Identifying Ordinal Similarities at Different Temporal Scales

Abstract

This study implements the permutation Jensen-Shannon distance as a metric for discerning ordinal patterns and similarities across multiple temporal scales in time series data. Initially, we present a numerically controlled analysis to validate the multiscale capabilities of this method. Subsequently, we apply our methodology to a complex photonic system, showcasing its practical utility in a real-world scenario. Our findings suggest that this approach is a powerful tool for identifying the precise temporal scales at which two distinct time series exhibit ordinal similarity. Given its robustness, we anticipate that this method could be widely applicable across various scientific disciplines, offering a new lens through which to analyze time series data.

Keywords: Jensen–Shannon divergence; chaotic semiconductor laser; delayed optical feedback; multiscale analysis; ordinal patterns; ordinal similarity; permutation Jensen–Shannon distance; permutation entropy; symbolic analysis; time series.

Figure A1

Running time to estimate the multiscale PJSD as a function of the time series length L for different orders D. The mean and standard deviation (presented as error bars) of the running time, from 100 independent estimations of the PJSD between two Gaussian white noise numerical realizations with order D and lags ${\tau }_1$ and ${\tau }_2$ between 1 and 20 (400 combinations), are plotted.

Figure 1

(a) Time series of MG system. Red crosses indicate sampling of $t_b=18·{10}^2\Delta t$ and blue empty circles sampling of $t_c=24·{10}^2\Delta t$. (b) Example of sequence extracted with sampling $t_b$. (c) Example of sequence extracted with sampling $t_c$.

Figure 2

PJSD estimations in logarithmic base 10 scale from two numerical realizations of Mackey–Glass oscillator operating in chaotic regime (${\tau }_S=30$) at different sampling intervals $t_b=18·{10}^2\Delta t$ and $t_c=24·{10}^2\Delta t$. Order D increases from 3 to 6 (from upper left to lower right plots), and lags ${\tau }_1$ and ${\tau }_2$ vary from 1 to 20. Time series of length $L=\mathrm{12,500}$ data points are considered in analysis.

Figure 3

Ordinal pattern probabilities with $D=3$ for the two numerical realizations of the Mackey–Glass oscillator operating in a chaotic regime (${\tau }_S=30$) at different sampling intervals $t_b=18·{10}^2\Delta t$ and $t_c=24·{10}^2\Delta t$. Particular choices of the symbolization lags ${\tau }_1$ and ${\tau }_2$ associated with these two signals are considered in each subplot.

Figure 4

The same as in Figure 3 but with $D=4$.

Figure 5

Distribution of 1000 estimated PJSD values from shuffled realizations of original MG time series for different orders D when lags ${\tau }_1$ and ${\tau }_2$ are equal to 4 and 3, respectively. Red vertical dashed line indicates estimated PJSD value for the original MG signals.

Figure 6

The same as in Figure 5 but for lags ${\tau }_1=20$ and ${\tau }_2=20$.

Figure 7

Scheme of experimental setup to study feedback dynamics. LD: laser diode; Circ: optical circulator; PC: polarization controller; Att: optical attenuator; Spl: one-by-two intensity splitter with $\Re =0.95$ and $(1-\Re )=0.05$ splitting ratios; →: optical isolator; and PD: photodiode.

Figure 8

PJSD estimations in logarithmic base 10 scale from two numerical realizations of LK equations ($L={10}^5$ data points) at different time scales but with equivalent statistical properties. Order D increases from 3 to 6 (from upper left to lower right plots), and lags ${\tau }_1$ and ${\tau }_2$ vary from 1 to 20. Numerical laser time series are subsampled to $10/\Gamma \simeq$ 28 ps prior to analysis.

Figure 9

Ordinal pattern probabilities with $D=3$ for the two numerical simulations of the LK equations. Particular choices of the symbolization lags ${\tau }_1$ and ${\tau }_2$ associated with these two signals are considered in each subplot.

Figure 10

The same as in Figure 9 but with $D=4$.

Figure 11

PJSD estimations in logarithmic base 10 scale from two experimentally obtained signals ($L={10}^5$ data points) with dynamical behaviors similar to numerical counterparts. Order D increases from 3 to 6 (from upper left to lower right plots), and lags ${\tau }_1$ and ${\tau }_2$ vary from 1 to 20. Experimental laser time series are acquired with sampling interval of 25 ps for analysis.

Figure 12

Ordinal patterns probabilities with $D=3$ for the two experimental signals. Particular choices of the symbolization lags ${\tau }_1$ and ${\tau }_2$ associated with these two signals are considered in each subplot.

Figure 13

The same as in Figure 12 but with $D=4$.