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plots enables us to analyze signals that are multivariate, nonlinear, nonstationary, and
noisy.
The global (large-scale) appearance of a RP can give hints on stationarity and
regularity, whereas local (small-scale) patterns are related to dynamical properties,
such as determinism [
38
]. Recent studies have shown that determinism, the percent-
age of recurrence points that form lines parallel to the main diagonal, reflects the
predictability of a dynamical system [
21
].
Given a recurrence matrix
R
with
N
N
entries generated by any of the intro-
duced recurrence plot variations, such as our proposed JCRP, we can compute the
determinism DET
×
(,
l
min
)
for a predefined
-threshold and a minimum diagonal line
length
l
min
as followed [
19
,
21
]:
l
=
l
min
l
·
P
(,
l
)
DET
(,
l
min
)
=
(12.5)
i
,
j
=
1
R
i
,
j
()
N
1
R
i
−
1
,
j
−
1
()
P
(,
l
)
=
−
i
,
j
=
1
×
1
R
i
+
l
,
j
+
l
()
−
l
−
1
×
R
i
+
k
,
j
+
k
()
(12.6)
k
=
0
where
P
(,
l
)
is the histogram of diagonal lines of length
l
with respect to a certain
neighborhood.
In general, processes with chaotic behavior cause none or short diagonals, whereas
deterministic processes cause relatively long diagonals and less single, isolated recur-
rence points [
21
,
37
]. In respect to JCRPs, diagonal lines usually occur when the
trajectory of two multivariate time series segments is similar according to a certain
threshold. Since we aim to measure the similarity between time series that contain
segments of similar trajectories at arbitrary positions, which in turn cause diagonal
line structures, we propose to use determinism as a similarity measure. According to
the introduced JCRP approach, a high DET value indicates high similarity or rather a
high percentage of multivariate segments with similar trajectory, whereas a relatively
low DET value suggests dissimilarity or rather the absence of similar multivariate
patterns.
However, data preprocessing like smoothing can introduce spurious line struc-
tures in a recurrence plot that cause high determinism value. In this case, further
criteria like the directionality of the trajectory should be considered to determine the
determinism of a dynamic system, e.g., by using iso-directional and perpendicular
RPs [
19
,
21
,
23
]. In contrast to traditional recurrence plots, perpendicular recur-
rence plots (PRPs) consider the dynamical evolution of only the neighborhoods in
the perpendicular direction to each phase flow, resulting in plots with lines of the
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