Energy-efficient recurrence quantification analysis

TL;DR

The paper tackles the energy burden of recurrence quantification analysis (RQA), whose standard RP-based implementation scales as $O(N^2)$ and is computationally expensive. It introduces two methods to avoid RP construction: direct histogram estimation from time series (RQA_woRP) and a sampling-based estimator (RQA_Samp) that uses $M$ random line structures to approximate the line-length distribution $P(\ell)$. These approaches enable computing RQA measures, such as $DET$, with substantial reductions in runtime, memory usage, and energy consumption while maintaining accuracy; benchmarks on the Rössler system demonstrate significant speedups and controllable error by adjusting $M$. The work broadens the practicality of recurrence-based methods, offering energy-efficient, scalable options for large-scale data analysis and machine-learning pipelines, and extends naturally to vertical-line and network-based recurrence measures.

Abstract

Recurrence quantification analysis (RQA) is a widely used tool for studying complex dynamical systems, but its standard implementation requires computationally expensive calculations of recurrence plots (RPs) and line length histograms. This study introduces strategies to compute RQA measures directly from time series or phase space vectors, avoiding the need to construct RPs. The calculations can be further accelerated and optimised by applying a random sampling procedure, in which only a subset of line structures is evaluated. These modifications result in shorter run times, less memory use and access, and lower overall energy consumption during analysis while maintaining accuracy. This makes them especially appealing for large-scale data analysis and machine learning applications. The ideas are not limited to diagonal line measures, but can likewise be applied to vertical line-based measures and to recurrence network measures. By lowering computational costs, the proposed strategies contribute to energy saving and sustainable data analysis, and broaden the applicability of recurrence-based methods in modern research contexts.

Figures (7)

• Figure 1: (A) Time series sequence and (B) corresponding recurrence plot. The sequence at time points 2 to 5 (orange) repeats four times within a small error $\varepsilon$ (here $\varepsilon = 0.25$) during the interval 8 to 11, forming a diagonal line with points $(i=2,j=8)$, $(i+1=3,j+1=9)$, $(i+2=4,j+2=10)$, and $(i+3=5,j+3=11)$ in the RP (orange points). The line is preceded by condition Eq. (\ref{['eq_start_condition']}), $\|\mathbf{x}(i-1) - \mathbf{x}(j-1)\| > \varepsilon$, and followed by condition Eq. (\ref{['eq_end_condition']}), $\|\mathbf{x}(i+\ell) - \mathbf{x}(j+\ell)\| > \varepsilon$ , (grey boxes in panel (B)), ensuring the lines start and end points. The length of the diagonal line ($\ell = 4$) can be measured in the RP or directly from the time series. The state at time $3$ recurs (within the $\varepsilon$ uncertainty, grey horizontal bar in panel (A)) at time $9$ and $12$ (blue dots), indicated by the points in the column $3$ (marked by blue box).
• Figure 2: Estimation error $DET_\text{sampled} - DET_\text{true}$ derived from sampled line length histogram for the Rössler system with 25,000 data points and for sampling size ranging from $M=10$ to $100,000$. The orange region indicates the error range of $\pm 10^{-4}$. A random sampling of only 500 line segments already results in minor errors smaller than $10^{-4}$.
• Figure 3: Computation times for the RQA implementations using RP (RQA_RP), without RP (RP_woRP), with a sampling scheme (RP_Samp), and using a recurrence microstates approximation for increasing data length $N$ (Rössler system, same parameters as in Fig. \ref{['fig_samplesize']}). The sampling size $M$ for RP_Samp and microstates is $M=4N = 100,000$, and for RP_Samp$^2$ it is $M=0.2N = 5,000$.
• Figure A1: Estimation error of DET derived from recurrence microstates approximation for the Rössler system with 25,000 data points and sampling size ranging from $M=10$ to $100,000$. The orange region indicates the error range of $\pm 10^{-4}$. To achieve estimation errors below $10^{-4}$, the sampling size must be at least $40,000$. Up to $M=5,000$, the estimated DET remains at $1.00$, resulting in the constant deviation.
• Figure A2: Estimation error $DET_\text{sampled} - DET_\text{true}$ for DET estimations using RQA_Samp, RQA_Samp$^2$, and recurrence microstates approximation for the Rössler system for increasing data length $N$. The sampling was $M=100,000$ for RQA_Samp and microstates RQA, and $M=5,000$ for RQA_Samp$^2$. Even for a very small sampling number, the RQA_Samp approach has small estimation errors $<10^{-4}$.