Introduction
Recurrence-based approaches have taken an important place in dynamical
system analysis. Related approaches have been used for several decades. The
basis of this analysis is finding recurrent points on a trajectory in the
phase space of a dynamical system. The first recurrence-based analysis method
was introduced by Poincaré as the method of the first recurrence
times . A Poincaré recurrence is the sequence of time
intervals between two visits of a trajectory at the same interval (or volume,
depending on the dimension of the trajectories).
Among the different approaches of investigating dynamical properties by
recurrence, the recurrence plot (RP) is a multifaceted and powerful approach
to study different aspects of dynamical systems. RPs were first introduced as
a visualization of recurrent states of phase space trajectories
, but then enriched by different quantification techniques
for characterizing dynamical properties, regime transitions, synchronization, and so on . In the study of complex systems, one of the most
important issues is finding dynamical transitions or regime changes.
Transitions in the dynamics can be detected by different RP-based measures,
which, in general, are very useful to study complex, real-world systems
. Examples of their successful
application in real-world systems can be found in life science
, earth science
, astrophysics
, and others .
The measures defined by the RP framework, called “recurrence quantification
analysis” (RQA), are based on point density and on the length
of diagonal and vertical line structures visible in the RP, being regarded as
alternative measures to quantify the complexity of physical systems. In order
to uncover their time-dependent behavior, RQA measures are often computed by
applying a sliding window on the time series, which then can be used to
identify dynamical transitions, such as period–chaos
transitions or chaos–chaos transitions .
Another popular method for analyzing complex systems is the complex network
approach . Complex network measurements are
useful for investigating and understanding the complex behavior of real-world
systems, such as social, computer , or brain networks
. The adjacency matrix of a complex network explains the
structure of the system and thus determines the links between the nodes of a
network. For unweighted and undirected networks, the adjacency matrix is
binary and symmetric, hence very similar to an RP. In our previous work, we
have shown that time series can be analyzed by complex networks by
identifying the RP by the adjacency matrix of a network
, forming so-called recurrence networks (RNs).
Complex network measures applied on RNs have been used to investigate
real-world systems, such as the climate system or the
cardiorespiratory system . RNs have been shown to be
more sensitive for the detection of periodic–chaos or chaos–periodic regime
transitions than some of the standard RQA measures .
Although recurrence-based methods are powerful tools to study complex
systems, they come with an important, non-trivial issue .
To identify recurrences, a spatial distance (or volume, depending on
the dimension of the system) in the phase space is usually used, and a sufficient
closeness between the trajectories is determined by applying a so-called
recurrence threshold ϵ to the distances .
Several approaches for selecting a meaningful threshold value
has been suggested . Of particular
interest are such methods that help one to overcome the problem of sliding-window-based analyses of systems with varying amplitude fluctuations (as coming from
different dynamical regimes or non-stationarities); e.g., those based on
normalizing time series or fixing recurrence density. However, in real-world
applications, time series are not usually smooth all the time. When
considering the time series by an RN representation, extreme points
(very high jumps or falls in the fluctuation of time series) in
the time series could break the connected components in the network
since the distance between an extreme point and other points
would be larger than the threshold value. The normalization
method would then result in non-optimal recurrence thresholds biasing the
recurrence analysis.
In this work, we will suggest a novel method of an adaptive threshold
selection based on the network's spectral properties .
We will present a comparison between the constant and the adaptive threshold
approach for detecting certain regime transitions (chaos–periodic or
periodic–chaos). Finally, we will demonstrate the novel approach for
analyzing lake-sediment-based paleoclimate variation.
Recurrence plots, recurrence networks, and the adaptive threshold
In the m dimensional phase space reconstruction of a time series, a state
is considered to be recurrent if its state vector falls into the
ϵ neighborhood of another state vector. Formally, for a given
trajectory xi (i = 1, …, N, xi ∈ Rm), the
recurrence plot R is defined as
Ri,j(ϵ)=Θϵ-xi-xj,i,j=1,…,N,
where N is the trajectory length, Θ(⋅) is the Heaviside
function, and |⋅| is the norm of the adopted phase
space . Thus, Ri,j = 1 if stated when times i and j
are recurrent, and Ri,j = 0 otherwise. The trajectory in the phase space
can be reconstructed via time delay embedding from a time series
{ui}i=1N :
xi=ui,ui+τ,…,ui+τ(m-1),
where m is the embedding dimension and τ is the embedding delay. The
embedding dimension m can be found by false nearest neighbors and the
delay τ by mutual information or auto-correlation .
The main diagonal of the RP, Ri,i = 1, represents the line of identity
(LOI). As we have mentioned, the RP is a symmetric, binary matrix. The
structures formed by line segments, which are parallel to the LOI in an RP,
characterize typical dynamical properties. We observe homogeneously
distributed recurrence points if the dynamics are white noise. If the system
is deterministic, diagonal line segments, which are parallel to the LOI, will
dominate. The dynamics are related to the length of the diagonal line
segments; chaotic dynamics cause mainly short line segments, but, conversely,
regular (periodic) dynamics cause long line segments. The RQA quantifies
this relation and can be used to detect transitions in the system's dynamics
.
Recurrence networks are based on the recurrence matrix,
Eq. (), which is an N × N matrix, where N is
the length of the phase space trajectory (the number of time steps). We now
consider these time steps as nodes of a network; if the nodes are
sufficiently close to each other – in other words, if the space vectors are
neighbors –, there is a link between them. In network theory, connections
between network nodes can be described with the adjacency matrix
A, with Ai,j = 1 if there is a link between nodes i
and j; otherwise, Ai,j = 0. To obtain the adjacency matrix from the recurrence
matrix, we discard self-loops in the recurrence matrix; i.e.,
Ai,j=Ri,j-δi,j,
where δi,j is the Kronecker delta (δi,j = 1 if i = j,
otherwise δi,j = 0).
The number of links at the ith node (the degree) is given by
ki = ∑jAij. In this paper, we use the eigenvalue spectrum of the
Laplacian matrix L to find an adaptive threshold ϵc,
where Li,j = δi,jki - Ai,j.
The crucial point in the paper is choosing the adaptive threshold for
calculating the RN. A threshold for recurrence-based methods should be
sufficiently small . Too
small ϵ causes very sparsely connected RN with many isolated
components; too large ϵ results in an almost completely connected
network. For data sets that are not smooth, choosing a reasonable,
small threshold could nevertheless result in unconnected recurrence network
components. These unconnected components would cause problems for some
complex network measures, since some of them need a connected network to be
computed for the entire network. For example, even if we have just one node
that is not connected to the network, the average path length will always be
infinite for the entire network. An even more important
motivation for avoiding isolated components in the RN is that the RN provides
a large amount of information about the dynamics of the underlying system,
although it contains only binary information. This has been demonstrated by
reconstructing time series from RPs . The
condition for reconstructing a time series from an RP is that all points are
connected by their neighborhoods; i.e., there are no isolated components. By
applying recurrence measures, we would like to quantify the dynamics encoded
by the RN. This can be ensured by the above-mentioned condition.
To find a sufficiently small threshold ϵ
that fulfills the desired condition of connected neighborhoods, we will use
the connectivity properties of the network. In particular, we choose the
value for ϵ that is the smallest one for the RN to be connected. In
order to find such an adaptive threshold, we start from very small values of
the threshold and vary the ϵ parameter until we get a connected
network. In order to apply this approach efficiently, we use an iterative bisection method in the simulations. The connectivity of a network
can be measured by the second-smallest eigenvalue λ2 of the
Laplacian matrix. If the network is connected, λ2 > 0
. We choose the adaptive threshold value as the minimum
value of the sequence of thresholds T = Ti, Ti+1, … when
the second minimum eigenvalue λ2 is positive:
ϵc=min(T)withT=Ti|∀i:λ2Ti>0.
Values ϵ below the critical value ϵc are indicating the
existence of unconnected components in the RN (Fig. ).
After that critical threshold, λ2 becomes positive and, if we
increase the threshold even more, the connectivity of the RN increases. By choosing
the critical point ϵc as the recurrence threshold, we ensure that
the RN will be connected by the smallest threshold possible.
Variation of the second-smallest eigenvalue of the Laplacian
λ2 due to changing threshold value, using the logistic map as an
illustrative example (control parameter a = 4.0). λ2 = 0 for
thresholds below a critical value ϵc, indicating the existence of
unconnected components in the RN. For ϵ > ϵc, there are no
unconnected components in the RN anymore. The adaptive threshold value for
this time series is ϵc ≈ 0.19.
Applications
Logistic map
As a first application, we compare some RN measures for first using the
adaptive and then the constant threshold approach by analyzing the logistic map:
xi+1=axi1-xi.
It is one of the most popular iterated maps, which has different regimes for
different control parameters a. The detection of the transitions of the
logistic map between these different regimes was previously studied with RP and RN
. The logistic map shows interesting
dynamics in the range of the control parameter a ∈ [3.5, 4.0], which is
studied here with a step size of Δa = 0.0005; for example, periodic and chaotic regimes, bifurcations, and inner and outer crises occur. We
compute a time series of length N = 5000 for each value of a. In order to
discard transients, we delete the first 2000 values, resulting in time
series consisting of 3000 values that have been used for all analyses of
the logistic map in this paper.
For the constant threshold selection method, we use the recurrence rate method
to choose a threshold value: a threshold is selected in such a way that the
recurrence rate (RR) is constant even for different time series with
different dynamics (e.g., different values of a) . In
this paper, we use RR = 5 % arbitrarily for further analysis.
(a) Lyapunov exponent and transitivity using (b) an adaptive threshold
and (c) a constant threshold for the logistic map. Dashed lines show certain
bifurcation points before the chaotic regime.
Now we compute the RNs by using the given threshold selection techniques
ϵ and ϵc for each control parameter a. We then calculate
transitivity, T, and betweenness centrality (BC) as the complex network
measures in order to detect the transitions from periodic–chaotic, chaotic–periodic states, bifurcations, and inner and outer crisis. The network
transitivity is given by
T=∑i,j,kAi,jAj,kAk,i∑i,j,kAk,iAk,j.
The average betweenness centrality of network is
BC=1N∑v∑s≠v≠tσst(v)σst,
where σst is the total number of shortest paths from node s to
node t, and σst(v) is number of the paths that pass through v.
As mentioned in the previous chapter, not all complex network measures can be
applied to a disconnected network. However, it would cause problems for
computing the measures on RNs calculated by using the constant threshold
technique, since the network could be disconnected. For instance, to compute
the average shortest path length or assortativity for an entire network, the
network must be connected. Disconnected nodes of the network could be
discarded from the calculation, but, in this case, we would lose information.
In the adaptive threshold case, we could calculate all these measurements on
the entire network, since the selection of the adaptive threshold ensures that
the recurrence network is connected.
Both threshold selection methods could detect transitions between dynamical
regimes (periodic–chaos or chaos–periodic). Transitivity gives large values
for the chaotic regime and small values for periodic. In the betweennesscentrality case, it is contrary to transitivity; large values for periodic
and small values for the chaotic regimes. Although the constant threshold selection detects the periodic
windows (chaos–period transitions) more sharply than
the adaptive threshold case, the transitivity, Tconstant, and
betweenness centrality, BCconstant, for the constant threshold
selection case (in the constant threshold case, in general, the threshold
is arbitrarily chosen by RR = 5 %) cannot distinguish between different
periodic dynamics; i.e., it cannot detect certain bifurcation points, such
as for period doublings at a ≈ 3.544, 3.564, and 3.84, for example. Conversely, in the adaptively chosen
threshold case, Tadaptive and BCadaptive are sensitive to
these bifurcations (Figs.
and ). Thus, using the adaptive threshold also allows
the detection of period–period transitions (i.e., the
study of bifurcation points, where the maximal Lyapunov exponent remains non-positive).
(a) Lyapunov exponent and betweenness centrality using (b) an adaptive
threshold and (c) a constant threshold for the logistic map. Dashed lines show
certain bifurcation points before the chaotic regime.
Application to paleoclimate record
The study of paleoclimate variation helps in understanding and evaluating
possible future climate change. Lake sediments provide valuable archives of
past climate variations.
In the following, we will focus on a well-dated high-resolution climate
archive from paleolake Lisan located beneath the archaeological site of
Masada in the Near East . The sediments from
the upper member were deposited (26–18 cal ka BP) when the lake reached its
highest stands . The sedimentary sequence
contains varves comprising seasonally deposited primary (evaporitic)
aragonite and silty detritus . The pure aragonite
sublaminae were precipitated from the upper layer of the lake during summer
evaporation. Their formation requires inflow of HCO3- ions into the
lake from the catchment area during winter floods that also
bring in silty detrital material. One detrital and overlying aragonite
sublaminae constitute a varve. Previous studies
indicate that small ice-rafting events
(denoted as a, b, c, and d), as well as prominent Heinrich events in
the North Atlantic, are associated with the eastern Mediterranean arid
intervals. The study of seasonal sublaminae yields evidence of decadal to century scale arid events that correlate with cooler temperatures at higher
latitudes. Analyses in the frequency domain indicate the presence of
periodicities centered at 1500, 500, 192, 139, 90,
and 50–60 years, suggesting a solar forcing on climate .
We use the yearly sampled pure aragonite proxy (CaCO3) from the paleolake
Lisan for our RN analysis (Fig. a). We use a time delay
embedding with dimension m = 3 and delay τ = 2 – these parameters have been
computed by a standard procedure using false nearest neighbors and mutual
information – for reconstructing the phase space.
To detect dynamical transitions in the paleoclimate data, we adopt a sliding
window of W data points with a step size of ΔW. RNs are computed one by one
for each window of the time series. We have chosen a sampling
window size of ΔT = 100 years, with 90 % overlap corresponding to a time
window size of W ≈ 100 data points (since there are some gaps in the
data, the number is not exactly 100). The time series' length is N = 7665 and the
total number of windows analyzed is
N-WΔW≈755.
Transitivity and betweenness centrality are then calculated within these
windows (Fig. b and c). As we have shown for the logistic map,
transitivity and betweenness centrality are both sensitive to detecting
transitions. Larger values of transitivity, T, refer to regular behavior,
whereas smaller values refer to more irregular dynamics in the considered window of
the time series.
The gray shaded horizontal band in Figs. b and c is the confidence
interval of the network measures. We apply a rather simple test
in order to see whether the characteristics of the dynamics at a certain time
statistically differ from the general characteristics of the
dynamics. In order to apply this test, we use the following approach. We
create surrogate data segments of length W by drawing data points randomly
from the entire time series, and we compute the RN and the network measures
from such a surrogate segment. We repeat this 10 000 times and have an
empirical test distribution of transitivity, T, and BC. A confidence interval is then estimated from these distributions by
their 0.05 and 0.95 quantiles.
(a) Aragonite (CaCO3) record from paleolake Lisan,
(b) transitivity, and (c) betweenness centrality results of RN using the adaptive
threshold. Abrupt changes in T and BC indicate transitions between
different climate regimes. Dry events in Lake Lisan (cooling of the higher
latitudes) are marked by blue bars and two interstadial peaks (warming) by
orange bars. The gray shaded band is the 90 % confidence interval for the
networks measures.
Previous studies had identified multiple climate
fluctuations in the varved Lisan record and correlated them with the
Greenland oxygen isotope data (indicative of temperature changes;
) and ice-rafting events in the North Atlantic
. The blue and orange vertical bars in Fig.
delineate periods of cooling and warming, respectively, in the higher latitudes
that resulted in drier and wetter episodes in the eastern Mediterranean.
The network measures, T and BC, both indicate abrupt transitions well
(Fig. b and c). In particular for T, the values jump between
high and low values. T reveals epochs of significantly low values at around
25.8–25.6, 25.2–25.1, 24.3–24.2, 24.0–23.9, 22.8–22.6, 22.3–22.1,
21.5–21.1, 21.7, 20.6–20.5, 20.1–19.9, 19.8–19.6, and 19.3–18.9 cal ka BP.
The periods 25.8–25.6, 22.3–22.1, 21.5–21.1, and 19.3–18.9 cal ka BP
correspond to the known Bond events d, c, b, and a, and the epoch
between 24.3 and 23.9 cal ka BP coincides with the Heinrich H2 event. During
the interstadial peaks “IS2” event at 23.8–23.7 and 23.3–23.2 cal ka BP, T shows
significant high values, almost reaching the value 1. BC exhibits
rather similar behavior of abrupt transitions like T, but with opposite
signs. A general observation is that low values in T can be found during dry
but high values during wet regimes, and that such regimes change abruptly.
A high transitivity value indicates a more regular deposition of aragonite,
and thus a more regular, or even periodic, climate variability. This could
be an indication for a dominant role of the (more or less periodic) solar
forcing via its influence on the temperature in the higher latitudes. During
phases of a colder North Atlantic, the solar forcing becomes less important,
but regional climate effects become more important and dominating, causing a more
complex, irregular climate variability, finally indicated by low values of T.
Combining the maxima of T and minima of BC, we can identify the above-mentioned periods of non-regular climate dynamics. Most of these periods
correspond to cold events; e.g., the Bond and Heinrich events, and the
found Lisan lake events L3 to L13 . Several regular
periods can be identified, some of which coinciding with the warm period
during the interstadial IS2. Few remaining periods of high or low regularity
have not yet been identified in the literature so far and call for further investigation.
The abrupt changes in T are available due to the adaptive threshold. By
using a constant threshold, T varies only slowly and more gradually. Defining
the time points of the climate regime shifts becomes more difficult in this case.
Conclusions
We have represented a novel method to choose a recurrence threshold adaptively
and compared it with the constant threshold selection technique. The selection
of recurrence thresholds for recurrence plots and recurrence networks is a
crucial step for these techniques. So far, the threshold had to be chosen
arbitrarily, taking into account different criteria and application cases, as
well as requiring some expertise. Here, we have proposed a novel technique to
determine such a threshold value automatically depending on the time series.
Such an adaptive threshold is directly derived from the topology of the
recurrence network. It is selected in such a way that the recurrence network
does not have unconnected components. We have discussed transitivity and
betweenness centrality measures of the complex network approach. Both
measures are related to the regularity of the dynamics.
Moreover, the proposed threshold selection can also be useful
for the recurrence quantification analysis. A systematic investigation of the
different threshold selections remains to be looked into in the future.
We have compared the novel adaptive threshold selection with the arbitrarily
selected threshold by applying them to the logistic map. Although both
methods distinguish the dynamical regimes clearly, the adaptively chosen
threshold approach detects many more bifurcations in particular, such as
period doubling. Such bifurcations are important characteristics of the
dynamical systems, since these bifurcations route to chaos from periodicity.
Moreover, we have used our approach to investigate a paleoclimate proxy
record from the paleolake Lisan representing the climate variability in the
Near East between 27 and 18 cal ka BP. Both transitivity and betweenness
centrality measures clearly identified transitions between wet and dry (and
vice versa) periods by an abrupt decrease of dynamical regularity, perhaps
due to a reduced solar influence. Our method identified some transitions
that have not been known so far from the literature and require further
investigation; e.g., by analyzing other proxy records from this region. By
choosing the adaptive threshold, we have been able to identify the
transitions more clearly than by using the arbitrary selected threshold approach.