Discovery of Driving Behavior Patterns - Smart Information Systems: Computational Intelligence for Real-Life Applications - page 316

R i , j () = Θ( −||

d

x i − x j || )

∈ R , , = ...

x i i j 1 n (12.1)

where x is a time series of length n , the Heaviside function. One

of the most crucial parameters of RPs is the recurrence threshold

|| · || a norm and Θ

, which influences
the formation of line structures [ 21 ]. In general, the recurrence threshold should be
chosen in a way that noise corrupted observations are filtered out, but at the same
time a sufficient number of recurrence structures are preserved. As a rule of thumb,
the recurrence rate should be approximately one percent with respect to the size of
the plot. For quasiperiodic processes, it has been suggested to use the diagonal line
structures to find the optimal recurrence threshold. However, changing the threshold
does not preserve the important distribution of recurrence structures [ 23 ].
A general problem with standard thresholding methods is that an inappropriate
threshold or laminar states cause thick diagonal lines, which basically corresponds
to redundant information. Schultz et al. [ 31 ] have proposed a local minima-based
thresholding approach, which can be performed without choosing any particular
threshold and yields in clean RPs of minimized line thickness. But this approach
comes with some side effects, e.g., bowed lines instead of straight diagonal lines.
Furthermore, it is important to discuss the definition of recurrences, because
distances can be calculated using different norms [ 18 ]. Although the L 2 -norm is
used in most cases, the L ∞ -norm is sometimes preferred for relatively large time
series with high computational demand [ 23 ].
Although traditional RPs only regard one trajectory, we can extend the concept
in a way that allows us to study the dynamics of two trajectories in parallel [ 22 ].
A cross recurrence plot (CRP) shows all those times at which a state in one dynamical
system occurs in a second dynamical system. In other words, the CRP reveals all
the times when the trajectories of the first and second time series, x and y , visits
roughly the same area in the phase space. The data length, n and m , of both systems
can differ, leading to a nonsquare CRP matrix [ 19 , 21 ].

CR x , y
i

d

j () = Θ( −||

x i − y j || ) x i , y j ∈ R , i = 1 ... n , j = 1 ... m (12.2)

,

For the creation of a CRP, both trajectories, x and y , have to present the same
dynamical system with equal state variables because they are in the same phase
space. The application of CRPs to absolutely different measurements, which are not
observations of the same dynamical system, is rather problematic and requires some
data preprocessing with utmost carefulness [ 21 ].
In order to test for simultaneously occurring recurrences in different systems,
another multivariate extension of RPs was introduced [ 22 ]. A joint recurrence plot
(JRP) shows all those times at which a recurrence in one dynamical system occurs
simultaneously with a recurrence in a second dynamical system. With other words,
the JRP is the Hadamard product of the RP of the first system and the RP of the
second system. JRPs can be computed from more than two systems. The data length
of the considered systems has to be the same. [ 19 , 21 ].