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multivariate patterns that (re)occur in pairwise compared time series. In dependence
on JCRPs and known RQAmeasures, such as determinism, we define a
R
ecu
RR
ence
plot-based (RRR) distance measure, which reflects the proportion of time series seg-
ments with similar trajectories or recurring patterns, respectively.
In order to demonstrate the practicability of our proposed recurrence plot-based
distance measure, we conduct experiments on both synthetic time series and real-
life vehicular sensor data [
32
,
33
,
35
]. The results show that, unlike commonly used
(dis)similarity functions, our proposed distance measure is able to (i) determine clus-
ter centers that preserve the characteristics of the data sequences and, furthermore,
(ii) identify prototypical time series that cover a high amount of recurring patterns.
The rest of the chapter is organized as follows. In Sect.
12.2
, we state the general
problem being investigated. Related work is discussed in Sect.
12.3
. Subsequently,
we introduce traditional recurrence plots as well as various extensions in Sect.
12.4
.
Recurrence quantification analysis and corresponding measures are discussed in
Sect.
12.5
. Our proposed recurrence plot-based distance measure and respective
evaluation criteria are introduced in Sect.
12.6
. Possible ways to reduce the com-
putational complexity of our introduced distance measure are offered in Sects.
12.7
and
12.8
. Our experimental results are presented and discussed in Sect.
12.9
. In addi-
tion, Sect.
12.10
presents BestTime, a platform-independent Matlab application with
graphical user interface, which enables us to find representative that best comprehend
the recurring temporal patterns contained in a certain time series dataset. Finally, we
conclude with future work in Sect.
12.11
.
12.2 Problem Statement
Car manufacturers aim to optimize the performance of newly developed engines
according to operational profiles that characterize recurring driving behavior. To
obtain real-life operational profiles for exhaust simulations, Volkswagen (VW) col-
lects data from test drives for various combinations of driver, vehicle, and route.
Given a set
X
={
X
1
,
X
2
,...,
X
t
}
of
t
test drives, the challenge is to find a
subset of
k
prototypical time series
Y
={
Y
1
,...,
Y
k
}∈
X
that best comprehend
the recurring (driving behavior) patterns found in set
X
. Test drives are represented
d
is
a
d
-dimensional feature vector summarizing the observed measurements at time
i
.
A
pattern S
as multivariate time series
X
=
(
x
1
,...,
x
n
)
of varying length
n
, where
x
i
∈ R
=
(
x
s
,...,
x
s
+
l
−
1
)
of
X
=
(
x
1
,...,
x
n
)
is a subsequence of
l
consecu-
tive time points from
X
, where
l
≤
n
and 1
≤
s
<
s
+
l
−
1
≤
n
. Assuming two time
series
X
=
(
x
1
,...,
x
n
)
and
Y
=
(
y
1
,...,
y
m
)
with patterns
S
=
(
x
s
,...,
x
s
+
l
−
1
)
and
P
=
(
y
p
,...,
y
p
+
l
−
1
)
of length
l
, we say that
S
and
P
are
recurring patterns
ₒ R
+
is a (dis)similarity function
of
X
and
Y
if
d
(
S
,
P
)
≤
, where and
d
:
X
×
X
and
is a certain similarity threshold. Note that recurring patterns of
X
and
Y
may
occur at arbitrary positions and in different order.
Since we aim to identify
k
prototypical time series that (i) best represent the set
X
X
and (ii) are members of the set
, one can employ the
k
-medoid clustering algorithm.
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