>Methods to study complex systems

related to PIK research on Time Series Analysis and Complex Networks

Recurrence plots

Recurrence Plot

Recurrence plots (RPs) provide an alternative way to study various aspects of complex systems, such regime transitions, classification, detection of time-scales, synchronisation, and coupling detection (RP bibliography). Main contributions have been in bivariate extensions (cross RPs) and coupling analysis, new measures of complexity, significance assessments of the RP based results, spatial extensions, parameter selection, RPs for irregularly sampled data and for extreme events data, or complex network based quantification.

  • N. Marwan, M. C. Romano, M. Thiel, J. Kurths: Recurrence Plots for the Analysis of Complex Systems, Physics Reports, 438(5-6), 237-329 (2007). DOI:10.1016/j.physrep.2006.11.001
  • N. Marwan, J. F. Donges, Y. Zou, R. V. Donner, J. Kurths: Complex network approach for recurrence analysis of time series, Physics Letters A, 373(46), 4246-4254 (2009). DOI:10.1016/j.physleta.2009.09.042
  • N. Marwan: How to avoid potential pitfalls in recurrence plot based data analysis, International Journal of Bifurcation and Chaos, 21(4), 1003-1017 (2011). DOI:10.1142/S0218127411029008
  • N. Marwan, S. Schinkel, J. Kurths: Recurrence plots 25 years later – Gaining confidence in dynamical transitions, Europhysics Letters, 101, 20007 (2013). DOI:10.1209/0295-5075/101/20007
  • N. Marwan: Challenges and perspectives in recurrence analyses of event time series, Frontiers in Applied Mathematics and Statistics, 9, 1129105 (2023). DOI:10.3389/fams.2023.1129105

Complex networks

Complex networks provide a powerful approach to investigate extended and spatio-temporal systems, such as the climate by climate networks. Moreover, they offer an alternative way for a recurrence based time-series analysis by recurrence networks.

  • K. Rehfeld, N. Marwan, S. F. M. Breitenbach, J. Kurths: Late Holocene Asian summer monsoon dynamics from small but complex networks of paleoclimate data, Climate Dynamics, 41(1), 3-19 (2013). DOI:10.1007/s00382-012-1448-3
  • N. Boers, B. Bookhagen, N. Marwan, J. Kurths, J. Marengo: Complex networks identify spatial patterns of extreme rainfall events of the South American Monsoon System, Geophysical Research Letters, 40(16), 4386-4392 (2013). DOI:10.1002/grl.50681
  • N. Marwan, J. Kurths: Complex network based techniques to identify extreme events and (sudden) transitions in spatio-temporal systems, Chaos, 25, 097609 (2015). DOI:10.1063/1.4916924
  • Y. Zou, R. V. Donner, N. Marwan, J. F. Donges, J. Kurths: Complex network approaches to nonlinear time series analysis, Physics Reports, 787, 1-97 (2019). DOI:10.1016/j.physrep.2018.10.005

Special time-series analysis methods for special problems

Absolute Age

Special problems require especially adopted methods of time-series analysis. For example, proxy records in Earth sciences are often irregularly sampled and come with uncertainties in the dating points. Approaches for considering such dating uncertainties in the subsequent analysis and methods for correlation analysis of irregularly sampled time series have been developed. Such approaches can be helpful for the reconstruction of palaeoclimate complex networks.

  • K. Rehfeld, N. Marwan, J. Heitzig, J. Kurths: Comparison of correlation analysis techniques for irregularly sampled time series, Nonlinear Processes in Geophysics, 18(3), 389-404 (2011). DOI:10.5194/npg-18-389-2011
  • S. F. M. Breitenbach, K. Rehfeld, B. Goswami, J. U. L. Baldini, H. E. Ridley, D. Kennett, K. Prufer, V. V. Aquino, Y. Asmerom, V. J. Polyak, H. Cheng, J. Kurths, N. Marwan: COnstructing Proxy-Record Age models (COPRA), Climate of the Past, 8, 1765-1779 (2012). DOI:10.5194/cp-8-1765-2012
  • D. Eroglu, F. H. McRobie, I. Ozken, T. Stemler, K.-H. Wyrwoll, S. F. M. Breitenbach, N. Marwan, J. Kurths: See-saw relationship of the Holocene East Asian-Australian summer monsoon, Nature Communications, 7, 12929 (2016). DOI:10.1038/ncomms12929
  • I. Ozken, D. Eroglu, S. F. M. Breitenbach, N. Marwan, L. Tan, U. Tirnakli, J. Kurths: Recurrence plot analysis of irregularly sampled data, Physical Review E, 98, 052215 (2018). DOI:10.1103/PhysRevE.98.052215
  • B. Goswami, N. Boers, A. Rheinwalt, N. Marwan, J. Heitzig, S. F. M. Breitenbach, J. Kurths: Abrupt transitions in time series with uncertainties, Nature Communications, 9, 48 (2018). DOI:10.1038/s41467-017-02456-6
  • N. Marwan, T. Braun: Power spectral estimate for discrete data, Chaos, 33(5), 053118 (2023). DOI:10.1063/5.0143224

>Complexity in applications

Climate and palaeoclimate

The study of palaeoclimate from proxy records is helpful for a better understanding of the climate system. Information based on lake sediments or speleothemes can be used to study complex interrelationships or past climate transitions. We are also participating in the coordinated scientific research in the Blessberg Cave, Thuringia.

Cover Science 2012
  • J. F. Donges, R. V. Donner, M. H. Trauth, N. Marwan, H. J. Schellnhuber, J. Kurths: Nonlinear detection of paleoclimate-variability transitions possibly related to human evolution, Proceedings of the National Academy of Sciences, 108(51), 20422-20427 (2011). DOI:10.1073/pnas.1117052108
  • Science cover story: D. J. Kennett, S. F. M. Breitenbach, V. V. Aquino, Y. Asmerom, J. Awe, J. U. L. Baldini, P. Bartlein, B. J. Culleton, C. Ebert, C. Jazwa, M. J. Macri, N. Marwan, V. Polyak, K. M. Prufer, H. E. Ridley, H. Sodemann, B. Winterhalder, G. H. Haug: Development and Disintegration of Maya Political Systems in Response to Climate Change, Science, 338(6108), 788-791 (2012). DOI:10.1126/science.1226299
  • N. Boers, B. Bookhagen, H. M. J. Barbosa, N. Marwan, J. Kurths, J. A. Marengo: Prediction of extreme floods in the eastern Central Andes based on a complex networks approach, Nature Communications, 5, 5199 (2014). DOI:10.1038/ncomms6199
  • T. Westerhold, N. Marwan, et al.: An astronomically dated record of Earth's climate and its predictability over the last 66 million years, Science, 369(6509), 1383–1387 (2020). DOI:10.1126/science.aba6853
  • M. Singh, R. Krishnan, B. Goswami, A. D. Choudhury, P. Swapna, R. Vellore, A. G. Prajeesh, N. Sandeep, C. Venkataraman, R. V. Donner, N. Marwan, J. Kurths: Fingerprint of volcanic forcing on the ENSO-Indian monsoon coupling, Science Advances, 6, eaba8164 (2020). DOI:10.1126/sciadv.aba8164

Cardiovascular systems

Besides the main focus on climate related problems, recurrence properties of the cardiovascular system are studied, e.g., to early detect ventricular tachycardia or preeclampsia, or to investigate the coupling mechanisms in the cardio-respiratory system.

  • N. Marwan, N. Wessel, U. Meyerfeldt, A. Schirdewan, J. Kurths: Recurrence Plot Based Measures of Complexity and its Application to Heart Rate Variability Data, Physical Review E, 66(2), 026702 (2002). DOI:10.1103/PhysRevE.66.026702
  • G. M. Ramírez Ávila, A. Gapelyuk, N. Marwan, H. Stepan, J. Kurths, T. Walther, N. Wessel: Classifying healthy women and preeclamptic patients from cardiovascular data using recurrence and complex network methods, Autonomic Neuroscience, 178(1-2), 103-110 (2013). DOI:10.1016/j.autneu.2013.05.003
  • N. Marwan, Y. Zou, N. Wessel, M. Riedl, J. Kurths: Estimating coupling directions in the cardio-respiratory system using recurrence properties, Philosophical Transactions of the Royal Society A, 371(1997), 20110624 (2013). DOI:10.1098/rsta.2011.0624

EEG analysis

Further interest in life science is related to EEG analysis, aiming at the detection of event related potentials or early signatures of epileptic seizures.

  • N. Marwan, A. Meinke: Extended recurrence plot analysis and its application to ERP data, International Journal of Bifurcation and Chaos, 14(2), 761-771 (2004). DOI:10.1142/S0218127404009454
  • S. Schinkel, N. Marwan, J. Kurths: Order patterns recurrence plots in the anaylsis of ERP data, Cognitive Neurodynamics, 1(4), 317-325 (2007). DOI:10.1007/s11571-007-9023-z
  • S. Schinkel, N. Marwan, J. Kurths: Brain signal analysis based on recurrences, Journal of Physiology-Paris, 103(6), 315-323 (2009). DOI:10.1016/j.jphysparis.2009.05.007
  • E. J. Ngamga, S. Bialonski, N. Marwan, J. Kurths, C. Geier, K. Lehnertz: Evaluation of selected recurrence measures in discriminating pre-ictal and inter-ictal periods from epileptic EEG data, Physics Letters A, 380(16), 1419–1425 (2016). DOI:10.1016/j.physleta.2016.02.024

3D image analysis

3D microCT image of trabecular bone

Methods to investigate complexity in 3D have been applied to study structural changes in trabecular bone, such as occurring during osteoporosis or space flights.

  • N. Marwan, P. Saparin, J. Kurths: Measures of complexity for 3D image analysis of trabecular bone, European Physical Journal – Special Topics, 143(1), 109-116 (2007). DOI:10.1140/epjst/e2007-00078-x
  • N. Marwan, J. Kurths, J. S. Thomsen, D. Felsenberg, P. Saparin: Three dimensional quantification of structures in trabecular bone using measures of complexity, Physical Review E, 79(2), 021903 (2009). DOI:10.1103/PhysRevE.79.021903
  • T. Schmah, N. Marwan, J. S. Thomsen, P. Saparin: Long range node-strut analysis of trabecular bone microarchitecture, Medical Physics, 38(9), 5003-5011 (2011). DOI:10.1118/1.3622600

>Cave research


Scientific research in caves is performed to explore and survey newly discovered cave parts, but also to collect data for the palaeoclimate studies (samples, monitoring). Cave research is focused on caves in Switzerland (research with isaak), but also in India, Caucasus, Kosovo, and Germany.

  • N. Marwan: Cave Blisters in der Oberländerhöhle (M3)/ Découverte de blisters dans la Oberländerhöhle (M3), Stalactite, 50(2), 103-105 (2000).
  • N. Marwan: Das Karstgebiet des Bol'soj Thac, Abhandlungen und Berichte des Naturkundemuseums Görlitz, 79(1), 55-84 (2007).
  • S. Breitenbach, N. Marwan, G. Wibbelt: Weißnasensyndrom in Nordamerika – Pilzbesiedlung in Europa, Nyctalus, 16(3), 172-179 (2011).
  • one of the first web presentations of speleology was the speleo server east

>Projects & grants