Recurrence Plots At A Glance
Definition
Recurrence plot - a recurrence plot (RP) is a graph which shows all those times
at which a state of the dynamical system recurs. With other words,
the RP reveals all the times when the phase space trajectory
visits roughly the same area in the phase space.
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Natural processes can have a distinct recurrent
behaviour, e.g. periodicities (as seasonal or
Milankovich cycles), but also irregular cyclicities (as El Niño Southern
Oscillation). Moreover, the recurrence
of states, in the meaning that states are arbitrary close after some time,
is a fundamental property of deterministic dynamical systems
and is typical for nonlinear or chaotic systems. The recurrence of
states in nature has been known for a long time and
has also been discussed in early publications
(e.g. recurrence phenomena in cosmic-ray intensity, Monk, 1939).
Eckmann et al. (1987) have introduced a tool which
can visualize the recurrence of states
xi in a phase
space. Usually,
a phase space does not have a dimension (two or three) which allows
it to be pictured. Higher dimensional phase spaces can
only be visualized by projection into the two or three
dimensional sub-spaces. However, Eckmann's tool
enables us to investigate the m-dimensional phase space
trajectory through a two-dimensional representation of its
recurrences. Such recurrence of a state at
time i at a different time
j is marked within a
two-dimensional squared matrix with ones and zeros dots
(black and white dots in the plot), where
both axes are time axes.
This representation is called recurrence
plot (RP).
Such an RP can be mathematically expressed as
Ri, j =
Θ
( εi −
|| xi − xj||),
xi
∈ ℜm,
i, j=1…N,
where N is the number of considered states
xi,
εi is a threshold distance,
|| ⋅ ||
a norm and Θ( ⋅ ) the Heaviside function.
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(A) Segment of the phase space trajectory of the Lorenz system
(for standard parameters r=28,
σ=10, b=8/3;
Lorenz, 1963) by using
its three components and (B) its corresponding recurrence plot.
A point of the trajectory at j
which falls into the neighbourhood (gray circle in (A)) of a
given point at i is considered
as a recurrence point (black point on the trajectory in (A)).
This is marked with a black point in the RP at the location
(i, j). A
point outside the neighbourhood (small circle in (A)) causes a white point
in the RP. The radius of the neighbourhood for the RP is
ε=5.
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Structures in Recurrence Plots
The initial purpose of RPs is the visual inspection of
higher dimensional phase space trajectories. The view on RPs gives
hints about the time evolution of these trajectories.
The advantage of RPs is that they can also be applied to
rather short and even nonstationary data.
The RPs exhibit characteristic large scale and small
scale patterns.
The first patterns were denoted by Eckmann et al. (1987) as typology
and the latter as texture. The typology
offers a global impression which can be characterized as
homogeneous, periodic, drift and disrupted.
- Homogeneous RPs are typical of stationary and autonomous systems in which
relaxation times are short in comparison with the time spanned by
the RP. An example of such an RP is that of a random time series.
- Oscillating systems have RPs with diagonal oriented, periodic recurrent structures
(diagonal lines, checkerboard structures). For quasi-periodic systems,
the distances between the diagonal lines are different.
However, even for those oscillating systems whose oscillations are not easily
recognizable, the RPs can be used in order to find their oscillations.
- The drift is caused
by systems with slowly varying parameters. Such slow (adiabatic)
change brightens the RP's upper-left and lower-right corners.
- Abrupt changes in the dynamics as well as extreme events
cause white areas or bands in the RP.
RPs offer an easy possibility to find and to assess extreme and rare
events by using the frequency of their recurrences.
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Characteristic typology of recurrence plots:
(A) homogeneous (uniformly distributed noise), (B) periodic (super-positioned
harmonic oscillations), (C) drift (logistic map corrupted
with a linearly increasing term)
and (D) disrupted (Brownian motion).
These examples illustrate how different RPs can be.
The used data have the length 400 (A, B, D) and
150 (C), respectively;
no embeddings are used; the thresholds are ε=0.2 (A, C, D) and
ε=0.4 (B).
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The closer inspection of the RPs reveals small scale structures (the texture) which
are single dots,
diagonal lines as well as vertical and horizontal
lines (the combination of vertical and horizontal lines obviously forms
rectangular clusters of recurrence points).
- Single, isolated recurrence points can occur if
states are rare, if they do not persist for any time or if they
fluctuate heavily. However, they are not a unique sign of chance or
noise (for example in maps).
- A diagonal line
Ri+k, j+k = 1 (for
k=1…l,
where l is the length of the diagonal line)
occurs when a segment of the trajectory runs parallel to another segment,
i.e. the trajectory visits the same region of the phase space at
different times. The length of this diagonal line is determined by
the duration of such similar local evolution of the trajectory segments.
The direction of these diagonal structures can differ. Diagonal
lines parallel to the LOI (angle π) represent the parallel running of
trajectories for the same time evolution. The diagonal structures
perpendicular to the LOI represent the parallel running with contrary
times (mirrored segments; this is often a hint for an inappropriate
embedding). Since the
definition of the Lyapunov exponent uses the time of the parallel
running of trajectories, the relationship between the diagonal
lines and the Lyapunov exponent is obvious.
- A vertical (horizontal) line
Ri,j+k = 1 (for
k=1…v,
where v is the length of the vertical line)
marks a time length in which a state does not change or changes very slowly.
It seems, that the state is trapped for some time. This is a typical
behaviour of laminar states (intermittency).
These small scale
structures are the base of a quantitative analysis of the RPs.
Summarizing the last mentioned points we can establish the
following list of observations and give the corresponding
qualitative interpretation:
| Observation
| Interpretation
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Homogeneity
| the process is obviously stationary
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Fading to the upper left and lower right corners
| nonstationarity; the process contains a trend or drift
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Disruptions (white bands) occur
| nonstationarity; some states are rare or far from the normal;
transitions may have occurred
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Periodic/ quasi-periodic patterns
| cyclicities in the process; the time distance between periodic
patterns (e.g. lines) corresponds to the period; long diagonal lines with
different distances to each other reveal a quasi-periodic process
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Single isolated points
| heavy fluctuation in the process;
if only single isolated points occur, the process may be an uncorrelated random or
even anti-correlated process
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Diagonal lines (parallel to the LOI)
| the evolution of states is similar at different times; the process could be deterministic;
if these diagonal lines occur beside single isolated points, the process could be chaotic (if
these diagonal lines are periodic, unstable periodic orbits can be retrieved)
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Diagonal lines (orthogonal to the LOI)
| the evolution of states is similar at different times but with reverse time;
sometimes this is a sign for an insufficient embedding
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Vertical and horizontal lines/clusters
| some states do not change or change slowly for some time;
indication for laminar states
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Long bowed line structures
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the evolution of states is similar at different epochs
but with different velocity; the dynamics of the system
could be changing (but note: this is not fully valid
for short bowed line structures) |
The visual interpretation of RPs requires some experience. The study of
RPs from paradigmatic systems gives a good introduction into characteristic
typology and texture. However,
their quantification offers a more objective way for the investigation
of the considered system. With this quantification, the RPs have become
more and more popular within a growing group of scientists who
use RPs and their quantification techniques for data analysis
(a search with the Scirus search engine in spring 2003
reveals over 200
journal published works and approximately 700
web published works about RPs).
» Recurrence plots in SciTopics
» Recurrence plots in Wikipedia
» Further definitions of recurrence plots (Google)
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as well as N. Marwan: Encounters With Neighbours -
Current Developments Of Concepts Based On Recurrence Plots
And Their Applications, Ph.D. Thesis, University of Potsdam,
ISBN 3-00-012347-4.
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