d Tw Algorithm

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Gene expression time series are expected to vary not

only in terms of expression amplitudes, but also in terms

of time progression since biological processes may unfold

with different rates in response to different experimental

conditions or within different organisms and individuals.

What is Special about Time Series Data?

time


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i

i+2

i

i i

timetime

Why Dynamic Time Warping?

Any distance (Euclidean, Manhattan,

… !hich aligns the i"th point on onetime series !ith the i"th point on the

other !ill produce a poor similarity

score#

A non"linear (elastic alignment

produces a more intuiti$e similaritymeasure, allo!ing similar shapes to

match e$en i% they are out o% phase in

the time a&is#


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j s

i s

m

'

n'

me Series B

Time Series A

Warping unction

pk

ps

p1

To find the best alignment

between A and B one needs

to find the path through the

grid

P = p1, … , ps , … , pk

ps ) (i s , j s

which minimizes the total

distance between them.

P is called a warping function.


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Time"*ormali+ed Distance Measure

D( A , B )

⋅

∑

∑

=

=

k

s

s

k

s

s s

w

w pd

'

'

(

d ( p s: distance between i s and j s

P

minarg

w s -: weighting coefficient.

Best alignment path between A

and B :

Time-normalized distance between

A and B :

P 0 ) ( D( A ,

B #

j s

i s

m

'

n'

me Series B

Time Series A

pk

ps

p1


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.ptimisations to the DTW Algorithm

The number of possible warping

paths through the grid is

exponentially explosive!

Restrictions on the warping function:

• monotonicity

• continuity• boundary conditions

• warping window

• slope constraint.

reduction of thesearch space

j s

i s

m

'

n'

me Series B

Time Series A


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/estrictions on the Warping unction

Monotonicity : i s-1 0 i s and j s-1 0 j s.

The alignment path does not go bac

in time" index.

Continuity : i s 1 i s-1 0 ' and j s 1 j s-1 0 '.

The alignment path does not #ump in

time" index.

Guarantees that features are not

repeated in the alignment.

Guarantees that the alignment does

not omit important features.

i

j

i

j


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Boundary Conditions: i 1 ) ', i k ) n and j 1 ) ', j k ) m.

The alignment path starts at the bottom

left and ends at the top right.

Warping Window : i s 1 j s 0 r , where r - is the window length.

$ good alignment path is unliely to

wander too far from the diagonal.

Guarantees that the alignment does not

consider partially one of the se%uences.

Guarantees that the alignment does not

try to sip different features and gets

stuc at similar features.

n

m

i

j

(',' i

j

r


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Slope Constraint : ( j s p 1 j s- / ( i s p 1 i s- 0 p and ( i sq 1 i s- / ( j sq 1 j s- 0 q , where q 3 -

is the number of steps in the x &direction and p 3 - is the number of steps in the y&

direction. $fter q steps in x one must step in y and vice versa: S ) p / q ∈4- , ∝5#

'revents that very short parts of the se%uences

are matched to very long ones.

The alignment path should not be too steep or

too shallow.

i

j ≤ p ≤ q


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The 6hoice o% the Weighting 6oe%%icient

D( A , B ) #

(

min

'

'

⋅

∑

∑

=

=k

s

s

k

s

s s

P

w

w pd

Time&normali(ed distance between A and B :

complicates

optimisation

⋅∑

=

k

s

s s P

w pd C '

(min'

D( A , B )

∑=

=k

s

swC

'

)eeing a weighting coefficient function which

guarantees that:

can be solved by use of dynamic programming.

is independent of the warping function. Thus

*eighting +oefficient efinitions

• Symmetric form

w s ) (i s 1 i s"' 7 ( j s 1 j s-',

then C = n 7 m.

• Asymmetric form

w s ) (i s 1 i s-',

then C = n#

-r e%uivalently,

w s ) ( j s 1 j s-',

then C = m#


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nitial condition: g(',' ) 2d(','#

'&e%uation:

g(i , j 1 ' 7 d(i , j

g(i , j ) min g(i 1 ', j 1 ' 7 2d(i , j #

g(i 1 ', j 7 d(i , j

*arping window: j 1 r 0 i 0 j 7 r #

Time&normali(ed distance:

D( A , B ) g(n, m 8 C

C = n 7 m.

Symmetric DTW Algorithm

(!arping !indo!, no slope constraint

j

m

'

n' i

g(','

g(n, m

i ) j 7 r

i ) j " r

''

2

me Series B

Time Series A


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nitial condition: g(',' ) d(','#

'&e%uation:

g(i , j 1 ' 7 d(i , j

g(i , j ) min g(i 1 ', j 1 ' 7 d(i , j #

g(i 1 ', j 7 d(i , j

*arping window: j 1 r 0 i 0 j 7 r #

Time&normali(ed distance:

D( A , B ) g(n, m 8 C

C = n 7 m.

9ua+i"symmetric DTW Algorithm

(!arping !indo!, no slope constraint

j

m

'

n' i

g(','

g(n, m

i ) j 7 r

i ) j " r

''

'

me Series B

Time Series A


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DTW Algorithm< E&ample

0!"# 0!"$ 0!"% 0!"& 0!&' 1!(% 1!') 0!)0 0!0 0!&(

0!"" 0!(1 0!"$ 0!"& 0!&$ 1!(& 1!$1 0!%1 0!0' 0!1"

0 !

) 0

0 !

) %

0 !

# 1

0 !

% "

0 !

1 #

0 ! #

#

1 !

( $

0 !

$ )

0 !

) &

0 !

) "

0 !

) '

0 !

' &

0 ! #

$

1 !

( #

0!0& -#-= -#-> -#'' -#' -#@ -#@ -#=> -#B -#BB

-#-@ 0!0$ -#-B -#-> -#'' -#2 -#@ -#= -#B@ -#BB

-#-B -#-B 0!0) -#-C -#'' -#2 -#=- -#B- -#B= -#B>

-#-> -#-> -#-> 0!0" -#'- -#' -#@C -#=C -#B2 -#B=

-#' -#' -#' -#'2 0!0" -#2B -#@- -#@C -#@ -#@

-#2C -#2C -#2B -#2= -#'B 0!1" 0!&' 0!&% 0!'1 -#B>

-#=' -#=' -#@ -#@ -#= -#'C -#2' -# -#@' 0!$(

Time Series B

Time Series A

Euclidean distance bet!een $ectors

d Tw Algorithm

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