Variations of Recurrence Plots
In the original definition of the RPs, the neighbourhood is a ball
(i.e. L2-norm
is used) and its radius
is chosen in such a way that it contains a fixed amount of states
xj
(Eckmann et al., 1987). With such a neighbourhood,
the radius εi
changes for each
xi
(i=1…N)
and
Ri, j ≠
Rj, i
because the
neighbourhood of xi
does not have to be the same as that of xj.
This property leads to an asymmetric RP, but all columns of
the RP have the same recurrence density.
We denote this neighbourhood as fixed amount of nearest neighbours (FAN).
However, the most commonly used neighbourhood is that with a
fixed radius
εi=ε, ∀ i.
For RPs
this neighbourhood was firstly used by Zbilut et al. (1991). A fixed
radius means that
Ri, j =
Rj, i
resulting in a symmetric RP.
The type of neighbourhood that should be used depends on the
application. Especially in applications of the
cross recurrence plots, the neighbourhood with a FAN will
play an important role.
In the literature further variations of the recurrence plots
have been proposed:
Iwanski and Bradley (1998) have defined a variation of an RP with a corridor
threshold [εin,εout],
Ri, j =
Θ ( || xi − xj|| − εin)
⋅
Θ ( εout − || xi − xj||).
Those points xj
are considered to be recurrent that fall into the
shell with the inner radius εin
and the outer radius εout.
The authors have suggested to use this kind of RPs in order to study “interesting structures”
in the RP. An advantage of such a corridor thresholded recurrence plot
is its increased robustness against recurrence points coming from
the tangential motion. However,
the threshold corridor removes the inner points in broad diagonal lines, which
results in two lines instead of one. These RPs are, therefore, not suitable
for a quantification analysis.
The usage of a shell as a neighbourhood can be found in an algorithm for
computing Lyapunov exponents from experimental time series (Eckmann et al, 1986).
Choi et al. (1999) have introduced the perpendicular recurrence plot
Ri, j =
Θ ( ε − || xi − xj||)
⋅
δ ( ∂txi ( xi − xj)).
Here, δ is the Delta function. This recurrence plot contains
only those points xj
that fall into the neigbourhood of
xi and lie in the
(m−1)-dimensional
subspace that is perpendicular to the phasespace trajectory
at xi.
These points correspond locally to those lying on a
Poincaré section.
This criterion cleans up the RP more from recurrence points based on the tangential
motion than the previous corridor thresholded RPs.
The authors have shown the
increased efficiency of the perpendicular RPs for their application on
estimation of the largest Lyapunov exponent.
Using this kind of an RP, the finding of unstable periodic orbits (if they exist)
is more robust.
In a similar direction goes the iso-directional recurrence plot,
introduced by Horai et al. (2002),
Ri, j =
Θ ( ε − ||
(xi+T − xi)
−
(xj+T − xj)
||).
Such recurrence points are related with neighboured trajectories which run parallel and
in the same direction. Horai introduced an additional iso-directional neighbours plot,
which is simply the product between the common recurrence plot and the iso-directional
recurrence plot
Ri, j =
Θ ( ε − ||
xi
−
xj
||)
⋅
Θ ( ε − ||
(xi+T − xi)
−
(xj+T − xj)
||).
The computation of this special recurrence plot is simpler than that of
a perpendicular recurrence plot. Although the cleaning the RP from
false recurrences is better than in a common recurrence plot, it does not
reach the quality of a perpendicular recurrence plot. A disadvantage is the
additional parameter T which has to be thoughtfully determined before
(however, it seems that this parameter has to be related with the embedding
delay τ).
The RP contains, finally, tests of all states with each other, which
results in N2 tests for
N considered states. Still, it is also
possible to test each state with a predefined amount k
of subsequent states (Zbilut et al., 1991; Koebbe and Mayer-Kress, 1992;
Atay and Altintas, 1999)
Ri, j =
Θ ( ε − || xi − xi−i0+j−1||),
i=1…N−k,
j=1…k.
This reveals an (N−k) × k-matrix
which does not have to be square.
The y-axis represents the time distances
to the following recurrence points but
not their absolute time. All diagonal oriented structures in the common RP are now
projected to the horizontal orientation. For i0=0,
the LOI,
which was the diagonal line in the common RP, is now the horizontal line on the
x-axis.
With non-zero i0 the RP contains recurrences of a
certain state only in the predefined time interval after time
i0 (Koebbe and Mayer-Kress, 1992).
This representation of recurrences may be more intuitive than the RPs usually are
because the consecutive
states are not oriented diagonally. However, such an RP represents only the first
(N−k) states. Mindlin and Gilmore (1992)
have proposed the close returns plot
which is, in fact, such an RP exactly for one dimension. Using this kind of
RP, a first quantification approach of RPs (or “close returns plots”) can be found
(“close returns histogram”, recurrence times).
It has been used for the investigation of periodic orbits and topological properties of strange
attractors (Lathrop and Kostelich, 1989;Tufillaro et al. 1990; Mindlin and Gilmore, 1992).
Instead plotting the recurrences with black points, the
distances
Di, j =
|| xi − xi−i0−1||
between the states
xi and
xj can be plotted.
Although this is not a real recurrence plot,
it is sometimes called global recurrence plot
(Webber Jr., 2003) or unthresholded recurrence plot
(Iwanski and Bradley, 1998). However, it should be termed distance plot.
This representation can also help in studying phase space trajectory.
Moreover, it may help to find an appropriate threshold value ε.
The windowed and meta recurrence plots have been suggested
as means of investigating an external force or the nonstationarity in a system
(Manuca and Savit, 1996; Casdagli, 1997). The first ones are obtained by covering
an RP with w × w-sized squares (windows) and by
averaging the recurrence points that are contained in
these windows (Casdagli, 1997). Consequently, a windowed recurrence plot
is an Nw × Nw-matrix,
where Nw is the
floor-rounded N/w,
and consists of values which are not limited to zero and one (this suggests
a colour-encoded representation). These values
correspond with the cross correlation sum
Ci, j =
(1/w2)
∑wi, j=1
Ri+(i−1)w, j+(j−1)w,
i, j=1…N/w
between sections in x with
length w and starting at
(i−1)w+1
and (j−1)w+1
(for cross-correlation integral cf. Kantz and Schreiber 1994).
The meta recurrence plot as it has been
defined by Casdagli (1997) is a distance matrix derived from
the cross correlation sum,
Di, j =
(1/εm)
(
Ci, i +
Cj, j −
2 Ci, j
)
By applying a further threshold value to
Di, j
(analogous to recurrence plots), a black-white dotted representation
is also possible.
Manuca and Savit (1996) have gone one step further. They have used quotients from the
cross correlation sum to form a meta phase space.
From this meta phase space a recurrence or non-recurrence
plot is created, which can be used to characterize the nonstationarity
in time series. For a sufficient explanation the work of Manuca and Savit (1996)
is recommended.
Instead of using the spatial closeness between phase space trajectories,
order patterns recurrence plots use order patterns π
for the definition
of a recurrence. An order pattern π
of dimension m is defined by the
discrete order sequence of the data series xi and has
length m. For
m = 3
we have, e. g., six order patterns:
Using order
patterns, the data series xi will be symbolized by order pattern:
xi,
xi-τ1, …,
xi-τm → πi.
The order patterns recurrence plot is then defined by the pairwise test
of order patterns (Groth, 2005):
Ri, j =
δ ( πi, πj).
Such a recurrence plot represent those times, when specific rank order sequences
in the system recur. Its main advantage is its much better robustness against
non-stationary data.
Instead of using the spatial closeness between phase space trajectories,
isometric recurrence plots (Sabelli, 2001) test only for similar lengths
of phase space vectors
Ri, j =
Θ ( ε − |
||xi||
−
||xj||
|).
This kind of recurrence was proposed for the study of "bios".
However, it is very sensitive to changes in the absolute
amplitude (e.g., a normalisation to mean zero would change
such recurrence). Therefore, it is also very sensitive to noise.
Such an RP contains much more recurrence points than the other
variants. Moreover, many diagonal lines with slopes –45°
will appear, indicating non-causal recurrences. Therefore,
this kind of RP is not suggested.
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Examples of various defined RPs for a section of
the x-component of the Lorenz system
(sampling time Δ t = 0.03):
(A) RP computed by using the L∞-norm,
(B) RP computed by using the L1-norm,
(C) RP computed by using the L2-norm,
(D) RP computed by using a fixed amount of nearest neighbours (FAN),
(E) RP computed by using a threshold corridor
[εin,εout],
(F) perpendicular RP (L2-norm),
(G) distance plot (unthresholded RP, L2-norm),
(H) order patterns RP (m = 3, τ1, 2, 3 = 9) and
(I) RP, where the y-axis represents the relative time
distances to the next recurrence points but not their absolute time (“close returns
plot”, L2-norm).
Except for (F), (G) and (H), the
parameter ε is chosen in such a way that the recurrence point
density (RR) is approximately the same. The embedding parameters
(m=5 and τ=5) correspond
to an appropriate time delay embedding. Please click in the figure to
find out the differences in detail!
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The selection of a specific variant from this variety of RPs depends on the problem
and on the kind of data. Perpendicular
RPs are highly recommended for the quantification analysis based on diagonal
structures, whereas corridor thresholded RPs are not suitable for this task.
Windowed RPs are appropriate for the visualization of the long range behaviour
of rather long data sets.
If the recurrence behaviour for the states
xi within a predefined section
{xi+i0, … ,
xi+i0+k}
of the phase space trajectory is of special interest,
an RP with a horizontal LOI will be practical.
It should be emphasized again that the recurrence of states is an
important feature. Beside the recurrence plots, there are some other methods
that use recurrences.
For example, the recurrence in the phase space is
used for the recurrence time statistics
(Kac, 1947; Gao, 1999; Balakrishnan et al., 2000),
first return map (Lathrop and Kostelich, 1989),
space time separation plot (Provenzale et al., 1992) or
as a measure for nonstationarity (Kennel, 1997; Rieke et al., 2002;
closely related to the recurrence time statistics).
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Current Developments Of Concepts Based On Recurrence Plots
And Their Applications, Ph.D. Thesis, University of Potsdam,
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