Local Minima-Based Recurrence Plots for Continuous Dynamical Systems

Schultz, Aaron P.; Zou, Yong; Marwan, Norbert; Turvey, M. T.

doi:10.1142/s0218127411029045

Cited by 19 publications

(21 citation statements)

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“…The similarity threshold, ǫ, is a parameter that determines whether vectors in the phase space are identified as recurrent, i.e., as revisiting the same area of the phase space. Choosing an appropriate value for ǫ can be difficult [34] and investigating the impact of ǫ on RQA output is an active research topic [35,36]; in the present work, we use a nearest neighbor approach that does not require an explicit ǫ to be set.…”

Section: Jrqamentioning

confidence: 99%

Comparison of Cross-Correlation and Joint-Recurrence Quantification Analysis Based Methods for Estimating Coupling Strength in Non-linear Systems

Tolston

Funke

Shockley

2020

Front. Appl. Math. Stat.

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Time-delay stability (TDS) analysis is a method for quantifying interactions in multivariate systems by identifying stable temporal relationships in time series data [1]. This method has been used to create network representations of complex systems. As originally presented, the TDS method relies on cross-correlation-a linear analysis that is restricted to estimating relationships between unidimensional time series, and which, by itself, often does not adequately characterize interactions between many non-linear complex systems of theoretical and practical interest. Thus, modifying TDS so that it relies on joint recurrence quantification analysis (JRQA), an intrinsically non-linear multidimensional framework, and then comparing the ability of the two approaches to detect interactions in non-linear systems is an important task. In the present work, we first show how TDS can be extended using JRQA, a method which is capable of multidimensional assessment of relationships in non-linear systems. In our application of JRQA, we introduce a modification in the form of a weighting factor that accounts for the truncation of time series that results from time-delayed JRQA. We also modify TDS by correcting for a bias in the method and show how analogs of recurrence-based metrics can also be obtained for TDS. We evaluate how TDS results obtained with JRQA compare to those obtained with cross-correlation for known dynamics of coupled non-linear oscillators and from unknown dynamics of multivariate behavioral signals measured from dyads performing a joint problem-solving task. We conclude that TDS using cross-correlation provides results that are comparable to those obtained with JRQA at a much-reduced computational cost.

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Section: Jrqamentioning

confidence: 99%

Comparison of Cross-Correlation and Joint-Recurrence Quantification Analysis Based Methods for Estimating Coupling Strength in Non-linear Systems

Tolston

Funke

Shockley

2020

Front. Appl. Math. Stat.

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“…Nevertheless, pitfalls and artifacts are not unusual when constructing RP such as those occurring with the wrong choice of embedding parameters or high threshold (ε). [7][8][9] Thick lines, for instance, could easily occur in continuous dynamical systems due to the temporal correlation of the phase space trajectory, and, hence causing the RP to contain redundant information. 8 Such thick lines artificially increase the number of diagonal lines and even introduce vertical lines that lead to artifacts in RQA such as an increase in DET and LAM .…”

Section: Introductionmentioning

confidence: 99%

“…[7][8][9] Thick lines, for instance, could easily occur in continuous dynamical systems due to the temporal correlation of the phase space trajectory, and, hence causing the RP to contain redundant information. 8 Such thick lines artificially increase the number of diagonal lines and even introduce vertical lines that lead to artifacts in RQA such as an increase in DET and LAM . With regard to these thick lines, Schultz et al suggested a local minima based RP with additional threshold ε to overcome this issue and demonstrated the benefits on the Lorenz system as an example.…”

Section: Introductionmentioning

confidence: 99%

“…Phase space trajectories of the Lorenz series dx dt , with σ = 10, ρ = 28, β = 8 3 : (a) without noise and (b) with noise (SNR = 20 dB); (c) zoomin view of both signal trajectories; and (d) extracted phase space distance (yaxis) between a selected point at the phase space trajectory and all other trajectory points by time lag (x-axis) and corresponding local minima (blue and red points, indicating local minima of no noise and noisy Lorenz, respectively). The shaded orange region refers to the region where local minima with threshold (ε) are used for RP with the LMT approach.…”

Section: Introductionmentioning

confidence: 99%

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Extended recurrence plot and quantification for noisy continuous dynamical systems

Wendi

Marwan

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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One main challenge in constructing a reliable recurrence plot (RP) and, hence, its quantification [recurrence quantification analysis (RQA)] of a continuous dynamical system is the induced noise that is commonly found in observation time series. This induced noise is known to cause disrupted and deviated diagonal lines despite the known deterministic features and, hence, biases the diagonal line based RQA measures and can lead to misleading conclusions. Although discontinuous lines can be further connected by increasing the recurrence threshold, such an approach triggers thick lines in the plot. However, thick lines also influence the RQA measures by artificially increasing the number of diagonals and the length of vertical lines [e.g., Determinism ( ) and Laminarity ( ) become artificially higher]. To take on this challenge, an extended RQA approach for accounting disrupted and deviated diagonal lines is proposed. The approach uses the concept of a sliding diagonal window with minimal window size that tolerates the mentioned deviated lines and also considers a specified minimal lag between points as connected. This is meant to derive a similar determinism indicator for noisy signal where conventional RQA fails to capture. Additionally, an extended local minima approach to construct RP is also proposed to further reduce artificial block structures and vertical lines that potentially increase the associated RQA like LAM. The methodology and applicability of the extended local minima approach and equivalent measure are presented and discussed, respectively.

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“…An alternative solution could be to use a different kind of recurrence criterion, e.g. local minima, as suggested by Schultz et al [2011].…”

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confidence: 99%

Editorial

Kulkarni¹,

Marwan

Parrott

et al. 2011

Int. J. Bifurcation Chaos

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Researchers in different subject areas are tackling similar questions that require a complex systems approach: Can we develop indicators that serve as warning signals for impending regime shifts or critical thresholds in the system behavior? What is the characteristic observation scale that allows for an optimal description of the system dynamics in space and time? How can the resilience (return time to equilibrium) and stability (resistance to external forcing) of a system subjected to disturbance regimes be quantified? Can we derive generalities on how natural and man-made systems develop in time? Amongst interacting units of the system, which ones are the keystones for its global functioning? In this context, recurrence plots are useful tools as they provide a common language for the study of complex systems [Webber, Jr. et al., 2009]. Furthermore, since the method does not require fitting a specific model to data, recurrence plots logically preserve a maximum of the information that is embedded in the sequence of observations. Recurrence plot based methods are also powerful visual and quantitative tools for pattern detection and, as such, continue to birth novel research hypotheses as well as challenge established ones. The ability of these methods to cope with nonlinear and transient behaviors, as well as observational noise is of particular interest to experimentalists and empiricists.The recurrence plot (RP) was introduced more than twenty years ago in order to visualize the dynamics of complex systems by their recurrences. The first natural extension was the quantification of small-scale structures in an RP by means of recurrence quantification analysis (RQA). RQA has turned out to be a powerful tool for distinguishing different types of dynamics, detecting dynamical transitions, or describing changes in the complexity of spatial objects [Marwan et al., 2007;Marwan, 2008].

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Local Minima-Based Recurrence Plots for Continuous Dynamical Systems

Cited by 19 publications

References 18 publications

Comparison of Cross-Correlation and Joint-Recurrence Quantification Analysis Based Methods for Estimating Coupling Strength in Non-linear Systems

Comparison of Cross-Correlation and Joint-Recurrence Quantification Analysis Based Methods for Estimating Coupling Strength in Non-linear Systems

Extended recurrence plot and quantification for noisy continuous dynamical systems

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