Analysis of crackling noise using the maximum-likelihood method: Power-law mixing and exponential damping

Crackling noise can be initiated by competing or coexisting mechanisms. These mechanisms can combine to generate an approximate scale invariant distribution that contains two or more contributions. The overall distribution function can be analyzed, to a good approximation, using maximum-likelihood methods and assuming that it follows a power law although with nonuniversal exponents depending on a varying lower cutoff. We propose that such distributions are rather common and originate from a simple superposition of crackling noise distributions or exponential damping.

Figure 1

(a) Histogram corresponding to a set of $N=5000$ data points generated according to a power law with an exponent $\alpha =2$, normalized above a minimum value $E_{min}=0.1$ a.u. ($-20$ dB). (b) ML estimation of the exponent as a function of the varying cutoff $E_0$. The dashed line corresponds to the exact exponent $\epsilon =\alpha =2$ and the continuum lines to the estimation of the error bar from Eq. (12).

Figure 2

(a) Histogram corresponding to a set of $N=50000$ values generated according to a mixture of power laws. 40% of the data correspond to an exponent $\alpha =1.5$ and 60% of the data to an exponent $\beta =2$. The two power-law distributions are normalized above a minimum value $E_{min}=0.1$ a.u. ($-20$ dB). (b) ML estimation of the exponent as a function of the varying cutoff $E_0$. The dashed line corresponds to the estimation from Eq. (17). The two horizontal lines indicate the exponents of the mixture.

Figure 3

(a) Histogram corresponding to a set of $N=5000$ values generated according to a mixture of power laws. 30% of the data correspond to an exponent $\alpha =1.5$ and 70% of the data to an exponent $\beta =2$. The two power-law distributions are normalized above a minimum value $E_{min}=0.1$ a.u. ($-20$ dB). (b) ML estimation of the exponent as a function of the cutoff $E_0$. The dashed line corresponds to the estimation from Eq. (17). The two horizontal lines indicate the exponents of the mixture.

Figure 4

(a) Histogram corresponding to a set of $N=5000$ data generated according to exponentially damped power laws with $\alpha =2.0$ and three different values of the exponentially damping scale as indicated by the legend. The power-law distributions are normalized above a minimum value $E_{min}=0.1$ a.u. ($-20$ dB). (b) ML estimation of the exponent as a function of the varying cutoff $E_0$. The dashed lines correspond to the estimations from Eq. (25).

Figure 5

(a) Histogram corresponding to a set of $N=5000$ data generated according to an exponentially predamped power law with $\alpha =2.0$ and three different values of the cutoff as indicated by the legend. The power-law distributions are normalized above a minimum value $E_{min}=0.1$ a.u. ($-20$ dB). (b) ML estimation of the exponent as a function of the cutoff $E_0$. The dashed lines correspond to the theoretical estimations from Eq. (29).

Figure 6

Summary of the behaviors of the ML fitted exponent as a function of the varying cutoff $E_0$ studied in this paper: the continuum line corresponds to a pure power law. The dashed line corresponds to a mixture of power laws, the dotted line corresponds to an exponentially damped power law, and the dotted-dashed line to a predamped power law.

Figure 7

(a) Histogram corresponding to a set of $N=58036$ AE signals. (b) ML estimation of the exponent as a function of the cutoff $E_0$. In this experimental case, energy units are aJ. The horizontal dashed lines correspond to the exponents of the two components of the mixture as proposed in Ref. [12].