Recurrence Pattern Correlation (RPC)

• Recurrence Pattern Correlation (RPC) is a framework that quantifies geometric motif alignment in recurrence plots using weighted spatial statistics.
• RPC leverages adaptable motif selection and localized statistics to surpass traditional RQA, capturing transient and invariant dynamics.
• It has practical applications in detecting unstable periodic orbits in systems like the logistic map and Lorenz system, distinguishing chaotic from stable structures.

Recurrence Pattern Correlation (RPC) is a methodological and conceptual framework developed for quantifying the alignment between geometric patterns observed in recurrence plots (RPs) and motifs of arbitrary shape, scale, and lag. RPC extends the scope of Recurrence Quantification Analysis (RQA) beyond simple global statistics, such as diagonal or vertical line counting, allowing for the identification and measurement of localized structures that encode the fundamental dynamics and invariants of complex nonlinear systems (Marghoti et al., 15 Aug 2025). By leveraging weighted spatial statistics—specifically, adaptations of Moran’s I—RPC reveals detailed information on phase-space organization, periodic orbit skeletons, and coupling phenomena in dynamical systems.

1. Conceptual Foundations and Mathematical Definition

The central object analyzed by RPC is the recurrence plot, constructed from an N-point time series embedded in a metric space. The RP is the binary matrix $R_{ij}$ given by

$$R_{ij} = \Theta(\epsilon - \Vert x_i - x_j \Vert) ,$$

where $\Theta$ is the Heaviside function, $\epsilon$ is a positive recurrence threshold, and $x_i$ are reconstructed phase space states.

RPC generalizes the measurement of pattern correlations using a spatial weight matrix $w_{\Delta i, \Delta j}$, which parameterizes the structure of the desired motif (diagonal, vertical, horizontal, or arbitrary), resulting in the RPC quantifier

$$\mathrm{RPC} = \frac{1}{W} \sum_{i \ne j} \sum_{i' \ne j'} w_{\Delta i, \Delta j} \frac{ (R_{i, j} - rr)(R_{i', j'} - rr) }{ rr(1-rr) } ,$$

where $rr$ is the global recurrence rate, and $W$ is the normalization over nonzero weights (Marghoti et al., 15 Aug 2025).

A variant, local RPC ($\ell \mathrm{RPC}$), restricts the computation to neighborhoods, providing time-resolved or column-resolved assessments of pattern correlation.

2. Methodological Innovations

Unlike traditional RQA measures—which focus on counting lines of fixed orientation and length (e.g., determinism DET, laminarity LAM)—RPC allows selection of motifs via $w_{\Delta i, \Delta j}$, enabling the targeting of specific geometric structures:

• Vertical lines: $$w_{\Delta i, \Delta j} = \delta_{\Delta i, 0} \delta_{\Delta j, k}$$ for delay $k$.
• Diagonal lines: $$w_{\Delta i, \Delta j} = \delta_{|\Delta i|, |\Delta j|}$$.
• Arbitrary motifs: $w$ constructed to highlight spatial shapes relevant to particular phase-space trajectories.

RPC is invariant under translation in RP coordinates, handling both global and localized motifs, and its detection sensitivity scales with the sparseness and structure of the RP.

3. Key Applications in Dynamical Systems

RPC reveals system-specific structures often missed by global RQA metrics:

Logistic Map

RPC with vertical motif weight uncovers unstable periodic orbits (UPOs) within the bifurcation diagram, with peaks in $\ell \mathrm{RPC}$ corresponding to period windows and their associated return times.

Standard Map

Applying RPC to the Standard map, the method distinguishes chaotic regions from stable islands via motif-targeted weights. Positive RPC values signal strong motif alignment in stability windows; negative values denote anti-correlation in fully chaotic or mixing regimes.

Lorenz '63 System

In the Lorenz system, RPC tracks the durations and positions of UPOs in three-dimensional phase space. Vertical motif correlation produces distinctive peaks at specific lags, matching analytic orbit timings. Localized RPC visualizes phase-space geometry, distinguishing "single wing" from "double wing" trajectory segments with high fidelity (Marghoti et al., 15 Aug 2025).

Stochastic Processes

For Gaussian white noise and AR(1) models, RPC correctly yields near-zero or modest values, demonstrating low structure and baseline recurrence properties, contrasting sharply with deterministic nonlinear systems.

4. Advantages, Limitations, and Computational Considerations

Advantages

• Motif Flexibility: RPC quantifies correlation to any user-defined pattern.
• Local Sensitivity: $\ell \mathrm{RPC}$ captures nonstationary or transient pattern arrangements.
• Adaptive Behavior: Sensitivity is higher in sparse RPs; RPC can reveal anti-alignment in dense RPs, facilitating nuanced interpretation across different threshold regimes.

Limitations

• Computational Intensity: The motif-weighted double sum is expensive, particularly for large N or broad weight matrices. Practical use mandates weight windowing and parallelization.
• Parametric Sensitivity: Results depend on careful specification of $\epsilon$ and motif structure; misparameterization may confound dynamical signatures.
• Interpretation of Negative RPC: Negative values represent anti-alignment, requiring careful dynamical context (e.g., motif avoidance due to deterministic rules or phase-space exclusion).

5. Significance for Nonlinear Dynamics and Phase-Space Analysis

RPC enables reconstruction of organizing skeletons—such as unstable periodic manifolds or synchronization regimes—by correlating recurrence structures with explicit geometric patterns. In chaotic systems, RPC peaks mark intrinsic timescales and organize recurrent patterns relative to phase-space separation. The method bridges qualitative visual RP inspection and quantitative dynamical analysis, providing tools to dissect mixing, invariant manifolds, and other dynamical invariants.

A relevant implication is that RPC can generalize to bivariate or multivariate recurrence plots (e.g., Cross Recurrence Plots) and quantification of directional couplings, suggesting extensions for analyzing complex networks, synchrony, and causality in coupled nonlinear systems.

6. Intersection with Traditional RQA and Future Research Directions

Whereas RQA focuses on line statistics for determinism and transition detection, RPC exploits arbitrary patterns, enriching the palette of recurrence analysis. The motif-driven approach complements entropy, laminarity, and trapping-time measures, and is especially powerful when the geometry of the attractor or phase-space skeleton is of interest.

Future research will address algorithmic optimizations, motif learning, and extensions to multivariate and spatio-temporal RPs. Integrations with machine learning methodologies—such as pattern dictionary learning or motif-based clustering—promise to harness RPC for anomaly detection and classification tasks in high-dimensional and nonstationary temporal data.

7. Summary Table: RPC Versus Standard RQA Measures

Metric	Pattern Sensitivity	Localization	Typical Application
RQA (DET/LAM)	Fixed (lines only)	Global	Predictability, transitions
RPC (global)	Arbitrary (motif-based)	Global	UPO detection, phase-space
RPC (local)	Arbitrary	Local/column	Time-resolved structure

RPC establishes a quantifiable, motif-driven paradigm for analyzing recurrence structures in nonlinear dynamical systems, with demonstrated utility in uncovering intrinsic phase-space organization, invariant manifolds, and dynamical transitions (Marghoti et al., 15 Aug 2025).