Analyzing Multivariate Dynamics Using Cross-Recurrence Quantification Analysis (CRQA), Diagonal-Cross-Recurrence Profiles (DCRP), and Multidimensional Recurrence Quantification Analysis (MdRQA) - A Tutorial in R

Abstract

This paper provides a practical, hands-on introduction to cross-recurrence quantification analysis (CRQA), diagonal cross-recurrence profiles (DCRP), and multidimensional recurrence quantification analysis (MdRQA) in R. These methods have enjoyed increasing popularity in the cognitive and social sciences since a recognition that many behavioral and neurophysiological processes are intrinsically time dependent and reliant on environmental and social context has emerged. Recurrence-based methods are particularly suited for time-series that are non-stationary or have complicated dynamics, such as longer recordings of continuous physiological or movement data, but are also useful in the case of time-series of symbolic data, as in the case of text/verbal transcriptions or categorically coded behaviors. In the past, they have been used to assess changes in the dynamics of, or coupling between physiological and behavioral measures, for example in joint action research to determine the co-evolution of the behavior between individuals in dyads or groups, or for assessing the strength of coupling/correlation between two or more time-series. In this paper, we provide readers with a conceptual introduction, followed by a step-by-step explanation on how the analyses are performed in R with a summary of the current best practices of their application.

Keywords: R; RQA; cross-recurrence quantification analysis; diagonal cross-recurrence profile; multidimensional recurrence quantification analysis; tutorial.

FIGURE 1

Illustration of recurrence of letters in the sequence “ABCDDABCDD” (A), and cross-recurrence of letters in the sequence “ABCDDABCDD” with “DDEFGABCDD” (B). The black squares in the matrices indicate the recurrence of a letter, and white spaces indicate the absence of recurrence. The distribution of recurrent points on the recurrence plot can be quantified to yield statistics of the repetitive patterns in a sequence.

FIGURE 2

Display of the data. 3D-plot (a.k.a. phase-space) of the Lorenz system (A); component time-series for the x, y, and z-axis of the Lorenz system (B–D).

FIGURE 3

Phase-space reconstruction and estimation of embedding parameters. Example of phase-space reconstruction via the method of time-delayed embedding using the x-dimension of the Lorenz attractor. To that end, the original series (A) is plotted against itself at a certain delay d (d = 9 in this case; see panel D and the main text). This is done m–1 times, where m–1 is the number of additional surrogate dimensions needed in order to arrive at the correct dimensionality of the source system. Then, the original time-series and its two surrogates are plotted against each other, resulting in a reconstructed phase-space (B), which is topologically isomorphic to the phase-space of the actual source system (see Figure 2A). The delay parameter d can be estimated using the average mutual information (AMI), where the first local minimum of that function provides a good estimate for d (C), and the embedding parameter m can be estimated using the false nearest-neighbor (FNN) function, where the first local minimum (or the point at which the function becomes stable) provides a good estimate for m (D).

FIGURE 4

Main steps of selecting and conducting the analyses. Please note that this is only a gross overview over the main steps – for specific problems, best practice, or tuning of parameters, please consult the respective sections on these steps in this article.

FIGURE 5

Schematic of embedding two time-series in a single phase-space (A), and charting cross-recurrence between coordinates of the two time-series in a cross-recurrence plot (B). Note that the indices on the x- and y-coordinates in panel B here do not display the values of the associated time-series, but the order in which these values appear in the time-series (i.e., the gray 10 is the 10th coordinate of the gray coordinate-series in the phase-space in panel A).

FIGURE 6

Doing recurrence analysis using cross-recurrence: Display of cross-recurrence plots resulting when the same time-series data (lorData$z) is added for both inputs as plotted directly from the output of the crqa()-function (A). However, recurrence plots are conventionally oriented so that time at lag0 runs along the main diagonal from the lower-left to the upper-right. Hence, the resulting cross-recurrence plot need to be rotated by 90° (B).

FIGURE 7

Display of cross-recurrence plots and resulting CRQA measures the time-series pairs lorData$x with lorData$y (A), lorData$x with lorData$z (B), and lorData$y with lorData$z (C).

FIGURE 8

Illustration of how to compute a diagonal recurrence profile from a cross-recurrence plot. The solid black line on the tilted cross-recurrence plot marks the LoS. The dotted black and red lines show the width of lags ±5 and ±10 around the LoS, respectively. The line graph in the back of the figure illustrates the summation of recurrence points for the lags – the diagonal cross-recurrent profile.

FIGURE 9

Recurrence profiles of the pairwise cross-recurrence analyses of the 1-dimensional x-y-z time-series from the Lorenz system. Plot of drcp_xy$profile (negative lags = y lagging behind) (A). Plot of drcp_xz$profile (negative lags = z lagging behind) (B). Plot of drcp_yz$profile (negative lags = z lagging behind) (C).

FIGURE 10

Recurrence profiles of the pairwise cross-recurrence analyses of the properly embedded time-series from the Lorenz system. Plot of the diagonal cross-recurrence profile of lorData$x and lorData$y, with the percentage of recurrence points plotted on the y-axis, and the number of lags around the LoS on the x-axis.

FIGURE 11

Phase-space reconstruction through 3-D embedding of the individual time-series of the Lorenz-system (A–C) and the phase-space portrait of the actual 3-dimensional Lorenz-system (D).

FIGURE 12

Recurrence plot and results of the MdRQA3 analysis for the time-series lorData$x, lorData$y, and lorData$z.

FIGURE 13

Recurrence plots and MdRQA2 results for the three dyads lorData$x, lorData$y (A), lorData$x, lorData$z (B) and lorData$y, lorData$z (C) of the group.