Recurrence plot statistics and the effect of embedding

Abstract

Recurrence plots provide a graphical representation of the recurrent patterns in a timeseries, the quantification of which is a relatively new field. Here we derive analytical expressions which relate the values of key statistics, notably determinism and entropy of line length distribution, to the correlation sum as a function of embedding dimension. These expressions are obtained by deriving the transformation which generates an embedded recurrence plot from an unembedded plot. A single unembedded recurrence plot thus provides the statistics of all possible embedded recurrence plots. If the correlation sum scales exponentially with embedding dimension, we show that these statistics are determined entirely by the exponent of the exponential. This explains the results of Iwanski and Bradley (Chaos 8 [1998] 861-871) who found that certain recurrence plot statistics are apparently invariant to embedding dimension for certain low-dimensional systems. We also examine the relationship between the mutual information content of two timeseries and the common recurrent structure seen in their recurrence plots. This allows time-localized contributions to mutual information to be visualized. This technique is demonstrated using geomagnetic index data; we show that the AU and AL geomagnetic indices share half their information, and find the timescale on which mutual features appear.

Distinguishing the noise and attractor strength of coordinated limb movements using recurrence analysis

“…The down-sampled time series of both the right and left wrists were converted to unit interval ranges to eliminate any differences in scale (Shockley 2002). To obtain meaningful quantifications of %REC and L max it is important to correctly determine the CRQA input parameters of time delay, embedding dimension, and the threshold radius r (Gao and Hauqing 2000;March et al 2005;Zbilut et al 2000Zbilut et al , 2002.…”

Modelling game sports as complex systems – application of recurrence analysis to golf and soccer

“…However, there are discussions on whether an embedding is necessary for the RP construction [35,39]. In such cases the RP construction is conducted with tau = 1 and m = 1.…”

“…On the one hand, Iwanski and Bradley [35] criticize that embedding can lead to correlations which are often hard to understand. They argue that unembedded RPs contain the same information as embedded ones [35,39]. On the other hand, in this soccer application 20×2 matrices were used to describe the system's states instead of one-dimensional measurements as in golf.…”

Proposal of Two Measures of Complexity Based on Lempel-Ziv for Dynamic Systems: An Application for Manufacturing Systems

“…According to [2] the flexible and automatized manufacture systems under certain operation conditions show conditions of chaotic behavior. In [24,25] the relation between chaos and the manufacturing systems is studied. For Fuertes et al [26] the chaotic behavior is a form of complex behavior; nevertheless the use of chaos behavior to characterize the dynamic complexity is restricted to systems with chaotic behavior.…”

“…Donner et al [24] propose the analysis of Recurrence Quantification Analysis (RQA) to characterized the dynamic of a manufacturing network. March et al [25] have studied the relation between the dimension of inlay and the measures that gives the RQA for some systems of low dimension which are not modified regarding the inlay dimension. However, Hasson et al [27] present the relation of the sensitivity of the RQA variables to the inlay dimension.…”