Multivariate Joint Recurrence Quantification Analysis: detecting coupling between time series of different dimensionalities

TL;DR

Multivariate Joint Recurrence Quantification Analysis (MvJRQA) extends MdRQA and JRQA to quantify coupling between time series that differ in dimensionality, without reducing multivariate signals to a common dimension. The method builds a recurrence plot for each system using MdRQA, forms a Joint Recurrence Plot by element-wise multiplication, and derives the Joint Recurrence Coupling Indicator (JRCI) from $JRR$ and $RR$ to quantify coupling strength, with theoretical baselines from the identical systems model and the random null model. The authors validate MvJRQA on four model systems spanning linear and nonlinear dynamics and apply it to EEG and 2D/3D eye-tracking data, demonstrating higher coupling when dimensionality differs more (3D eye movements) and offering guidance on choosing target recurrence rate $RR_T$ and tolerances for empirical data. They compare against MdRQA, show that MvJRQA remains sensitive across diverse dynamics, discuss limitations under extreme coupling, and provide practical recommendations and open-source tools for applying MvJRQA in neuroscience and other complex systems contexts.

Abstract

One challenge with the analysis of complex systems and the interaction between such systems is that they are composed of different numbers of components, or simply the fact that a different number of observables is available for each system. The challenge is how to analyze the interaction of two systems which are not described by the same number of variables. Here, we present multivariate joint recurrence quantification analysis (MvJRQA), a recurrence-based technique that allows to analyze coupling properties between multivariate datasets that differ in dimensionality (i.e., number of observables) and type of data (such as nominal or interval-scaled, for example). First, we introduce the methods, and test it on simulated data from linear and nonlinear systems. Then we apply it to an empirical dataset of EEG and eye tracking data. We introduce the joint recurrence coupling indicator (JRCI) as a measure to assess and compare coupling between systems. Finally, we discuss practical issues regarding the application of the method.

Figures (15)

• Figure 1: Conceptual overview of constructing a multidimensional joint recurrence plot. Two systems, shown in the columns 'System 1' and 'System 2,' have times series shown in the top row: Time series. For each system a columns shows a symbolic representation, while the other column shows a graphical example. The first step, Embedding, places the time series into a higher dimensional Phase space, where the rows in $\mathbf{X}$ and $\mathbf{Y}$ represent points in phase space. Then MdRQA is used to construct individual Recurrence plots that are combined into a Joint recurrence plot.
• Figure 2: Model systems and example time series. A linear system with one stochastic process coupled to two correlated stochastic processes via weight $k$ (A). A periodic signal driving a set of two coupled logistic maps via coupling parameter $\eta$ (B). The canonical Lorenz system driving a harmonic oscillator via a coupling parameter $c$ (C). The Lorenz-96 system with 5 dimensions driving a harmonic oscillator via a coupling parameter $\kappa$ (C). Colors indicate the driving (green) and driven (purple) systems.
• Figure 3: Overview of results for the model systems.Top row: model systems as in Figure \ref{['fig:model-timeseries']}. For each value of coupling strength, 100 independent models were run with different initial conditions. Middle row:$JRR/RR$ is seen to increase monotonically with coupling between systems for a range of fixed subsystem $RR$. Bottom row: Plots of JRCI for different values of the coupling show ability to correctly order systems by coupling strength with weakly interacting systems closer to the random null model and non-interacting systems consistent with the null model for low $RR$. Error bars indicate bootstrap estimates of 95% confidence intervals.
• Figure 4: MvJRQA compared to MdRQA for the model systems. For each value of coupling strength, 100 independent models were run with different initial conditions to generate a time series of length 500. The error bars indicate bootstrap estimates of 95% confidence intervals. For the linear system (A), both, MvJRQA and MdRQA show increased $JRR$ or $RR$ with increased coupling. However, MdRQA does not pick up coupling in $RR$ for the three nonlinear systems (B--D).
• Figure 5: Extreme coupling. As a method to detect coupling in sets of time series, MvJRQA is compromised in situations of extreme coupling. For models A, C, and D we see a leveling-off of $JRR/RR$ with extreme coupling values, which is somewhat less pronounced for the linear systems (A), particularly when fixed $RR$ is small. For the coupled logistic maps system (B), extreme coupling leads to a significant drop and renewed increase in $JRR/RR$ with increase of coupling strength to extreme values.