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Historical Review of Recurrence Plots
Norbert Marwan
TL;DR
The paper reviews how recurrence plots developed from a visualization of recurrences into a quantified and extensible method used across many disciplines. It traces methodological, theoretical, software, and application milestones, while noting limits in the available application-field statistics. The review reports growing usage, including 728 downloads in November 2005–May 2008.
Problem
The historical development of recurrence plots, their quantification, theoretical foundations, and expanding scientific applications needed to be summarized after 20 years.
Method
The paper synthesizes methodological, theoretical, software, and application milestones in recurrence plots and recurrence quantification analysis.
Results
728 downloads were recorded between November 2005 and May 2008, demonstrating increasing popularity and demand for a corresponding MATLAB toolbox.
Takeaways & Limitations
Recurrence methods expanded from nonlinear time-series visualization into tools for quantification, cross-system analysis, spatial structures, and interdisciplinary research.
Takeaways & Limitations
Application-field statistics were not claimed to be complete or optimally categorized because field selection could be arbitrary and multiple choices were allowed.
Abstract
from arXiv · showhide
In the last two decades recurrence plots (RPs) were introduced in many different scientific disciplines. It turned out how powerful this method is. After introducing approaches of quantification of RPs and by the study of relationships between RPs and fundamental properties of dynamical systems, this method attracted even more attention. After 20 years of RPs it is time to summarise this development in a historical context.
1 Introduction
Recurrence ideas predated recurrence plots, but modern computation made it possible to study recurrences systematically in simulations, measurements, and pairwise data comparisons. The recurrence plot emerged in chaos theory as a visualization of repeated states in higher-dimensional phase space.
- Recurrences were studied in mathematics as a fundamental property of conservative dynamical systems.
- Powerful computers enabled numerically costly recurrence studies of models such as the Lorenz system and real measurements.
- Similarity matrices represented pairwise similarities across all combinations of a data series.
- Recurrence-based methods can handle non-stationary, non-linear, and relatively short data series.
- Recurrence plots applied similarity matrices to compare all possible states along higher-dimensional phase-space trajectories.
2 The birth of the recurrence plot
Eckmann et al.'s 1987 use of similarity matrices to visualize recurrences in higher-dimensional phase space is regarded as the birth of modern recurrence plots and their quantification. A related close returns plot soon offered a more intuitive representation by comparing only specified past and future times.
- 1987 is considered the birth of recurrence plots and their quantification as a modern nonlinear data-analysis tool.
- Close returns plots, introduced independently by different authors no later than 1992, compare a given time into the past and future.
- Close returns plots can be more intuitive for beginners because recurrence-plot line structures run parallel to the x-axis.
3 Recurrence quantification analysis
Recurrence quantification analysis converted recurrence plots from primarily visual displays into quantitative descriptions of their structures. Subsequent work added software, time-dependent analysis, theoretical links, and refined recurrence constructions.
- Recurrence quantification analysis: Recurrence quantification analysis initially measured recurrence-point density and diagonal-line-length histograms to reduce subjective visual interpretation.
- Recurrence quantification analysis: Core RQA measures included recurrence rate, determinism, maximal line length, divergence, Shannon entropy, and trend.
- Recurrence quantification analysis: Freely available RQA Software, VRA, and TISEAN helped users compute recurrence plots or their measures.
- Recurrence quantification analysis: Time-dependent RQA calculated measures in windows moved along the main diagonal to study their evolution and detect transitions.
- Recurrence quantification analysis: By the mid-1990s, scientific awareness of recurrence plots increased, as shown by continuously rising publication numbers from 1996 to 2004.
- Recurrence quantification analysis: Theoretical studies connected recurrence-plot structures with reconstructed data, K2 entropy, and information dimension.
- Recurrence quantification analysis: Perpendicular and iso-directional recurrence plots added geometric or directional conditions to recurrence definitions for divergence-related analysis.
4 Extensions for the recurrence plot and quantification analysis
Recurrence plots and their quantification expanded beyond univariate time-series analysis through cross-system comparisons, new complexity measures, and applications to spatial structures. These extensions broadened both the method's scope and its interpretive tools.
- Extensions for the recurrence plot and quantification analysis: Cross recurrence plots test simultaneous similar states in two systems and support analysis of deterministic signals and inter-system relations.
- Extensions for the recurrence plot and quantification analysis: The CRP Toolbox for MATLAB provided a platform-independent collection of recurrence-plot tools and measures.
- Extensions for the recurrence plot and quantification analysis: Laminarity and trapping time used vertical recurrence-plot structures to detect chaos-chaos transitions.
- Extensions for the recurrence plot and quantification analysis: In bio-informatics, recurrence methods were applied to spatial series and structures, including distance matrices without a pre-existing series.
- Extensions for the recurrence plot and quantification analysis: Sampling-rate-dependent gaps in recurrence plots can help detect slight frequency changes that standard spectral analysis does not show.
5 Theoretical basis and dynamical invariants
During the 2000s, recurrence-plot research developed theoretical links to dynamical invariants, synchronization, and alternative recurrence representations, while revealing sampling-related information in oscillatory signals.
- Romano and Thiel justified recurrence-threshold choices for noisy observations and analytically described recurrence plots of noise.
- Joint recurrence plots test simultaneous recurrences across systems and support detection of general synchronization.
- Delay-based recurrence-quantification measures can detect phase synchronization, including in non-phase-coherent oscillators, and indicate coupling direction.
- Order-pattern recurrence plots use local rank order instead of spatial phase-space information, helping address changing amplitudes such as drift.
- Sampling rate can create large apparent gaps in recurrence plots that help detect slight frequency changes invisible to standard spectral analysis.
- Recurrence plots can be treated as adjacency matrices, enabling topological analysis of complex networks and graphs using recurrence quantification analysis.
6 The spreading application fields
Recurrence plots spread from early life-science applications into many disciplines, with publication activity and CRP Toolbox downloads increasing while life sciences remained the largest reported application field.
- Since 2005, more than 50 recurrence-plot publications appeared annually across physiology, biology, earth sciences, acoustics, engineering, materials, finance, economics, chemistry, and physics.
- The CRP Toolbox usage statistics estimate application-field distributions from downloads since 2003, although field categories and user selections were not fully standardized.
- 728 downloads were recorded from November 2005 to May 2008, compared with 383 from May 2003 to October 2005.
- Between the two periods, engineering applications increased from 15% to 18%, while earth-science applications decreased from 15% to 12%.
- In the later period, life sciences led usage with 275 downloads, followed by engineering with 131 and earth sciences with 89.
7 Outlook
The authors close by noting a 2008 Google April Fools’ hoax that invoked recurrence plots in a purported technology for predicting future internet content.
- Google’s April 1, 2008 press release described the fictional gDay search technology as using recurrence plots and fuzzy measure analysis.
- The authors present the hoax as a curious sign that recurrence plots were becoming widely known and accepted.
- The hoax claimed gDay could model internet content 24 hours ahead, including share prices, sports results, and news events.