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JR
x
,
y
i
x
y
x
y
j
(
,
)
=
Θ(
−||
x
i
−
x
j
||
)
·
Θ(
−||
y
i
−
y
j
||
)
(12.3)
,
d
1
d
2
x
i
∈ R
,
y
j
∈ R
,
i
,
j
=
1
...
n
Such joint recurrence plots have the advantage that the individual measurements
can present different observables with different magnitudes or range. They are often
used for the detection of phase synchronization [
19
,
21
].
Since this work aims at clustering test drives, which involves pairwise
(dis)similarity comparisons of multivariate time series, we propose a combination of
joint and cross recurrence plot, namely (JCRP) joint cross recurrence plot. A JCRP
shows all those times at which a multivariate state in one dynamical system occurs
simultaneously in a second dynamical system.
JCR
x
,
y
i
1
k
1
x
i
y
j
||
)
×···×
Θ(
k
x
i
y
j
||
)
j
(
,...,
)
=
Θ(
−||
−
−||
−
(12.4)
,
d
x
i
,
y
j
∈ R
,
i
=
1
...
n
,
j
=
1
...
m
For the creation of a JRCP both trajectories,
x
and
y
, need to have the same
dimensionality or number of parameters
d
, but can have different length,
n
and
m
.
We shall see that JCRPs are very useful, because they enable us to compare two mul-
tivariate systems with the same set of observables that can have different magnitudes.
In other words, the introduced JCR notation allows us to determine an
-threshold
for each individual parameter, which is advantageous for observables with different
variance. A toy example for JCRPs is given in the following:
dfcghGATHERSPEEDlmknhDECELERATEghfkd
rsqtpACCELERATORxywzvBRAKEPEDALtvswr
x
=
kdhfSLOWDOWNglbkchdgfGATHERSPEEDnkml
tpsBRAKEPEDALzrysxtwvACCELERATORxtwv
y
=
Assume two multivariate time series
x
and
y
which comprise the speed and accel-
erator signal recorded during different car drives. Both time series contain multivari-
ate states or rather string sequences that occur in both systems, as demonstrated in
Fig.
12.1
a. The corresponding JCRP of
x
and
y
, as illustrated in Fig.
12.1
b, shows
the times at which a multivariate state occurs simultaneously in both systems. Fur-
thermore, the diagonal line structure in Fig.
12.1
b reveals that both trajectories run
through a similar region in phase space for a certain time interval. With other words,
both systems contain the same multivariate pattern, which represents that the driver
hits the 'ACCELERATOR' pedal and the vehicle simultaneously 'GATHERSPEED'.
In Sect.
12.5
, we discuss how to interpret single recurrence points and diagonal line
structures, and explain how to use them to define a distance measure for time series
with certain distortions or invariance.
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