Point-occurrence self-similarity in crackling-noise systems and in other complex systems

Corral, √Ålvaro

doi:10.1088/1742-5468/2009/01/p01022

Cited by 16 publications

(19 citation statements)

References 67 publications

(111 reference statements)

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“…Intriguingly, a qualitatively similar scaling phenomenology is observed in renormalized renewal processes with diverging mean interval sizes [21]. The random deletion of points (that, together with a rescaling of time, constitutes the renormalization procedure) is analogous to the raising of a threshold.…”

Section: Summary and Discussionmentioning

confidence: 56%

The perils of thresholding

et al. 2015

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The thresholding of time series of activity or intensity is frequently used to define and differentiate events. This is either implicit, for example due to resolution limits, or explicit, in order to filter certain small scale physics from the supposed true asymptotic events. Thresholding the birth-death process, however, introduces a scaling region into the event size distribution, which is characterized by an exponent that is unrelated to the actual asymptote and is rather an artefact of thresholding. As a result, numerical fits of simulation data produce a range of exponents, with the true asymptote visible only in the tail of the distribution. This tail is increasingly difficult to sample as the threshold is increased. In the present case, the exponents and the spurious nature of the scaling region can be determined analytically, thus demonstrating the way in which thresholding conceals the true asymptote. The analysis also suggests a procedure for detecting the influence of the threshold by means of a data collapse involving the threshold-imposed scale.

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Section: Summary and Discussionmentioning

confidence: 56%

The perils of thresholding

et al. 2015

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“…This is obvious if one intuitively knows the properties of the Poisson process. But it is not only that the Poisson process is invariant under the transformation (7), the results also tells us that the Poisson process is the only marked renewal process invariant under such transformation [13,15]. And, even more, it is easy to show that it is also an attractor for any marked renewal process with waiting-time density with a finite mean (and whose Laplace transform exists), see Ref.…”

Section: A Linear Rescaling and Trivial Poissonian Fixed Pointmentioning

confidence: 99%

“…And, even more, it is easy to show that it is also an attractor for any marked renewal process with waiting-time density with a finite mean (and whose Laplace transform exists), see Ref. [13].…”

Section: A Linear Rescaling and Trivial Poissonian Fixed Pointmentioning

confidence: 99%

“…These scaling laws, Eqs. (9), (12), and (13), are very useful in practice, as they have been shown to show up in several natural hazards, as for instance, earthquakes [20]. However, the scaling function F is not provided by an exponential function alone; rather, it contains a decreasing power-law for small to intermediate times, with exponent ν < 1.…”

Section: A Linear Rescaling and Trivial Poissonian Fixed Pointmentioning

confidence: 99%

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Scaling in the timing of extreme events

Corral

2015

Chaos, Solitons & Fractals

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Extreme events can come either from point processes, when the size or energy of the events is above a certain threshold, or from time series, when the intensity of a signal surpasses a threshold value. We are particularly concerned by the time between these extreme events, called respectively waiting time and quiet time. If the thresholds are high enough it is possible to justify the existence of scaling laws for the probability distribution of the times as a function of the threshold value, although the scaling functions are different in each case. For point processes, in addition to the trivial Poisson process, one can obtain double-power-law distributions with no finite mean value. This is justified in the context of renormalization-group transformations, where such distributions arise as limiting distributions after iterations of the transformation. Clear connections with the generalized central limit theorem are established from here. The non-existence of finite moments leads to a semi-parametric scaling law in terms of the sample mean waiting time, in which the (usually unkown) scale parameter is eliminated but not the exponents. In the case of time series, scaling can arise by considering random-walk-like signals with absorbing boundaries, resulting in distributions with a power-law "bulk" and a faster decay for long times. For large thresholds the moments of the quiet-time distribution show a power-law dependence with the scale parameter, and isolation of the latter and of the exponents leads to a non-parametric scaling law in terms only of the moments of the distribution. Conclusions about the projections of changes in the occurrence of natural hazards lead to the necessity of distinguishing the behavior of the mean of the distribution with the behavior of the extreme events.

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“…where this equation is just the inversion of Eq. (8). If u is uniformly distributed between 0 and 1, this yields a power law defined between a i and b i with exponentτ i , but note that the concrete data values are different from the original ones, due to the reshuffling.…”

Section: Permutational Test Without a Common Power-law Rangementioning

confidence: 99%

Testing universality in critical exponents: The case of rainfall

2016

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One of the key clues to consider rainfall as a self-organized critical phenomenon is the existence of power-law distributions for rain-event sizes. We have studied the problem of universality in the exponents of these distributions by means of a suitable statistic whose distribution is inferred by several variations of a permutational test. In contrast to more common approaches, our procedure does not suffer from the difficulties of multiple testing and does not require the precise knowledge of the uncertainties associated to the power-law exponents. When applied to seven sites monitored by the Atmospheric Radiation Measurement Program the tests lead to the rejection of the universality hypothesis, despite the fact that the exponents are rather close to each other. We discuss the reasons of the rejection.

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Point-occurrence self-similarity in crackling-noise systems and in other complex systems

Cited by 16 publications

References 67 publications

The perils of thresholding

The perils of thresholding

Scaling in the timing of extreme events

Testing universality in critical exponents: The case of rainfall

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