Scaling in the Timing of Extreme Events

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.Comment: Submitted to a Chaos, Solitons and Fractals special issue on Extreme Event

Fitting and goodness-of-fit test of non-truncated and truncated power-law distributions

Power-law distributions contain precious information about a large variety of processes in geoscience and elsewhere. Although there are sound theoretical grounds for these distributions, the empirical evidence in favor of power laws has been traditionally weak. Recently, Clauset et al. have proposed a systematic method to find over which range (if any) a certain distribution behaves as a power law. However, their method has been found to fail, in the sense that true (simulated) power-law tails are not recognized as such in some instances, and then the power-law hypothesis is rejected. Moreover, the method does not work well when extended to power-law distributions with an upper truncation. We explain in detail a similar but alternative procedure, valid for truncated as well as for non-truncated power-law distributions, based in maximum likelihood estimation, the Kolmogorov-Smirnov goodness-of-fit test, and Monte Carlo simulations. An overview of the main concepts as well as a recipe for their practical implementation is provided. The performance of our method is put to test on several empirical data which were previously analyzed with less systematic approaches. The databases presented here include the half-lives of the radionuclides, the seismic moment of earthquakes in the whole world and in Southern California, a proxy for the energy dissipated by tropical cyclones elsewhere, the area burned by forest fires in Italy, and the waiting times calculated over different spatial subdivisions of Southern California. We find the functioning of the method very satisfactory.Comment: 26 pages, 9 figure

Variability of North Atlantic hurricanes: seasonal versus individual-event features

Tropical cyclones are affected by a large number of climatic factors, which translates into complex patterns of occurrence. The variability of annual metrics of tropical-cyclone activity has been intensively studied, in particular since the sudden activation of the N Atl in the mid 1990's. We provide first a swift overview on previous work by diverse authors about these annual metrics for the NAtl basin, where the natural variability of the phenomenon, the existence of trends, the drawbacks of the records, and the influence of global warming have been the subject of interesting debates. Next, we present an alternative approach that does not focus on seasonal features but on the characteristics of single events [Corral et al Nature Phys 6, 693, 2010]. It is argued that the individual-storm power dissipation index (PDI) constitutes a natural way to describe each event, and further, that the PDI statistics yields a robust law for the occurrence of tropical cyclones in terms of a power law. In this context, methods of fitting these distributions are discussed. As an important extension to this work we introduce a distribution function that models the whole range of the PDI density (excluding incompleteness effects at the smallest values), the gamma distribution, consisting in a power-law with an exponential decay at the tail. The characteristic scale of this decay, represented by the cutoff parameter, provides very valuable information on the finiteness size of the basin, via the largest values of the PDIs that the basin can sustain. We use the gamma fit to evaluate the influence of sea surface temperature (SST) on the occurrence of extreme PDI values, for which we find an increase around 50 % in the values of these basin-wide events for a 0.49 degC SST average difference. ...Comment: final version available soon in the 1st author's web, http://www.crm.cat/Researchers/acorral/Pages/PersonalInformation.asp

Log-log Convexity of Type-Token Growth in Zipf's Systems

It is traditionally assumed that Zipf's law implies the power-law growth of the number of different elements with the total number of elements in a system - the so-called Heaps' law. We show that a careful definition of Zipf's law leads to the violation of Heaps' law in random systems, and obtain alternative growth curves. These curves fulfill universal data collapses that only depend on the value of the Zipf's exponent. We observe that real books behave very much in the same way as random systems, despite the presence of burstiness in word occurrence. We advance an explanation for this unexpected correspondence

Universal Earthquake-Occurrence Jumps, Correlations with Time, and Anomalous Diffusion

Spatiotemporal properties of seismicity are investigated for a worldwide (WW) catalog and for Southern California in the stationary case (SC), showing a nearly universal scaling behavior. Distributions of distances between consecutive earthquakes (jumps) are magnitude independent and show two power-law regimes, separated by jump values about 200 km (WW) and 15 km (SC). Distributions of waiting times conditioned to the value of jumps show that both variables are correlated in general, but turn out to be independent when only short or long jumps are considered. Finally, diffusion profiles reflect the shape of the jump distribution.Comment: Short pape

Testing Universality in Critical Exponents: the Case of Rainfall

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 test lead to the rejection of the universality hypothesis, despite the fact that the exponents are rather close to each other