Dynamics of complex systems inferred from multivariate time series

Els Weinans

Research output: Thesisinternal PhD, WU


Complex systems are systems whose behaviour arises from the interaction between different elements. The dynamics of complex systems are highly unpredictable and are characterized by interactions on different scales and nonlinear responses. In chapter 1 I describe how these properties complicate the analysis of complex systems. Additionally, I explain that the analysis of complex systems requires a range of approaches, such as simplistic models, realistic models, experiments, and time series analysis. The latter is the focus of this thesis.

Two popular tools to analyze complex systems are resilience indicators and complexity indicators. In chapter 2 I address the use of these indicators, with applications to systems related to human physiology. I explore how resilience indicators can be used to infer a systems capacity to recover from small perturbations and can be used to signal upcoming `tipping points' related to some diseases, such as atrial fibrillation or depression. Complexity indicators can be used to infer when a system loses responsiveness and can be used to infer a loss of complexity which is related to old age and some diseases such as congestive heart failure. I propose that the variables that constitute `the human system' have different functions. Some variables aim for homeostasis, i.e. they seek an equilibrium. For these variables, their capacity to recover seems like a good statistic to quantify their functioning. Other variables aim for high responsiveness. For these variables, their complexity seems like a good statistic to quantify their functioning.

Currently, the most popular methods to infer a system's resilience are autocorrelation and variance of a time series. As these quantification tools are based on single time series, it is not always clear how to adapt these tools to multivariate complex systems. In Chapter 3 I propose a novel usage of a known statistical tool called Min/Max Autocorrelation Factors (MAF). This tool was developed as an alternative to PCA, but instead of finding the direction of highest variance (as with PCA), it finds the direction of the highest autocorrelation. I propose that this direction of highest autocorrelation can be used to tell which perturbation in the system is most dangerous, in the sense that perturbations in this direction will lead to the slowest recovery of the system. Furthermore, if the system is subject to tipping points, this ‘dangerous direction’ will likely also be the direction where the system can most easily shift to another state.

An obvious next question is what this future state might look like. This question is addressed in chapter 4. We use the fact that most complex behaviour, such as oscillations or reactivity, arise from delayed negative feedbacks. Therefore, positive feedbacks, which are at the core of mutualistic networks, are expected to give rise to relatively simple dynamics. We find that this relative simplicity allows for extrapolation of the direction of lowest resilience, found by PCA in this chapter, to predict the future state after a tipping point has passed. Therefore, this tool is a valuable addition to our toolbox to analyse complex systems as it provides a way to predict not just when something is about to happen, but also what might happen.

A clear comparison between different multivariate indicators of resilience is lacking. Therefore in chapter 5 I investigate how different methods relate to one another, if there are methods that are preferred over others and under which conditions the different methods are expected to correctly predict an upcoming tipping point. I demonstrate that there is not one best indicator to warn for an upcoming transition, but that instead it depends on the scenario that the system is subject to. Furthermore, this chapter demonstrates that all methods become unreliable when not all variables can be observed. As this scenario is extremely relevant for empirical studies, where one can never be sure that all variables are included, this suggests a cautious interpretation of all work on multivariate resilience indicators.

In chapters 6-7 I explore the applications of time series analysis tools for complex systems to two real world datasets. In chapter 6, I systematically analyze word-use in millions of books from 1850-2019. I demonstrate that there are two dominant modes of change. The first captures the general trends of world popularity over time. The second mode disentangles human nature related words, such as pronouns and emotions, from words related to rational decision and procedures. I demonstrate that the ‘rational’ words show a steady increase from 1850-1980, followed by a drop. The words related to humans/emotions show the opposite behaviour. We propose that the current increase of sentiment laden words, could be a reaction to several decades of rational thinking. The fact that the strong increase in sentiment laden words is accompanied by an increase in the use of facebook, suggests that this shift from rational thinking to intuitive thinking is strengthened by the increasing popularity of social media.

In chapter 7 I use climate data of the past 800.000 years to infer causal links in the carbon cycle. I use sediment cores to determine Ba/Fe (a proxy for biological productivity), δ18O (a proxy for climate and ice cover), and δ13C (a proxy for ocean ventilation) and ice cores to determine dust and CO2 (a proxy for climate and alkalinity). One mystery of the glacial-interglacial cycles is their saw-tooth shape of slow cooling and rapid warming, which hints at the existence of nonlinear processes in the system, such as a feedback loop. As all possible links have been described, it is hard to pinpoint the dominant drivers. Here, I demonstrate that a causal detection method based on Taken's theorem (convergent cross-mapping) can elucidate causal links in the system and results in one dominant causal loop from ocean ventilation to biological productivity to climate back to ocean ventilation. This loop forms a potential explanation for the shape of the glacial-interglacial cycles.

In chapter 8 I reflect on the findings of the previous chapters, bring up some scientific considerations that I learned about while performing the work presented here, and share ideas for future studies.

Original languageEnglish
QualificationDoctor of Philosophy
Awarding Institution
  • Wageningen University
  • Scheffer, Marten, Promotor
  • van de Leemput, Ingrid, Co-promotor
  • van Nes, Egbert, Co-promotor
Award date18 Oct 2021
Place of PublicationWageningen
Print ISBNs9789463958882
Publication statusPublished - 18 Oct 2021


Dive into the research topics of 'Dynamics of complex systems inferred from multivariate time series'. Together they form a unique fingerprint.

Cite this