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Exact Indexing of Dynamic Time WarpingExact Indexing of Dynamic Time Warping

Dr Eamonn KeoghDr Eamonn Keogh

University of California RiversideUniversity of California Riverside

Computer cience ! Engineering DepartmentComputer cience ! Engineering Department


2/16

What are Time eries"What are Time eries"

# time series is a collection of o$servations made se%uentially in time time series is a collection of o$servations made se%uentially in time&

'ots of useful information can $e o$tained $y measuring time series data'ots of useful information can $e o$tained $y measuring time series dataover times&over times&

Time series occur in virtually every medical( scientific and $usinessesTime series occur in virtually every medical( scientific and $usinessesdomaindomain

)inding out the similarity $et*een t*o time series is the heart of many)inding out the similarity $et*een t*o time series is the heart of many

time series data mining applicationstime series data mining applications

pattern that are commonly $eing classifiedpattern that are commonly $eing classified


3/16

What are the challenges of *or+ing *ith Time eriesWhat are the challenges of *or+ing *ith Time eries

data"data"

su$,ective notion of similaritysu$,ective notion of similarity

How do we define similarityHow do we define similarity

large amount of datalarge amount of datadifferent type of data formatdifferent type of data formatHow do we search quicklyHow do we search quickly

#ny solutions availa$le"#ny solutions availa$le"

We need a method that allows an elastic shifting of the time axis, toWe need a method that allows an elastic shiftin

g of the time axis, to

accommodate sequences which are similar, but out of time phaseaccommodate s

equences which are similar, but out of time phase

Euclidean DistanceEuclidean Distancemost popular approach for defining similarity and indexing of time seriesmost popular approach for defining similarity and indexing of time series

data&data&a very $rittle distance approach *hich cannot index time series accuratelya very $rittle distance approach *hich cannot index time series accurately

among t*o different time phases&among t*o different time phases&

Dynamic Time WarpingDynamic Time Warping$ase on dynamic programming *hich proved to $e a very relia$le method&$ase on dynamic programming *hich proved to $e a very relia$le method&does not o$ey the triangular ine%uality& This has resisted attempts at exactdoes not o$ey the triangular ine%uality& This has resisted attempts at exact

indexing&indexing&-.-. performance on large data$ase may $e a limitation&/performance on large data$ase may $e a limitation&/


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mean error ratemean error rate speedspeed

Euclidean Distance 0etricEuclidean Distance 0etric 1&23451&2345 66

Dynamic Time Warping 7DTW8Dynamic Time Warping 7DTW8 1&129:1&129: 24162416

The result proved the relia$ility of DTW and motivates the necessity of introducingThe result proved the relia$ility of DTW and motivates the necessity of introducing

techni%ue to index DTWtechni%ue to index DTW

Comparison of t*o approachesComparison of t*o approaches

What are the challenges of *or+ing *ith Time eriesWhat are the challenges of *or+ing *ith Time eries

data"data" cont.cont.

Times Series ATimes Series A

Times Series BTimes Series B

Classification experiment on Cylinder;


5/16

shifting of time axisshifting of time axis

t*o series in different time phaset*o series in different time phase

What is Dynamic Time Warping "What is Dynamic Time Warping "

DTW is $eing used in different area li+e chemical engineering( patternDTW is $eing used in different area li+e chemical engineering( pattern

matching( $ioinformatics( & & &matching( $ioinformatics( & & &

What is Time Warping"What is Time Warping"

=iven> t*o se%uences=iven> t*o se%uencesxx??((xx

22(&&&((&&&(xx

nnandand yy

??((yy

22(&&&((&&&(yy

mm

Wanted> align t*o se%uence $ase on a commonWanted> align t*o se%uence $ase on a common

time;axistime;axis

t'o time series and &t'o time series and &

length n and m respecti"el*length n and m respecti"el* an (n+m) matrix is constr$ctedan (n+m) matrix is constr$cted

to store the distance bet'eento store the distance bet'een

items in and .items in and .

the res$lt alignmentthe res$lt alignment

optimal 'arping path

#ligning time series *ith Dynamic @rogramming 0atrix#ligning time series *ith Dynamic @rogramming 0atrix


6/16

What is Dynamic Time Warping"What is Dynamic Time Warping" cont.cont.

Demonstration of computing the 0inimal Editing Distance> http>AAisl&ira&u+a&deAspeechCourseAslidesAdt*AeditdistAappletAapplet&html

In the matrix( there are many *arping paths that satisfyIn the matrix( there are many *arping paths that satisfythe three $asic constraints&the three $asic constraints&

GoalGoal :: How can we find a path that givesHow can we find a path that givesthe minimal overall distance the minimal overall distance

form$la of d*namic programming

(i&,) - d(qi&c,) min/ (i#1&j#1) & (i#1&j) & (i&j#1)

ThereBre three $asic constraints for timeThereBre three $asic constraints for time

*arping*arping


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What is =lo$al @ath Constraints "What is =lo$al @ath Constraints "# path sho$ld be close to diagonal# path sho$ld be close to diagonal#in theor*& it limits 'arping path b* ho' far it ma*in theor*& it limits 'arping path b* ho' far it ma*

sta* from the diagonalsta* from the diagonal

#in practice& it constrains the range of indices in thein practice& it constrains the range of indices in the 'arping path'arping path

WhyWhyusingusingglo$alglo$alconstraints "constraints " # speed $p the T2 distance calc$lation# speed $p the T2 distance calc$lation

(red$ces the search effort from(red$ces the search effort from OO((nn33) to) to OO((nn))))

# to a"oid a relati"el* small section of one se4$ence# to a"oid a relati"el* small section of one se4$ence

maps onto a relati"el* large section of anothermaps onto a relati"el* large section of another

se4$ence.se4$ence.

o* to speed up the calculation of DTW"o* to speed up the calculation of DTW"

#pproximate the time series *ith some compressed or do*n sampled #pproximate the time series *ith some compressed or do*n sampled

representation( and do DTW on the ne* representation&representation( and do DTW on the ne* representation&

Q

C

n11

p

w1

wk

i

j

olutionolution>> !ower "ounding #easure with Global $ath %onstraint!ower "ounding #easure with Global $ath %onstraint

*arping *indo**arping *indo*


8/16

What is 'o*er


9/16

o* to speed up the calculation of DTW"o* to speed up the calculation of DTW" cont.cont.

@roposed lo*er $ounding measure@roposed lo*er $ounding measure > ' '


10/16

o* to index Dynamic Time Warping"o* to index Dynamic Time Warping"

@iece*ise Constant #pproximation 7@##8@iece*ise Constant #pproximation 7@##8

Represent the time series as a se%uence of $ox $asis functions& Represent the time series as a se%uence of $ox $asis functions&

Each $ox is in same lengthEach $ox is in same length

;ed$cing the time series from;ed$cing the time series from nn dimensions todimensions to NN

dimensions& the data is di"ided intodimensions& the data is di"ided into NN e4$al si is

red$ced to 1> dimensionsred$ced to 1> dimensions

Why using @## "Why using @## "

time series data ma* incl$de h$ndreds to tho$sands items&time series data ma* incl$de h$ndreds to tho$sands items&

this 'ill rapidl* degrade the performance of indexing.this 'ill rapidl* degrade the performance of indexing.

1> dimension time series 'ill be reasonabl* handled b* m$lti#1> dimension time series 'ill be reasonabl* handled b* m$lti#

dimensional index str$ct$re.dimensional index str$ct$re.

a 'a* is needed to f$rther red$ce the dimension of lo'era 'a* is needed to f$rther red$ce the dimension of lo'er

bo$nd b* !B?@eoghbo$nd b* !B?@eogh

8AA is the most efficient techni4$e among other approaches8AA is the most efficient techni4$e among other approaches7Wavelets( )ourier Transforms( #daptive @iece*ise Constant #pproximation87Wavelets( )ourier Transforms( #daptive @iece*ise Constant #pproximation8


11/16

there are t'o time series data sets ( and ) in length n& both are being di"ided into Nthere are t'o time series data sets ( and ) in length n& both are being di"ided into Ndimension. is a candidate se4$ence. is a 4$er* se4$ence.dimension. is a candidate se4$ence. is a 4$er* se4$ence.

approximate the minim$m bo$nding rectangle (;) in each dimension of candidateapproximate the minim$m bo$nding rectangle (;) in each dimension of candidate

se4$ence se4$ence

approximate the max (9) and min (!) point in each dimension of 4$er* se4$ence approximate the max (9) and min (!) point in each dimension of 4$er* se4$ence

b* $sing !B?8AAb* $sing !B?8AA

o* to index Dynamic Time Warping"o* to index Dynamic Time Warping" cont.cont.

0odified @## to index time *arped %ueries '


12/16

define adefine a 0IHDIT7(R80IHDIT7(R8f$nction that ret$rns a lo'er bo$nding meas$re of the distancef$nction that ret$rns a lo'er bo$nding meas$re of the distancebet'een a 4$er* & and ;& 'ere ; is a inim$m Bo$nding ;ectangle (B;) of .bet'een a 4$er* & and ;& 'ere ; is a inim$m Bo$nding ;ectangle (B;) of .

o* to index Dynamic Time Warping"o* to index Dynamic Time Warping" cont.cont.

0odified @## to index time *arped %ueries0odified @## to index time *arped %ueries

h2

'L2

l5

UL5

ULM

lM

reduced dimensionreduced dimension

original dimensionoriginal dimension


13/16

o* to search time series *ith DTW "o* to search time series *ith DTW "

K;Hearest Heigh$or earch #lgorithmK;Hearest Heigh$or earch #lgorithm

What is K;HH earch ;What is K;HH earch ; @NNSearch(&@)@NNSearch(&@)""

4$er* se4$ence and desired n$mber of @ time series neighbors4$er* se4$ence and desired n$mber of @ time series neighbors

from a set from a set

priorit* 4$e$e is being $sed for storing the index in an increasingpriorit* 4$e$e is being $sed for storing the index in an increasing

order of distanceorder of distance

Rangeearch #lgorithmRangeearch #lgorithm

What is Rangeearch #lgorithmWhat is Rangeearch #lgorithm # ;angeSearch(&5&T)C# ;angeSearch(&5&T)C

ans'ering a range 4$eriesans'ering a range 4$eries

a classic ;#tree#st*le rec$rsi"e search algorithma classic ;#tree#st*le rec$rsi"e search algorithm


14/16

Experimental EvaluationExperimental Evaluation

Evaluation among three lo*er $ounding measuresEvaluation among three lo*er $ounding measures 7'


15/16

ConclusionConclusion

This paper override the traditional $elieve of Ndynamic timeThis paper override the traditional $elieve of Ndynamic time

*arping &&&cannot $e speeded up $y indexing*arping &&&cannot $e speeded up $y indexing

o*ever( it $ased on t*o assumptiono*ever( it $ased on t*o assumption

; $oth time series data are in the same length; $oth time series data are in the same length

77o$t of time phase is allo'edo$t of time phase is allo'ed88 ; index se%uence *hen *arping path is constrained; index se%uence *hen *arping path is constrained

77Bo$ndar* conditions& ontin$it*& onotonicit*& Dlobal constraintBo$ndar* conditions& ontin$it*& onotonicit*& Dlobal constraint88

The proposed approach is state of the art in terms of efficiencyThe proposed approach is state of the art in terms of efficiency

and flexi$ility& It may $enefit the matching of 2 and 4 dimensionaland flexi$ility& It may $enefit the matching of 2 and 4 dimensional

shapes&shapes&


16/16

#c+no*ledge#c+no*ledge

Dr Eamonnn KeoghDr Eamonnn Keogh

(omp$ter Science E 5ngineering epartment& 9ni"ersit* of alifornia(omp$ter Science E 5ngineering epartment& 9ni"ersit* of aliforniaF ;i"erside& ;i"erside&A G3=31F ;i"erside& ;i"erside&A G3=31))

Exact Indexing of Dynamic time WarpingExact Indexing of Dynamic time Warping

# Tutorial on Indexing and 0ining Time eries Data# Tutorial on Indexing and 0ining Time eries Data

http://www.cs.ucr.edu/~eamonn/tutorial_on_time_series.ppthttp://www.cs.ucr.edu/~eamonn/tutorial_on_time_series.ppt

Carnegie 0ellon UniversityCarnegie 0ellon University

#utomatic peech Recognition#utomatic peech Recognition

http://werner.ira.uka.de/speechCoursehttp://werner.ira.uka.de/speechCourse

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