OFFPRINT Recurrence plots 25 years later — Gaining confidence in dynamical transitions

Recurrence-plot–based time series analysis is widely used to study changes and transitions in the dynamics of a system or temporal deviations from its overall dynamical regime. However, most studies do not discuss the significance of the detected variations in the recurrence quantification measures. In this letter we propose a novel method to add a confidence measure to the recurrence quantification analysis. We show how this approach can be used to study significant changes in dynamical systems due to a change in control parameters, chaos-order as well as chaoschaos transitions. Finally we study and discuss climate transitions by analysing a marine proxy record for past sea surface temperature. This paper is dedicated to the 25th anniversary of the introduction of recurrence plots. Copyright c © EPLA, 2013 Introduction. – In the first November issue of EPL in 1987, Eckmann et al. proposed the recurrence plot as a tool to easily get insights into even high-dimensional dynamical systems [1,2]. Over the last 25 years, their paper has “led to an active field, with many ramifications (these authors) certainly had not anticipated” [3]. Starting from the visual concept of recurrence plots (RPs), different statistical and quantification approaches have been added, like recurrence quantification (RQA), dynamical invariants from RPs, and recurrence networks [2,4–6]. Twenty-five years after Eckmann’s seminal paper, RPs and related methods are widely accepted tools for data analysis in various disciplines, as in physics [7] and chemistry [8], but also for real-world systems as in life science [9,10], engineering [11,12], earth science [13], or finance and economy [14–16]. This interdisciplinary success is not only caused by the attractive appearance of RPs but also by the simplicity of the method [17]. Based on RPs, we can study the dynamics, transitions, or synchronisation of complex systems [1,2,5]. In particular, such transitions can be uncovered from a changing recurrence structure. The different aspects of recurrences can be inferred by measures of complexity, also known as recurrence quantification analysis (RQA). Although these measures are often applied to real data and interpreted as indicators of a change of the system’s dynamics, a statistical evaluation of the results was not yet satisfiably addressed. An early attempt has suggested to use a specific model class (e.g., autocorrelated noise) corresponding to the nullhypothesis and then testing the RQA results against such models [18]. For a general test of how significant the value of certain RQA measures (in particular determinism DET and laminarity LAM) is, a test distribution was derived using binomial distributions [19]. In order to compare time-dependent RQA measures of different observations, a bootstrap approach was introduced [20]. However, we still miss a method which can derive the important significance level of dynamical transitions within one dynamical system as indicated by RQA. Without providing some statement on the confidence of RQA results, any conclusions drawn from RQA might remain questionable [21]. In this letter we propose a method which calculates the confidence level for the most important, line-based RQA measures. We pick up the idea of bootstrapping [20] and