Development of advanced time series analysis techniques

Research background

Real world processes are complex and appear to be chaotic, i.e., their predictability is limited. Nonlinear theory offers new approaches for the development of modern data analysis techniques which can shed light on specific features in the dynamics of complex systems where standard or linear methods are not able to study them.

Example: Challenges in palaeoclimate research

Recent projects are collecting and extending the data archive of the palaeoclimate. New high precision records, e.g., such from stalagmites, offer new insights into the climate system or extreme events in the past that have not been available before on such high-resolution time-scales. A reliable interpretation requires statistical analysis and comparison with many environmental variables.

Moreover, such geological archives come with specific problems, such as time uncertainties and irregular sampling – challenges that hamper the application of standard methods or introducing nontrivial biases in the final results. Moreover, the definition of extreme climate events on the basis of geological archives is difficult, as it depends strongly on the time scale and since geological archives usually aggregate and integrate the environmental conditions over a longer period of time. There is a big need in methods which can be applied on such kind of data to derive reliable information about climate variability and the distribution of extremes while properly taking into account uncertainties.

Methods

We are developing and adopting methods of non-linear time series analysis for direct application to data with irregular sampling and with special focus on time uncertainties. This includes simple methods like kernel-based correlations, but also more complex methods like conditional mutual information, edit distance and recurrence based measures. We further develop methods for studying recurrence properties of extreme events data. New definitions of recurrence and extremes are in development for the particular consideration of time uncertainties. The recurrence concept is under extension to study spatio-temporal data. Moreover, we work on the development of methods for studying couplings, interrelations, and causal directions within extended systems. This research is also a basis for the reconstruction of complex networks and networks of networks.

Research Highlights

Development of time series analysis approaches and statistical association methods (correlation and mutual information) for the direct application on irregularly sampled time series. This development is the core for the subsequent investigation of regime changes and dynamical transitions, as well as spatio-temporal investigation of palaeo-climate data by complex networks (palaeo-climate networks).

• 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).
• 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).
• 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, 12929p. (2016).

Several approaches for the investigation of direct and indirect links, causal couplings, or coincidences of special events have been developed, critically tested, and finally applied to climate data, answering questions about differences in the spatial patterns of extreme monsoonal rainfall in India and South America, interrelations between the East Asian Summer monsoon and Indian Summer monsoon, or causal relationships between global warming, solar and volcanic activity, and human impact.

• J. H. Feldhoff, R. V. Donner, J. F. Donges, N. Marwan, J. Kurths: Geometric signature of complex synchronisation scenarios, Europhysics Letters, 102(3), 30007 (2013).
• J. Runge, V. Petoukhov, J. F. Donges, J. Hlinka, N. Jajcay, M. Vejmelka, D. Hartman, N. Marwan, M. Paluš, J. Kurths: Identifying causal gateways and mediators in complex spatio-temporal systems, Nature Communications, 6, 8502 (2015).
• A. M. T. Ramos, A. Builes-Jaramillo, G. Poveda, B. Goswami, E. E. N. Macau, J. Kurths, N. Marwan: Recurrence measure of conditional dependence and applications, Physical Review E, 95, 052206 (2017).
• D. A. Smirnov, S. F. M. Breitenbach, G. Feulner, F. A. Lechleitner, K. M. Prufer, J. U. L. Baldini, N. Marwan, J. Kurths: A regime shift in the Sun–Climate connection with the end of the Medieval Climate Anomaly, Scientific Reports, 7, 11131 (2017).

Introduction of a framework for dealing with time uncertainties in palaeo-climate proxy records (COPRA). This approach was the basis, e.g., for the discussion about the socio-economic climate impact in the Mayan civilization.

• 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).
• 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).
• 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).
• S. F. M. Breitenbach, B. Plessen, S. Waltgenbach, R. Tjallingii, J. Leonhardt, K. P. Jochum, H. Meyer, B. Goswami, N. Marwan, D. Scholz: Holocene interaction of maritime and continental climate in Central Europe: New speleothem evidence from Central Germany, Global and Planetary Change, 176, 144–161 (2019).

Further extending of recurrence analysis is an ongoing topic. The combination with the complex network approach has provided additional measures of complexity for the study of dynamical transitions in complex systems (recurrence networks). We have extended this approach by multiplex recurrence networks to study multivariate data, by new approaches for analysing irregularly sampled time series and data with uncertainties, by multiscale concepts, and by using alternative recurrence definitions for extreme events or spatio-temporal data. The critical and systematic study of the recurrence plot concept results in better understanding of this method and led to powerful correction schemes.

• N. Marwan, S. Schinkel, J. Kurths: Recurrence plots 25 years later – Gaining confidence in dynamical transitions, Europhysics Letters, 101, 20007 (2013).
• 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).
• 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).
• D. Eroglu, N. Marwan, M. Stebich, J. Kurths: Multiplex recurrence networks, Physical Review E, 97, 012312p. (2018).
• 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).
• K. H. Kraemer, N. Marwan: Border effect corrections for diagonal line based recurrence quantification analysis measures, Physics Letters A, 383(34), 125977 (2019).

With Toolboxes for Complex Systems we provide a compilation of innovative methods for modern nonlinear data analysis. These methods were developed during our scientific research and cover recurrence analysis, causality investigations, filter procedures, time series analysis for irregularly sampled time series, etc.

Part of this toolbox are, e.g., the CRP Toolbox for MATLAB (providing many tools for recurrence analysis), or the pyunicorn package for Python (providing recurrence and complex network analysis tools).