Detection of dynamical regime transitions with lacunarity as a multiscale recurrence quantification measure

Abstract

We propose lacunarity as a novel recurrence quantification measure and illustrate its efficacy to detect dynamical regime transitions which are exhibited by many complex real-world systems. We carry out a recurrence plot-based analysis for different paradigmatic systems and nonlinear empirical data in order to demonstrate the ability of our method to detect dynamical transitions ranging across different temporal scales. It succeeds to distinguish states of varying dynamical complexity in the presence of noise and non-stationarity, even when the time series is of short length. In contrast to traditional recurrence quantifiers, no specification of minimal line lengths is required and geometric features beyond linear structures in the recurrence plot can be accounted for. This makes lacunarity more broadly applicable as a recurrence quantification measure. Lacunarity is usually interpreted as a measure of heterogeneity or translational invariance of an arbitrary spatial pattern. In application to recurrence plots, it quantifies the degree of heterogeneity in the temporal recurrence patterns at all relevant time scales. We demonstrate the potential of the proposed method when applied to empirical data, namely time series of acoustic pressure fluctuations from a turbulent combustor. Recurrence lacunarity captures both the rich variability in dynamical complexity of acoustic pressure fluctuations and shifting time scales encoded in the recurrence plots. Furthermore, it contributes to a better distinction between stable operation and near blowout states of combustors.

Trends in recurrence analysis of dynamical systems

“…Although the quantification of RPs has its roots in the early 1990s, there are still some aspects that require innovative ideas for quantifying the apparently different visual impression of RPs. Inspired by the research on fractal geometries, the lacunarity measure was adopted to RPs [90]. It characterises the homogeneity of the RP and allows to detect characteristic time scales, such as periodicities or extended laminar regimes (Fig.…”

“…2) and multifractal Gaussian noise), as well as a RP with characteristic temporal scales (Rössler system). Technical details can be found in [90] In data analysis, it can be important to identify the times of a specific dynamical behaviour. This corresponds to a segmentation of the phase space.…”

Optimal state space reconstruction via Monte Carlo decision tree search

“…with • a norm, ε a recurrence threshold, and Θ the Heaviside function. There are numerous ideas of how to quantify a RP [84,85]. Some statistics are based on the distribution of recurrence points, some on the diagonal line structures, some on the vertical structures, and it is also possible to use complex-network measures, when interpreting R (subtracting the main diagonal) as an adjacency matrix A = R − 1 of a recurrence network (RN) [86].…”

Optimal State Space Reconstruction via Monte Carlo Decision Tree Search

“…with • a norm, ε a recurrence threshold, and Θ the Heaviside function. There are numerous ideas of how to quantify a RP [67,7]. Some statistics are based on the distribution of recurrence points, some on the diagonal line structures, some on the vertical structures, and it is also possible to use complex-network measures, when interpreting R (subtracting the main diagonal) as a an adjacency matrix A = R − ✶ of a recurrence network (RN) [99].…”