1.3 Bursty phenomena and thresholds
1.3.2 Bursty phenomena
We now focus our attention on certain phenomena that we dub, for lack of a better term, bursty phenomena. These are characterized by a tendency to occur in “bursts” of activity, combining periods of inactivity or very weak activity with periods of high-intensity activity, and the fact that such periods can last from very short times to extremely long ones, with all intermediate scales of duration and intensity being possible. Perhaps a good example is the case of rainfall: In plain words, sometimes it rains only for 5 minutes, sometimes it rains for an hour, and sometimes it rains for the whole day. We also know, from our every-day experience, that the intensity of rain seems to vary as well: it can go from very light rain where we might not even need an umbrella, to extremely intense rain that causes floods and, in some extreme cases, has devastating consequences. Besides the example of rainfall (Andrade et al.,1998;Peters et al.,2002;Peters and Christensen,2002, 2006; Peters et al., 2010; Deluca and Corral, 2014), other examples of bursty phenomena in geosciences are earthquakes (Sornette and Sornette,1989; Davidsen and Kwiatek,2013;
Kagan, 2010; Lippiello et al.,2012), and hurricanes (Corral, 2010); and –at least at a qualitative level– also solar flares (Baiesi et al., 2006;Boffetta et al.,1999; Paczuski et al., 2005), volcanic eruptions (Grasso and Bachélery,1995), rock avalanches (Turcotte and Malamud,2004) and forest fires (Malamud et al.,2005;Corral et al.,2008).
To put this qualitative description in a sound quantitative framework, let us say that the signal a(t) corresponds to the rain rate in a given site, in units of e.g. millimeters per hour, mm/h. A typical plot of the rain rate over a long period of time is show in Figure 1.8. Notice that the signal a(t), the rain rate in this example, is defined for all times, even if it takes the value 0when it does not rain. Indeed, the bursty nature of rain makes it natural to think in terms of rain events rather than on a continuous, ongoing rain-rate time series: loosely speaking, a rain event starts “when it starts raining”, and ends “when it ends raining”. So it is all up to what is considered rain and what is not, which is actually a very delicate matter25. In any case, let us assume that we agree on how to define the start and the end of the rain events. Then it is easy to transform a rain rate time-series a(t) into a series of “rain events”, which we shall index by i= 1,2, . . ., and which are characterized by a series of start-times and end-times, say {t1, t2, . . .}and {t01, t02, . . .} respectively, so that thei-th rain event starts at time ti and ends at time t0i. Then typically two observables are defined for each rain event: the event duration τi and the event size si,
τi ≡t0i−ti; si ≡ Z t0i
ti
a(t)dt (1.38)
where the integration is generally substituted by a discrete sum if one deals with real datasets. The observable τ corresponds to the duration, in units of time, of one rain event, while scorresponds to the rain depth, i.e., the total volume of rain per unit of
25For instance, it might be difficult to distinguish very weak rain from moisture
1.3 Bursty phenomena and thresholds 27
Figure 1.8 A real example of a rain-rate time series, shown at three different scales. The x-axis, time, is measured in minutes, while the rain rate in they-axis is in units of mm/h. The dataset has a resolution of 1 minute. Missing points correspond to a recorded value of 0.
area. Now, assuming that we have collected rain data over a long period of time26, and that we have an agreement in how to define the start and end of the rain events, we can construct a very large collection of rain events and, therefore, of associated durations{τi} and sizes{si}. Ifsandτ are regarded as random variables, then{τi}and{si}constitute samples of sandτ, and we can estimate the probability distribution ofsand τ,
P(s)ds'Prob[s≤si < s+ds] (1.39) P(τ)dτ 'Prob[τ ≤τi < τ+dτ]. (1.40) In the examples of bursty phenomena mentioned above, it turns out that P(s) and P(τ) display scale-invariant properties, in the form of scaling laws as defined in §1.1.2. The fact that e.g.rainfall seems to display a scale-invariant distribution of event sizes, durations, or dry spells27, see Figure 1.9, is considered interesting in the general complex systems
26There are datasets that span a period of over 10 years, with a 1 minute resolution of rain rate
27This observable, which we have not defined, measures the timebetweensubsequent events.
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Figure 1.9 Data collapse of probability density functions of rain event sizes (left) and durations (right). Colors correspond to data from different stations in Catalonia (NE Spain), from the database maintained by the Agència Catalana de l’Aigua. Reproduced from Deluca and Corral (2014).
framework, but it is of particular interest for advocates of the theory of Self-Organized Criticality (SOC), see (Bak,1996;Jensen,1998;Pruessner,2012) and many others. In short, SOC tries to give a good explanation of how and why the “critical point”, a concept borrowed from the theory of critical phenomena, see §1.1.1, would be reached in other systems where some some analogies to critical phenomena have been drawn28.
In this Thesis, however, we will not analyze real-world dataset of the aforementioned phenomena, nor will we enter the discussion of the case of rainfall or any other phenomena being a real-world example of SOC, or any other theory, on the basis of the possible scale invariance of some observables. Our interest will be in the thresholding procedure, which we introduce in what follows, and which we will analyze at a purely theoretical level. The findings of (Font-Clos et al.,2015, §3.3) provide a “warning”, if one wishes, of the unexpected consequences that thresholding might have; in particular, the possibility that spurious exponents are measured due to the introduction of the threshold. This is discussed at depth in (Font-Clos et al.,2015, §3.3), but let us emphasize now how it relates with what we have explained so far: if e.g. rainfall is conjectured to be in some sense analogous to a critical phenomenon, for instance in the SOC-sense, then the scaling exponents are of capital importance, specially if one envisages to establish the existence universality classes as well. But we will see the threshold can, on the one hand, disturb the value of measured exponents in non-negligible ways; but it also is, on the other hand, an unavoidable step in the analysis of many real-world datasets.
28The issue is the following: while with ferromagnetic materials the critical temperatureTcis obviously not reached spontaneously,i.e., we must heat the material to its critical temperature; it seems that with other systems the “critical point” is reached spontaneously, that is, the systems somehow poses itself into a “critical state”. Obviously, this needs a sound explanation as to how it happens, and the theory of SOC gives one possibility for that.
1.3 Bursty phenomena and thresholds 29