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Fig. 6.
Translation of sequence
A
.
2 range from 0 to 1. Therefore we
can define the distance function between two sequences as follows:
So the similarity functions
S
1and
S
Definition 6.
Given
δ, ε
and two sequences
A
and
B
we define the following
distance functions:
D
1(
δ, ε, A, B
)=1
− S
1(
δ, ε, A, B
)
and
D
2(
δ, ε, A, B
)=1
− S
2(
δ, ε, A, B
)
Note that
D
1and
D
2are
symmetric
.
LCSS
δ,ε
(
A, B
) is equal to
LCSS
δ,ε
(
B, A
) and the transformation that we use in
D
2 is translation
which preserves the symmetric property.
By allowing translations, we can detect similarities between movements
that are parallel, but not identical. In addition, the
LCSS
model allows
stretching and displacement in time, so we can detect similarities in move-
ments that happen with different speeds, or at different times. In Figure 6
we show an example where a sequence
A
matches another sequence
B
after
a translation is applied.
The similarity function
1,
because: (i) now we can detect parallel movements, (ii) the use of
normal-
ization
does not guarantee that we will get the best match between two
time-series. Usually, because of the significant amount of noise, the average
value and/or the standard deviation of the time-series that are being used in
the normalization process can be distorted leading to improper translations.
S
2 is a significant improvement over the
S
3.3.
Differences between
DTW
and
LCSS
Time Warping and the
LCSS
share many similarities. Here, we argue that
the
LCSS
is a better similarity function for correctly identifying noisy