Indexing

Multi-Dimensional

Time-Series

with

Support

for

Multiple

Distance

Measures

Although

most

time-series

data

mining

research

has

concentrated

on

providing

solutions

for

a

single

distance

function

in

this

work

we

motivate

the

need

for

a

single

index

structure

that

can

support

multiple

distance

measures

Our

specific

area

of

interest

is

the

efficient

retrieval

and

analysis

of

trajectory

similarities

Trajectory

datasets

are

very

common

in

environmental

applications

mobility

experiments

video

surveillance

and

are

especially

important

for

the

discovery

of

certain

biological

patterns

Our

primary

similarity

measure

is

based

on

the

Longest

Common

Subsequence

-LRB-

LCSS

-RRB-

model

that

offers

enhanced

robustness

particularly

for

noisy

data

which

are

encountered

very

often

in

real

world

applications

However

our

index

is

able

to

accommodate

other

distance

measures

as

well

including

the

ubiquitous

Euclidean

distance

and

the

increasingly

popular

Dynamic

Time

Warping

-LRB-

DTW

-RRB-

While

other

researchers

have

advocated

one

or

other

of

these

similarity

measures

a

major

contribution

of

our

work

is

the

ability

to

support

all

these

measures

without

the

need

to

restructure

the

index

Our

framework

guarantees

no

false

dismissals

and

can

also

be

tailored

to

provide

much

faster

response

time

at

the

expense

of

slightly

reduced

precision/recall

The

experimental

results

demonstrate

that

our

index

can

help

speed-up

the

computation

of

expensive

similarity

measures

such

as

the

LCSS

and

the

DTW

H.

-LSB-

Database

Management

-RSB-

Database

Applications

Data

Mining

In

this

work

we

present

an

efficient

and

compact

external

memory

index

for

fast

detection

of

similar

trajectories

Trajectory

data

are

prevalent

in

diverse

fields

of

interest

such

as

meteorology

GPS

tracking

wireless

applications

video

tracking

-LSB-

-RSB-

and

motion

capture

-LSB-

-RSB-

Recent

advances

in

mobile

computing

sensor

and

GPS

technology

have

made

it

possible

to

collect

large

amounts

of

spatiotemporal

data

and

The

research

of

this

author

was

supported

by

NSF

ITR

NSF

CAREER

NSF

IIS

and

NRDRP

Permission

to

make

digital

or

hard

copies

of

all

or

part

of

this

work

for

personal

or

classroom

use

is

granted

without

fee

provided

that

copies

are

not

made

or

distributed

for

profit

or

commercial

advantage

and

that

copies

bear

this

notice

and

the

full

citation

on

the

first

page

To

copy

otherwise

or

republish

to

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specific

permission

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fee

SIGKDD

August

Washington

DC

USA

Copyright

ACM

there

is

increasing

interest

in

performing

data

analysis

tasks

over

such

data

-LSB-

-RSB-

In

mobile

computing

users

equipped

with

mobile

devices

move

in

space

and

register

their

location

at

different

time

instances

to

spatiotemporal

databases

via

wireless

links

In

environmental

information

systems

tracking

animals

and

weather

conditions

is

very

common

and

large

datasets

can

be

created

by

storing

locations

of

observed

objects

over

time

Human

motion

data

generated

by

tracking

simultaneously

various

body

joints

are

also

multidimensional

trajectories

In

this

field

of

computer

graphics

fundamental

operations

include

the

clustering

of

similar

movements

leading

to

a

multitude

of

applications

such

as

interactive

generation

of

motions

-LSB-

-RSB-

Spatiotemporal

data

are

also

produced

by

migrating

particles

in

biological

sciences

where

the

focus

can

be

on

the

discovery

of

subtle

patterns

during

cellular

mitoses

-LSB-

-RSB-

In

general

any

dataset

that

involves

storage

of

multiple

streams

-LRB-

attributes

-RRB-

of

data

can

be

considered

and

treated

as

a

multidimensional

trajectory

One

very

common

task

for

such

data

is

the

discovery

of

objects

that

follow

a

certain

motion

pattern

for

purposes

of

clustering

or

classification

The

objective

here

is

to

efficiently

organize

trajectories

on

disk

so

that

we

can

quickly

answer

k-Nearest-Neighbors

-LRB-

kNN

-RRB-

queries

A

frequent

obstacle

in

the

analysis

of

spatiotemporal

data

is

the

presence

of

noise

which

can

be

introduced

due

to

electromagnetic

anomalies

transceiver

problems

etc.

Another

impediment

is

that

objects

may

move

in

a

similar

way

but

at

different

speeds

So

we

would

like

our

similarity

model

to

be

robust

to

noise

support

elastic

and

imprecise

matches

Choosing

the

Euclidean

distance

as

the

similarity

model

is

unrealistic

since

its

performance

degrades

rapidly

in

the

presence

of

noise

and

this

measure

is

also

sensitive

to

small

variations

in

the

time

axis

We

concentrate

on

two

similarity

models

the

first

is

an

extension

of

Dynamic

Time

Warping

for

higher

dimensions

We

note

that

DTW

has

been

used

so

far

for

one-dimensional

time

series

Here

we

present

a

formulation

for

sequences

of

arbitrary

dimensions

The

second

distance

measure

is

a

modification

of

the

Longest

Common

Subsequence

-LRB-

LCSS

-RRB-

specially

adapted

for

continuous

values

Both

measures

represent

a

significant

improvement

compared

to

the

Euclidean

distance

However

LCSS

is

more

robust

than

DTW

under

noisy

conditions

-LSB-

-RSB-

as

figure

shows

Euclidean

matching

completely

disregards

the

variations

in

the

time

axis

while

DTW

performs

excessive

matchings

therefore

distorting

the

true

distance

between

sequences

The

LCSS

produces

the

most

robust

and

intuitive

correspondence

between

points

By

incorporating

warping

in

time

as

a

requirement

to

Euclidean

Matching

Time

Warping

Longest

Common

Subsequence

Figure

A

lucid

example

about

the

quality

matching

of

the

LCSS

compared

to

other

distance

functions

The

Euclidean

distance

performs

an

inflexible

matching

while

the

DTW

gives

many

superfluous

and

spurious

matchings

in

the

presence

of

noise

our

model

our

algorithms

are

automatically

challenged

with

quadratic

execution

time

Moreover

these

flexible

functions

are

typically

non-metric

which

makes

difficult

the

design

of

indexing

structures

To

speed

up

the

execution

of

a

similarity

function

one

can

devise

a

low

cost

upper

bounding

function

-LRB-

since

the

LCSS

model

captures

the

similarity

which

is

inversely

analogous

to

the

distance

-RRB-

We

utilize

a

fast

prefiltering

scheme

that

will

return

upper

bound

estimates

for

the

LCSS

similarity

between

the

query

and

the

indexed

trajectories

In

addition

to

providing

similarity

measures

that

guarantee

no

false

dismissals

we

also

propose

approximate

similarity

estimates

that

significantly

reduce

the

index

response

time

Finally

we

show

that

the

same

index

can

support

other

distance

measures

as

well

Our

technique

works

by

splitting

the

trajectories

in

multidimensional

MBRs

and

storing

them

in

an

R-tree

For

a

given

query

we

construct

a

Minimum

Bounding

Envelope

-LRB-

MBE

-RRB-

that

covers

all

the

possible

matching

areas

of

the

query

under

warping

conditions

This

MBE

is

decomposed

into

MBRs

and

then

probed

in

the

R-tree

index

Using

the

index

we

can

discover

which

trajectories

could

potentially

be

similar

to

the

query

The

index

size

is

compact

and

its

construction

time

scales

well

with

the

trajectory

length

and

the

database

size

therefore

our

method

can

be

utilized

for

massive

datamining

tasks

The

main

contributions

of

the

paper

are

We

present

the

first

external

memory

index

for

multidimensional

trajectories

that

supports

multiple

distance

functions

-LRB-

such

as

LCSS

DTW

and

Euclidean

-RRB-

without

the

need

to

rebuild

the

index

We

give

efficient

techniques

for

upper

-LRB-

lower

-RRB-

bounding

and

for

approximating

the

LCSS

-LRB-

DTW

-RRB-

for

a

set

of

trajectories

We

incorporate

these

techniques

in

the

design

of

an

efficient

indexing

structure

for

the

LCSS

and

the

DTW

We

provide

a

flexible

method

that

allows

the

user

to

specify

queries

of

variable

warping

length

and

the

technique

can

be

tuned

to

optimize

the

retrieval

time

or

the

accuracy

of

the

solution

There

has

been

a

wealth

of

papers

that

use

an

L

p

distance

family

function

to

perform

similarity

matching

for

time-series

Work

on

multidimensional

sequences

can

be

found

in

-LSB-

-RSB-

However

they

support

only

Euclidean

distance

which

as

mentioned

in

the

introduction

can

not

capture

flexible

similarities

Although

the

vast

majority

of

database/data

mining

research

on

time

series

data

mining

has

focused

on

Euclidean

distance

virtually

all

real

world

systems

that

use

time

series

matching

as

a

subroutine

use

a

similarity

measure

which

allows

warping

In

retrospect

this

is

not

very

surprising

since

most

real

world

processes

particularly

biological

processes

can

evolve

at

varying

rates

For

example

in

bioinformat-ics

it

is

well

understood

that

functionally

related

genes

will

express

themselves

in

similar

ways

but

possibly

at

different

rates

Because

of

this

DTW

is

used

for

gene

expression

data

mining

-LSB-

-RSB-

Dynamic

Time

Warping

is

a

ubiquitous

tool

in

the

biometric/surveillance

community

It

has

been

used

for

tracking

time

series

extracted

from

video

-LSB-

-RSB-

classifying

handwritten

text

-LSB-

-RSB-

and

even

fingerprint

indexing

-LSB-

-RSB-

While

the

above

examples

testify

to

the

utility

of

a

time

warped

distance

measure

they

all

echo

the

same

complaint

DTW

has

serious

scalability

issues

Work

that

attempted

to

mitigate

the

large

computational

cost

has

appeared

in

-LSB-

-RSB-

and

-LSB-

-RSB-

where

the

authors

use

lower

bounding

measures

to

speed

up

the

execution

of

DTW

However

the

lower

bounds

can

be

loose

approximations

of

the

original

distance

when

the

data

are

normalized

In

-LSB-

-RSB-

a

different

approach

is

used

for

indexing

Time

Warping

by

using

suffix

trees

Nonetheless

the

index

requires

excessive

disk

space

-LRB-

about

times

the

size

of

the

original

data

-RRB-

The

flexibility

provided

by

DTW

is

very

important

however

its

efficiency

deteriorates

for

noisy

data

since

by

matching

all

the

points

it

also

matches

the

outliers

distorting

the

true

distance

between

the

sequences

An

alternative

approach

is

the

use

of

Longest

Common

Subsequence

-LRB-

LCSS

-RRB-

which

is

a

variation

of

the

edit

distance

The

basic

idea

is

to

match

two

sequences

by

allowing

them

to

stretch

without

rearranging

the

order

of

the

elements

but

allowing

some

elements

to

be

unmatched

Using

the

LCSS

of

two

sequences

one

can

define

the

distance

using

the

length

of

this

subsequence

-LSB-

-RSB-

In

-LSB-

-RSB-

an

internal

memory

index

for

the

LCSS

has

been

proposed

It

also

demonstrated

that

while

the

LCSS

presents

similar

advantages

to

DTW

it

does

not

share

its

volatile

performance

in

the

presence

of

outliers

Closest

in

spirit

to

our

approach

is

the

work

of

-LSB-

-RSB-

which

however

only

addresses

time-series

The

author

uses

constrained

DTW

as

the

distance

function

and

surrounds

the

possible

matching

regions

by

a

modified

version

of

a

Piecewise

Approximation

which

is

later

stored

as

equi-length

MBRs

in

an

R-tree

However

by

using

DTW

such

an

approach

is

susceptible

to

high

bias

of

outliers

Also

the

fixed

MBR

size

-LRB-

although

simplifies

the

index

operations

-RRB-

can

lead

to

degenerate

approximations

of

the

original

sequence

Moreover

the

embedding

of

the

envelope

in

the

indexed

sequences

can

slow

the

index

construction

time

and

limit

the

user

's

query

capabilities

to

a

predefined

warping

length

The

use

of

LCSS

as

our

primary

similarity

measure

lends

itself

to

a

more

natural

use

of

the

R-tree

where

the

similarity

estimates

are

simply

computed

by

calculating

the

MBR

intersection

areas

Since

the

index

is

not

constructed

for

a

specific

warping

window

the

user

can

pose

queries

with

variable

warping

length

The

purpose

of

this

paper

is

to

reconcile

the

best

of

both

worlds

We

provide

a

framework

that

can

support

in

the

same

index

the

LCSS

DTW

and

Euclidean

distance

functions

The

only

aspect

that

changes

is

the

different

representation

of

the

query

for

each

distance

measure

In

this

section

we

present

details

of

how

the

Dynamic

Time

Warping

and

the

LCSS

model

can

be

extended

to

describe

the

similarity

between

trajectories

Dynamic

Time

Warping

for

trajectories

We

describe

an

extension

in

of

the

original

DTW

function

as

described

by

Berndt

and

Clifford

-LSB-

-RSB-

Let

A

and

B

be

two

trajectories

of

moving

objects

with

size

n

and

m

respectively

where

A

-LRB-

-LRB-

a

x

a

y

-RRB-

-LRB-

a

x

n

a

y

n

-RRB-

-RRB-

and

B

-LRB-

-LRB-

b

x

b

y

-RRB-

-LRB-

b

x

m

b

y

m

-RRB-

-RRB-

For

a

trajectory

A

let

Head

-LRB-

A

-RRB-

-LRB-

-LRB-

a

x

a

y

-RRB-

-LRB-

a

x

n

a

y

n

-RRB-

-RRB-

Definition

The

Time

Warping

between

sequences

A

and

B

is

DT

W

-LRB-

A

B

-RRB-

L

p

-LRB-

-LRB-

a

x

n

a

y

n

-RRB-

-LRB-

b

x

m

b

y

m

-RRB-

-RRB-

min

-LCB-

DTW

-LRB-

Head

-LRB-

A

-RRB-

Head

-LRB-

B

-RRB-

-RRB-

DT

W

-LRB-

Head

-LRB-

A

-RRB-

B

-RRB-

DT

W

-LRB-

A

Head

-LRB-

B

-RRB-

-RRB-

-RCB-

-LRB-

-RRB-

where

L

p

is

any

p-Norm

Using

dynamic

programming

and

constraining

the

matching

region

within

the

time

required

to

compute

DTW

is

O

-LRB-

-LRB-

n

m

-RRB-

-RRB-

In

order

to

represent

an

accurate

relationship

of

distances

between

sequences

with

different

lengths

the

quantity

in

equation

is

normalized

by

the

length

of

the

warping

path

The

extension

to

n

dimensions

is

similar

In

figure

we

show

an

example

of

time

warping

for

two

trajectories

LCSS

model

for

trajectories

The

original

LCSS

model

refers

to

sequences

we

must

therefore

extend

it

to

the

case

In

addition

the

LCSS

paradigm

matches

discrete

values

however

in

our

model

we

want

to

allow

a

matching

when

the

values

are

within

a

certain

range

in

space

and

time

-LRB-

note

that

like

this

we

also

avoid

distant

and

degenerate

matchings

-RRB-

Definition

Given

an

integer

and

a

real

number

we

define

the

LCSS

-LRB-

A

B

-RRB-

as

follows

LCSS

-LRB-

A

B

-RRB-

if

A

or

B

is

empty

LCSS

-LRB-

Head

-LRB-

A

-RRB-

Head

-LRB-

B

-RRB-

-RRB-

if

a

x

n

b

x

m

and

a

y

n

b

y

m

and

n

m

max

-LRB-

LCSS

-LRB-

Head

-LRB-

A

-RRB-

B

-RRB-

LCSS

-LRB-

A

Head

-LRB-

B

-RRB-

-RRB-

-RRB-

otherwise

X

movement

Time

Y

movement

Figure

The

support

of

flexible

matching

in

spatiotemporal

queries

is

very

important

However

we

can

observe

that

Dynamic

Time

Warping

matches

all

points

-LRB-

so

the

outliers

as

well

-RRB-

therefore

distorting

the

true

distance

In

contrast

the

LCSS

model

can

efficiently

ignore

the

noisy

parts

where

sequences

A

and

Head

-LRB-

A

-RRB-

are

defined

similarly

as

before

The

constant

controls

the

flexibility

of

matching

in

time

and

constant

is

the

matching

threshold

is

space

The

aforementioned

LCSS

model

has

the

same

O

-LRB-

-LRB-

n

m

-RRB-

-RRB-

computational

complexity

as

the

DTW

when

we

only

allow

a

matching

window

in

time

-LSB-

-RSB-

The

value

of

LCSS

is

unbounded

and

depends

on

the

length

of

the

compared

sequences

We

need

to

normalize

it

in

order

to

support

sequences

of

variable

length

The

distance

derived

from

the

LCSS

similarity

can

be

defined

as

follows

Definition

The

distance

D

expressed

in

terms

of

the

LCSS

similarity

between

two

trajectories

A

and

B

is

given

by

D

-LRB-

A

B

-RRB-

LCSS

-LRB-

A

B

-RRB-

min

-LRB-

n

m

-RRB-

-LRB-

-RRB-

Even

though

imposing

a

matching

window

can

help

speed

up

the

execution

the

computation

can

still

be

quadratic

when

is

a

significant

portion

of

the

sequence

's

length

Therefore

comparing

a

query

to

all

the

trajectories

becomes

intractable

for

large

databases

We

are

seeking

ways

to

avoid

examining

the

trajectories

that

are

very

distant

to

our

query

This

can

be

accomplished

by

discovering

a

close

match

to

our

query

as

early

as

possible

A

fast

pre-filtering

step

is

employed

that

eliminates

the

majority

of

distant

matches

Only

for

some

qualified

sequences

will

we

execute

the

costly

-LRB-

but

accurate

-RRB-

quadratic

time

algorithm

This

philosophy

has

also

been

successfully

used

in

-LSB-

-RSB-

There

are

certain

preprocessing

steps

that

we

follow

The

trajectories

are

segmented

into

MBRs

which

are

stored

in

an

Rtree

T.

Given

a

query

Q

we

discover

the

areas

of

possible

matching

by

constructing

its

Minimum

Bounding

Envelope

-LRB-

M

BE

Q

-RRB-

M

BE

Q

is

decomposed

into

MBRs

that

are

probed

in

the

index

T.

Based

on

the

MBR

intersections

similarity

estimates

are

computed

and

the

exact

LCSS

-LRB-

or

DTW

-RRB-

is

performed

only

on

the

qualified

trajectories

The

above

notions

are

illustrated

in

figure

and

we

explain

in

detail

how

they

can

be

applied

for

the

LCSS

case

in

the

sections

that

follow

E.

LCSS

Upper

Bound

Estimate

A.

Query

Q

C.

Envelope

Splitting

B.

Query

Envelope

D.

Sequence

MBRs

Figure

An

example

of

our

approach

-LRB-

in

for

clarity

-RRB-

A

query

is

extended

into

a

bounding

envelope

which

in

turn

is

also

split

into

the

resulting

MBRs

Overlap

between

the

query

and

the

index

MBRs

suggest

areas

of

possible

matching

Bounding

the

Matching

Regions

Let

us

first

consider

a

time-series

and

let

a

sequence

A

be

-LRB-

a

x

a

x

n

-RRB-

Ignoring

for

now

the

parameter

we

would

like

to

perform

a

very

fast

LCSS

match

between

sequence

A

and

some

query

Q.

Suppore

that

we

replicate

each

point

Q

i

for

time

instances

before

and

after

time

i

The

envelope

that

includes

all

these

points

defines

the

areas

of

possible

matching

Everything

outside

this

envelope

can

never

be

matched

pts

pts

Q

A

Figure

The

Minimum

Bounding

Envelope

-LRB-

MBE

-RRB-

within

in

time

and

in

space

of

a

sequence

Everything

that

lies

outside

this

envelope

can

never

be

matched

We

call

this

envelope

the

Minimum

Bounding

Envelope

-LRB-

MBE

-RRB-

of

a

sequence

Also

once

we

incorporate

the

matching

within

in

space

this

envelope

should

extent

above

and

below

the

original

envelope

-LRB-

figure

-RRB-

The

notion

of

the

bounding

envelope

can

be

trivially

extended

in

more

dimensions

where

M

BE

-LRB-

-RRB-

for

a

trajectory

Q

-LRB-

-LRB-

q

x

q

y

-RRB-

-LRB-

q

x

n

q

y

n

-RRB-

covers

the

area

between

the

following

time-series

EnvLow

MBE

-LRB-

-RRB-

EnvHigh

where

EnvHigh

-LSB-

i

-RSB-

max

-LRB-

Q

-LSB-

j

-RSB-

epsilon

-RRB-

i

j

EnvLow

-LSB-

j

-RSB-

min

-LRB-

Q

-LSB-

j

-RSB-

epsilon

-RRB-

i

j

The

LCSS

similarity

between

the

envelope

of

Q

and

a

sequence

A

is

defined

as

LCSS

-LRB-

M

BE

Q

A

-RRB-

n

i

if

A

-LSB-

i

-RSB-

within

envelope

otherwise

For

example

in

figure

the

LCSS

similarity

between

M

BE

Q

and

sequence

A

is

as

indicated

in

the

figure

This

value

represents

an

upper

bound

for

the

similarity

of

Q

and

A

We

can

use

the

M

BE

Q

to

compute

a

lower

bound

on

the

distance

between

trajectories

Lemma

For

any

two

trajectories

Q

and

A

the

following

holds

D

-LRB-

M

BE

Q

A

-RRB-

D

-LRB-

Q

A

-RRB-

Proof

-LRB-

Sketch

-RRB-

D

-LRB-

M

BE

Q

A

-RRB-

LCSS

-LRB-

M

BE

Q

A

-RRB-

min

-LRB-

Q

A

-RRB-

therefore

it

is

sufficient

to

show

that

LCSS

-LRB-

M

BE

Q

A

-RRB-

LCSS

-LRB-

Q

A

-RRB-

This

is

true

since

M

BE

Q

by

construction

contains

all

possible

areas

within

and

of

the

query

Q.

Therefore

no

possible

matching

points

will

be

missed

The

previous

lemma

provides

us

with

the

power

to

create

an

index

that

guarantees

no

false

dismissals

However

this

lower

bound

refers

to

the

raw

data

In

the

sections

that

follow

we

will

split

the

M

BE

of

a

trajectory

into

a

number

of

Minimum

Bounding

Rectangles

-LRB-

MBRs

-RRB-

to

accommodate

their

storage

into

a

multidimensional

R-tree

We

will

show

that

the

above

inequality

still

holds

between

trajectory

MBRs

The

MBR

generation

procedure

is

orthogonal

to

our

approach

since

any

segmentation

methodology

can

be

applied

to

our

framework

Therefore

the

description

of

the

potential

MBR

generation

methods

-LRB-

and

of

our

implementation

choice

-RRB-

will

be

delayed

until

later

Suppose

that

we

have

an

index

with

the

segmented

trajectories

and

the

user

provides

a

query

Q

Our

goal

is

the

discovery

of

the

k

closest

trajectories

to

the

given

query

according

to

the

LCSS

similarity

A

prefiltering

step

will

aid

the

quick

discovery

of

a

close

match

to

the

query

helping

us

discard

the

distant

trajectories

without

using

the

costly

quadratic

algorithm

Therefore

in

this

phase

we

compute

upper

bound

estimates

of

the

similarity

between

the

query

and

the

indexed

sequences

using

their

MBRs

Below

we

describe

the

algorithm

to

find

the

closest

trajectory

to

a

given

query

Input

Query

Q

Index

I

with

trajectory

MBRs

Method

Output

Most

similar

trajectory

to

Q.

Box

Env

constructM

BE

-LRB-

Q

-RRB-

Vector

V

Q

CreateM

BRs

-LRB-

Env

-RRB-

V

Q

contains

a

number

of

boxes

Priority

queue

P

Q

P

Q

keeps

one

entry

per

trajectory

sorted

according

to

the

similarity

estimate

for

each

box

B

in

V

Q

V

I.intersectionQuery

-LRB-

B

-RRB-

V

contains

all

trajectory

MBRs

that

intersect

with

B.

if

Method

Exact

upper

bound

P

Q

computeL-SimilarityEstimates

-LRB-

V

B

-RRB-

else

approximate

P

Q

computeV-SimilarityEstimates

-LRB-

V

B

-RRB-

BestSoF

ar

Best

while

P

Q

not

empty

E

PQ.top

if

E.estimate

BestSoF

ar

break

else

D

computeLCCS

-LRB-

Q

E

-RRB-

exact

if

D

BestSoF

ar

BestSoF

ar

D

Best

E

Report

Best

The

above

algorithm

can

be

adjusted

to

return

the

kNN

sequences

simply

by

comparing

with

the

k

th

bestSoF

ar

match

Next

we

examine

the

possible

similarity

estimates

Some

of

them

guarantee

that

will

find

the

best

match

-LRB-

they

lower

bound

the

original

distance

or

upper

bound

the

original

similarity

-RRB-

while

other

estimates

provide

faster

but

approximate

results

Similarity

Estimates

Here

we

will

show

how

to

compute

estimates

of

the

LCSS

similarity

based

on

the

geometric

properties

of

the

trajectory

MBRs

and

their

intersection

An

upper

bound

estimate

is

provided

by

the

length

of

the

MBR

intersection

and

an

approximate

estimate

is

given

as

a

parameter

of

the

intersecting

volume

To

formalize

these

notions

first

we

present

several

operators

Then

we

will

use

these

operators

to

derive

the

estimates

Estimates

for

the

LCSS

Each

trajectory

T

can

be

decomposed

into

a

number

of

MBRs

The

i

th

MBR

of

T

consists

of

six

numbers

M

T

i

-LCB-

t

l

t

h

x

l

x

h

y

l

y

h

-RCB-

Now

let

us

define

the

operators

-LRB-

c

-RRB-

t

-LRB-

p

-RRB-

t

and

V

between

two

MBRs

M

P

i

and

M

R

j

belonging

to

objects

P

and

R

respectively

-LRB-

c

-RRB-

t

-LRB-

M

P

i

M

R

j

-RRB-

Intersection

t

where

M

R

j

x

l

M

P

i

x

l

M

R

j

x

h

and

M

R

j

x

l

M

P

i

x

h

M

R

j

x

h

and

M

R

j

y

l

M

P

i

y

l

M

R

j

y

h

and

M

R

j

y

l

M

P

i

y

h

M

R

j

y

h

or

similarly

by

rotating

M

R

j

M

P

i

Therefore

this

operator

computes

the

time

intersection

of

two

MBR

when

one

fully

contains

the

other

in

the

x

y

dimensions

-LRB-

p

-RRB-

t

-LRB-

M

P

i

M

R

j

-RRB-

Intersection

t

otherwise

V

-LRB-

M

P

i

M

R

j

-RRB-

Intersection

t

Intersection

x

Intersection

y

We

can

use

upper

bound

or

approximate

estimates

for

the

similarity

Common

Volume

Intersection

The

Intersection

of

MBRs

is

fully

contained

within

one

MBR

Intersection

between

two

MBRs

time

y

x

Figure

Top

left

Intersection

recorded

in

list

L

t

partial

Top

right

Intersection

recorded

in

list

L

t

complete

Bottom

left

Percentage

of

Volume

Intersection

kept

in

L

V

Upper

bound

estimates

-LRB-

L-similarity

estimate

-RRB-

Such

estimates

are

computed

using

the

following

data-structures

The

list

L

t

complete

an

element

L

-LRB-

P

-RRB-

of

which

is

defined

as

L

-LRB-

P

-RRB-

m

n

M

Q

m

-LRB-

c

-RRB-

t

M

P

n

where

Q

is

a

query

and

P

is

a

trajectory

in

the

index

So

the

list

stores

for

each

trajectory

the

total

time

that

its

MBRs

intersected

with

the

query

's

MBRs

We

record

into

this

list

only

the

intersections

where

a

query

MBR

is

fully

contained

in

all

spatial

dimensions

by

a

trajectory

MBR

-LRB-

or

vice

versa

it

is

equivalent

See

figure

top

right

-RRB-

The

list

L

t

partial

an

element

L

-LRB-

P

-RRB-

of

which

is

defined

as

L

-LRB-

P

-RRB-

m

n

M

Q

m

-LRB-

p

-RRB-

t

M

P

n

This

list

records

for

each

sequence

the

total

intersection

in

time

for

those

query

MBRs

that

are

not

fully

contained

within

the

x

y

dimensions

by

the

trajectory

MBRs

-LRB-

or

vice

versa

Figure

top

left

-RRB-

Regarding

a

query

Q

for

any

trajectory

P

the

sum

of

L

t

complete

-LRB-

P

-RRB-

L

t

partial

-LRB-

P

-RRB-

will

provide

an

upper

bound

on

the

similarity

of

P

and

Q

The

reason

for

the

distinction

of

the

L-similarity

estimate

in

two

separate

lists

derives

from

the

fact

that

the

estimates

stored

in

list

L

t

partial

can

significantly

overestimate

the

LCSS

similarity

If

one

wishes

to

relax

the

accuracy

in

favor

of

enhanced

performance

it

is

instructive

to

give

a

weight

w

p

to

all

estimates

in

list

L

t

partial

Even

though

now

we

may

miss

the

best

match

to

our

query

we

are

going

to

find

a

close

match

in

less

time

This

weighted

approach

is

used

when

we

are

seeking

for

approximate

but

very

good

quality

answers

however

it

will

not

be

explained

further

due

to

space

limitations

Approximate

estimates

-LRB-

V-similarity

estimate

-RRB-

This

second

estimate

is

based

on

the

intersecting

volume

of

the

MBRs

This

type

of

estimates

are

stored

in

list

L

V

Any

element

L

V

-LRB-

P

-RRB-

of

list

L

V

records

similarity

estimates

between

trajectory

P

and

query

Q

based

on

the

total

volume

intersection

between

the

MBRs

of

P

and

Q.

L

-LRB-

P

-RRB-

length

-LRB-

P

-RRB-

m

n

M

Q

m

V

M

P

n

M

Q

m

V

M

Q

m

t

where

M

V

denotes

the

volume

of

MBR

M

and

M

t

its

length

on

the

time

axis

The

L-similarity

overestimates

the

LCSS

between

two

sequences

A

and

B

and

so

it

can

be

deployed

for

the

design

of

an

index

structure

Lemma

The

use

of

the

L-similarity

estimate

upper

bounds

the

LCSS

similarity

between

two

sequences

A

and

B

and

therefore

does

not

introduce

any

false

dismissals

The

V-similarity

estimate

can

be

used

for

approximate

query

answering

Even

though

it

does

not

guarantee

the

absence

of

false

dismissals

the

results

will

be

close

to

the

optimal

ones

with

high

probability

Also

because

this

estimate

provides

a

tighter

approximation

to

the

original

distance

we

expect

faster

response

time

Indeed

as

we

show

in

the

experimental

section

the

index

performance

is

boosted

while

the

error

in

similarity

is

frequently

less

then

Estimates

for

the

DTW

When

the

distance

function

used

is

the

Time

Warping

using

the

index

we

obtain

a

lower

bound

of

the

actual

distance

In

this

case

we

have

the

inverse

situation

from

the

LCSS

instead

of

calculating

the

degree

of

overlap

between

the

MBRs

of

the

indexed

trajectories

and

the

query

we

evaluate

the

distance

between

the

MBRs

The

overall

distance

between

the

MBRs

underestimates

the

true

distance

of

the

trajectories

and

no

false

dismissals

are

introduced

Using

the

MBRs

we

can

also

calculate

upper

bound

estimates

on

the

distance

which

had

n't

been

exploited

in

previous

work

-LSB-

-RSB-

Sequences

with

lower

bound

larger

than

the

smallest

upper

bound

can

be

pruned

With

this

additional

prefiltering

step

we

can

gain

on

average

an

additional

speedup

in

the

total

execution

time

Due

to

space

limitations

only

a

visual

representation

of

this

approach

is

provided

in

figure

Given

a

multidimensional

time-series

-LRB-

or

an

MBE

-RRB-

our

objective

is

to

minimize

the

volume

of

the

sequence

using

k

MBRs

Clearly

the

best

approximation

of

a

trajectory

-LRB-

or

an

MBE

-RRB-

using

a

fixed

number

of

MBRs

is

the

set

of

MBRs

that

completely

contain

the

sequence

and

minimize

the

volume

consumption

We

can

show

the

following

lemma

Lemma

Minimizing

the

volume

of

the

Minimum

Bounding

Envelope

minimizes

the

expected

similarity

approximation

error

Three

different

approaches

are

considered

k-Optimal

We

can

discover

the

k

MBRs

of

a

sequence

that

take

up

the

least

volume

using

a

dynamic

programming

algorithm

that

requires

O

-LRB-

n

k

-RRB-

time

-LRB-

-LSB-

-RSB-

-RRB-

where

n

is

the

length

of

the

given

sequence

Since

this

approach

is

not

reasonable

for

large

databases

we

are

motivated

to

consider

approximate

and

faster

solutions

Equi-Split

This

technique

produces

MBRs

of

fixed

length

l

It

is

a

simple

approach

with

cost

linear

in

the

length

of

a

sequence

However

in

pathological

cases

increasing

the

number

of

splits

can

result

to

larger

space

utilization

therefore

the

choice

of

the

MBR

length

becomes

a

critical

parameter

-LRB-

see

figure

for

an

example

-RRB-

A.

Query

Q

B.

Query

Envelope

C.

Envelope

Splitting

D.

Sequence

MBRs

E.

MINDIST

-LRB-

Q

R

-RRB-

F.

MAXDIST

-LRB-

Q

R

-RRB-

Figure

A

visual

intuition

of

the

DTW

indexing

technique

-LRB-

the

one-dimensional

case

is

shown

for

clarity

-RRB-

The

original

query

-LRB-

A

-RRB-

is

enclosed

in

a

minimum-bounding

envelope

-LRB-

B

-RRB-

like

the

LCSS

approach

The

MBE

is

split

into

its

MBRs

using

equi

or

greedy

split

-LRB-

fig

-LRB-

C

-RRB-

-RRB-

The

candidate

sequences

in

the

database

have

their

MBRs

stored

in

the

index

-LRB-

D

-RRB-

Between

the

query

and

any

sequence

in

the

index

the

minimum

and

maximum

distance

can

be

quickly

determined

by

examining

the

distance

between

the

MBRs

and

the

query

's

bounding

envelope

as

represented

by

the

arrows

in

-LRB-

E

-RRB-

and

-LRB-

F

-RRB-

Greedy-Split

The

Greedy

approach

is

our

implementation

choice

in

this

paper

Initially

we

assign

an

MBR

to

each

of

the

n

sequence

points

and

at

each

subsequent

step

we

merge

the

consecutive

MBRs

that

will

introduce

the

least

volume

consumption

The

algorithm

has

a

running

time

of

O

-LRB-

nlogn

-RRB-

We

can

see

a

sketch

of

the

method

in

fig.

Al-ternatively

instead

of

assigning

the

same

number

of

splits

to

all

objects

according

to

our

space

requirements

we

can

assign

a

total

of

K

splits

to

be

distributed

among

all

objects

This

method

can

provide

better

results

since

we

can

assign

more

splits

for

the

objects

that

will

yield

more

space

gain

Also

this

approach

is

more

appropriate

when

one

is

dealing

with

sequences

of

different

lengths

The

complexity

of

this

approach

is

O

-LRB-

K

N

logN

-RRB-

for

a

total

of

N

objects

-LRB-

-LSB-

-RSB-

-RRB-

Input

A

spatiotemporal

trajectory

T

and

an

integer

k

denoting

the

number

of

final

MBRs

For

i

n

compute

the

volume

of

the

MBR

produced

by

merging

T

i

and

T

i

The

results

are

stored

in

a

priority

queue

While

M

BRs

k

Using

the

priority

queue

merge

the

pair

of

consecutive

MBRs

that

yield

the

smallest

increase

in

volume

Delete

the

two

merged

MBRs

and

insert

the

new

one

in

the

priority

queue

Output

A

set

of

MBRs

that

cover

T

Figure

The

greedy

algorithm

for

producing

k

MBRs

that

cover

the

trajectory

T

After

a

trajectory

is

segmented

the

MBRs

can

be

stored

in

a

Using

the

greedy

split

each

additional

split

will

always

lead

to

smaller

-LRB-

or

equal

-RRB-

volume

-LRB-

figure

-RRB-

A

similar

greedy

split

algorithm

is

also

used

for

splitting

the

MBE

of

the

query

trajectory

Q.

-LRB-

a

-RRB-

Equi-Split

MBRs

Gain

-LRB-

b

-RRB-

Equi-Split

MBRs

Gain

-LRB-

c

-RRB-

Greedy-Split

Gain

-LRB-

d

-RRB-

Greedy-Split

Gain

Figure

-LRB-

a

-RRB-

MBRs

produced

using

equi-Split

The

volume

gain

over

having

MBR

is

-LRB-

b

-RRB-

Segmenting

into

MBRs

decreases

the

volume

gain

to

So

disk

space

is

wasted

without

providing

a

better

approximation

of

the

trajectory

-LRB-

c

-RRB-

MBRs

using

greedy-Split

The

volume

gain

over

having

MBR

is

-LRB-

d

-RRB-

Every

additional

split

will

yield

better

space

utilization

Segmentation

into

MBRs

increases

volume

gain

to

The

application

of

the

Minimum

Bounding

Envelope

only

on

the

query

suggests

that

user

queries

are

not

confined

to

a

predefined

and

rigid

matching

window

The

user

can

pose

queries

of

variable

warping

in

time

In

some

datasets

there

is

no

need

to

perform

warping

since

the

Euclidean

distance

performs

acceptably

-LSB-

-RSB-

In

other

datasets

by

using

the

Euclidean

distance

we

can

find

quickly

some

very

close

matches

while

using

warping

we

can

distinguish

more

flexible

similarities

So

we

can

start

by

using

a

query

with

-LRB-

no

bounding

envelope

-RRB-

and

increase

it

progressively

in

order

to

find

more

flexible

matches

-LRB-

figure

-RRB-

Therefore

our

framework

offers

the

unique

advantage

that

multiple

distance

functions

can

be

supported

in

a

single

index

The

index

sequences

have

been

segmented

without

any

envelope

applied

on

them

and

never

have

to

be

adjusted

again

For

different

measures

the

aspects

that

change

are

the

creation

of

the

query

envelope

and

the

type

of

operation

between

MBRs

In

order

to

pose

queries

based

on

Euclidean

distance

we

follow

the

steps

The

query

is

segmented

with

no

envelope

applied

on

it

The

minDist

and

maxDist

estimators

for

the

Euclidean

distance

are

derived

by

calculating

the

distance

between

the

query

and

index

MBRs

just

like

in

the

DTW

case

Index

Trajectory

Query

Figure

By

incorporating

the

bounding

envelope

on

the

query

our

approach

can

support

Euclidean

distance

constrained

or

full

warping

This

is

accomplished

by

progressively

expanding

the

MBE

In

this

section

we

compare

the

effectiveness

of

various

splitting

methods

and

we

demonstrate

the

superiority

of

our

lower

bounding

technique

-LRB-

for

the

DTW

-RRB-

compared

to

other

proposed

lower

bounds

We

describe

the

datasets

we

used

and

present

comprehensive

experiments

regarding

the

index

performance

for

the

two

similarity

estimates

In

addition

we

evaluate

the

accuracy

of

the

approximate

estimates

All

experiments

conducted

were

run

on

an

AMD

Athlon

Ghz

with

GB

RAM

and

GB

of

hard

drive

ASL

Buoy

Sensor

Video

Track

Flutter

Marine

Mammals

Word

Tracking

Random

Walk

Video

Track

Figure

Datasets

used

for

testing

the

efficiency

of

various

MBR

generation

methods

MBR

Generation

Comparison

The

purpose

of

our

first

experiment

is

to

test

the

space

consumption

of

the

presented

MBR

generation

methods

We

have

used

eight

datasets

with

diverse

characteristics

in

order

to

provide

objective

results

We

evaluate

the

space

consumption

by

calculating

the

Average

Volume

Gain

-LRB-

AvgV

olGain

-RRB-

which

is

defined

as

the

percentage

of

volume

when

using

i

MBRs

over

the

volume

when

using

only

MBR

normalized

by

the

maximum

gain

provided

over

all

methods

-LRB-

for

various

number

of

splits

-RRB-

Random

Equi

Greedy

Dataset

Average

Volume

Gain

Figure

The

greedy-split

MBR

generation

algorithm

presents

the

highest

volume

gain

by

producing

MBRs

that

consume

consistently

less

space

over

a

number

of

datasets

and

for

diverse

number

of

generated

MBRs

DATASET

EQ

GR

EQ

GR

EQ

GR

EQ

GR

LB-Kim

LB-Yi

LCSS

DTW

ASL

Marine

Word

Random

Table

Some

indicative

results

of

how

close

our

similarity

estimates

are

to

the

exact

value

-LRB-

for

and

splits

-RRB-

For

all

datasets

the

greedy-split

approach

provides

the

closest

similarity

estimates

to

the

actual

similarity

AvgV

olGain

is

a

number

between

and

where

higher

numbers

indicate

increased

volume

gain

-LRB-

or

less

space

consumption

-RRB-

against

the

competitive

methods

In

figure

we

observe

the

average

volume

gain

for

the

eight

datasets

The

greedy-split

algorithm

produced

MBRs

that

took

at

least

half

the

space

compared

to

equi-split

The

equi-split

offers

slightly

better

results

than

producing

MBRs

at

random

positions

The

volume

gain

of

greedy-split

was

less

only

for

the

buoy

sensor

which

is

a

very

busy

and

unstruc-tured

signal

This

experiment

validates

that

our

choice

to

use

the

greedy-split

method

was

correct

Since

the

indexed

MBR

trajectories

will

take

less

space

we

also

expect

tighter

similarity

estimates

therefore

fewer

false

positives

Tightness

of

Bounds

In

table

we

show

how

close

our

similarity

estimates

are

-LRB-

for

LCSS

and

DTW

-RRB-

to

the

actual

similarity

between

sequences

Numbers

closer

to

indicate

higher

similarity

to

the

value

returned

by

the

exact

algorithm

To

our

best

knowledge

this

paper

introduces

the

first

upper

bounding

technique

for

the

LCSS

For

DTW

there

have

been

a

few

approaches

to

provide

a

lower

bound

of

the

distance

we

refer

to

them

as

LB-Kim

-LSB-

-RSB-

and

LB-Yi

-LSB-

-RSB-

These

lower

bounds

originally

referred

to

time-series

here

we

extend

them

in

more

dimensions

in

order

to

provide

unambiguous

results

about

the

tightness

of

our

estimates

Note

that

the

previously

proposed

methods

operate

on

the

raw

data

Our

approach

can

still

provide

tighter

estimates

while

operating

only

on

the

trajectory

MBRs

Using

the

raw

data

our

experiments

indicate

that

we

are

consistently

times

better

than

the

best

alternative

approach

However

since

our

index

operates

on

the

segmented

time-series

we

only

report

the

results

on

the

MBRs

The

greedy-split

method

approximates

the

similarity

consistently

tighter

than

the

equi-split

In

table

only

the

results

for

of

the

query

's

length

are

reported

but

similar

results

are

observed

for

increasing

values

of

It

is

evident

from

the

table

that

using

our

method

we

can

provide

very

tight

lower

bounds

of

the

actual

distance

Matching

Quality

We

demonstrate

the

usefulness

of

our

similarity

measures

in

a

real

world

dataset

The

Library

of

Congress

maintains

thousands

of

handwritten

manuscripts

and

there

is

an

increasing

interest

to

perform

automatic

transcribing

of

these

documents

Given

the

multiple

variations

of

each

word

and

due

to

the

manuscript

degradations

this

is

a

particularly

challenging

task

and

the

need

for

a

flexible

and

robust

distance

function

is

essential

We

have

applied

the

LCSS

and

DTW

measures

on

word

Figure

Results

for

a

real

world

application

reported

for

each

query

using

Dynamic

Time

Warping

to

match

features

extracted

from

scanned

manuscript

words

images

extracted

from

a

page

scanned

manuscript

time-series

features

have

originally

been

extracted

for

each

word

Here

we

maintain

the

least

correlated

time-series

features

and

treat

each

word

as

a

trajectory

In

figure

we

observe

the

results

using

DT

W

for

various

word

queries

The

results

are

very

good

showing

high

accuracy

even

for

similarly

looking

words

Analogous

results

have

been

obtained

using

the

LCSS

Index

performance

We

tested

the

performance

of

our

index

using

the

upper

bound

and

the

approximate

similarity

estimates

and

compared

it

to

the

sequential

scan

Because

of

limited

space

the

majority

of

the

figures

record

the

index

performance

using

the

LCSS

as

a

similarity

measure

The

performance

measure

used

is

the

total

computation

time

required

for

the

index

and

the

sequential

scan

to

return

the

nearest

neighbor

for

the

same

one

hundred

queries

For

the

linear

scan

one

can

also

perform

early

termination

of

the

LCSS

-LRB-

or

the

DTW

-RRB-

computation

Therefore

the

LCSS

execution

can

be

stopped

at

the

point

where

one

is

sure

that

the

current

sequence

will

not

be

more

similar

to

the

query

than

the

bestSoFar

We

call

this

optimistic

linear

scan

Pessimistic

linear

scan

is

the

one

than

does

not

reuse

the

previously

computed

similarity

values

and

can

be

an

accurate

time

estimate

when

the

query

match

resides

at

the

end

of

the

dataset

We

demonstrate

the

index

performance

relative

to

Dataset

size

Time

Ratio

Compared

to

Linear

Scan

Optimistic

Pessimistic

Linear

Scan

Dataset

size

Time

Ratio

Compared

to

Linear

Scan

Optimistic

Pessimistic

Linear

Scan

Dataset

size

Time

Ratio

Compared

to

Linear

Scan

Optimistic

Pessimistic

Linear

Scan

Figure

Index

performance

For

small

warping

windows

the

index

can

be

up

to

times

faster

than

sequential

scan

without

compromising

accuracy

The

gray

regions

indicate

the

range

of

potential

speedup

Dataset

size

Time

Ratio

Compared

to

Linear

Scan

Optimistic

Pessimistic

Linear

Scan

Dataset

size

Time

Ratio

Compared

to

Linear

Scan

Optimistic

Pessimistic

Linear

Scan

Dataset

size

Time

Ratio

Compared

to

Linear

Scan

Optimistic

Pessimistic

Linear

Scan

Figure

Using

the

approximate

similarity

estimates

the

response

time

can

be

more

than

times

faster

both

types

of

linear

scan

because

this

provides

a

realistic

upper

or

lower

bound

on

the

index

speedup

The

dataset

we

used

contained

trajectories

Taking

under

consideration

that

the

average

trajectory

size

is

around

points

this

resulted

to

a

database

with

more

than

million

points

The

trajectories

have

been

normalized

by

subtracting

the

average

value

in

each

direction

of

movement

All

data

and

queries

can

be

obtained

by

emailing

the

first

author

Mixed

two-dimensional

Time-series

-LRB-

-RRB-

This

second

dataset

consists

of

time-series

of

variable

length

ranging

from

less

than

points

to

over

points

The

dataset

is

comprised

by

the

aggregation

of

the

eight

datasets

we

used

for

comparing

the

MBR

generation

methods

Since

the

total

number

of

these

trajectories

is

less

than

we

have

used

them

as

seeds

to

generate

increasingly

larger

datasets

We

create

multiple

copies

of

the

original

trajectories

by

incorporating

the

following

features

Addition

of

small

variations

in

the

original

trajectory

pattern

Addition

of

random

compression

and

decompression

in

time

The

small

variations

in

the

pattern

were

added

by

interpolating

peaks

of

Gaussian

noise

using

splines

In

this

manner

we

are

able

to

create

the

smooth

variations

that

existed

in

the

original

datasets

Results

on

the

upper

bound

Estimates

The

index

performance

is

influenced

be

three

parameters

the

size

of

the

dataset

the

warping

length

-LRB-

as

a

percentage

of

the

query

's

length

-RRB-

and

the

number

of

splits

For

all

experiments

the

parameter

-LRB-

matching

in

space

-RRB-

was

set

to

of

the

query

which

provided

good

and

intuitive

results

Dataset

size

In

figure

we

can

observe

how

the

performance

of

the

index

scales

with

the

database

size

-LRB-

for

various

lengths

of

matching

window

-RRB-

We

record

the

index

response

time

relative

to

both

optimistic

and

pessimistic

linear

scan

Therefore

the

gray

region

in

the

figures

indicates

the

range

of

possible

speedup

It

is

evident

that

the

early

termination

feature

of

the

sequential

scan

can

significantly

assist

its

performance

The

usefulness

of

an

index

becomes

obvious

for

large

dataset

sizes

where

the

quadratic

computational

cost

dominates

the

I/O

cost

of

the

index

For

these

cases

our

approach

can

be

up

to

times

faster

than

linear

scan

In

figure

we

also

demonstrate

the

pruning

power

of

the

index

as

a

true

indicator

-LRB-

not

biased

by

any

implementation

details

-RRB-

about

the

efficacy

of

our

index

Using

the

index

we

perform

times

fewer

LCSS

computations

than

the

linear

scan

We

observe

similar

speedup

when

using

the

DTW

as

the

distance

function

in

figure

Parameter

The

index

performance

is

better

for

smaller

warping

lengths

-LRB-

parameter

-RRB-

The

experiments

record

the

performance

for

warping

from

to

of

the

query

's

length

Increasing

values

signify

larger

bounding

envelopes

around

the

query

therefore

larger

space

of

search

and

less

accurate

similarity

estimates

The

graphs

suggest

that

an

index

can

not

not

be

useful

under

full

warping

-LRB-

when

the

data

are

normalized

-RRB-

Number

of

Splits

Although

greater

number

of

MBRs

for

each

trajectory

implies

better

volume

utilization

nonetheless

more

MBRs

also

lead

to

increased

I/O

cost

When

we

are

referring

to

x

splits

it

means

that

we

have

assigned

a

total

of

-LRB-

n

i

-LRB-

T

i

-RRB-

-RRB-

splits

for

all

sequences

T

i

In

our

figures

we

provide

the

splits

scenario

for

the

MBRs

which

offers

better

performance

than

and

splits

since

for

the

last

two

cases

the

I/O

cost

negates

the

effect

of

the

better

query

approximation

The

index

space

requirements

for

splits

is

less

than

a

quarter

of

the

dataset

size

Results

on

the

approximate

Estimates

Here

we

present

the

index

performance

when

the

volume

intersections

of

the

MBRs

are

used

as

estimates

of

the

Dataset

size

Ratio

of

LCSS

performed

by

the

index

Pruning

Power

compared

to

Linear

Scan

splits

splits

splits

Linear

Scan

Figure

Each

gray

band

indicates

-LRB-

for

a

certain

warping

window

-RRB-

the

percentage

of

LCSS

computations

conducted

by

the

index

compared

to

linear

scan

Dataset

size

Average

similarity

Error

Similarity

Error

splits

Figure

Using

the

V-similarity

estimate

we

can

retrieve

answers

faster

with

very

high

accuracy

The

LCSS

similarity

is

very

close

-LRB-

-RRB-

to

the

exact

answer

returned

by

the

sequential

scan

Dataset

size

Time

Ratio

Compared

to

Linear

Scan

Optimistic

Pessimistic

Linear

Scan

Figure

Index

Performance

using

DTW

as

the

distance

measure

-LRB-

-RRB-

We

can

observe

up

to

times

speedup

ilarity

and

the

results

are

shown

in

figure

We

observe

that

using

this

approximate

similarity

estimate

our

index

performance

is

boosted

up

The

use

of

the

V-similarity

estimate

leads

to

more

tight

approximations

of

the

original

similarity

compared

to

the

L-similarity

estimate

however

now

we

may

miss

finding

the

best

match

Naturally

comes

the

question

of

the

quality

of

the

results

We

capture

this

by

calculating

the

absolute

difference

between

the

similarity

of

the

best

match

returned

by

the

index

and

the

best

match

found

by

the

sequential

scan

for

each

query

Then

we

average

the

results

over

a

number

of

queries

q

Therefore

the

Average

Similarity

Error

-LRB-

ASE

-RRB-

is

ASE

q

q

i

-LRB-

BestMatch

index

BestM

atch

exhaustive

-RRB-

The

results

are

shown

in

figure

We

can

see

that

the

similarity

returned

by

the

V-similarity

estimate

is

approxi-mately

within

of

the

actual

similarity

-LRB-

splits

used

-RRB-

Therefore

by

providing

two

similarity

estimates

the

user

can

decide

for

the

trade-off

between

the

expedited

execution

time

and

the

quality

of

results

Since

by

using

the

latter

estimator

we

can

significantly

increase

the

performance

of

the

index

this

is

the

approach

we

recommend

for

mining

large

datasets

In

this

paper

we

have

presented

an

external

memory

indexing

method

for

discovering

similar

multidimensional

time-series

The

unique

advantage

of

our

approach

is

that

it

can

accommodate

multiple

distance

measures

The

method

guarantees

no

false

dismissals

and

depicts

a

significant

execution

speed

up

for

the

LCSS

and

DTW

compared

to

sequential

scan

We

have

shown

the

tightness

of

our

similarity

estimates

and

demonstrated

the

usefulness

of

our

measures

for

challenging

real

world

applications

We

hope

that

our

effort

can

act

as

a

bridge

between

metric

and

non-metric

functions

as

well

as

a

tool

for

understanding

better

their

strengths

and

weaknesses

In

the

future

we

plan

to

investigate

the

combination

of

several

heuristics

in

order

to

provide

even

tighter

estimates

Acknowledgements

We

would

like

to

thank

Margrit

Betke

for

providing

us

the

Video

Track

I

and

II

datasets

We

also

feel

obliged

to

T.

Rath

and

R.

Manmatha

for

kindly

providing

the

manuscript

words

dataset

-LSB-

-RSB-

J.

Aach

and

G.

Church

Aligning

gene

expression

time

series

with

time

warping

algorithms

In

Bioinformatics

Volume

pages

-LSB-

-RSB-

O.

Arikan

and

D.

Forsyth

Interactive

motion

generation

from

examples

In

Proc

of

ACM

SIGGRAPH

-LSB-

-RSB-

Z.

Bar-Joseph

G.

Gerber

D.

Gifford

T.

Jaakkola

and

I.

Simon

A

new

approach

to

analyzing

gene

expression

time

series

data

In

Proc

of

RECOMB

pages

-LSB-

-RSB-

D.

Berndt

and

J.

Clifford

Using

Dynamic

Time

Warping

to

Find

Patterns

in

Time

Series

In

Proc

of

KDD

Workshop

-LSB-

-RSB-

M.

Betke

J.

Gips

and

P.

Fleming

The

camera

mouse

Visual

tracking

of

body

features

to

provide

computer

access

for

people

with

severe

disabilities

In

IEEE

Transactions

on

Neural

Systems

and

Rehabilitation

Engineering

Vol

No.

-LSB-

-RSB-

G.

Das

D.

Gunopulos

and

H.

Mannila

Finding

Similar

Time

Series

In

Proc

of

the

First

PKDD

Symp.

pages

-LSB-

-RSB-

D.

Gavrila

and

L.

Davis

Towards

model-based

tracking

and

recognition

of

human

movement

a

multi-view

approach

In

Int

Workshop

on

Face

and

Gesture

Recognition

-LSB-

-RSB-

M.

Hadjieleftheriou

G.

Kollios

V.

Tsotras

and

D.

Gunopulos

Efficient

indexing

of

spatiotemporal

objects

In

Proc

of

EDBT

-LSB-

-RSB-

T.

Kahveci

A.

Singh

and

A.

Gurel

Similarity

searching

for

multi-attribute

sequences

In

Proc

of

SSDBM

-LSB-

-RSB-

E.

Keogh

Exact

indexing

of

dynamic

time

warping

In

Proc

of

VLDB

-LSB-

-RSB-

E.

Keogh

and

S.

Kasetty

On

the

need

for

time

series

data

mining

benchmarks

A

survey

and

empirical

demonstration

In

Proc

of

SIGKDD

-LSB-

-RSB-

S.

Kim

S.

Park

and

W.

Chu

An

index-based

approach

for

similarity

search

supporting

time

warping

in

large

sequence

databases

In

In

Proc

of

ICDE

-LSB-

-RSB-

Z.

Kov

acs-Vajna

A

fingerprint

verification

system

based

on

triangular

matching

and

dynamic

time

warping

In

IEEE

Transactions

on

PAMI

Vol

No.

-LSB-

-RSB-

S.-L

Lee

S.-J

Chun

D.-H

Kim

J.-H

Lee

and

C.-W

Chung

Similarity

Search

for

Multidimensional

Data

Sequences

Proc

of

ICDE

pages

-LSB-

-RSB-

S.

Park

W.

Chu

J.

Yoon

and

C.

Hsu

Efficient

Searches

for

Similar

Subsequences

of

Different

Lengths

in

Sequence

Databases

In

Proc

of

ICDE

pages

-LSB-

-RSB-

T.

Rath

and

R.

Manmatha

Word

image

matching

using

dynamic

time

warping

In

Tec

Report

Center

for

Intelligent

Information

Retrieval

University

of

Massachusetts

Amherst

-LSB-

-RSB-

J.

F.

Roddick

and

K.

Hornsby

Temporal

Spatial

and

Spatio-Temporal

Data

Mining

-LSB-

-RSB-

M.

Shimada

and

K.

Uehara

Discovery

of

correlation

from

multi-stream

of

human

motion

In

Discovery

Science

-LSB-

-RSB-

R.

E.

Valdes-Perez

and

C.

A.

Stone

Systematic

detection

of

subtle

spatio-temporal

patterns

in

time-lapse

imaging

ii

particle

migrations

In

Bioimaging

-LRB-

-RRB-

pages

-LSB-

-RSB-

M.

Vlachos

G.

Kollios

and

D.

Gunopulos

Discovering

similar

multidimensional

trajectories

In

Proc

of

ICDE

-LSB-

-RSB-

B.-K

Yi

H.

V.

Jagadish

and

C.

Faloutsos

Efficient

retrieval

of

similar

time

sequences

under

time

warping

In

Proc

of

ICDE

pages

-LSB-

-RSB-

Y.

Zhu

and

D.

Shasha

Query

by

humming

a

time

series

database

approach

In

Proc

of

SIGMOD

trajectory
similarity
distance function
longest common subsequence
dynamic time warping
trajectories
indexing
measure

dynamic time warping (dtw)
longest common subsequence (lcss)