Recurrence plot analysis of nonstationary data: The understanding of curved patterns

Recurrence plots of the calls of the Nomascus concolor (Western black crested gibbon) and Hylobates lar (White-handed gibbon) show characteristic circular, curved, and hyperbolic patterns superimposed to the main temporal scale of the signal. It is shown that these patterns are related to particular nonstationarities in the signal. Some of them can be reproduced by artificial signals like frequency modulated sinusoids and sinusoids with time divergent frequency. These modulations are too faint to be resolved by conventional time-frequency analysis with similar precision. Therefore, recurrence plots act as a magnifying glass for the detection of multiple temporal scales in slightly modulated signals. The detected phenomena in these acoustic signals can be explained in the biomechanical context by taking in account the role of the muscles controlling the vocal folds.

Figure 1

Analysis of the call of a Nomascus concolor gibbon. In the time domain the signal shows some amplitude modulation, while the spectrogram exhibits a line corresponding to one isolated almost constant frequency without pronounced harmonics. The inspection of the RP shows white and textured boxes. The white ones are due to amplitude variations in the signal, in textured ones the existence of two times scales is evident, one related to the carrier frequency, the other formed by hyperbolic macropatterns converging toward the LOI.

Figure 2

Analysis of the call of Hylobates lar. In the spectrogram a slight frequency modulation is visible. The strong harmonics are related to the non-sinusoidal nature of the signal which is artificially imposed by the cutoff of the amplitudes by the recording. In the RP two interesting patterns can be identified: curved and concentric gapped patterns appear periodically in the RP, while white circular regions are due to the frequency shift inside the signal.

Figure 3

Reproduction of the patterns observed in the vocalization of Nomascus concolor by the signal $\cos (2\pi 1000t+100\pi t^2)$. The RP was obtained using $(D_E=3,T=3,ϵ=0.08)$. From the inspection two different scales emerge: the lower is relative to the macrostructures which build the periodic hyperbolic patterns, the higher is relative to the aggregation of short parallel lines building the macrolines. The signal has a frequency shift of $40\mathrm{Hz}$.

Figure 4

A magnified zone of the RP of the Nomascus concolor gibbon. The parallel lines forming the macropatterns have a spatial period of 11 time samples, related to the pitch of the signal.

Figure 5

Reproduction of “gapped” patterns observed in the call of Hylobates lar by the signal $\cos \mathbf{(}2\pi 800t+10\sin \left(2\pi 8t\right)\mathbf{)}$. The RP was computed using $D_E=3$, $T=3$, and $ϵ=0.05$.