=32, t4=128

n Evaluation

q Compared FTW with LB_PAA (the best so far) q Mainly computation time

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Outline of experiments

n Speed vs db size n Speed vs warping scope W n Effect of filtering n Effect of varying-length data sequences

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Search Performance

n Itakura Parallelogram

P Q p1 pi pN q1 qj qM

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Search Performance

n Wall clock time as a function of data set size n Temperature

FTW is up to 50 times faster!

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Search Performance

n Wall clock time as a function of data set size n Fintime

FTW is up to 40 times faster!

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Search Performance

n Wall clock time as a function of data set size n RandomWalk

FTW is up to 40 times faster! More effective as the size grows

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Outline of experiments

n Speed vs db size n Speed vs warping scope W n Effect of filtering n Effect of varying-length data sequences

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Search Performance

n Sakoe-Chiba Band

P Q p1 pi pN q1 qj qM

W1

P Q p1 pi pN q1 qj qM

W2

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Search Performance

n Wall clock time as a function of warping scope n Temperature

FTW is up to 220 times faster!

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Search Performance

n Wall clock time as a function of warping scope n Fintime

FTW is up to 70 times faster!

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Search Performance

n Wall clock time as a function of warping scope n RandomWalk

FTW is up to 100 times faster!

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Outline of experiments

n Speed vs db size n Speed vs warping scope W n Effect of filtering n Effect of varying-length data sequences

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Effect of filtering

n Most of data sequences are excluded by coarser

approximations (t4=128 and t3=32)

q Using multiple granularities has significant advantages

Frequency of approximation use

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Outline of experiments

n Speed vs db size n Speed vs warping scope W n Effect of filtering n Effect of varying-length sequences

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Difference in Sequence Lengths

n 5 sequence data sets

Random(2048,0): length 2048 +/- 0 Random(2048,32): length 2048 +/- 16 Random(2048,64), Random(2048,128), Random(2048,256)

Outperform by 2+ orders of magnitude LB_PAA can not handle

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Overview

n Introduction n Related work n Main ideas n Experimental results n Conclusions

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Conclusions

n

Design goals:

  • 1. Fast
  • 2. No false dismissals
  • 3. No restriction on the sequence length
  • 4. Support for any, as well as for no

restriction on “warping scope”

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Conclusions

n

Design goals:

  • 1. Fast (up to 220 times faster)
  • 2. No false dismissals
  • 3. No restriction on the sequence length
  • 4. Support for any, as well as for no

restriction on “warping scope”

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Page Accesses

n Sequential scan of feature data should boost

performance (speed-up factors SF=5, SF=10)

PAds: page accesses for data sequences PAfd: page accesses for feature data

ds fd SF

PA SF PA PA + =

details