Discovery of Driving Behavior Patterns - Smart Information Systems: Computational Intelligence for Real-Life Applications - page 320

similar width without spreading out in various directions. Removing spurious widths
makes it more reasonable to define line-based quantification measures, such as diver-
gence and determinism [ 3 ]. Another solution is to estimate the entropy by looking
at the distribution of the diagonal lines [ 23 ]. The entropy is based on the probability
p (, l ) that diagonal lines structures with certain length l and similarity occur in

the recurrence matrix [ 19 , 21 ], and can be computed as follows:

N

ENTR (, l min ) =− p (, l ) ln p (, l ) (12.7)

l = l min

Recurrence plots (RPs) and corresponding recurrence quantification analysis
(RQA) measures have been used to detect transitions and temporal deviations in
the dynamics of time series. Since detected variations in RQA measures can easily
be misinterpreted, Marwan et al. [ 20 ] have proposed to calculate a confidence level
to study significant changes. They formulated the hypothesis that the dynamics of a
system do not change over time, and therefore the RQA measures obtained by the
sliding window technique will be normally distributed. Consequently, if the RQA
measures are out of a predefined interquantile range, an observation can be considered
significantly. Detecting changes in dynamics by means of RQA measures obtained
from a sliding window have been proven to be useful in real-life applications such
as comparing traffic flow time series under fine and adverse weather conditions [ 37 ].
Since recurrence plot-based techniques are still a rather young field in nonlinear
time series analysis, systematic research is necessary to define reliable criteria for
the selection of parameters, and the estimation of RQA measures [ 23 ].

12.6 Recurrence Plot-Based Distance

According to our formalization of joint cross recurrence (JCR) in Eq. 12.4 and the
denotation of the determinism (DET) in Eq. 12.5 , we can define our RecuRRence
Plot-based (RRR) distance measure as follows:

RRR (, l min ) = 1 − DET (, l min ) (12.8)

Since the DET value ranges from 0 to 1, depending on the proportion of diagonal
line structures found in a JCR plot, the RRR distance is 0 if the trajectory of both
dynamical systems is identical and 1 if there are no similar patterns at any position
in time.
Although our proposed RRR distance measure can be used as a subroutine for
various time series mining tasks, this work primarily focuses on clustering. Our aim
is to group a set of t unlabeled time series T into k clusters C with centroids Z .
In order to evaluate the performance of the time series clustering with respect to
our RRR distance, we suggest to quantify the number of similar patterns that recur