Power Law frequency statistics.

The Scaling or Power Law (PL) distributions play an important role in describing

non-linear complex systems. They are often used to describe many natural and social

phenomena and specifically to analyse both the intensity and the intervals between

environmental disturbances. Rainfall in Bermuda has been also study in this work under

PL scaling concept which is related to the scale-invariance of the time series distribution.

A time series present scaling behaviour if its parameters are similarly related over a wide

range of sizes or scales exhibiting the same statistical characteristics. In this case the

intensity of precipitation amounts are taking into account. The Scaling, Power Law o

Pareto distributions play an important role in describing non-linear complex systems.

Scaling or Power Law relationships arise commonly as probability or frequency-

size distributions (in this study probability of reaching certain thresholds of rainfall

intensity or observing certain daily accumulated rainfall (mmd-1_{)), and are characterized}

by the form

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Where C is a constant and the value f(x) is proportional to some power of the input x

(accumulated daily rain) (White et al. 2008). In this case accumulated daily rain. The

logarithmic transformation of this function becomes a line and the slope of the resultant

straight line gives an estimation of the scale exponent α (White et al. 2008). This

behaviour occurs for values of the variable x higher than a given threshold that are in the

tail of the distribution. The presence of a cross over or break point in a PL suggests the

existence of possible critical phenomena associated with transition phases (McGarry et

al. 2002, Sornette 2006, Scheffer et al. 2009) where environmental properties of a certain

phenomenon are probably changing rapidly (Olsson et al. 1993). The change in some

parameter can modify the properties of the whole system.

Scaling framework allows the comparison of analogous phenomena and the

characterization of regions over a similar environment. A time series present scaling

behaviour if its parameters are similarly related over a wide range of sizes or scales

exhibiting the same statistical characteristics at any scale and are not associated with a

particular one. Thus, the presence of scaling invariance in a given process, is a

characteristic associated with PL distributions (Sornette 2004, Newman et al. 2006). Such

process is scale free, considering scale as the spatial and temporal dimension of the

phenomenon. In other words, a change in scale does not alter the statistical behaviour of

the system. Furthermore, systems that behaves as PL evolve far from equilibrium and are

frequently high dissipative.

Several methods have been proposed to prove this PL behaviour mainly at the end

of the tails distributions and when, as in this study, the length of the time series is not

enough long. Due to its heavy-tail, the PL distribution suggests that extremely large

values occur at higher frequency than in other distributions, such as the normal or

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from large, extreme events (Stumpf & Porter 2012). Therefore, a PL at the end, long tail

distribution, indicates that the frequencies or probabilities to find extreme phenomena or

rare events, far away from average values are higher than those ones obtained from the

classic statistics. A Power Law behaviour also suggests the presence of a non-linear

complex system (Savaglio & Carbone 2000) where different processes occur

simultaneously, showing several behaviour degrees. The whole behaviour of the system

is function of all the elements which conform it and have a strong non-linear relationship

(Goodwin 1994, Amaral & Ottino 2004).

In recent years it has been shown that a wide number of nature phenomena follow

PL distributions (Schroeder 1991, Newman 2005). Even in extreme natural hazards e.g.

earthquakes (Mega et al. 2003), floods (Mega et al. 2003, Malamud & Turcotte 2006),

landslides (Li et al. 2011) or forest fires (Weiguo et al. 2006). Many atmospheric

variables, like rainfall, also follows a power law distribution, at least in the tails of the

distributions. The presence of power laws has also been suggested as the fingerprint of

systems that show self-organized criticality (SOC) (Bak 1996). A P.L behaviour in the

empirical data values or in the time interval between them may show the existence of rare

underlying mechanisms or processes like feedback loops, random network, self-

organization or phase transitions (West et al. 1997, Barabási & Albert 1999, Newman

2005, Newman et al. 2006).

One important limitation of this tool related to the occurrence of the power–law

behaviour at the tail of the distribution (Stumpf & Porter 2012) where increases the

uncertainty on the exponent value estimation. On the other hand, other distributions that

sometimes offer a best data fit than scaling laws are the lognormal (Mitzenmacher 2004),

the stretched exponential (Laherrere & Sornette 1998) and other truncated PL (Burroughs

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the analysis. This is quite consistent with much of the literature on scaling in ecologic,

geophysics or economics systems. Furthermore, recently scaling laws have been shown

to be present over a large range of scales (Virkar & Clauset 2012, Clauset 2009). In spite

of the fact that model performance regarding to prediction has improved significantly

over the last few years, the complexity of some phenomena, extraordinarily non-linear

ones. This makes the understanding of the underlying dynamics of such events much

more difficult.

In this study a histogram of the daily accumulated rainfall and a log-log plot were

performed on the numerical series obtained from differences between daily rainfall

intensity.