Nonlin. Processes Geophys., 18, 545–562, 2011 www.nonlin-processes-geophys.net/18/545/2011/ doi:10.5194/npg-18-545-2011 © Author(s) 2011. CC Attribution 3.0 License. Nonlinear Processes in Geophysics Identification of dynamical transitions in marine palaeoclimate records by recurrence network analysis J. F. Donges1,2 , R. V. Donner1,3 , K. Rehfeld1,2 , N. Marwan1 , M. H. Trauth4 , and J. Kurths1,2 1 Potsdam Institute for Climate Impact Research, P.O. Box 601203, 14412 Potsdam, Germany of Physics, Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany 3 Institute for Transport and Economics, Dresden University of Technology, Würzburger Str. 35, 01187 Dresden, Germany 4 Department of Geosciences, University of Potsdam, Karl-Liebknecht-Str. 24, 14476 Potsdam, Germany 2 Department Received: 25 May 2011 – Revised: 11 August 2011 – Accepted: 27 August 2011 – Published: 5 September 2011 Abstract. The analysis of palaeoclimate time series is usually affected by severe methodological problems, resulting primarily from non-equidistant sampling and uncertain age models. As an alternative to existing methods of time series analysis, in this paper we argue that the statistical properties of recurrence networks – a recently developed approach – are promising candidates for characterising the system’s nonlinear dynamics and quantifying structural changes in its reconstructed phase space as time evolves. In a first order approximation, the results of recurrence network analysis are invariant to changes in the age model and are not directly affected by non-equidistant sampling of the data. Specifically, we investigate the behaviour of recurrence network measures for both paradigmatic model systems with non-stationary parameters and four marine records of long-term palaeoclimate variations. We show that the obtained results are qualitatively robust under changes of the relevant parameters of our method, including detrending, size of the running window used for analysis, and embedding delay. We demonstrate that recurrence network analysis is able to detect relevant regime shifts in synthetic data as well as in problematic geoscientific time series. This suggests its application as a general exploratory tool of time series analysis complementing existing methods. 1 Introduction Palaeoclimate proxy data representing past variations of environmental conditions can be obtained from various types of geological archives distributed over the Earth’s surface. The study of time series of such proxies, i.e. data that encode the Correspondence to: J. F. Donges ([email protected]) temporal variability of physical, chemical, biological or sedimentological properties, is a major source of information fostering our understanding of the functioning of the complex Earth system in the past, present, and future. However, nonequidistant sampling, uncertain age models, multi-scale, and multi-stable state variability as well as relatively high noise levels render the study of these proxy records a challenging problem for time series analysis. Methods used for time series analysis can be roughly classified as linear or nonlinear. On the one hand, linear methods are based on the evaluation of certain classical statistical characteristics and assume the presence of an underlying linear stochastic process with eventually some superimposed deterministic (e.g. periodic) components (Brockwell and Davis, 1991, 2002; Hamilton, 1994). Prominent examples that are frequently used for the analysis of realworld time series, including such obtained from geological archives (Schulz and Stattegger, 1997; Schulz and Mudelsee, 2002; Mudelsee et al., 2009; Rehfeld et al., 2011), are correlation functions and power spectra. On the other hand, nonlinear methods follow a dynamical systems point of view, implicitly assuming the presence of certain types of deterministic behaviour (Abarbanel, 1996; Kantz and Schreiber, 1997; Donner and Barbosa, 2008). The vast majority of existing linear or nonlinear methods of time series analysis relies on the quantification of patterns of temporal dependences between observations x(t) made at different times t, i.e. aims to quantify functional relationships of the form X x(t) = f (x(t − τ ),τ,t) + η(t), (1) τ >0 where f (x,τ,t) is a general deterministic function, and {η(t)} is a stochastic process (often assumed to be fully uncorrelated, i.e. δ-correlated, in time). For a stationary system, the functional dependence f does not explicitly depend Published by Copernicus Publications on behalf of the European Geosciences Union and the American Geophysical Union. 546 J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis on time t. In standard linear methods of time series analysis, f is often assumed to be a linear function; in this case, the parameters of f encode linear temporal correlations. More generally, one may consider arbitrary (i.e. not explicitly specified) deterministic relationships f , which may be characterized using concepts such as mutual information (Kantz and Schreiber, 1997). In the following, we will refer to methods of time series analysis that are based on the quantification of temporal interrelationships between observations, e.g. correlation and mutual information functions or power spectra, as correlative methods. These clearly depend on how well the observation points are specified. In particular, in case of a nonuniform sampling of the considered time series, estimates of even simple linear characteristics can often not be expressed in a straightforward analytical way. For example, if one wishes to avoid interpolation (which leads to additional uncertainties), power spectra can be estimated using harmonic regression of the data (e.g. by means of the LombScargle periodogram; Lomb, 1976; Scargle, 1982), projection methods (Foster, 1996a,b), or a variety of alternative approaches (Babu and Stoica, 2010; Rehfeld et al., 2011). However, in the specific case of palaeoclimate data where typically not even the exact timing of the individual observations is sufficiently well known (Telford et al., 2004), correlative methods can have strong conceptual disadvantages. In contrast to this large class of methods (which characterise time series from a more or less rigorous statistical point of view), alternative concepts such as fractal dimensions and generalisations thereof have been first developed in different mathematical disciplines and later applied to the characterisation of the properties of certain dynamical systems (Sprott, 2003). Statistical estimates of such measures can be obtained by a variety of different approaches, most of which take into account the spatial arrangement of observations in the (possibly reconstructed) phase space. From this perspective, the mentioned methods do not directly require knowledge about the timing of observations, i.e. can be considered as non-correlative or geometric methods, since they rely on geometric attractor properties in phase space rather than on dynamical information. In the case of palaeoclimate data with uncertain age models, geometric methods may provide a considerable alternative for statistical analysis. However, as a particular disadvantage, we note that the proper estimation of fractal dimensions usually requires a considerably larger amount of data than necessary for most correlative methods (Sprott, 2003), which are typically not available in palaeoclimatology. Some fundamental relationships between the geometric properties of attractors in phase space (e.g. Hausdorff and box dimensions) and important invariants of the associated dynamics (e.g. Lyapunov exponents) are known to exist (Chlouverakis and Sprott, 2005). Note that certain measures of dimensionality include both geometric and dynamical information, i.e. all Rényi dimensions Dq for q > 1 Nonlin. Processes Geophys., 18, 545–562, 2011 including the information dimension D1 (Sprott, 2003). However, besides fractal dimension estimates based on attractor topology there are only very few suitable and purely geometric methods available. Recently, it has been suggested to characterise the mutual proximity relationships of all pairs of state vectors from the sampled attractor in phase space by means of complex network methods (Zhang and Small, 2006; Yang and Yang, 2008; Xu et al., 2008; Marwan et al., 2009; Donner et al., 2010a). Among others, the concept of recurrence networks (RNs) (Marwan et al., 2009; Donner et al., 2010a,b) has been proven particularly useful for this purpose. Since such complex network representations of time series take only spatial information into account, they can be considered as important examples of geometric methods of time series analysis. RNs provide a set of nonlinear measures characterising the complexity of dynamical systems (Donner et al., 2010a, 2011a), e.g. allowing to distinguish periodic from chaotic dynamics. While recent findings demonstrate close interrelationships between certain RN properties and fractal dimensions (Donner et al., 2011b), the graph-theoretical measures can often be estimated with high confidence from much shorter time series than fractal dimensions. This warrants their application as a tool for windowbased analysis of non-stationary data (Marwan et al., 2009; Donner et al., 2011a). In contrast to the aforementioned approaches, transition networks (Nicolis et al., 2005) and visibility graphs (Lacasa et al., 2008) are correlative methods in the sense that they depend explicitly on the temporal ordering of observations. When considering network-based methods of time series analysis, only RNs (Marwan et al., 2009; Donner et al., 2011a; Hirata et al., 2011) and visibility graphs (Elsner et al., 2009) have been used to analyse geoscientific data. So far RN analysis is the only network-based technique that has been applied to investigate palaeoclimate proxy records. In this work, we discuss the application of RNs to studies of palaeoclimate records, with a special focus on the identification of structural changes in the dynamics that are not easily found when relying on simple linear statistics. As a benchmark example, we will mainly utilise three marine records of aeolian dust flux from Northern Africa during the last 5 Myr (million years) (Trauth et al., 2009; Marwan et al., 2009; Donner et al., 2011a; Donges et al., 2011). In Sect. 2, we present a detailed description of the considered data sets, the necessary preprocessing steps, and the general idea of RNs and their quantitative analysis. Application to typical nonlinear model systems with a systematic drift of the control parameters in Sect. 3 suggests that network statistics are well suited for identifying dynamical transitions from finite time series. Finally, in Sect. 4, we describe the results of our investigations obtained for the different palaeoclimate time series and discuss their robustness with respect to the fundamental parameters of our method. www.nonlin-processes-geophys.net/18/545/2011/ J.J.F.F.Donges transitions in marine marine palaeoclimate palaeoclimaterecords recordsby byRN RNanalysis analysis Dongesetetal.: al.:Identification Identificationof of dynamical dynamical transitions transitions in in dynamical marine palaeoclimate records by RN analysis 3 547 3 45°N 45°N Site Site 967 967 30°N 30°N Site Site659 659 15°N 15°N Site 721/722 Site 721/722 0°0° 15°W 15°W 0°0° 15°E 15°E 30°E 30°E 45°E 60°E 60°E Fig. 1. Map displaying the locations of the three ODP drilling sites et al., deMenocal, considered indisplaying this studythe (Tiedemann Fig.1.1.Map Mapdisplaying the locations of of the the 1994; three ODP drilling sites Fig. locations three drilling1995, sites 2004; Larrasoaña et al., 2003). considered in this study (Tiedemann et al., 1994; deMenocal, considered in this study (Tiedemann et al., 1994; deMenocal, 1995, 1995, 2004;Larrasoaña Larrasoañaetetal., al.,2003). 2003). 2004; 2 Data and methods Description of the data 2 Data Data andmethods methods 2 2.1 and Marine records of terrigenous 2.1 Description Description thedata datadust flux from North Africa are 2.1 ofofthe an important source of information on the long-term aridifiMarinerecords records terrigenous dust flux from North North Africa are cation of the continent during theflux Plio-Pleistocene (Trauth Marine ofofterrigenous dust from Africa are N aridifian important source of information on the long-term 2009). Continuous time series {x = x(t )} et al., sami i=1aridifian important source of information on thei long-term cationatoftimes the continent during theanPlio-Pleistocene pled {ti }, where i isthe index variable (Trauth and N cation of the continent during Plio-Pleistocene (Trauth et al., 2009). Continuous time series {xthree = x(t )}NN the number of samples, are available from sediment et al., 2009). Continuous time series {xii = x(tii )}i=1 i=1 sampled at 659 times(Atlantic t , where i isoffshore an index variable West and cores: ODP Ocean subtropical sampled at times ti ,i where i is an index variable and N the number of samples, are available from three sediAfrica) (Tiedemann et al., 1994), ODP 721/722 (Arabian N the number of samples, are available from three sediment (deMenocal, cores: ODP 659 (Atlantic subtropical Sea) 1995, 2004),Ocean and offshore ODP 967 (Eastern ment cores: ODP 659 (Atlantic Ocean offshore subtropical West Africa) (Tiedemann et al., 1994), ODP (Fig. 721/722 (AraMediterranean Sea) (Larrasoaña et al., 2003) 1). In adWest Africa) (Tiedemann et al., 1994), ODP 721/722 (Ara18 bian Sea) 1995, 2004),(δand 967from (Eastern dition, the (deMenocal, benthic oxygen isotope O)ODP record ODP bian Sea) (deMenocal, 1995, 2004),al., and ODP(Fig. 9671). (Eastern Mediterranean Sea) (Larrasoaña 2003) adsite 659 (Tiedemann et al., 1994)etwill be studied as a In proxy Mediterranean Sea) (Larrasoaña et al., 182003) (Fig. 1). In addition, the benthic oxygen (δ18 O) record from ODP for variations in global iceisotope volume, which can be assumed dition, thea(Tiedemann benthic oxygen isotope (δ O) record as from ODP sitehave 659 et al., 1994) studied a proxy to considerable impact onwill the be continental aridificasite 659 (Tiedemann et al., 1994) will be studied as a proxy for variations in global ice volume,ofwhich assumed tion via a southward displacement climatecan andbevegetation for in series globalare ice volume, which assumed to variations haveAll a considerable impact onin the continental 2. can be aridificazones. time shown Fig. totion have a considerable impact on the continental aridificavia a southward displacement of climate and vegetation tion viaDetrending aAll southward displacement climate zones. time series are shown inofFig. 2. and vegetation 2.2 zones. All time series are shown in Fig. 2. 2.2 considered Detrending All time series {xi } show a nonlinear trend of in2.2 Detrending creasing amplitude and variance towards the present. This All considered time series {xiaridification } show a nonlinear trend intrend reflects the successive of North andofEast creasing and variance towards the present. This All considered series {xi }ofshow a nonlinear trendglacial of inAfrica andamplitude thetime intensification Northern hemisphere trend reflects the aridification and East creasing amplitude and variance towards the present. This cycles during the successive Plio-Pleistocene (Trauthof etNorth al., 2009). To Africa and the intensification of Northern hemisphere glacial trend reflects the successive aridification of North East prevent corruption of the results of our analysis andand significyclesand during the Plio-Pleistocene (Trauth et al., 2009). To Africa the of Northern glacial cance test dueintensification to this nonlinear trend, wehemisphere attempt to remove prevent corruption of the results of our analysis and significycles during thebyPlio-Pleistocene et al.,of2009). To it to first order subtracting from(Trauth xi the mean a sliding cance test nonlinear trend, we attempt to remove prevent corruption of the results of our analysis and signifiwindow ofdue sizeto Wthis (t ) centered at t for all time points t , i.e. D i i i it to first ordertoby subtracting xiwe theattempt mean oftoa remove sliding cance test due this nonlinearfrom trend, bW (t )/2c D iat t for all time points t , X of sizebyW1subtracting D (ti ) centered i it window to first order from xii the mean of a sliding x̂i.e., xi+j , (2) i = xi − window of2bW sizeDW (ti )+centered (tiD)/2c 1 j =−bW at(t t)/2c i for all time points ti , D i i.e., bWD (ti )/2c X 1 where window sizexi+j WD,, (2) x̂i = xifor − a chosen detrendingbW D (ti )/2c 2bWD (t1i )/2c + 1 X j=−bWD (ti )/2c xi+j , (2) x̂i = xi − 2bWD (ti )/2c + 1 j=−bWD (ti )/2c www.nonlin-processes-geophys.net/18/545/2011/ Fig. 2. Plio-Pleistocene variability of (A) δ 18 O at ODP site 659 18 atflux (Tiedemann et al., 1994),variability and of terrigenous from North Fig. ofof(A) δ 18δdust O ODP sitesite 659659 Fig.2.2. Plio-Pleistocene Plio-Pleistocene variability (A) O at ODP et al., 1994), (C)from 721/722 Africa at ODP sites (B) 659 (Tiedemann (Tiedemann et al., 1994), and of terrigenous dust flux from North (Tiedemann et al., 1994), and of terrigenous dust flux North (deMenocal, 1995, 2004), and (D) 967 et (Larrasoaña et (C) al., 2003). Africa sites (Tiedemann al.,al., 1994), (C) 721/722 AfricaatatODP ODP sites(B) (B)659 659 (Tiedemann et 1994), 721/722 (deMenocal, 2004), and (D) et al., 2003). The horizontal red bars in panel indicate two consecutive (deMenocal,1995, 1995, 2004), and(A) (D)967 967(Larrasoaña (Larrasoaña et al.,recur2003). ∗= The horizontal in panel consecutive recurrence windowsred of bars length W ∗ =(A) 410indicate kyr andtwo mutual offset 1W The horizontal red bars in panel (A) indicate two consecutive∗recur∗ rence of the length W = 410 kyr and mutual offset ∆W = 41 kyrwindows as used in analysis of Sect. 4 and in Figs. 9–12. rence windows of length W ∗ = 410 kyr and mutual offset ∆W ∗ = 41 kyr as used in the analysis of Sec. 4 and in Figs. 9–12. 41 kyr as used in the analysis of Sec. 4 and in Figs. 9–12. 2(i − 1) for i < WD , WD for WD ≤ i ≤ N − WD , WD (ti ) = where for achosen detrending size WD , 2(N − for iwindow > N −W D. W where for a choseni)detrending window size , (3) D That is, the effective detrending window size decreases to− 1) boundaries, for i < W 2(iseries’ wards the time resulting in x̂1 = x̂N = 0. D, 2(i − 1) for i < W W for W ≤ iD ≤,N − Wof (3)or W (t ) = This approach D avoids theDcomplication D , locally D simple i W for W ≤ i ≤ N − W , (3) W (t ) = D D D D i 2(N − i) for i > N − W . globally fitting D performing high higher-order polynomials or 2(N − i) for i > N − W . D pass filtering given irregular sampling and uncertain dating of measurements to remove the nonlinear trend. Since RN analThat is, our the method effectiveofdetrending size decreases toysis as choice is awindow non-correlative technique That is, the effective detrending window size decreases wards the time series’ boundaries, resulting in x̂ = x̂ = 0.to1 N and its results are permutation invariant (Sect. 2.6), spuriwards the time series’ boundaries, resulting in x̂ = x̂ = 0. This simple approachwhich avoidsmay the be complication of locally 1 N or ous autocorrelations introduced by the sliding This simple approach avoids the complication of locally globally higher-order polynomials orproblem performing highwindow fitting detrending do not pose a serious here. Weor globally higher-order polynomials or performing highpass filtering irregular sampling dating of will showfitting in given Sect. 4.2 that our resultsand are uncertain robust with respect measurements to remove the nonlinear trend. Since RN analpass filtering given irregular sampling and uncertain dating to a large range of reasonable choices of WD . Except of theof ysis as our method of choice a non-correlative technique measurements to remove the is nonlinear Since RNdata. analdetrending, no further preprocessing wastrend. applied to the and its results are permutation invariant (Sec. 2.6), ysis as our method of resample choice is the a non-correlative technique Particularly, we do not time series to spurious obtain an autocorrelations which introduced thethe sliding winand itsspaced resultsrecord are permutation invariant (Sec. 2.6), spurious evenly inmay the be time domain, by since necessary dow detrending dowhich not pose serious problem here. We will autocorrelations mayathe be introduced the sliding wininterpolation would corrupt results of thebyfurther analysis show in Sec. 4.2 that our results are robust with respect dow do not(see, posee.g. a serious problem here. Wetowill Rehfeld et al., 2011). to be detrending performed below ashow largeinrange choicesare of robust WD . Except of the to Sec.of4.2reasonable that our results with respect detrending, no further preprocessing was applied to the a large range of reasonable choices of WD . Exceptdata. of the 2.3 Embedding Particularly, do not resample the time to obtain detrending, we no further preprocessing wasseries applied to the an data. evenly spacedwe record in the time domain, since the necessary Particularly, do not resample series Univariate time series often reflectthe thetime dynamics oftoaobtain higher-an interpolation would corrupt results of thesince further analysis evenly spaced record in thethe time thesome necessary dimensional complex system as domain, viewed through obto be performed below (see, e.g., Rehfeld et al. (2011)). interpolation would corrupt the results of the further analysis servation function. In typical situations it is possible to to be performed below (see, e.g., Rehfeld et al. (2011)). Nonlin. Processes Geophys., 18, 545–562, 2011 4 548 J.J.F.F.Donges marine palaeoclimate palaeoclimaterecords recordsby byRN RNanalysis analysis Dongesetetal.: al.:Identification Identificationof of dynamical dynamical transitions in marine reconstruct the phase space trajectory using time-delay em2.3 Embedding bedding, i.e. considering state vectors the dynamics of a higherUnivariate time series often reflect (m,τ ) yi = x̂i , x̂i+τ ,..., x̂i+(m−1)τ (4) dimensional complex system as viewed through some obinstead of the univariate observations x̂i themselves (Packard servation function. In typical situations it is possible to reet al., 1980; Takens, 1981). Due to the time-delay finite lengthembedof the construct the phase space trajectory using available time series, state the index i is now restricted to the range ding, i.e., considering vectors i = 1,...,N − (m − 1)τ . The embedding parameters embed to be appropriately de(m,τ ) dimension m and delay τ have yiding = x̂i , x̂i+τ ,..., x̂i+(m−1)τ (4) termined from the available data, e.g. using approaches such et al., 1992) and avas the of false (Kennel instead the nearest-neighbours univariate observations x̂i themselves (Packard and Swinney, 1986) metherage mutual information (Fraser et al., 1980; Takens, 1981). Due to the finite length of the ods, respectively. Although there are good reasons for apavailable time series, the index i is now restricted to the range plying embedding techniques, it is known that this approach i = 1,...,N − (m − 1)τ . The embedding parameters embedalsodimension has conceptual and induce spurious ding m anddisadvantages delay τ have to bemay appropriately deterstructures in recurrence plots and corresponding misleading mined from the available data, e.g., using approaches such as recurrence quantification analysis (RQA) theresults false of nearest-neighbours (Kennel et al., 1992) and (Thiel averet al., 2006). In contrast, many important dynamical inage mutual information (Fraser and Swinney, 1986) methods, variants can be estimated from non-embedded time series as respectively. Although there are good reasons for applying well, especially using recurrence plot-based methods (Thiel embedding techniques, it is known that this approach also has et al., 2004a). From here on we will use the simplified notaconceptual disadvantages and may induce spurious structures tion y i for reconstructed state vectors and assign to them the in recurrence plots and corresponding misleading results of ages ti , respectively. recurrence quantification analysis (RQA) (Thiel et al., 2006). While the standard approaches for determining the optiIn contrast, many important dynamical invariants can be esmum embedding parameters typically provide feasible retimated non-embedded time series the as well, especially sults infrom the case of many applications, situation is conusing recurrence plot-based methods (Thiel et al., 2004a). siderably more challenging for palaeoclimate records: on From herehand, on we will useembedding the simplified notation yi equally for rethe one traditional methods require constructed state vectors to them of thethe ages ti , respaced observations, soand thatassign interpolation available spectively. data might become necessary with all corresponding concepWhereas the standard approaches forindetermining optual disadvantages. On the other hand, the presencethe of dattimum embedding parameters typically provide feasible reing uncertainties, even such interpolation is hardly possible sults the case applications, the situation is conandin would lead of to many an enormous enhancement of uncertainty siderably more challenging in the embedded record. for palaeoclimate records: On the one hand, traditional embedding methods Given these methodological difficulties werequire attemptequally a comspaced observations, so that interpolation promise: (i) the embedding dimension mof= the 3 isavailable a tradedata necessary with all series corresponding concepoff might given become the relatively short time forbidding larger tual disadvantages. On the(Eckmann other hand,etinal., the1992; presence of datembedding dimensions Kantz and ing uncertainties, interpolation is hardly possible Schreiber, 1997)even and such the underlying high-dimensional dynamics estimated by the false nearest-neighbours criteand wouldaslead to an enormous enhancement of uncertainty al., 1992; Marwan et al., 2009). (ii) Us(Kennel etrecord. inrion the embedded ing a Gaussian kernel-based of we the attempt autocorrelation Given these methodologicalestimator difficulties a comfunction adapted to irregularly sampled time promise: (i) The embedding dimension m =series 3 is (Rehfeld a tradeal., 2011), we find that the autocorrelation of all fourlarger time offet given the relatively short time series forbidding series has decayed markedly after 10 kyr (Fig. 3). Hence, embedding dimensions (Eckmann et al., 1992; Kantz and we choose1997) the delay to cover approximately the same time Schreiber, and τthe underlying high-dimensional dy∗ = 10 kyr for all considered records, i.e. scale τ namics as estimated by the false nearest-neighbours criterion τ =(Kennel bτ ∗ /h1Tetic,al., 1992; Marwan et al., 2009). (ii) Us(5) ing a Gaussian kernel-based estimator of the autocorrelation where h1T i is the average sampling time (Table 1) and bxc function adapted to irregularly sampled time series (Rehfeld denotes the integer part of x. This yields τ1 = 2 for ODP et al., 2011), we find that the autocorrelation of all four time site 659, τ2 = 5 for site 721/722, and τ3 = 27 for site 967. A series has decayed markedly after 10 kyr (Fig. 3). Hence, promising technique for consistent embedding of irregularly we choose the delay τ to cover approximately the same time sampled time series is based on Legendre polynomials (Gibscale τ ∗ = 10 kyr for all considered records, i.e. son et al., 1992) and should be explored in future studies. τ = bτ ∗ /h∆T ic, Nonlin. Processes Geophys., 18, 545–562, 2011 (5) Fig. 3. Linear autocorrelation functions C(τ ) for (A) the δ 18 O Fig. 3. atLinear autocorrelation functions C(τ ) for (A)ODP the sites δ 18 O record ODP site 659 and the dust flux records from record at (C) ODP site 659and and(D) the967. dustThe fluxautocorrelation records from functions ODP sites (B) 659, 721/722, (B) (C) 721/722, (D) 967. The autocorrelation functions were659, estimated using aand Gaussian kernel-based estimator (Rehfeld were usingtoairregularly Gaussian sampled kernel-based estimator et al., estimated 2011) adapted data (solid line) (Rehfeld and diet al., 2011) adapted to irregularly sampled data and directly from time series linearly interpolated to a (solid regularline) sampling h1T i (dash-dotted with sampling line).to aFor the Gaussian rectly from timetime series linearly interpolated regular sampling kernel-based we iused the recommended optimum bandwith samplingestimator time h∆T (dash-dotted line). For the Gaussian i/4 (Rehfeld width h = h1T et al., where h is the standard kernel-based estimator we used the2011), recommended optimum banddeviation the i/4 Gaussian kernel. width h =ofh∆T (Rehfeld et al., 2011), where h is the standard deviation of the Gaussian kernel. 2.4 Windowed analysis where h∆T i is the average sampling time (Tab. 1). This For detecting structural changes encoded by yields τ1 = 2 for ODP site 659, in τ2the = 5dynamics for site 721/722, and the time series, we slide a window over the embedded record ∗ τ3 = 27 for site 967 corresponding to τ = 10 kyr. A promis{y } and perform the subsequent analysis for the data coningi technique for consistent embedding of irregularly samtained in each window separately. However, the records unpled time series is based on Legendre polynomials (Gibson der study are quite heterogeneous with respect to their basic et al., 1992) and should be explored in future studies. sampling properties (Table 1). The average sampling time h1T iWindowed differs widely across the records. In order to assure 2.4 analysis comparability of our results uncovered from the different timedetecting series, the most natural approach to chooseencoded windows For structural changes in theisdynamics by ∗ in units of time. However, this approach of atime fixedseries, size W the we slide a window over the embedded record has}two disadvantages: The exact timing ti of the available {y i and perform the subsequent analysis for the data conobservations not known as is the However, case for most tained in eachiswindow separately. the geological records unproxy records, and due to the non-uniform sampling rates, der study are quite heterogeneous with respect to their badifferent windows would contain different amounts of data. sic sampling properties (Tab. 1). The average sampling time While the latter is not across problematic for statistical teststoagainst h∆T i differs widely the records. In order assure homogeneity of the distribution of values in different wincomparability of our results uncovered from the different dows, a quantitative comparison of statistical characteristics time series, the most natural approach is to choose windows of the associated RNs (see Sect. 2.5) is not possible. Thereof a fixed size W ∗ in units of time. However, this approach fore, in the following, we will proceed in a different way by has two disadvantages: The exact timing ti of the available prescribing both the window size W and step size 1W for observations is not known as is the case for most geological RN analysis measured in units of sampling points. In order proxy records, and due to the non-uniform sampling rates, to derive W and 1W from the desired quantities in units of different∗ windows∗would contain different amounts of data. time, W and 1W , we divide by the average sampling time, While the latter is not problematic for statistical tests against ic,distribution of values in different winW = bW ∗ /h1T (6) homogeneity of the ∗ dows, a quantitative comparison of statistical characteristics 1W = b1W /h1T ic. (7) of the associated RNs (see Sec. 2.5) is not possible. There∗ In turn, thefollowing, actual window sizeproceed W (ti ) is by the fore, in the we will in determined a different way by average sampling time in the size-W window centred around prescribing both the window size W and step size ∆W for RN analysis measured in units of sampling points. In order www.nonlin-processes-geophys.net/18/545/2011/ J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis to derive W and ∆W from the∗ desired quantities in units t . For a particular choice of W and the associated step size ofi time, W ∗ and ∆W ∗ , we divide by the average sampling 1W ∗ in units of time, the resulting values of W and 1W , time, the mean window widths, and step sizes as well as the corre∗ sponding deviations are given in Table 1. W = bWstandard /h∆T ic, (6) ∗ approach for determining the window size deThe simple ∆W = b∆W /h∆T ic, (7) scribed above guarantees that the windows cover approxwhere the integer partallofrecords x. In turn, actual imatelybxc thedenotes same time span for and the positions window sizetime W ∗ (t the average sampling within the series. While mostby sampling intervals take i ) is determined time in the centred around outliers, ti . For which a parvalues closesize-W to the window mean, there are distinct ticular choicecorrespond of W ∗ andtothe associated step to size ∆W ∗ in most likely missing data due incomplete units time, the resulting values of of Wthe andsediment ∆W , the mean coreof recovery, hiata or disturbances such as 4). Nevertheless, the standard deviation of turbidites (Fig. and window widths, step sizes as well as the corresponding windowdeviations size σ (W ∗are ) is given still small in comparison to the averstandard in Tab. 1. hW ∗ i for age (Table 1), which the suggests thatsize statisThewindow simple size approach determining window detical characteristics computed windows still scribed above guarantees thatfor thedifferent windows cover can approxbe quantitatively a reasonable way.and positions imately the same compared time spanin for all records µ Formally, theseries. data series } within the µ-th window, within the time While{ymost sampling intervals take i N−W c, is given by µ = 1,2,...,b values close to 1W the mean, there are distinct outliers, which mostµ likely correspond to missing data due to incomplete {y i recovery, } = {y (µ−1)1W core hiata+ior}, disturbances of the sediment such(8) as turbidites (Fig. 4A). Nevertheless, the standard deviation of where from here on i = 1,...,W . We use the window’s midwindow size σ(∆W ∗ ) is still small in comparison to the avpoint’s timing erage window size h∆W ∗ i (Tab. 1), which suggests that statistical characteristics computed for different windows can t µ = t(µ−1)1W (9) +bW/2c still be quantitatively compared in a reasonable way. toFormally, attach an age the series scalar network measures g µ calculated the to data {yiµ } within the µ-th window, the data withinby the µ-th window. µfrom = 1,2,..., is given µ {y2.5 {y(µ−1)∆W network +i }, i } =Recurrence analysis (8) where from here on i = 1,...,W . We use the window’s midRecurrence point’s timingin phase space is a basic property of complex dynamical systems. Since the seminal work of Poincaré (1890), tµit=ist(µ−1)∆W (9) known that under rather general conditions, dynamical +bW/2c systems tend to return arbitrarily close to their previous states to attach an age to the scalar network measures g µ calculated in the long-term limit. In the last decades, the recurrence from the data within the µ-th window. property has attracted considerable interest, since it has been shown that essential information about the main dynamical 2.5 Recurrence network analysis properties is contained in the temporal pattern of mutual recurrences of state (Thiel et aal., 2004b; Robinson and Thiel, Recurrence inaphase space is basic property of complex dy2009). Particularly, the visual representations of recurrence namical systems. Since the seminal work of Poincaré (1890), (Eckmann et al., 1987; al., 2007) have found itplots is known that under ratherMarwan general etconditions, dynamical wide use, which are most commonly expressed by a binary systems tend to return arbitrarily close to their previous states matrix inrecurrence the long-term limit. In the last decades, the recurrence µ µ µ property attracted interest, since it has been Rij (ε) =has 2(ε − ky −considerable y j k), (10) shown that essentiali information about the main dynamical properties in theε temporal pattern threshold of mutualand rewhere foristhecontained µ-th window, is the recurrence currences of a state (Thiel et al., 2004b; In Robinson and Thiel, 2(·) denotes the Heaviside function. the following we 2009). the visual of recurrence use theParticularly, supremum norm k · k torepresentations measure distances in the reconstructed phase space of the considered (see plots (Eckmann et al., 1987; Marwan et al., observable 2007) haveyfound Fig. use, 5 for which examples). The appropriate of thebyimportant wide are most commonly choice expressed a binary parameter matrix ε is discussed below. recurrence It turned out that recurrence plots show distinct line strucµ Rtures, Θ(ε −length kyiµ − distribution yjµ k), (10) ij (ε) = whose can be used for defining suitable of complexity terms of threshold RQA, or and for where for measures the µ-th window, ε is theinrecurrence Θ(·) denotes the Heaviside function. In the following we www.nonlin-processes-geophys.net/18/545/2011/ 549 5 Fig. 4. (A) Probability distribution (PDF) p(1T ) of the sampling intervals of the three dust flux records according to their established Fig. 4. (A) Probability distribution (PDF) p(∆T ) of the sampling age models (ODP sites 659: solid line, 721/722: dash-dotted, 967: intervals of the three dust flux records according to their established dashed). The distribution for the δ 18 O record at ODP site 659 is viage models (ODP sites 659: solid line, 721/722: dash-dotted, 967: sually almost indistinguishable from18that of the corresponding dust dashed). The distribution for the δ O record at ODP site 659 is viflux record and therefore not shown. The PDFs were estimated ussually almost indistinguishable fromhthat the)(N corresponding dust ing a Gaussian kernel with bandwidth = σof (1T − 1)−1/5 (Taflux record and therefore not shown. The PDFs were estimated ble 1) following Scott’s rule (Scott, 1982). (B, C, D) Temporal using a Gaussian kerneltimes with for bandwidth = σ(∆T )(N − 1)−1/5 variation of the sampling the three hdust flux records. (Tab. 1) following Scott’s rule (Scott, 1982). (B,C,D) Temporal variation of the sampling times for the three dust flux records. estimating dynamical invariants such as correlation dimension, 2nd-order Rényi entropy, or generalised mutual inforuse the(Marwan supremum norm k · kIntothe measure in the remation et al., 2007). contextdistances of palaeoclimate constructed phase space considered observable y (see research, recurrence plots of andtheRQA have been successfully Fig. 5 for The appropriate applied forexamples). tracing dynamical changes choice (Trauthofetthe al.,important 2003; parameter ε is2003) discussed below. records with different ageMarwan et al., and aligning It turned that recurrence plots show line strucdepth modelsout (Marwan et al., 2002). RQAdistinct is a correlative tures, whose distribution be used forondefining method of time length series analysis, as it can explicitly relies temsuitable measures complexity in terms of RQA, poral structures in theofform of diagonal and vertical lines.or for estimating as correlation dimenRecently, dynamical it has beeninvariants suggestedsuch to approach recurrence sion, 2nd-order Rényi entropy, generalised informatrices from a complex networkor perspective by mutual identifying mation (Marwan et al., 2007). In the context of palaeoclimate µ µ Aresearch, (11) ij (ε) − δij plots and RQA have been successfully ij (ε) = Rrecurrence for tracing dynamical changes (Trauth et al., 2003; (δapplied ij denoting Kronecker’s delta) with the adjacency matrix Marwan et al., 2003) and aligningtorecords with different ageof a complex network associated the underlying time sedepth models et(Marwan et al., 2002). RQA is a 1correlative ries (Marwan al., 2009; Donner et al., 2010a) . This method of time series analysis, as it explicitly relies on tem1 Note that similar approaches can also be found in other poral structures in the form of diagonal and vertical lines. Recently, it has been suggested to approach recurrence Nonlin. Processes Geophys., 18, 545–562, 2011 550 J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis 6 J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis Table 1. Basic properties of the analysed palaeoclimate time series. N is the number of samples contained in the time series, h1T i the mean sampling interval, and σ (1T ) the standard deviation of sampling intervals. For a desired window size W ∗ = 410 kyr and step Table 1. ∗Basic properties of theinanalysed time W series. N isgive the the number of samples contained thesize time h∆T size 1W = 41 Sect. 4.1palaeoclimate for RN analysis), and 1W corresponding window and in step (inseries, numbers ofi kyr (as chosen ∗ ∗ the ∗ ) the the mean sampling σ(∆T ) theeffective standardwindow deviation sampling intervals. desired window size W = 410deviations kyr and step observations), W ∗ interval, and 1Wand average andofstep size, and σ (W ∗ ) For and aσ (1W associated standard (in ∗ size = 41 kyr (as chosen in Sec. 4.1 for RN analysis), W and ∆W give the corresponding window and step size (in numbers of units∆W of time). observations), hW ∗ i and h∆W ∗ i the average effective window and step size, and σ(W ∗ ) and σ(∆W ∗ ) the associated standard deviations ∗ (in units of time). h1T i σ (1T ) N W 1W W σ (W ∗ ) 1W ∗ σ (1W ∗ ) A (kyr) (kyr) σ(∆T ) (kyr) 4.282.88 2.88 4.102.69 2.69 1.811.52 1.52 0.360.31 0.31 B 1400 Time (kyr) Time (kyr) 1300 1200 1100 1000 1000 (kyr) W ∆W hW ∗ i (kyr) 400.37 95 95 9 9 400.37 100 10 408.16 100 10 408.16 401.97 226226 22 22 401.97 1139 113 409.10 1139 113 409.10 C 2350 3350 2250 3300 2200 2150 1400 3250 3200 3150 3100 2050 1200 1300 Time (kyr) σ(∆W ∗ ) (kyr) 4.36 3.22 6.09 7.51 3400 2300 2100 1100 (kyr) (kyr) (kyr) σ(W ∗ ) (kyr) h∆W ∗ i (kyr) 46.5846.5838.37 4.36 38.37 33.0933.0941.25 3.22 41.25 62.6662.6639.29 6.09 39.29 78.0478.0440.67 7.51 40.67 Time (kyr) N 18 h∆T i (kyr) ODP 659 δ 18 O ODP 659 1170δ O 4.281170 ODP 659 dust ODP 659 1221dust 4.101221 ODP 721/722 dust ODP 721/722 dust 2757 1.812757 ODP 967 dust ODP 967 dust 8417 0.368417 3050 2100 2200 Time (kyr) 2300 3100 3200 3300 Time (kyr) 3400 Fig. 5. Recurrence plots (equivalently adjacency matrices of the RNs shown in Fig. 6) obtained from the dust flux record at ODP site 659, ∗ 41 kyr, and embedding parameters centred around (A)plots 1.2, (B) 2.2, and (C) 3.2 Myr BP, usingof window sizeshown W ∗ =in410 size 1W Fig. 5. Recurrence (equivalently adjacency matrices the RNs Fig.kyr, 6) step obtained from = the dust flux record at ODP site 659, ∗ ∗ m = 3, around τ = 10(A) kyr.1.2, ε was a data-adaptive way towindow yield a size fixedW edge 0.05∆W for ∗each centred (B) chosen 2.2, andin(C) 3.2 Myr BP, using =density 410 kyr,ρ(ε) step=size = 41window. kyr, and embedding parameters m = 3, τ ∗ = 10 kyr. ε was chosen in a data-adaptive way to yield a fixed edge density ρ(ε) = 0.05 for each window. analogy implies that each sampled state vector is assigned a vertex infrom the aRN, wherenetwork two vertices are linked if the cormatrices complex perspective by identifying responding state vectors are recurrent, i.e. mutually close, µ Ainµij (ε) = Rspace δij 6). According to the conventions(11) phase (Fig. of ij (ε) − Sect. 2.3, each vertex i in the µ-th window has an age (δµij denoting Kronecker’s delta) with the adjacency matrix t = t(µ−1)1W +i attached to it. To simplify the notation ofi a complex network associated to the underlying time sewhen defining network measures, we will drop the1window ries (Marwan et al., 2009; Donner et al., 2010a) . This index µ in the following. analogy implies that each sampled state vector is assigned The edge density a vertex in the RN, where two vertices are linked if the corresponding state X are recurrent, i.e., mutually close, 1 vectors = space (Fig. 6). Aij (ε) inρ(ε) phase According to the conventions(12) of W (W − 1) i,j Sec. 2.3, each vertex i in the µ-th window has an age tµi = t(µ−1)∆W +i attached to it. To simplify the notation when measures which fraction of the maximum theoretically posdefining network measures, we will drop the window index sible number W (W − 1)/2 of undirected edges is present in µ in the following. the RN, where the number of vertices W is determined by The edge density the chosen recurrence window size. ρ(ε) is equivalent to the 1 in X recurrence rate traditional RQA. Aij (ε) (12) ρ(ε) = W (W − 1) i,j applications of data analysis, such as dengeoscientifically relevant drograms in agglomerative cluster analysis, or nonlinear decompo1 Note that similar approaches can also be found in other geosition of multivariate data using isometric feature mapping (Gámez scientifically relevant applications of data analysis, such as dendroet al., 2004). grams in agglomerative cluster analysis, or nonlinear decomposition of multivariate data using isometric feature mapping (Gámez etNonlin. al., 2004). Processes Geophys., 18, 545–562, 2011 The properties of the resulting RNs (parameterised by the single which parameter ε) have been shown totheoretically trace structures measures fraction of the maximum posin phase space corresponding to dynamically ob-in sible number W (W − 1)/2 of undirected edgesinvariant is present jects (Donner et al., 2010a, 2011b) as well as changes in the RN, where the number of vertices W is determinedthe by dynamical arbitrary timeρ(ε) series (Marwan ettoal., the chosen behaviour recurrenceofwindow size. is equivalent the 2009; Donner 2011a). RQA. For detecting bifurcations in recurrence rateetinal., traditional The properties of the retime series, global-scale network characteristics of complex sulting RNs (parameterised by the single parameter ε) have Bocnetwork theory are ofstructures main interest (Newman, been shown to trace in phase space 2003; correspondcaletti et al., 2006; Costa et al., 2007). Here we will focus ing to dynamically invariant objects (Donner et al., 2010a,onin the following measures: press) as well four as changes in the dynamical behaviour of arbi- trary time series (Marwan et al., 2009; Donner et al., 2011). i. Transitivity T : the transitivity For detecting bifurcations in time series, global-scale netP work characteristics of complex network theory are of main Aj k Aki i,j,k Aij 2003; interest (Newman, Boccaletti et al., 2006; da Costa T = P (13) A A ki kj et al., 2007). Here we will focus on the following four meai,j,k sures: of an unweighted and undirected network characterises (i) the Transitivity T : The transitivity overall probability that two randomly chosen neighP bours of an also randomly chosen vertex are connected i,j,k Aij Ajk Aki T = P 2003). In case of RNs, T serves as a mea(13) (Newman, Aki Akj i,j,k sure for the regularity of the dynamics as encoded in the RN’s mesoscopic (Donner etcharacterises al., 2010a). of an unweighted and structure undirected network the overall probability that two randomly chosen neighbourswww.nonlin-processes-geophys.net/18/545/2011/ of an also randomly chosen vertex are connected J. F. Donges et al.:et Identification of of dynamical marinepalaeoclimate palaeoclimate records by RN analysis J. F. Donges al.: Identification dynamicaltransitions transitions in in marine records by RN analysis B A 7 551 C Fig. 6. Recurrence networks obtained from the dust flux record at ODP site 659, centred around (A) 1.2, (B) 2.2, and (C) 3.2 Myr BP and Fig. 6. Recurrence networks obtained from the dust flux record at ODP site 659, centred around (A) 1.2, (B) 2.2, and (C) 3.2 Myr BP and µ corresponding to the recurrence plots of Fig. 5. Vertex color indicates the age ti associated to single state vectors µ (from blue [= old] corresponding to the recurrence plots of Fig. 5. Vertex color indicates the age tµi associated to single state vectors µ (from blue [=old] to red to red [=young]). [= young]).TheThe two-dimensional visualisation been with obtained with the software package using placement a force-directed two-dimensional graph graph visualisation has beenhas obtained the software package GUESS using aGUESS force-directed placement algorithm (http://graphexploration.cond.org). It is important to note that in this visualisation, node positions are determined algorithm (http://graphexploration.cond.org). It is important to note that in this visualisation, node positions are determined by the aforementioned algorithm and do notand correspond to a projection of the of node coordinates in the reconstructed three-dimensional phase space. by the aforementioned algorithm do not correspond to a projection the node coordinates in the reconstructed three-dimensional phase space. Specifically, regular dynamics (e.g. on a periodic orbit) (Newman, 2003). In case of RNs,values T serves is typically characterised by higher of as thea meatransure for the regularity of the dynamics as encoded in sitivity T than chaotic dynamics. T can furthermore the RN’s mesoscopic structure (Donner et al., 2010a). be interpreted as aregular globaldynamics measure(e.g., of theonunderlying atSpecifically, a periodic ortractive set’s effective dimensionality d (Donner et al., bit) is typically characterised by higher values of the d 2011b), i.e. the theoretical resultdynamics. is T = (3/4) ustransitivity T than chaotic T canwhen furthering themore supremum norm as in aphase Forofcontinuousbe interpreted globalspace. measure the underlying attractive effective dimensionality d (Donner time systems, this set’s implies T = 3/4 for a periodic orbit d in for press), i.e., the theoreticalHowever, result is T for = (3/4) and T et<al., 3/4 chaotic dynamics. small when using the supremum norm in phase space. For numbers of vertices (state vectors) W as used in this systems, implies T =from 3/4 these for a work continuous-time the estimated values of Tthiswill deviate periodic orbit and T < 3/4 for chaotic dynamics. Howtheoretical expectations (Donner et al., 2011b). ever, for small numbers of vertices (state vectors) W as in thiswith workshort the estimated values of T will deviWhenused dealing time series (segments) as it from expectations et al., is theate case in these this theoretical work, transitivity is a(Donner more robust in press). measure than the related global clustering coefficient C (Watts and Strogatz, 1998; Newman, 2003), since the latter gives relatively more weight to sparsely sampled regions in phase low degree When dealing withspace short (vertices time serieswith (segments) as it k) (Donner al.,in2010a, 2011a). is the et case this work, transitivity is a more robust measure than the related global clustering coefficient ii. Average path length L: the average path length C (Watts and Strogatz, 1998; Newman, 2003), since the latter gives relatively more weight to sparsely samL = lpled (14) ij i,j regions in phase space (vertices with low degree k) (Donner et al., 2010a, 2011). is defined as the mean value of the shortest path lengths lij between all mutually reachable pairs of vertices (i,j ) (measured in terms of geodesic graph distance, i.e. the minimum number of edges that have to be traversed on any path connecting the vertices i and j ) (Watts and www.nonlin-processes-geophys.net/18/545/2011/ Strogatz, 1998; Newman, 2003). A pair of vertices (i,j ) (ii) Average length L: The average path length is calledpath mutually reachable if there exists at least one path connecting i and j . Since for comparable values of ε, the average distances along different types of orbits L = hlij ii,j (14) typically differ significantly, changes in L can be used as sensitive indicators of dynamical transitions (Maret al., 2009; Donner etthe al.,shortest 2010a).path lengths iswan defined as the mean value of lij between all mutually reachable pairs of vertices iii. (i,j) Assortativity complex networkgraph is called assortative (measuredR: in aterms of geodesic distance, if vertices tend to connect preferentially to vertices i.e., the minimum number of edges that have to be traP with a similar number connections (degreei kand i = j) j Aij ). versed on any path of connecting the vertices On theand other hand, 1998; it is called disassortative if vertices of (Watts Strogatz, Newman, 2003). A pair of vertices (i,j) isprefer called to mutually high degree link toreachable vertices if ofthere low exists degree, and atvice leastversa one path connecting i andThis j. Since for compa(Newman, 2002). assortativity property rable values of ε, the average distances along different can be quantified by the Pearson correlation coefficient types of orbits typically differ significantly, changes in D E2 L can be used as sensitive indicators of dynamical tran1P 1 k k A − (k + k ) i j ij i j sitions (Marwan et al., 2009; Donner et al., 2010a). j >i L 2 i,j R= E2 1 2 2 )A − 1 (k + k ) (k + k ij i j j >i j L 2 is calledi,jassor(iii) Assortativity R:2 Ai complex network D 1P (15) tative if vertices tend to connect preferentially to verbetween the ,kjconnections of the vertices onkboth tices of a similar numberkiof (degree i = ends Pdegrees P ). On other called disassortative if ofj A allijL = thej >i Aijhand, edgesit is (i,j ), where vertices of high degree prefer to link to vertices of low X1 degree, 2002). This assorta1 and vice versa1 (Newman, 2 (ki + kj ) = i,j L j >i 2 (ki + kj )Aij (16) is the mean of the average edge end-point degree (ki + kj )/2 (Costa et al., 2007). In the RN context, R can be considered as a measure for the local continuity of Nonlin. Processes Geophys., 18, 545–562, 2011 552 J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis the phase space density of state vectors (Donner et al., 2010a). be viewed as a test against stationarity of the higher-order geometrical properties of the time series that are quantified by qualitatively different RN measures. iv. Network diameter D: the network diameter 2.7 D = max lij i,j (17) is the maximum geodesic (shortest-path) distance between all mutually reachable pairs of vertices in the network (Newman, 2003). From this definition, there are obvious relationships with the average path length L, which are expected to lead to strong correlations between both measures (Donner et al., 2010a). In order to apply RNs in a sliding window analysis, a reference framework is necessary. Here, we consider a dataadaptive choice of ε that guarantees for a fixed edge density ρ of 5 %, which has been found a reasonable choice in previous studies (Donner et al., 2010b). One should note, however, that even with this choice the characteristics of RNs can only be compared in a meaningful way if the network size W is kept fixed (see Sect. 2.1). Among the considered complex network measures, T and R are mainly affected by finite-sample problems otherwise, whereas L and D explicitly depend on ε and W (Donner et al., 2010a). 2.6 Significance test We perform a relatively simple statistical test of whether the network characteristics in a certain time interval differ significantly from the general network characteristics expected given the phase space distribution of state vectors yi from the whole detrended and embedded record and a certain recurrence window size W . The corresponding null hypothesis is that the network measures observed for a certain window are consistent with being calculated from a random draw of W state vectors from the prescribed phase space distribution induced by the complete detrended time series. We can justly assume a thus randomised embedded time series without losing essential information, because all network measures g(·) considered here are permutation-invariant when considering a fixed subset of state vectors y 1 ,...,y W . More formally, g(y 1 ,...,y W ) = g(y π(1) ,...,y π(W ) ) for arbitrary permutations π. A similar test for RQA measures requires a more advanced method (Schinkel et al., 2009). In order to create an appropriate null-model, we use the following approach: (i) draw randomly W state vectors from the embedded time series (corresponding to the window size chosen for the original data), (ii) construct a RN from this set of state vectors, and (iii) calculate the network measures of interest. Repeating this procedure sufficiently many times, we obtain a test distribution for each of the network measures and estimate its 0.05 and 0.95 quantiles that can be interpreted as 90 % confidence bounds. The proposed significance test can Nonlin. Processes Geophys., 18, 545–562, 2011 Implementation We implemented the above described methods using the programming language Python (van Rossum and Drake, 2006), the packages NumPy (Oliphant, 2006) and SciPy (Jones et al., 2011) as well as embedded C++ code. Complex network measures have been calculated employing the Python package igraph (Csárdi and Nepusz, 2006). 3 Dynamical transitions in model systems To validate the proposed methodology for detecting transitions in time series based on RNs, we apply it to the logistic map and the Lorenz system with drifting bifurcation parameter as paradigmatic examples of discrete and continuous-time dynamical systems, respectively. While step-like changes of bifurcation parameters have already been studied for discrete (Marwan et al., 2009) and continuous-time dynamical systems (Zou et al., 2010; Donner et al., 2011a), here we are particularly interested in the effect of transients, which are expected to be present in real-world systems and, hence, data extracted from them. We will check whether the global network quantifiers described above are able to detect transitions in the system’s dynamics induced by bifurcations due to a slowly changing control parameter. For this purpose we are specifically looking for time intervals (or equivalently values of the bifurcation parameter) where one or more of the considered network quantifiers undergo sudden changes. This requires taking into account the measures’ interpretation in terms of dynamical systems theory (Sect. 2.5). Furthermore, we will study how their performance and the level of resolved detail depend on the window size W . This analysis particularly shows that the window sizes W chosen for the RN analysis of terrigenous dust flux records (Table 1) are indeed appropriate for detecting bifurcations. 3.1 Logistic map We iterate the logistic map xi+1 = ri xi (1 − xi ) ri+1 = ri + 1r (18) while varying the bifurcation parameter linearly from r1 = 3.8 to rM = 3.9 in M = 10 000 equidistant steps setting 1r = 1 × 10−5 (Fig. 7), similar to Trulla et al. (1996). We analyse the resulting time series {xi } without embedding or detrending. The transition from chaotic √ to 3-periodic dynamics after an interior crisis at r = 1 + 8 ≈ 3.8284 (Wackerbauer et al., 1994) is clearly displayed by all four measures. As expected from theoretical considerations for discrete-time www.nonlin-processes-geophys.net/18/545/2011/ J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis 10 J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis Transitivity Average path length Assortativity Diameter 553 Fig. 7. (A) Transitivity T Transitivity , (B) average length L, (C) and(D) (D) diameter for varying recurrence Fig. 7. (A) T , (B)path average path length L, assortativity (C) assortativityR, R, and diameter D forD varying recurrence window sizewindow W for thesize W for logistic map (Eq.drifting (18)) withbifurcation drifting bifurcation parameter r (see text) andinitial initial condition x1 x =10.7. W was linearly linearly in the interval 18) with parameter r (see text) and condition = 0.7. W varied was varied in the interval the logistic map (Eq. [100,600], the recurrence window step size wastofixed ∆W 10 time steps.No No embedding embedding was usedused and the set to ε =set 0.05σ [100,600], the recurrence window step size was fixed 1Wto = 10=time steps. was andthreshold the threshold to ε = 0.05σ (Marwan et al., 2009), where σ denotes the standard deviation of the time series segment within the recurrence window. Vertical dashed lines (Marwan et al., 2009), where σ denotes the standard deviation of the time series segment within the recurrence window. Vertical dashed indicate the critial values of r discussed in the main text. lines indicate the critial values of r discussed in the main text. danger of confusing statisticaletfluctuations with proper dysystems (Marwan et al., 2009; Donner al., 2010a, 2011b), namical changes substantially. Time series from T and R abruptly increase to their maximum value of 1geological folarchives are typically characterised by a variety of different lowing this transition, whereas at the same time L and types of nonstationarities, including (i) changes in theDlongsharply decrease theiror minimum value of proxies, 1. Among all termtomean variance of the recorded (ii) variations in the Tamplitudes almost periodic variability the four measures, and R of most clearly detect the components termi(e.g., such attributed to Milankovich-type variations caused nation of the period-doubling cascade following the period-3 by periodic changes in the Earth’s orbit), or (iii) even mulbehaviour at the accumulation point r ≈ 3.849, while T ,and L intimodal behaviour (e.g., transitions between glacial and D highlightterglacial the merger of the three periods). All subsequently these three typesformed of nonstationarities contained our data 2 and Trauth et al. (2009)). chaotic bands atarethe interiorincrisis at r(Fig. ≈ 3.857 (Wackerbauer While these different phenomena can be analysed usinginmore et al., 1994). The latter transition is only weakly visible specific methodological approaches, we propose RN analyR. Additionally, much fine-structure is resolved by the netsis as a general exploratory tool for detecting time intervals work measures,containing e.g. a narrow window . 3.89 bechanges period-4 in the dominating typeat of rdynamical haviour. In the following, we will illustrate the robustness that is most clearly indicated by an increased transitivity T across all W . Generally, the transitions appear more and more blurred as W increases, which is due to the growing number of samples from both periodic and chaotic dynamical regimes contained in the recurrence windows when sliding over the bifurcation point. In consequence, some of the narrow periodic windows appearing for r < 3.83 and r > 3.86 are only visible for small recurrence window sizes W . As a rule of thumb, we can expect a periodic/chaotic window of width wr embedded within a chaotic/periodic background to be detectable if wr & W 1r. www.nonlin-processes-geophys.net/18/545/2011/ of this approach for feature the four is marine in a clear Another notable that records both Lintroduced and D show Sec. 2.1 and briefly discuss the possible climatological backtendency to increase with growing W in the chaotic paramground of the observed dynamical changes. eter ranges (Fig. 7b and d). This is theoretically expected, since measures are extensive, i.e. they depend explic4.1 both Time-dependence of network properties itly and nonlinearly on the number of vertices W in the RN consider the four space marine distribution palaeoclimate records forWe a general phase of stateembedvectors as inded in a three-dimensional reconstructed phase space with a duced by chaotic dynamics ∗ (Donner et al., 2010a). In contime delay of approximately τ = 10 kyr, resulting in the emtrast, L and D do not changeinwith theanperiodic bedding parameters described Sec. W 2.3.inFor initial in-windows, most notably in the large period-3 window the kyr logistic map spection, we use recurrence windows of size W ∗of = 410 with7b a mutual offset of can subsequent windows ∆W ∗ = 41 kyr. (Fig. and d). We explain this behaviour by recalling Note that the latter two parameter choices correspond to that for discrete-time systems in a p-periodic regime, the RN those used in previous work on the ODP site 659 dust flux reduces to a setetof fully connected components record (Marwan al.,p 2009; Donner et al., 2011). The selec- (Donner et tion al., of 2010a). Following the definitions in Sect. 2.5, this in both parameters results from a compromise between turn leads to L = D = 1 in any periodic regime and independent of W . 3.2 Lorenz system To illustrate the performance of windowed RN analysis for detecting transitions in continuous-time dynamical systems, we consider the Lorenz system with a time-dependent bifurcation parameter r = r(t), Nonlin. Processes Geophys., 18, 545–562, 2011 554 J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis 11 J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis Transitivity Average path length Assortativity Diameter Fig. 8. (A) T , (B) average pathL, length (C) assortativity and(D) (D) diameter for for varying recurrence window window size W forsize W for Fig. 8. (A) Transitivity T ,Transitivity (B) average path length (C)L,assortativity R,R,and diameterD D varying recurrence the (Eq. Lorenz system (19)) with drifting bifurcation parameter initial condition condition (x0(x ,y0,y ,z ) = (10,10,10). Because we are the Lorenz system 19) with(Eq. drifting bifurcation parameter r (seer (see text)text) andandinitial 0 00 ,z0 ) = (10,10,10). Because we are interested in the performance of our method for scalar time series, we chose the x-component of the trajectory sampled with sampling time interested in the performance of our method for scalar time series, we chose the x-component of the trajectory sampled with sampling time ∆t = 0.05 and embed it with embedding dimension m = 3 and delay τ = 15. W was varied linearly in the interval [100,600], the recurrence 1t = 0.05 and embed it with embedding dimension m = 3We and delay τ = 15. W was varied linearly in the interval [100,600], window step size was fixed to ∆W = 10 samples. varied the recurrence threshold ε to yield a fixed edge density ρ = 0.05 (Donner etthe al.,recurrence window step size2010b). was fixed to dashed 1W =lines 10 indicate samples. varied threshold Vertical the We critial valuesthe of r recurrence discussed in the main text.ε to yield a fixed edge density ρ = 0.05 (Donner et al., 2010b). Vertical dashed lines indicate the critial values of r discussed in the main text. high temporal resolution of the finally produced RN mea- which has not been considered in the aforementioned stud- sures (small W ∗ , ∆W ∗ ) and larger statistical ies. As we will show in the following, the main features reconfidence in d 8 ∗ detect at r ≈ changed 163.5, rwhen ≈ 164.5 and results (large − W z) ). −The choice critical others covered by ourweaker analysistransitions are not qualitatively (x,y,z) = the 10(y − x),x(r y,xy − ofz W. ∗ is more(19) dt 3 directly influences the r ≈ than that of ∆W ∗ , because the former applying detrending. However, thisbe step appearswhen relevant 166. Note that one has to careful comparing number of vertices W in the RNs via Eq. (6). Since a formal these in other kinds of statistical analyses, e.g., fordistinct estimating results to bifurcation studies where realisations While the system is evolving, r increases linearly from r0 = criterion for determining an optimal choice of W ∗ and ∆W ∗ spectrograms or time-dependent coefficients of autoregresthe processes, Lorenz system with fixed parameter r (not varying in so far, we study the500, robustness since the data show considerable long-term 160 at time t0 =is0nottoavailable rf = 170 at time tf = i.e. of our results of sive ∗ time) are studied (e.g. Donner et al., 2011a), since transients with respect to variations in the more critical parameter W trends in both mean and variance (Fig. 2). rf −inrSec. 0 4.2. influence thethe results and(Fig. cannot excluded by construcRegarding transitivity 9), webefind a synchronous r(t) = r0 + (t − t0 ). (20) behaviour two geographically distinct ODP when of r the is continuously varied inrecords time. at However, our tf − tWe 0 additionally apply local detrending by removing the tion ∗ sites 659 721/722 during Pliocene Myret BPal. (1996) long-term average taken over windows of WD = 500 kyr, results areand consistent withthethe work(∼5.3-2.6 of Trulla (before present)) and Early Pleistocene (2.6-1.0 Myr BP)2 , For consistencywhere with the analysis of scalar palaeoclimate who observed that in transient scenarios bifurcations may time series to be performed below, we use an embedding of 2 appear for“Early larger bifurcation than in their nonHere Pleistocene” does notparameters refer to any of the archetypthe x-component time series for RN analysis without prior(21) ∗ WD = bWD /h∆T ic, ical stagesequivalents. (Upper, Middle and Lower Pleistocene). of Its timing 2.6transient The dependence the results on the detrending (see caption of Fig. 8 for details). The RN mearecurrence window size W is more pronounced than that desures indicate two major transitions towards increasingly irscribed above for the logistic map. This is likely due to the regular dynamics at r ≈ 161 and r ≈ 166.5 (Fig. 8). The forfact that transients play a larger role in continuous-time sysmer possibly reflects an initial transient due to the chosen tems like the Lorenz model than in discrete-time systems. initial condition. The latter agrees well with the major shift from periodic (large T , large L and D for continuous-time systems; Donner et al., 2010a, 2011b; Zou et al., 2010) to 4 Dynamical transitions in palaeoclimate records chaotic (small T , small L and D) behaviour which is present in the Lorenz system’s non-transient bifurcation scenario at Our studies in the previous section demonstrated that RN r ≈ 166 (Barrio and Serrano, 2007; Donner et al., 2011a). On analysis can be meaningfully applied for detecting dynama shorter time scale, the path-based measures L and D among ical transitions in non-stationary time series from different Nonlin. Processes Geophys., 18, 545–562, 2011 www.nonlin-processes-geophys.net/18/545/2011/ 12palaeoclimate records J. F. by Donges et al.: Identification of555 dynamical tran J. F. Donges et al.: Identification of dynamical transitions in marine RN analysis model systems by applying this kind of analysis to running windows. This is a necessary, but not sufficient condition for ensuring the feasibility of RN analysis for detecting regime shifts in palaeoclimate records as well. However, the application of our simple significance test (Sect. 2.6) diminishes the danger of confusing statistical fluctuations with proper dynamical changes substantially. Time series from geological archives are typically characterised by a variety of different types of nonstationarities, including (i) changes in the long-term mean or variance of the recorded proxies, (ii) variations in the amplitudes of almost periodic variability components (e.g. such attributed to Milankovich-type variations caused by periodic changes in the Earth’s orbit), or (iii) even multimodal behaviour (e.g. transitions between glacial and interglacial periods). All these three types of nonstationarities are contained in our data (Fig. 2 and Trauth et al., 2009). While these different phenomena can be analysed using more specific methodological approaches, we propose RN analysis as a general exploratory tool for detecting time intervals containing changes in the dominating type of dynamical behaviour. In the following, we will illustrate the robustness of this approach for the four marine records introduced in Sect. 2.1 and briefly discuss the possible climatological background of the observed dynamical changes. 4.1 Time-dependence of network properties We consider the four marine palaeoclimate records embedded in a three-dimensional reconstructed phase space with a time delay of approximately τ ∗ = 10 kyr, resulting in the embedding parameters described in Sect. 2.3. For an initial inspection, we use recurrence windows of size W ∗ = 410 kyr with a mutual offset of subsequent windows of 1W ∗ = 41 kyr. Note that the latter two parameter choices correspond to those used in previous work on the ODP site 659 dust flux record (Marwan et al., 2009; Donner et al., 2011a). The selection of both parameters results from a compromise between high temporal resolution of the finally produced RN measures (small W ∗ , 1W ∗ ) and larger statistical confidence in the results (large W ∗ ). The choice of W ∗ is more critical than that of 1W ∗ , because the former directly influences the number of vertices W in the RNs via Eq. (6). Since a formal criterion for determining an optimal choice of W ∗ and 1W ∗ is not available so far, we study the robustness of our results with respect to variations in the more critical parameter W ∗ in Sect. 4.2. We additionally apply local detrending by removing the long-term average taken over windows of WD∗ = 500 kyr, where WD = bWD∗ /h1T ic, (21) which has not been considered in the aforementioned studies. As we will show in the following, the main features recovered by our analysis are not qualitatively changed when applying detrending. However, this step appears relevant in www.nonlin-processes-geophys.net/18/545/2011/ Fig. 9. Evolution of RN transitivity T for (A) the δ 18 O record from ODP 659, andofthe flux records sites (B) 659, Fig. 9.site Evolution RNdust transitivity T forfrom (A) ODP the δ 18 O record from (C) 721, (D) 967. reveals in theODP regularity ODP siteand 659, and theTdust flux changes records from sites of (B)African 659, (C) climate during the Plio-Pleistocene for the latter three records. Here 721, and (D) 967. T reveals changes in the regularity of African cliwe used a detrending window size WD∗ = 500 kyr, recurrence winmate during the Plio-Pleistocene. Here we used a detrending window size W ∗∗= 410 kyr and step size 1W ∗ = 41 kyr, embedding window size W ∗ = 410 kyr dow size W = 500 kyr, recurrence dimension mD= 3 and delay τ ∗ = 10 kyr. The recurrence thresh∗ and step size ∆W = 41 kyr, embedding dimension m = 3 and deold ε ∗was chosen adaptively to yield a fixed edge density ρ = 0.05. lay τ = 10 kyr. The recurrence threshold ε was chosen adaptively The grey bars represent the 5 % and 95 % quantiles with respect to to yield a fixed edgeobtained density from ρ = 0.05. grey barsofrepresent the test distribution 10 000The realisations our null-the 5% and 95% quantiles with respect to the test distribution obtained model for each record separately. Vertical dashed lines indicate the from 10,000 realisations of our null-model for each record sepadetected epochs of transitions discussed in the main text. rately. Vertical dashed lines indicate the detected epochs of transitions discussed in the main text. other kinds of statistical analyses, e.g. for estimating spectrograms or time-dependent of autoregressive including two periods of coefficients extraordinarily large values ofproT at cesses, since the data show considerable long-term trends in about 3.45-3.05 and 2.2-2.1 Myr BP, related to pronounced both mean variance (Fig. 2). 6B,C. The first of these peclusters of and vertices shown in Figs. we find asuppressed synchronous Regarding the transitivity (Fig. riods results from a time interval 9), of strongly and behaviour of the two geographically distinct records ODP2), almost constant dust flux in the Mid Pliocene (seeatFig. sites 721/722 during the Pliocene while659 the and latter one coincides with a period(∼5.3–2.6 of almostMyr periBP; before present) and Early Pleistocene (2.6–1.0 Myr odic2 Milankovich-type variations (Trauth et al., 2009). We BP) , including two periods of extraordinarily large values note that it is known (and empirically understood) that both of T at about 3.45–3.05 and 2.2–2.1 Myr BP, related to protypes of dynamics typically lead to large values of T (Marnounced clusters of vertices shown in Figs. 6b and c. The wan et al., 2009; Donner et al., 2010a, in press; Zou et al., first of these periods results from a time interval of strongly 2010), so that this result is consistent with theoretical expecsuppressed and almost constant dust flux in the Mid Pliocene tations. thethe Early Pleistocene, the with signatures at both (see Fig.During 2), while latter one coincides a period of sites decouple from each other, which could be the result almost periodic Milankovich-type variations (Trauth et al.,of an enhancement theitatmospheric Walker circulation (Rav2009). We note of that is known (and empirically underelo et al., 2004). For the last about 1.5 Myr, the variations of transitivity become more similar ODP sites 721/722 2 Here “Early Pleistocene” does notbetween refer to any of the archetypandstages 967, (Upper, particularly highlighting the Mid Pleistocene tranical Middle and Lower Pleistocene). Its timing 2.6– sition and 0.7 Myr BP (Fig.but6A), which corre1.0 Myrbetween BP is not1.2 motivated stratigraphically, climatologically, i.e. by the onset of the Mid-Pleistocene transition around 1.0 Myr 1.0 Myr BP is not motivated stratigraphically, but climatologically, BP. i.e., by the onset of the Mid-Pleistocene transition around 1.0 Myr BP. Nonlin. Processes Geophys., 18, 545–562, 2011 Fig. recor (B) 6 clima icanc spon riodi (Fig. light tocen varia at ab mate for a Th of b ent c nous ity re An i pecta syste lar lo to th signi lated at ar tent sugg clim 900 and i J. F. Donges et al.: Identification of dynamical transitions in marine p of dynamical556 transitions in marine by RNofanalysis J. F.palaeoclimate Donges et al.:records Identification dynamical transitions in marine palaeoclimate records by RN analysis rd from 59, (C) can cling win410 kyr and deaptively sent the btained d sepatransi- of T at ounced ese peed and Fig. 2), st peri9). We at both (Maret al., expecat both sult of n (Ravions of 21/722 e trancorre- gically, 1.0 Myr Tab evol and (D) unde mar Fig. 10. Evolution of RN average path length L for (A) the δ 18 O record from ODP site 659, and the dust flux records from ODP sites Fig. 10. Evolution of RN average path length L for (A) the δ 18 O (B) 659, (C) 721, and (D) 967, indicating transitions in African record from ODP site 659, and the dust flux records from ODP sites climate dynamics during the Plio-Pleistocene. Parameters, signifi(B) 659, (C) 721, and (D) 967, indicating transitions in African cance test, and vertical lines are the same as in Fig. 9. climate dynamics during the Plio-Pleistocene. Parameters, significance test, and vertical lines are the same as in Fig. 9. stood) that both types of dynamics typically lead to large values of T (Marwan et al., 2009; Donner et al., 2010a, sponds a change in the dominating pe2011b; to Zou et al., 2010), so that this Milankovich-type result is consistent riodicity. The results obtained for the average path length with theoretical expectations. During the Early Pleistocene, L (Fig. 10) are mostly with these findings, also highthe signatures at bothconsistent sites decouple from each other, which lighting the Mid Pliocene, Early Pleistocene, and Mid Pleiscould be the result of an enhancement of the atmospheric tocene periods (Ravelo with changes the long-term flux Walker as circulation et al., in 2004). For the lastdust about variability. Specifically, L tends to show significant peaks 1.5 Myr, the variations of transitivity become more similar at abruptODP change points between regular and morehighlighterratic clibetween sites 721/722 and 967, particularly mate variability, as indicated by T (see Marwan et al. (2009) ing the Mid Pleistocene transition between 1.2 and 0.7 Myr for theoretical explanation of thistobehaviour). BP a(Fig. 6a), which corresponds a change in the dominating Milankovich-type periodicity. The from resultsthe obtained The oxygen isotope anomaly obtained analysis mostly consistent for benthic the average path lengthcharacterises L (Fig. 10) are of foraminifera a distinctively differwith these findings, also highlighting the Mid Pliocene, Early ent climatic parameter (i.e., global ice volume) than terrigePleistocene, and so Mid Pleistocene periods with changes in nous dust flux, that it can beasexpected that the variabilthe long-term dust flux variability. Specifically, L tends to ity recorded by this proxy differs from that of the dust flux. show significantofpeaks at abrupt change points betweenthis regAn inspection the RN properties indeed confirms exular and more erratic climate variability, as indicated by pectation. Specifically, the transitivity T does not show Tany et al. at (2009) for a theoretical explanation of (see Marwan systematic maxima all (indicating time intervals with reguthis behaviour). lar long-term dynamics) (Fig. 9A), which is in clear contrast oxygen dust isotope anomaly obtainedpath from the analysis to The the aeolian flux. The average length L shows of benthic foraminifera characterises a distinctively differ-resignificant maxima around 2.9 Myr BP (possibly being ent climatic parameter (i.e. global ice volume) than terrigelated to the intensification of Northern hemisphere glaciation nous dust flux, so thatbetween it can be1.8 expected variabilat around this time), and 1.3that MyrtheBP (consisity recorded by this proxy differs from that of the dust flux. tent with the corresponding results for the dust flux records, An inspection of the RN properties indeed confirms this exsuggesting a high-latitude mechanism behind the large-scale pectation. Specifically, the transitivity T does not show any climatic changes during this time period), and after about systematic maxima at all (Fig. 9a), which is in clear contrast 900 kyr BP (possibly resulting from the glacial terminations and inceptions with a rather long – roughly 100 kyr – periodNonlin. Processes Geophys., 18, 545–562, 2011 Fig. 11. Evolution of RN assortativity R for (A) the δ 18 O record from ODP site 659, and the dust flux records from ODP Fig. 11. Evolution of RN assortativity R for (A) the δ 18 Osites record (B) 659, (C) 721, and (D) 967 during the Plio-Pleistocene. Paramfrom ODP site 659, and the dust flux records from ODP sites (B) eters, significance test, and vertical lines are the same as in Fig. 9. 659, (C) 721, and (D) 967 during the Plio-Pleistocene. Parameters, significance test, and vertical lines are the same as in Fig. 9. to the aeolian dust flux. The average path length L shows significant maxima around 2.9 Myr BP (possibly being related to the intensification of Northern hemisphere glaciation at around this time), between 1.8 and 1.3 Myr BP (consistent with the corresponding results for the dust flux records, suggesting a high-latitude mechanism behind the large-scale climatic changes during this time period), and after about 900 kyr BP (possibly resulting from the glacial terminations and inceptions with a rather long – roughly 100 kyr – periodicity) (Fig. 10a). Figures 11 and 12 additionally show the time variability of the two other RN properties assortativity R and diameter D. Since the latter one is closely related to the average path length L (Donner et al., 2010a), the variability of both measures is very similar. Moreover, we also find some much weaker similarities between the temporal variability patterns of transitivity T and assortativity R, which are less pronounced, since both properties characterise less obviously related aspects of the network geometry in phase space. Specifically, the time interval of suppressed dust flux in ODP 659 and 721/722 during the Mid Pliocene results not only in an increased transitivity, but also a high assortativity. The latter feature can be explained by the fact that a relatively large12. cluster of state vectors representing this regime Fig. Evolution of RN diameter D for (A) thelaminar δ 18 O record from emerges theand network, is rather densely connected ODP site in 659, the dustwhich flux records from ODP sites (B) 659, 6c).and (D) 967 during the Plio-Pleistocene. Parameters, sig(Fig.721, (C) nificance test, andthat vertical lines are the same as in Fig. 9. We conclude the RN measures are not statistically independent in their time evolution (Table 2). For the ODP site 659 δ 18 O and dust flux records, the correlations between icity) (Fig. 10A). Figures www.nonlin-processes-geophys.net/18/545/2011/ 11 and 12 additionally show the time variability of t eter erag of b som abil are obv spac in O only The larg eme (Fig W inde site tran age man retic mor the 721 Fig. 11. Evolution of RN assortativity R for (A) the δ 18 O record from ODP site 659, and the dust flux records from ODP sites (B) 659, (C) 721, and (D) 967 during the Plio-Pleistocene. Parameters, significance test, and vertical lines are the same as in Fig. 9. D 0.37 0.74 0.35 1.00 T L R D T 1.00 0.65 0.61 0.23 J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate by RN 1.00 analysis 557 L 0.65 0.54 0.78 (D)records R 0.61 0.54 1.00 0.16 0.23 0.78 0.16 correlations 1.00 Table 2. Spearman’sDρ measuring rank-order in the time evolution of RN measures for (A) the ODP site 659 δ 18 O record, and the dust flux records from ODP sites (B) 659, (C) 721/722, and (D) 967. Significant correlations having a p-value of thethan two other and diamsmaller 0.05 underRN the properties assumption assortativity of uncorrelatedR data of the eter D. Since the latter one is closely related to the avsame length are marked in bold. Fig. 12. Evolution of RN diameter D for (A) the δ 18 O record from ODP site Evolution 659, and the dustdiameter flux records ODP (B) 659, Fig. 12. of RN D forfrom (A) the δ 18sites O record from (C) 721, (D)and 967the during Parameters, ODP siteand 659, dust the fluxPlio-Pleistocene. records from ODP sites (B)sig659, nificance linesthe arePlio-Pleistocene. the same as in Fig. 9. (C) 721, test, and and (D) vertical 967 during Parameters, significance test, and vertical lines are the same as in Fig. 9. transitivity T and assortativity R as well as between average path length icity) (Fig. 10A).L and diameter D as measured by Spearman’s ρ are most pronounced, which is consistent with theFigures 11 and 12 additionally show the time variability oretical expectations (Donner et al., 2010a). Correlations are more clearly developed between all four measures in case of the more highly sampled dust flux records from ODP sites 721/722 and 967 (Table 1). However, for all records the four measures can be considered sufficiently independent to justify including all of them for a broad and thorough nonlinear time series analysis of oxygen isotope and terrigenous dust flux variability. 4.2 Robustness of the results To assure the reliability and robustness of our results, we systematically study their dependence on the relevant algorithmic parameters of our method, in particular, the widths of the recurrence window (W ∗ ) and the detrending window (WD∗ ) as well as the embedding delay (τ ∗ ). In Figs. 13–15, the results of the significance test are presented as contours at two prescribed significance levels obtained from the observed measure’s quantiles with respect to the corresponding test distribution. Green contours represent the lower prescribed quantile (5 %), while black contours indicate the upper one (95 %). This implies that values of the measure under study enclosed by green contours can be considered as exceptionally low, while those lying within black contours are exceptionally large, recalling the interpretation of the applied null-model given in Sect. 2.6. It is, however, www.nonlin-processes-geophys.net/18/545/2011/ erage path length L (Donner et al., 2010a), the variability of both measures T is very L similar. RMoreover, D we also find some much weaker similarities between the temporal variT 1.00 −0.08 0.38 0.00 ability patterns of transitivity T and assortativity R, which (A) L −0.08 1.00 −0.06 0.92 are less pronounced, since both properties characterise not so R 0.38 −0.06 1.00 0.03 obviouslyDrelated of the network geometry in phase 0.00aspects0.92 0.03 1.00 space. Specifically, the time interval of suppressed dust flux T L R D in ODP 659 and 721/722 during the Mid Pliocene results not 1.00 transitivity, −0.05 but 0.12 0.03assortativity. only in anTincreased also a high −0.05 1.00 −0.08 The(B) latter L feature can be explained by the fact0.77 that a relatively R of0.12 −0.08 1.00 this0.23 large cluster state vectors representing laminar regime D the0.03 1.00 connected emerges in network,0.77 which is 0.23 rather densely T L R D (Fig. 6C). We conclude that the0.50 RN measures not statistically T 1.00 0.40 are 0.37 (C) L in0.50 1.00 0.37 0.74For the ODP independent their time evolution (Tab. 2). RO and 0.40dust flux 0.37records,1.00 0.35 site 659 δ 18 the correlations between D 0.37 0.74 0.35 transitivity T and assortativity R as well as1.00 between average path length D by SpearT L and diameter L R as measured D man’s ρ are most pronounced, which is consistent with theoT 1.00 0.65 0.61 0.23 retical expectations (Donner et al., 2010a). Correlations are (D) L 0.65 1.00 0.54 0.78 more clearly developed between all four measures in case of R 0.61 0.54 1.00 0.16 the more D highly sampled0.78 dust flux0.16 records 1.00 from ODP sites 0.23 721/722 and 967 (Tab. 1). However, for all records the four important to recognise that the null-hypothesis of stationarity has been tested pointwise, while physical significance requires the null-hypothesis to be rejected over a certain period of time, i.e. for several subsequent time points (Maraun et al., 2007). Therefore, certain line-like structures, particularly those seen in Fig. 15, are likely to reflect statistical fluctuations rather than physically significant dynamical transitions. In the following, we will only present the results for the ODP site 659 dust flux record. i. Recurrence window size W ∗ : as for the model systems in Sect. 3, we first discuss the sensitivity of our results to the changing width of the recurrence window W ∗ . The corresponding results for the four chosen RN measures are shown in Fig. 13. We recognise that the most significant features persist under varying W ∗ , although the relevant structures become broader and less significant for larger windows. This is to be expected since more and more data from time intervals not directly affected by the origin of specific network properties (e.g. a laminar phase in the dynamics) contribute to the longer windows. As the window width is increased linearly, conelike structures emerge (which is especially well visible for the Mid Pliocene transitivity maximum as the most Nonlin. Processes Geophys., 18, 545–562, 2011 558 J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis J. F. Donges et al.: Identification of dynamical transitions in marine palaeoclimate records by RN analysis Transitivity Average path length Assortativity Diameter 15 Fig. 13. Dependence of (A) transitivity T , (B)Taverage path path length L, (C) assortativity onthe therecurrence recurrence window Fig. 13. Dependence of (A) transitivity , (B) average length L, (C) assortativityR, R,and and(D) (D) diameter diameter DDon window ∗ = 41 kyr, the detrending window ∗ ∗ size W ∗ for the dust flux record from ODP site 659. The recurrence window step size is fixed to 1W size W for the dust flux record from ODP site 659. The recurrence window step size is fixed to ∆W = 41 kyr, the detrending window ∗ size to WD∗size = 500 kyr. and black contours correspond to theto5the % and 95 % quantiles thetest testdistribution distribution obtained from 500 kyr. Green and black contours correspond 5% and 95% quantileswith withrespect respect to the obtained from to W D =Green 10,000 realisations of our null-model. Other parameters embedding dimension anddelay delay ττ ∗ = 10 waswas chosen 10 000 realisations of our null-model. Other parameters were:were: embedding dimension mm ==3 3and 10kyr, kyr,the thethreshold threshold chosen to yield a fixed edgeρdensity ρ =The 0.05. The white indicating value” at the leftand andright rightmargins margins of because we we to yield a fixed edge density = 0.05. white bandsbands indicating “no “no value” at the left ofeach eachpanel panelappear appear because ∗ µ,W ∗ µ plot themeasures network measures at the mid-points the windows µ used for RN analysis(Sect. (Sec. 2.4). 2.4). As the mid-points of of thethe firstfirst (last) As the mid-points (last) t µ of ttheofwindows µ used for RN analysis plot the network g µ,W atg the mid-points ∗ window move further into the past (present) for increasing the white bands grow linearlyfor forlinearly linearly increasing increasing WW∗ ∗. . window move further into the past (present) for increasing W ∗, W the ,white bands grow linearly of time series analysis, the consideration of embedding with relevant feature). general, iswenecessary observeinthat fortoour properly chosenInparameters order obtain example the transitivity is most robust with respect to feasible results. changes of W ∗ , whereas the other network measures In contrast to other techniques, RNofanalysis does not may lose significance if this parameter our analysis characterise temporal interrelationships within analysed method is varied. We note, however, that the the periods records (although time information enters indirectly through of interest identified in Sect. 4.1 are robust for a wide embedding parameters, however, mostly on short time scales range of recurrence∗ window sizes, presenting a trade-off as typically mτ N h∆T i), but quantifies geometric propbetween of identified features (small ertiesgood of thelocalisation sampled dynamical system in its (reconstructed) windows) and reasonable statistical confidence phase space. The only implicit assumption is thatofthethe availcalculated network properties (large windows). able sample of observed state vectors {yiµ } represents the spatial distribution of the true state vectors in the (properly reconstructed) phase the underlying dynamical sysii. Detrending window sizespace WD∗ :ofregarding the dependence tem sufficiently well. of our observations on the choice of the detrending window, Fig. 14 shows that the general temporal variability pattern of the different network measures remains unchanged as WD∗ is altered, whereas the actual significance levels are more strongly influenced. In general, we can conclude, however, that the most significant time Nonlin. Processes Geophys., 18, 545–562, 2011 In this respect, our approach is very generally applicaundermoderate variations of WD∗ , which bleperiods and haspersist comparably requirements in termsisofparexpressed transitivity theticularly requestedwell number of datafor (i.e.,thewindows with during O(100) the Mid Pliocene. Together the factanalysis that RN data points are sufficient for awith reasonable of analysis nonstationary systems). case of palaeoclimate records, thisproof the three dust In flux records without detrending complementary way results for characterising duces consistent (Donges ettime al., series 2011) avoids this sugconceptual of other approaches due to influence uncertain on gests thatproblems trends do not have a significant agethe models and non-uniform sampling. E.g., the∗ results of i. N h1T outcomes of RN analysis as long as W RN analysis {g µ } are invariant to changes in the age model However, this should be checked in any particular appli{ti }, only the associated windows’ mid-points {tµ } change cation by comparing the results for the time series data with variations in {ti } (Eq. (9)). However, the aforemenbefore and after detrending. thenecessary results for tioned problems indirectly persist inNote termsthat of the undetrended time series are approximated by embedding of the data and have to be finally resolvedthose in cor-dish1T i,work played infuture Fig. work. 14 forWhile WD∗ ≈ sincefocussed RN analysis responding theNpresent on is invariant to nearly uniform translations of the data. iii. Embedding delay τ ∗ : our results are also seen to be robust with respect to reasonable variations of the embedding delay τ ∗ around the previously chosen delay time τ ∗ = 10 kyr (Fig. 15). However, for the www.nonlin-processes-geophys.net/18/545/2011/ J. F. Donges of dynamical transitionsofindynamical marine palaeoclimate records by RN analysis 16 et al.: Identification J. F. Donges et al.: Identification transitions in marine palaeoclimate records by RN analysis 559 Transitivity Average path length Assortativity Diameter Fig. 14. Dependence of (A) transitivity T , (B) average path length L, (C) assortativity R, and (D) diameter D on the detrending window Fig. 14. Dependence of (A) transitivity T , (B) average path length L, (C) assortativity R, and (D) D on the detrending window ∗ diameter ∗ size ∗ ∗ size WD∗ for the dust from ODP sitesite 659. sizeisisfixed fixedtoto 410 a step of ∗1W 41 kyr. for the flux dust record flux record from ODP 659.The Therecurrence recurrence window window size WW == 410 kyrkyr withwith a step size size of ∆W = 41 = kyr. WD Green andGreen blackand contours correspond to theto5 % 95 %95% quantiles the test testdistribution distribution obtained 10 000 realisations black contours correspond the and 5% and quantileswith withrespect respect to to the obtained fromfrom 10,000 realisations ∗ 10 kyr, the threshold was chosen to yield a fixed of our null-model. Other parameters were: were: embedding dimension mm == 3 and of our null-model. Other parameters embedding dimension 3 anddelay delayττ ∗ = = kyr, the threshold was chosen to yield a fixed edge = 0.05. In regions the black dashed results areinfluenced influenced by boundary since the the effective detrending edge density ρ density = 0.05.ρ In regions outsideoutside the black dashed lineslines thethe results are boundaryeffects, effects, since effective detrending window size Wto has to decrease time series’ (Eq. (3)). Thebands white indicating bands indicating “no value” at left the and left and right D (t) The white “no value” at the right margins window size WD (t) has decrease towards towards the timethe series’ limits limits (Eq. 3). ∗ µ,W ∗ at the µ of the margins of each panel we appear we plot measures the networkg measures g µ,Wmid-points at the mid-points tµ of the windows µ used for RN analysis(Sect. (Sec. 2.4). of each panel appear because plotbecause the network t windows µ used for RN analysis In contrast to Fig. 13 their width does not change W ∗ is fixed here. In contrast2.4). to Fig. 13 their width does not change as W ∗ is as fixed here. the technical of applying RNkyr analysis to palaeocliembedding delayaspects exceeding τ ∗ = 20 the results and mate time series, an in-depth discussion of the significance levels change considerably. Thisresults is ex-obtained for the three dust flux records in the light of addipected as for delays larger than 20 kyr, autocorrelational proxy records and palaeontological evidence is given tionsininDonges the time series do not decrease significantly anyet al. (subm.). more. In the case of the δ 18 O record from ODP site 659 they even increase again due to pronounced (obliquityAcknowledgements. This work has been financially supported by driven) Milankovich cycles with a period around 41Ministry kyr the Leibniz association (project ECONS) and the Federal (Fig.for3), so that the autocorrelation criterion for the Education and Research (BMBF) via the Potsdam Research ∗ does Cluster Georisk Environmental Change and Sustainchoice of τfor notAnalysis, apply here anymore. 5 ability (PROGRESS). JFD thanks the German National Academic Foundation for financial support. We thank Roger Grzondziel and Ciaron Linstead for help with the IBM iDataPlex Cluster at the Potsdam Institute for Climate Impact Research. Conclusions We have demonstrated that RN analysis allows detecting dynamical transitions in non-stationary model systems as well as real-world palaeoclimate data. Transitivity and average path length have been previously discussed as appropriate network properties indicating qualitative changes in the www.nonlin-processes-geophys.net/18/545/2011/ References dynamics of the underlying system. Here we have provided examples that also other global network measures such as asAbarbanel, H.D.I. [1996] Analysis of Observed Chaotic Data sortativity and network (Springer, New York). diameter trace qualitative changes in dynamical however, do not have a similarly Babu, P. &systems, Stoica, P. which, [2010] “Spectral analysis of nonuniformly sampled data –interpretation a review”, Dig. Sign. Proc. 20, straighforward in terms of 359-378. basic system propBarrio, Serrano, S. aforementioned [2007] “A three-parametric study of the erties asR.the& two other quantities. Lorenz model”, Physica D 229, 43-51. Our results show V., that the outcomes analysis Boccaletti, S., Latora, Moreno, Y., Chavez, of M. RN & Hwang, D. are quite[2006] robust if the fundamental parameters of the method “Complex networks: Structure and dynamics”, Phys. (deRep. 424, trending and175-308. recurrence window sizes, embedding delay) are Brockwell, P.J. a&reasonable Davis, R.A. [1991] to Time varied within range.Introduction Unlike for otherSeries methods and Forecasting (Springer, New York, 2nd ed). ofBrockwell, time series analysis, the consideration of embedding P.J. & Davis, R.A. [2002] Time Series: Theory and with properly chosen parameters is necessary in order to obtain Methods (Springer, New York, 2nd ed). feasible results. In contrast to other techniques, RN analysis does not characterise temporal interrelationships within the analysed records (although time information enters indirectly through embedding parameters, however, mostly on short time scales as typically mτ ∗ N h1T i), but quantifies geometric Nonlin. Processes Geophys., 18, 545–562, 2011 560 Donges et al.: Identification of dynamical transitions in marine J. F. DongesJ.etF.al.: Identification of dynamical transitions in marine palaeoclimate recordspalaeoclimate by RN analysisrecords by RN analysis 17 Transitivity Average path length Assortativity Diameter Fig. 15. Dependence of (A) transitivity T , (B)Taverage path path length L, (C) assortativity (D)diameter diameter embedding Fig. 15. Dependence of (A) transitivity , (B) average length L, (C) assortativityR, R, and and (D) DD on on thethe embedding delaydelay time time ∗= τ ∗ for the dust ODP 659. window size W∗∗==410 410kyr kyrwith with a step of ∗1W 41 kyr, τ ∗ forflux the record dust fluxfrom record fromsite ODP site The 659. recurrence The recurrence window sizeisisfixed fixed to W a step sizesize of ∆W = 41 kyr, ∗ to W ∗ kyr. kyr. Green and black contours correspondtoto the the 55% quantiles withwith respect to theto the the detrending window size the detrending window size to W and black contours correspond % and and95% 95 % quantiles respect D = 500Green D = 500 test distribution realisations our null-model. Verticalline-shaped line-shaped contours to correspond to statistical test distribution obtained obtained from 10from 000 10,000 realisations of ourof null-model. Vertical contoursare arelikely likely to correspond to statistical than physically significant time intervals text). Otherparameters parameters were: were: embedding dimension m =m3,= the3,threshold fluctuationsfluctuations rather thanrather physically significant time intervals (see(see text). Other embedding dimension the threshold was chosen to yield a fixed edge density ρ = 0.05. was chosen to yield a fixed edge density ρ = 0.05. K.E. & dynamical Sprott, J.C. [2005] “A comparison of correlaproperties Chlouverakis, of the sampled system in its (recontion and Lyapunov dimensions”, Physica D 200, 156-164. structed) phase The only implicit assumption is that Csárdi, space. G. & Nepusz, T. [2006] “The igraph software package µ the availableforsample observed stateInterJournal vectors {y } complexof network research”, Complex Systems i repreCX.18, 1695. sents the spatial distribution of the true state vectors in the Costa, L.F., Rodrigues, F.A., Travieso & Villas Boas, (properly da reconstructed) phase space of theG.underlying dy-P.R. [2007] “Characterization of complex networks: a survey of meanamical system sufficiently well. surements”, Adv. Phys. 56, 167242. Analysis in the Geosciences (Springer, Heidelberg). technical aspects of applying RN analysis to palaeoclimate Donner, R.V., Zou, Y., Donges, J.F., Marwan, N. & Kurths, J. time[2010a] series,“Recurrence an in-depth discussion of the results obtained for networks – a novel paradigm for nonlinear the three dust New fluxJ.records the light of additional proxy time series”, Phys. 12, in 033025. Donner,and R.V.,palaeontological Zou, Y., Donges, J.F., Marwan,isN.given & Kurths, J. records evidence in Donges “Ambiguities in recurrence-based complex network rep(2011). et al.[2010b] resentations of time series”, Phys. Rev. E 81, 015101(R). Donner, R.V., Small, M., Donges, J.F., Marwan, N., Zou, Y., Xiang, Acknowledgements. This work has been financially supported R. & Kurths, J. [2011] “Recurrence-based time series analysis by the Leibniz association (project ECONS) and the Federal by means of complex network methods”, Int. J. Bifurc. Chaos Ministry for Education and Research (BMBF) via the Potsdam 21, 1019-1046. Research for Georisk Analysis,J.F., Environmental Change and Donner, Cluster R.V., Heitzig, J., Donges, Zou, Y., Marwan, N. & Kurths, J. [in press] JFD “The thanks geometry chaotic dySustainability (PROGRESS). theofGerman National namics Foundation – A complexfor network perspective”, Phys. J. B,Roger Academic financial support.Eur.We thank doi:10.1140/epjb/e2011-10899-1 (online Grzondziel and Ciaron Linstead for helpfirst). with the IBM iDataPlex Eckmann, J.-P., Kamphorst, S.O. & Ruelle, D. [1987] “Recurrence Cluster at the Potsdam Institute for Climate Impact Research. plots of dynamical systems”, Europhys. Lett. 4, 973-977. P.B. approach [1995] “Plio-Pleistocene African Climate”, In this deMenocal, respect, our is very generally applica-Sci270, 53-59.moderate requirements in terms of ble and has ence comparably deMenocal, P.B. 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