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

Abstract

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.

Avalanche criticality during compression of porcine cortical bone of different ages

“…The power law hypothesis can be corroborated by evaluating the p-value of each estimation using a KolmogorovSmirnov test. In this evaluation the value of the ex- ponent fitted with the maximum likelihood method is forced, which yields an overestimation of the p-value [32]. Considering this bias, we propose to choose p-values of acceptance and rejection thresholds at 0.5 and 0.05, respectively.…”

Auditory Power-Law Activation Avalanches Exhibit a Fundamental Computational Ground State

“…3(b)]. In the figure, dotted lines of slope −1.5 indicate the regions across which a power-law test value p > 0.9 holds [31]. The power-law regime excludes input-specific effects to the left and finite size effects of the cochlea combined with stimulation specifics to the right.…”

Novel methods for predicting clinical outcome in neonates from clinical electroencephalography recordings

“…Another form of power-law distribution is the strictly-truncated power law, which is restricted to a range of data between both lower and upper cut-offs -a narrower range than the tail distributions fitted above (Deluca and Corral 2013). Verification of power-law fits were further tested via the use of strictly truncated power-laws fitted to burst distributions (Deluca and Corral, 2013).…”

“…Verification of power-law fits were further tested via the use of strictly truncated power-laws fitted to burst distributions (Deluca and Corral, 2013). These fits were used to identify self-consistent ranges of power-law scaling in areas and durations.…”

“…For each candidate range, the strictly truncated power-law was fitted using maximum likelihood estimates, and used to generate 50 synthetic datasets. Following the method of Deluca and Corral (2013), the fitted range that was the widest was chosen (maximizing the ratio of upper and lower cutoffs) that yielded a plausible fit, defined as requiring at least 20% of the synthetic datasets to deviate from the fit by at least as much as the data.…”