Multidimensional Recurrence Quantification Analysis (MdRQA) for the Analysis of Multidimensional Time-Series: A Software Implementation in MATLAB and Its Application to Group-Level Data in Joint Action

Abstract

We introduce Multidimensional Recurrence Quantification Analysis (MdRQA) as a tool to analyze multidimensional time-series data. We show how MdRQA can be used to capture the dynamics of high-dimensional signals, and how MdRQA can be used to assess coupling between two or more variables. In particular, we describe applications of the method in research on joint and collective action, as it provides a coherent analysis framework to systematically investigate dynamics at different group levels-from individual dynamics, to dyadic dynamics, up to global group-level of arbitrary size. The Appendix in Supplementary Material contains a software implementation in MATLAB to calculate MdRQA measures.

Keywords: MATLAB; MdRQA; Multidimensional Recurrence Quantification Analysis; correlation; dynamics; joint action; multidimensional time-series.

Figure 1

Illustration of phase-space reconstruction and resulting RP using a noisy sine-wave. A noisy sine-wave in the upper panel of (A) and its time-shifted copy (surrogate) in the lower panel of (A). Reconstructed phase-space portrait (B), obtained by plotting the original sine-wave ${\tilde{\text{V}}}_1$ against its time-delayed copy ${\tilde{\text{V}}}_2$. Resulting RP (C), where diagonal lines of recurrences (black dots on the plot) indicate that the sine-wave signal repeats itself at intervals of roughly 60 data points. The speckled appearance of the diagonal lines indicates that repetitions are not perfect (i.e., the presence of noise).

Figure 2

Illustration of the effect of the threshold parameter on the percentage of recurrence points in an RP. The upper panels (A–C) show the same phase-spaces as in Figure 1, but with an application of increasingly larger threshold, within which points in phase-space are counted as being recurrent, illustrated by a gray circle. The lower panels (A–C) show that the corresponding RP yield increasingly higher percentages of recurrence points, evident by the increasing thickness of the diagonal line patterns on the plots.

Figure 3

The time series for x (A), y (B), and z (C) obtained by numerical integration in the solution interval. The Lorenz attractor (G) is the phase space plot of x, y, and z shown in z -scored dimensions. The reconstructed attractor based on time delayed embedding of x, y, and z respectively, with D = 3 and τ = 4, is shown in (D–F) (also in z -scored dimensions). Finally recurrence plots, using a threshold T = 0.1 to define a recurrence, are shown for the reconstructed attractors, based on x (H), y (I), and z (J); as well as based on the original attractor (K) with a threshold T = 0.08.

Figure 4

Temporal dynamics (A) of a system of two (in red and blue) coupled van der Pol oscillators with ϵ₁ = 0.01. CRP (B) and MdRP (C) for the time series shown in (A) with D = 2, τ = 1, and T = 0.01 for both CRP and MdRP. (D–F) show the same, but for ϵ₁ = 0.02. In both cases ϵ₂ = 5ϵ₁.

Figure 5

Recurrence measures RR, DET, ADL, and LDL for CRQA (dashed lines) and MdRQA (solid lines) as a function of the coupling constant ϵ₁.

Figure 6

Multivariate JPR obtained by joining the individual RPs from Figures 3H–J (A). MdRP from Figure 3K (B). The plots convey a similar qualitative picture of the dynamics of the Lorenz system, with the main difference that the JRP has fewer points and fewer diagonal structures than the MdRP.

Figure 7

R² of a simple regression model using RR, DET, ADL, and LDL as predictors for (A) the number of successfully built origami boats and (B) the number of unsuccessful attempts during each of the five trials for average individual dynamics (MdRQA1), average dyadic dynamics (MdRQA2), and group-level dynamics (MdRQA3). As all regression models had the same number of degrees of freedom (predictor DF = 4, residual DF = 95), a significant model at α = 0.05 had to explain at least 9.6% of variance (R² = 0.096), i.e., all models with R² > 0.096 are significant at p < 0.05.

Figure 8

Scaling of average phase-space distance with phase-space dimensionality (each dimension is a z-scored random variable taken from a uniform distribution [0, 1]). (A) Shows the increase of average distance as a function of separately added dimensions, and (B) shows that the increase in average distance follows the square-root of the dimensionality of the phase-space. (C) Shows the increase of average distance as a function of separately number of embeddings via time-delayed surrogates of a single random variable, and (D) shows that the increase in average distance follows the square-root of phase-space dimensionality as well. Distances in both cases scale similarly, with $L_D={\left(L_{D+n}^2-2n\right)}^{1/2}$.