Recurrence Patterns Correlation

TL;DR

This paper introduces Recurrence Pattern Correlation (RPC), a Moran’s I–inspired metric for Recurrence Plots that targets localized, motif-like recurrence structures rather than global line counts. An additional local variant, ell_RPC, adapts to time-indexed neighborhoods to map geometric features directly onto state-space locations. RPC recovers traditional RQA signals for standard motifs and, critically, reveals the skeleton of nonlinear dynamics—such as unstable periodic orbits and their manifolds—in systems like the Logistic map, Standard map, and Lorenz '63. The framework provides a flexible, directionally aware tool for analyzing nonlinear recurrences across stochastic and chaotic regimes, with open-source implementation to enable broader adoption and motif inference.

Abstract

Recurrence plots (RPs) are powerful tools for visualizing time series dynamics; however, traditional Recurrence Quantification Analysis (RQA) often relies on global metrics, such as line counting, that can overlook system-specific, localized structures. To address this, we introduce Recurrence Pattern Correlation (RPC), a quantifier inspired by spatial statistics that bridges the gap between qualitative RP inspection and quantitative analysis. RPC is designed to measure the correlation degree of an RP to patterns of arbitrary shape and scale. By choosing patterns with specific time lags, we visualize the unstable manifolds of periodic orbits within the Logistic map bifurcation diagram, dissect the mixed phase space of the Standard map, and track the unstable periodic orbits of the Lorenz '63 system's 3-dimensional phase space. This framework reveals how long-range correlations in recurrence patterns encode the underlying properties of nonlinear dynamics and provides a more flexible tool to analyze pattern formation in recurrent dynamical systems.

Figures (7)

• Figure 1: Recurrence Patterns Correlation method key elements. From a (a) time series, we generate a (b) standard RP, and by choosing an appropriate (c) weight matrix, we can quantify to which structure the patterns within the RP are correlated. Black regions indicate entries $1$ while white regions indicate $0$. The gray index indicates the reference recurrence point at $\Delta i = \Delta j = 0$, which is not informative, thus set $\text{w}_{0,0}=0$.
• Figure 2: Dependence of RPC numerator terms on the recurrence rate ($rr$). For $rr < 0.5$, contributions from recurring combinations, related to $(1 - rr)^2$, dominate over those from non-recurring combinations, associated with $rr^2$. The opposite holds for $rr > 0.5$. Consequently, the RPC quantifier becomes more sensitive to structured recurrence patterns in sparse RPs and more sensitive to non-recurrences in denser RPs. This adaptive sensitivity enhances RPC’s ability to characterize varying recurrence regimes. The only negative contribution to the correlation measure is the association of recurrence with no recurrence.
• Figure 3: Recurrence Patterns Correlation (RPC) for paradigmatic dynamics and (a) representative recurrence motifs, and (b) as a function of the time series length. Here we consider a $1\%$ recurrence rate across all recurrence plots. In this scale, the recurring condition is highlighted over nonrecurring patterns. Each bar represents a specific weight matrix $\text{w}_{\Delta i, \Delta j}$, corresponding to: sides ($\text{w}_{\Delta i, \Delta j} = \delta_{\Delta i,0} \delta_{\Delta j,\pm 1} + \delta_{\Delta i, \pm 1} \delta_{\Delta j, 0}$), diagonals ($\text{w}_{\Delta i, \Delta j} = \delta_{\Delta i,\pm 1} \delta_{\Delta j,\pm 1}$), and anti-diagonals ($\text{w}_{\Delta i, \Delta j} = \delta_{\Delta i,\pm 1} \delta_{\Delta j,\mp 1}$). In the bottom panel, the solid line represents the mean over 20 trials, and the error bars indicate the corresponding standard deviation. In the top panel, we present results for time series with length $N=10000$.
• Figure 4: RPC for single structured time lags and representative dynamical systems as a function of the recurrence rate, we compare point-wise motifs on diagonal direction to DET (left) and point-wise motifs on horizontal direction to LAM (right). The analyzed systems are: (a-b) Gaussian white noise, (c-d) auto-regressive model with $\alpha = 0.8$, (e-f) auto-regressive model with $\alpha = 0.99$, (g-h) chaotic Logistic map, (i-j) chaotic Lorenz '63 system, and (k-l) periodic sine function.
• Figure 5: Logistic map bifurcation diagram along with $\ell\text{RPC}$. We consider the simple single recurrence pattern with lag $k$ along a vertical direction, $\text{w}_{\Delta i, \Delta j} = \delta_{\Delta i, 0} \delta_{\Delta j, k}$, emphasizing the return time to a phase space position. We detect UPO nearby to the trajectory, related to (a) period 1, (b) period 2, (c) period 3, and (d) period 4.