Visualization of coupling in time series by order recurrence plots

We introduce a method to visualize dependencies between two time series by applying the concept of cross recurrence plots to the local ordinal structure. We derive a measure of the coupling strength which is robust against observational noise, nonlinear distortion of the amplitude, and low-frequency trends. Connections to the instantaneous phase and the determination of phase coupling of two coupled Rössler systems in the standard and funnel regimes are shown. An application to electroencephalogram data demonstrates that the method is robust with respect to artifacts.

Figure 1

Order patterns at embedding dimension $D=3$.

Figure 2

Left: Delay embedding of a sine function and decomposition of the phase space by order patterns at dimension $D=3$. Right: Same plot with viewing angle in direction of the main diagonal.

Figure 3

ORP on different time series. (a) Bivariate Gaussian noise, (b) periodic functions with time-varying amplitudes and offsets, but constant period, (c) periodic functions with decreasing and increasing period, respectively, and (d) onset of an epileptic seizure on EEG channels F7 and Fz. All plots with embedding dimension $D=3$. The embedding delays are appropriately chosen as discussed in Sec. 2C.

Figure 4

Same ORP as in Fig. 3d but now as a function of $t$ and $\tau$ and with a longer range in time.

Figure 5

Time delay embedding of the Rössler system with $\vartheta =1.9$ and its decomposition by order patterns equivalent to Fig. 2.

Figure 6

ORP of the $x$ components of two coupled Rössler systems with detuned frequencies ${\omega }_{1,2}=1\pm 0.015$ at different coupling strength $k$ and corresponding recurrence rate $\mathrm{rr}\left(\tau \right)$.

Figure 7

Coupling indices on two coupled Rössler systems. (a) ${\rho }_{\pi }$ and (b) ${\rho }_{\mathcal{H}}$ as a function of coupling strength $k$ and detuning $\Delta \omega$. Coupling indices are gray coded, where black corresponds to no coupling. Time series length $L=2000$.

Figure 8

Coupling indices on two coupled Rössler systems with observational noise ${\sigma }_x∕{\sigma }_{\text{noise}}=1∕3$. (a) ${\rho }_{\pi }$ and (b) ${\rho }_{\mathcal{H}}$ as a function of coupling strength $k$ and detuning $\Delta \omega$. Coupling indices are gray coded, where black corresponds to no coupling. Time series length $L=2000$.

Figure 9

Upper panel: Projection of the funnel system onto the $x\text{−}y$ plane. Lower panel: Time delay embedding with $D=3$ and $\vartheta =0.9$. Parameters are $\omega =1$, $b=0.1$, and $c=8.5$.

Figure 10

ORP of two coupled funnel attractors at different coupling strength $k$ and corresponding recurrence rate $\mathrm{rr}\left(\tau \right)$. Parameters are ${\omega }_{1,2}=1\pm 0.015$, $a=0.29$, $b=0.1$, and $c=8.5$.

Figure 11

Coupling of two EEG channels (a)–(c) during onset of an epileptic seizure. (d) ORP with $\vartheta =2.7\mathrm{ms}$ and (e) resulting coupling index ${\rho }_{\pi }$. (f) ${\rho }_{\mathcal{H}}$ and (g) ${\rho }_{\mathrm{c}}$. All coupling indices with $L=2\mathrm{s}$. (h) Spectrogram $\mid \mathcal{F}(t,\omega )\mid$ of C3, where black indicates high values.

Figure 12

Frequency distributions of coupling indices before the seizure $t<40\mathrm{s}$ (thin line) and during the seizure $42<t<58\mathrm{s}$ (thick line) from Fig. 11.