Wavelet-Based Multiscale Intermittency Analysis: The Effect of Deformation

Abstract

Intermittency represents a certain form of heterogeneous behavior that has interest in diverse fields of application, particularly regarding the characterization of system dynamics and for risk assessment. Given its intrinsic location-scale-dependent nature, wavelets constitute a useful functional tool for technical analysis of intermittency. Deformation of the support may induce complex structural changes in a signal. In this paper, we study the effect of deformation on intermittency. Specifically, we analyze the interscale transfer of energy and its implications on different wavelet-based intermittency indicators, depending on whether the signal corresponds to a 'level'- or a 'flow'-type physical magnitude. Further, we evaluate the effect of deformation on the interscale distribution of energy in terms of generalized entropy and complexity measures. For illustration, various contrasting scenarios are considered based on simulation, as well as two segments corresponding to different regimes in a real seismic series before and after a significant earthquake.

Keywords: complexity; deformation; energy transfer; entropy; intermittency; wavelets.

Figure 1

From top to bottom: (a,b) simulated realizations of model (9), with (a) Cauchy and (b) Gaussian white noise; correspondingly in left and right columns, using Haar wavelet, (c,d) scalogram $W_x^2(a,b)$, (e,f) $LI{M}_x$ map, (g,h) threshold exceedance set for $LI{M}_x^2(a,b)>3$, and (i,j) $F_x$ curve.

Figure 2

From top to bottom: (a,b) simulated realizations of model (9), with (a) Cauchy and (b) Gaussian white noise; correspondingly in left and right columns, using Morlet wavelet, (c,d) scalogram $W_x^2(a,b)$, (e,f) $LI{M}_x$ map, (g,h) threshold exceedance set for $LI{M}_x^2(a,b)>3$, and (i,j) $F_x$ curve.

Figure 2

From top to bottom: (a,b) simulated realizations of model (9), with (a) Cauchy and (b) Gaussian white noise; correspondingly in left and right columns, using Morlet wavelet, (c,d) scalogram $W_x^2(a,b)$, (e,f) $LI{M}_x$ map, (g,h) threshold exceedance set for $LI{M}_x^2(a,b)>3$, and (i,j) $F_x$ curve.

Figure 3

Deformation $\Phi$ given by (10).

Figure 4

From top to bottom: (a) simulated signal realization x generated from model (9) with Cauchy white noise, and (b) its (‘level’-type) deformation $x[\Phi ]$; correspondingly in left and right columns, (c,d) scalogram $W^2(a,b)$, (e,f) $LIM$ map, (g,h) threshold exceedance set for $LI{M}^2(a,b)>3$, and (i,j) F curve.

Figure 5

From top to bottom: (a) simulated signal realization x generated from model (9) with Gaussian white noise, and (b) its (‘level’-type) deformation $x[\Phi ]$; correspondingly in left and right columns, (c,d) scalogram $W^2(a,b)$, (e,f) $LIM$ map, (g,h) threshold exceedance set for $LI{M}^2(a,b)>3$, and (i,j) F curve.

Figure 6

From top to bottom: (a) simulated signal realization x generated from the ARIMA(1,2,1) model obtained by double integration of model (9) with Cauchy noise, and (b) its (‘flow’-type) deformation $x\left[\tilde{\Phi }\right]$; correspondingly in left and right columns, (c,d) scalogram $W^2(a,b)$, (e,f) $LIM$ map, (g,h) threshold exceedance set for $LI{M}^2(a,b)>3$, and (i,j) F curve.

Figure 7

From top to bottom: (a) simulated signal realization x generated from the ARIMA(1,2,1) model obtained by double integration of model (9) with Gaussian noise, and (b) its (‘flow’-type) deformation $x\left[\tilde{\Phi }\right]$; correspondingly in left and right columns, (c,d) scalogram $W^2(a,b)$, (e,f) $LIM$ map, (g,h) threshold exceedance set for $LI{M}^2(a,b)>3$, and (i,j) F curve.

Figure 8

(a,b) Shannon entropy, (c,d) Rényi entropy of order $q=3$, and (e,f) Tsallis entropy of order $q=3$, displayed in blue color for original signal generated from model (9) with (a,c,e) Cauchy and (b,d,f) Gaussian white noise, and in red color for the corresponding (‘level’-type) deformation.

Figure 9

(a,b) Shannon entropy, (c,d) Rényi entropy of order $q=3$, and (e,f) Tsallis entropy of order $q=3$, displayed in blue color for original signal generated from the ARIMA(1,2,1) model obtained by double integration of model (9) with (a,c,e) Cauchy and (b,d,f) Gaussian white noise, and in red color for the corresponding (‘flow’-type) deformation.

Figure 10

Seismic signal of L’Aquila earthquake (6 April 2009).

Figure 11

From top to bottom: (a) segment 1, (b) segment 2 of seismic signal; correspondingly in left and right columns, (c,d) scalogram $W_x^2(a,b)$, (e,f) $LI{M}_x$ map, (g,h) threshold exceedance set for $LI{M}_x^2(a,b)>3$, and (i,j) $F_x$ curve.

Figure 12

From top to bottom: (a) segment 1 of seismic signal, and (b) its (‘level’-type) deformation; correspondingly in left and right columns, (c,d) scalogram $W_x^2(a,b)$, (e,f) $LI{M}_x$ map, (g,h) threshold exceedance set for $LI{M}_x^2(a,b)>3$, and (i,j) $F_x$ curve.

Figure 13

From top to bottom: (a) segment 2 of seismic signal, and (b) its (‘level’-type) deformation; correspondingly in left and right columns, (c,d) scalogram $W_x^2(a,b)$, (e,f) $LI{M}_x$ map, (g,h) threshold exceedance set for $LI{M}_x^2(a,b)>3$, and (i,j) $F_x$ curve.

Figure 14

(a,b) Shannon entropy, (c,d) Rényi entropy of order $q=3$, and (e,f) Tsallis entropy of order $q=3$, displayed in blue color for original seismic signal with (a,c,e) segment 1 and (b,d,f) segment 2, and in red color for the corresponding (‘level’-type) deformation.

Figure 15

Rényi-based generalized complexity under selected $(\alpha ,\beta )$ values, for (a,b) original seismic signal x, (a) segment 1 and (b) segment 2, and (c,d) for the corresponding (‘level’-type) deformation $x[\Phi ]$.

Figure 15

Rényi-based generalized complexity under selected $(\alpha ,\beta )$ values, for (a,b) original seismic signal x, (a) segment 1 and (b) segment 2, and (c,d) for the corresponding (‘level’-type) deformation $x[\Phi ]$.