Analysing palaeoclimate proxy time series using windowed recurrence network analysis (wRNA) has been shown to provide valuable information on past climate variability. In turn, it has also been found that the robustness of the obtained results differs among proxies from different palaeoclimate archives. To systematically test the suitability of wRNA for studying different types of palaeoclimate proxy time series, we use the framework of forward proxy modelling. For this, we create artificial input time series with different properties and compare the areawise significant anomalies detected using wRNA of the input and the model output time series. Also, taking into account results for general filtering of different time series, we find that the variability of the network transitivity is altered for stochastic input time series while being rather robust for deterministic input. In terms of significant anomalies of the network transitivity, we observe that these anomalies may be missed by proxies from tree and lake archives after the non-linear filtering by the corresponding proxy system models. For proxies from speleothems, we additionally observe falsely identified significant anomalies that are not present in the input time series. Finally, for proxies from ice cores, the wRNA results show the best correspondence to those for the input data. Our results contribute to improve the interpretation of windowed recurrence network analysis results obtained from real-world palaeoclimate time series.

Palaeoclimate proxy time series from archives such as trees, lakes, speleothems, or ice cores play
an important role in past climate reconstructions

Windowed recurrence network analysis (wRNA) has already been successfully used to detect dynamical anomalies
in time series from different palaeoclimate archives

Proxy system models are forward models that simulate the formation of a palaeoclimate proxy based on
the systemic understanding of that proxy

With this work, we want to contribute to a better understanding and an improved interpretation of results obtained with wRNA for palaeoclimate applications. We first introduce our analysis framework in Sect.

The first step when analysing data using recurrence-based approaches is to reconstruct the
higher-dimensional phase space of the system from the measured univariate time series

In practical applications, however, the box-counting dimension of the original attractor is usually
unknown and the data are finite and subject to noise such that the choice of the embedding parameters
plays an important role in the quality of the phase space reconstruction. In particular, the
embedding dimension can be estimated using the method of false nearest neighbours

We analyse the embedded time series

From the adjacency matrix, we can estimate various network properties. In the course of this work, we will restrict ourselves to the
network transitivity

For the analysis performed here, we repeat the wRNA for different
values of the window width

In order to identify dynamical anomalies from the resulting network transitivity,
we first perform a pointwise significance test using random shuffling surrogates. That is, for every window
width

As intrinsic correlations of the analysis results in both the time domain (due to the short offset

Here, we employ a data-adaptive null model using iterative amplitude-adjusted Fourier transform
surrogates of the original time series

Forward modelling of palaeoclimate proxies offers the possibility of gaining insights into the underlying processes that influence the sensitivity of a given proxy to climate variations and can thus be used to investigate characteristic properties of time series of different palaeoclimate archives and their implications for further analyses. We here use four models for typical proxies from tree rings, lake sediments, ice cores and speleothems, respectively, in order to test how well dynamical anomalies can be identified when applying wRNA to time series originating from those archives.

Generally, a proxy system model can be divided into an environment, a sensor, an archive and an observation
sub-model

Tree rings are one of the most important archives for palaeoclimate reconstructions of the last
millennium

If precipitation is given, the leaky bucket model

It should be noted that this model does not take into account juvenile tree growth. Real tree ring width data are of course subject to juvenile growth and the effect is usually subtracted from the measured data. Problems arising from this are thus disregarded in the model, which we will further address when discussing the results for the model.

To set the model parameters to realistic values, we use an exemplary real-world data set of a local tree ring width index
chronology from eastern Canada (

Climatic input variables and model parameters for the tree ring width model as derived from the eastern Canada data

Records from lake sediments are available from many regions worldwide and can provide information
about past temperatures and precipitation, depending on the regional boundary conditions and
measured proxy

BrGDGTs are produced by bacteria, and their degree of
cyclisation and methylation has been related to soil temperatures, lake pH, and also to
mean annual air temperatures

To tune the climatic input variables, we use the climatic setup corresponding to the one used for the tree
ring archive in eastern Canada, while for the
model parameters, we use typical values for lake sediments that are taken as a default in the PRYSM
implementation of the lake archive model

Climatic input variables and model parameters for the lake sediment model as derived from the eastern Canada data

Oxygen isotope fractions of speleothems have been shown to provide valuable insights into past climate
variability

The sensor model covers processes in the karst and the cave, while processes in the soil such as
evapotranspiration are neglected.
The model filters the

The parameters for the speleothem

Climatic input variables and model parameters for the speleothem

Proxy time series from ice cores have been used in a variety of contexts to study past climate variability
on different timescales

The sensor model corrects the isotopic composition of the precipitation for the altitude of the glacier by using the relation

To tune the climatic input variables and model parameters of the ice core

Climatic input variables and model parameters for the ice core

We now introduce the data sets that we use as input for the proxy system models. We first consider two stationary stochastic processes, namely Gaussian white noise (GWN) and an autoregressive process of order 1 (AR(1) process), to evaluate whether such input can lead to the detection
of dynamical anomalies in the proxy time series.
Then, we consider non-stationary versions of the two well-known
Rössler (ROS) and Lorenz (LOR) systems. For all those processes, time series of length

For the case of GWN, we draw

For the AR(1) process, we create a time series of length

In a next step, we use data from a non-stationary version of the Rössler system which exhibits
non-trivial and rich cascades of bifurcations despite its rather simple attractor topology.
The Rössler system is defined by the set of ordinary differential equations (ODEs)

We here use the two fixed parameters

From the Feigenbaum diagram, it becomes clear that we expect to detect alternating periods of lower- and higher-dimensional dynamics in the time series. In particular, we stress that we do not expect to detect the bifurcation points but periods of outstandingly high- or low-dimensional dynamics in between them as we use random shuffling surrogates for the pointwise significance test.

The Lorenz system shows a more complicated attractor topology than the Rössler system and was
originally introduced as a simple toy model for atmospheric convection
processes

The stationary Lorenz system has been found to exhibit a shift from periodic to chaotic dynamics at

Finally, we consider reconstructed temperature and precipitation data of the years 501–2000 CE
from the Last Millennium
Reanalysis project version 2

Before studying the different proxy system models, we note that applying some general filtering techniques such as
moving average filtering or exponential smoothing to the described stochastic
input time series on the one hand and the deterministic input time series on the other hand changes the
results of the windowed recurrence network analysis in different ways (not shown here for brevity). For the stochastic time series,
the filtering mostly changes the variability of the calculated network transitivity. In particular,
filtering an AR(1) process with moving averages produces extended and additional areawise significant
patches of high values of the network transitivity. In turn, for deterministic input time series, the
variability of the network transitivity remains rather robust under filtering. Furthermore, the results
for the network transitivity remain similar when adding white noise to the different input time series
up to signal-to-noise ratios of

We then turn to the output time series of the different proxy system models which combine linear
and non-linear transformations and filtering of the input time series. The different input and model
output time series and some remarks on their properties can be found in the Supplement.
Figures

Network transitivity (colour-coded) for GWN model input and output with an areawise significance test (contours).

For Gaussian white noise (Fig.

Same as in Fig.

In the case of the AR(1) process (Fig.

Same as in Fig.

For the non-stationary Rössler system (Fig.

Same as in Fig.

The input time series of the non-stationary Lorenz system (Fig.

Network transitivity (colour-coded) for Last Millennium Reanalysis input at

For the Last Millennium Reanalysis data temperature input (Fig.

Fraction of missed and falsely identified significant points in the different proxy models with respect to the corresponding reference input variables.

In order to quantify the effect of the proxy system models as non-linear filters of the input signal on the detection of areawise significant
points, Table

Taken together, the previous findings should raise awareness in the context of future applications of wRNA to palaeoclimate proxy time series, suggesting that interpretations of results obtained for individual records only may not be sufficiently robust for drawing substantiated conclusions. From a practical perspective, this calls for combining different time series from different proxies and/or archives from the same region to obtain further climatological knowledge from such kinds of analysis

In this paper, we have studied the suitability of windowed recurrence network analysis (wRNA) for detecting
dynamical anomalies in time series from different proxy archives. For this, we used proxy system
models that simulate the formation of proxy archives, such as tree rings, lake sediments,
speleothems, and ice cores, given some climatic input variables like temperature and precipitation.
We created artificial input time series with different properties and additionally used temperature and
precipitation data from the Last Millennium Reanalysis project

We first compared the results of wRNA for stochastic and deterministic input to corresponding results when filtering the time series using moving average filtering and exponential smoothing (results not shown). We found that filtering alters the variability of the network transitivity, with a bias towards additional and extended patches of areawise significant high transitivity values for stochastic input. The network transitivity of deterministic input seems to be rather robust under such filtering. When processing the input time series through the different proxy system models, these differences for stochastic and deterministic input were also apparent. In terms of areawise significant anomalies of the network transitivity, we found that time series of tree ring width and brGDGTs in lake sediments have problems with missing areawise significant points, while the isotopic composition of speleothems also exhibits falsely identified significant points of high values of the network transitivity probably related to the bias towards higher values of the network transitivity due to the particular filtering in the model. Time series of the isotopic composition of ice yield comparable results to the corresponding input, but also sometimes miss significant points.

Taken together, our results show the need for further study of the effects of different filtering
mechanisms on the results of the wRNA in order to interpret the results and draw reliable conclusions
when analysing real-world data. This particularly concerns data from speleothems, as they have been
studied quite often using windowed recurrence analysis (e.g.

Still, we want to stress that even then, providing a general recipe for interpreting the resulting
network transitivity is hardly possible in the (palaeo)climate context. Climate-related interpretations
always vary depending on the location and, thus, local boundary conditions have to
be taken into account. As motivated in Sect.

Future work should also include the study of alternative proxy system models within this framework. Results of proxy system models for both the same proxies (but with more detailed systemic understanding of the formation of the proxies, as for example a tree ring width model accounting for juvenile growth of the trees) and different proxy variables (in particular, for other lacustrine proxy variables) will complement the improved understanding of the suitability of wRNA for these types of time series and will advance the interpretation of the corresponding results. Also, sensitivity studies for the different model parameters are of interest to better interpret results obtained with wRNA for a given real-world data set. This concerns particularly the mean aquifer transit time of the speleothem model.

For the stochastic input time series as for example the isotope input of GWN, we found some areawise significant artefacts in single realisations. To improve the reliability of the results for the these processes, more realisations should be considered to confirm the results and to exclude the influence of random artefacts. As we applied an areawise significance test to identify dynamical anomalies, which reduces the number of false positives in the analysis results, this can also reduce the number of true positives and increase the number of false negatives independent of whether considering model input or output time series. In this regard, we also observed that time series with stronger autocorrelations in most cases show higher correlations for the wRNA results in the different domains and, thus, have more restrictive bounds of the areawise significance test (not shown).

Additionally, the study of properties of the analysed time series can serve as
a starting point to judge the suitability of wRNA for other data to be
analysed. In particular, the effect of filtering the time series with different non-linear filters prior
to the analysis as done within the different proxy system models can and should be studied more
systematically. Also, the theory of non-linear observability might give an interesting new
perspective on this as the filtering can be seen as creating a new observable, and the choice of observable
has already been shown to influence results of recurrence quantification analysis and recurrence network analysis

Exemplary Python code for the windowed recurrence network analysis including the areawise significance test is available at

The supplement related to this article is available online at:

JL and RVD designed the research. JL performed the numerical experiments and data analyses. JL and RVD discussed the results and wrote the manuscript.

The authors declare that they have no conflict of interest.

Calculations have been performed with the help
of the

This research has been supported by the Bundesministerium für Bildung und Forschung (BMBF) via the BMBF Young Investigators Group,

This paper was edited by Stéphane Vannitsem and reviewed by Dmitry Divine and Michel Crucifix.