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

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Abstract

Recently, Clauset, Shalizi, and Newman 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. We find the functioning of the method very satisfactory.

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References

  • Aban, I.B., M.M. Meerschaert, and A.K. Panorska (2006), Parameter estimation for the truncated Pareto distribution, J. Am. Stat. Assoc. 101,473, 270–277, DOI: 10.1198/016214505000000411.

    Article  Google Scholar 

  • Aschwanden, M.J. (2013), SOC systems in astrophysics. In: M.J. Aschwanden (ed.), Self-Organized Criticality Systems, Open Academic Press, Berlin, 439–478.

    Google Scholar 

  • Baiesi, M., M. Paczuski, and A.L. Stella (2006), Intensity thresholds and the statistics of the temporal occurrence of solar flares, Phys. Rev. Lett. 96,5, 051103, DOI: 10.1103/PhysRevLett.96.051103.

    Article  Google Scholar 

  • Bak, P. (1996), How Nature Works: The Science of Self-Organized Criticality, Copernicus, New York.

    Book  Google Scholar 

  • Bak, P., K. Christensen, L. Danon, and T. Scanlon (2002), Unified scaling law for earthquakes, Phys. Rev. Lett. 88,17, 178501, DOI: 10.1103/PhysRevLett. 88.178501.

    Article  Google Scholar 

  • Barndorff-Nielsen, O. (1978), Information and Exponential Families in Statistical Theory, John Wiley & Sons Inc., New York, 238 pp.

    Google Scholar 

  • Baró, J., and E. Vives (2012), Analysis of power-law exponents by maximum-likelihood maps, Phys. Rev. E 85,6, 066121, DOI: 10.1103/PhysRevE.85.066121.

    Article  Google Scholar 

  • Bauke, H. (2007), Parameter estimation for power-law distributions by maximum likelihood methods, Eur. Phys. J. B 58,2, 167–173, DOI: 10.1140/epjb/e2007-00219-y.

    Article  Google Scholar 

  • Ben-Zion, Y. (2008), Collective behavior of earthquakes and faults: continuum-discrete transitions, progressive evolutionary changes, and different dynamic regimes, Rev. Geophys. 46,4, RG4006, DOI: 10.1029/2008RG000260.

    Article  Google Scholar 

  • Boffetta, G., V. Carbone, P. Giuliani, P. Veltri, and A. Vulpiani (1999), Power laws in solar flares: Self-organized criticality or turbulence? Phys. Rev. Lett. 83,22, 4662–4665, DOI: 10.1103/PhysRevLett.83.4662.

    Article  Google Scholar 

  • Boguñá, M., and A. Corral (1997), Long-tailed trapping times and Lévy flights in a self-organized critical granular system, Phys. Rev. Lett. 78,26, 4950–4953, DOI: 10.1103/PhysRevLett.78.4950.

    Article  Google Scholar 

  • Bouchaud, J.-P., and A. Georges (1990), Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. Rep. 195,4–5, 127–293, DOI: 10.1016/0370-1573(90)90099-N.

    Article  Google Scholar 

  • Burroughs, S.M., and S.F. Tebbens (2001), Upper-truncated power laws in natural systems, Pure Appl. Geophys. 158,4, 741–757, DOI: 10.1007/PL00001202.

    Article  Google Scholar 

  • Burroughs, S.M., and S.F. Tebbens (2005), Power-law scaling and probabilistic forecasting of tsunami runup heights, Pure Appl. Geophys. 162,2, 331–342, DOI: 10.1007/s00024-004-2603-5.

    Article  Google Scholar 

  • Carrillo-Menéndez, S., and A. Suárez (2012), Robust quantification of the exposure to operational risk: Bringing economic sense to economic capital, Comput. Oper. Res. 39,4, 792–804, DOI: 10.1016/j.cor.2010.10.001.

    Article  Google Scholar 

  • Casella, G., and R.L. Berger (2002), Statistical Inference, 2nd ed., Duxbury Advanced Series, Duxbury Thomson Learning, Pacific Grove, 660 pp.

    Google Scholar 

  • Chicheportiche, R., and J.-P. Bouchaud (2012), Weighted Kolmogorov-Smirnov test: Accounting for the tails, Phys. Rev. E 86,4, 041115, DOI: 10.1103/Phys-RevE.86.041115.

    Article  Google Scholar 

  • Christensen, K., and N.R. Moloney (2005), Complexity and Criticality, Imperial College Press Advanced Physics Texts, Vol. 1, Imperial College Press, London.

    Book  Google Scholar 

  • Chu, J.-H., C.R. Sampson, A.S. Levine, and E. Fukada (2002), The Joint Typhoon Warning Center tropical cyclone best-tracks, 1945–2000, Naval Research Laboratory, Tech. Rep. NRL/MR/7540-02-16.

    Google Scholar 

  • Chu, S.Y.F., L.P. Ekström, and R.B. Firestone (1999), The Lund/LBNL Nuclear Data Search, Version 2.

    Google Scholar 

  • Clauset, A., C.R. Shalizi, and M.E.J. Newman (2009), Power-law distributions in empirical data, SIAM Rev. 51,4, 661–703, DOI: 10.1137/070710111.

    Article  Google Scholar 

  • Corpo Forestale dello Stato (2012), http://www.corpoforestale.it.

  • Corral, A. (2003), Local distributions and rate fluctuations in a unified scaling law for earthquakes, Phys. Rev. E 68,3, 035102, DOI: 10.1103/PhysRevE.68.035102.

    Article  Google Scholar 

  • Corral, A. (2004a), Long-term clustering, scaling, and universality in the temporal occurrence of earthquakes, Phys. Rev. Lett. 92,10, 108501, DOI: 10.1103/Phys-RevLett.92.108501.

    Article  Google Scholar 

  • Corral, A. (2004b), Universal local versus unified global scaling laws in the statistics of seismicity, Physica A 340,4, 590–597, DOI: 10.1016/j.physa.2004.05.010.

    Article  Google Scholar 

  • Corral, A. (2005), Comment on “Do earthquakes exhibit self-organized criticality?”, Phys. Rev. Lett. 95,15, 159801, DOI: 10.1103/PhysRevLett.95.159801.

    Article  Google Scholar 

  • Corral, A. (2006), Universal earthquake-occurrence jumps, correlations with time, and anomalous diffusion, Phys. Rev. Lett. 97,17, 178501, DOI: 10.1103/Phys-RevLett.97.178501.

    Article  Google Scholar 

  • Corral, A. (2008), Scaling and universality in the dynamics of seismic occurrence and beyond. In: A. Carpinteri and G. Lacidogna (eds.), Acoustic Emission and Critical Phenomena, Taylor and Francis, London, 225–244.

    Chapter  Google Scholar 

  • Corral, A. (2009a), Point-occurrence self-similarity in crackling-noise systems and in other complex systems, J. Stat. Mech. P01022, DOI: 10.1088/1742-5468/2009/01/P01022.

    Google Scholar 

  • Corral, A. (2009b), Statistical tests for scaling in the inter-event times of earthquakes in California, Int. J. Mod. Phys. B 23,28–29, 5570–5582, DOI: 10.1142/S0217979209063869.

    Article  Google Scholar 

  • Corral, A. (2010), Tropical cyclones as a critical phenomenon. In: J.B. Elsner, R.E. Hodges, J.C. Malmstadt, and K.N. Scheitlin (eds.), Hurricanes and Climate Change, Vol. 2, Springer, Heidelberg, 81–99, DOI: 10.1007/978-90-481-9510-7_5.

    Google Scholar 

  • Corral, A., and K. Christensen (2006), Comment on “Earthquakes descaled: On waiting time distributions and scaling laws”, Phys. Rev. Lett. 96,10, 109801, DOI: 10.1103/PhysRevLett.96.109801.

    Article  Google Scholar 

  • Corral, A., and F. Font-Clos (2013), Criticality and self-organization in branching processes: application to natural hazards. In: M. Aschwanden (ed.), Self-Organized Criticality Systems, Open Academic Press, Berlin, 183–228.

    Google Scholar 

  • Corral, A., and A. Turiel (2012), Variability of North Atlantic hurricanes: seasonal versus individual-event features. In: A.S. Sharma, A. Bunde, V.P. Dimri, and D.N. Baker (eds.), Extreme Events and Natural Hazards: the Complexity Perspective, Geopress, Washington, 111–125, DOI: 10.1029/2011GM001069.

    Chapter  Google Scholar 

  • Corral, A., L. Telesca, and R. Lasaponara (2008), Scaling and correlations in the dynamics of forest-fire occurrence, Phys. Rev. E 77,1 016101, DOI: 10.1103/PhysRevE.77.016101.

    Google Scholar 

  • Corral, A., A. Ossó, and J.E. Llebot (2010), Scaling of tropical-cyclone dissipation, Nature Phys. 6, 693–696, DOI: 10.1038/nphys1725.

    Article  Google Scholar 

  • Corral, A., F. Font, and J. Camacho (2011), Noncharacteristic half-lives in radioactive decay, Phys. Rev. E 83,6, 066103, DOI: 10.1103/PhysRevE.83.066103.

    Article  Google Scholar 

  • Corral, A., A. Deluca, and R. Ferrer-i-Cancho (2012), A practical recipe to fit discrete power-law distributions, arXiv:1209.1270.

    Google Scholar 

  • Czechowski, Z. (2003), The privilege as the cause of power distributions in geophysics, Geophys. J. Int. 154,3, 754–766, DOI: 10.1046/j.1365-246X.2003.01994.x.

    Article  Google Scholar 

  • Davidsen, J., and M. Paczuski (2005), Analysis of the spatial distribution between successive earthquakes, Phys. Rev. Lett. 94,4, 048501, DOI: 10.1103/Phys-RevLett.94.048501.

    Article  Google Scholar 

  • del Castillo, J. (2013), Exponential models, Lecture notes (unpublished).

    Google Scholar 

  • del Castillo, J., and P. Puig (1999), The best test of exponentiality against singly truncated normal alternatives, J. Am. Stat. Assoc. 94,446, 529–532, DOI: 10.1080/01621459.1999.10474147.

    Article  Google Scholar 

  • del Castillo, J., J. Daoudi, and I. Serra (2012), The full-tails gamma distribution applied to model extreme values, arXiv:1211.0130.

    Google Scholar 

  • Devroye, L. (1986), Non-Uniform Random Variate Generation, Springer-Verlag, New York.

    Google Scholar 

  • Dickman, R. (2003), Rain, power laws, and advection, Phys. Rev. Lett. 90,10, 108701, DOI: 10.1103/PhysRevLett.90.108701.

    Article  Google Scholar 

  • Durrett, R. (2010), Probability: Theory and Examples, 4th ed., Cambridge University Press, Cambridge.

    Book  Google Scholar 

  • Emanuel, K. (2005a), Divine Wind: the History and Science of Hurricanes, Oxford University Press, New York.

    Google Scholar 

  • Emanuel, K. (2005b), Increasing destructiveness of tropical cyclones over the past 30 years, Nature 436,7051, 686–688, DOI: 10.1038/nature03906.

    Article  Google Scholar 

  • Evans, M., N. Hastings, and B. Peacock (2000), Statistical Distributions, 3rd ed., John Wiley & Sons Inc., New York.

    Google Scholar 

  • Felzer, K.R., and E.E. Brodsky (2006), Decay of aftershock density with distance indicates triggering by dynamic stress, Nature 441,7094, 735–738, DOI: 10.1038/nature04799.

    Article  Google Scholar 

  • Freeman, M.P., and N.W. Watkins (2002), The heavens in a pile of sand, Science 298,5595, 979–980, DOI: 10.1126/science.1075555.

    Article  Google Scholar 

  • Geist, E.L., and T. Parsons (2008), Distribution of tsunami interevent times, Geophys. Res. Lett. 35,2, L02612, DOI: 10.1029/2007GL032690.

    Article  Google Scholar 

  • Goldstein, M.L., S.A. Morris, and G.G. Yen (2004), Problems with fitting to the powerlaw distribution, Eur. Phys. J. B 41,2, 255–258, DOI: 10.1140 /epjb/e2004-00316-5.

    Article  Google Scholar 

  • Gutenberg, B., and C.F. Richter (1944), Frequency of earthquakes in California, Bull. Seismol. Soc. Am. 34,4, 185–188.

    Google Scholar 

  • Hauksson, E., W. Yang, and P. Shearer (2012), Waveform relocated earthquake catalog for southern California (1981 to June 2011), Bull. Seismol. Soc. Am. 102,5, 2239–2244, DOI: 10.1785/0120120010.

    Article  Google Scholar 

  • Hergarten, S. (2002), Self-Organized Criticality in Earth Systems, Springer, Berlin.

    Book  Google Scholar 

  • Jarvinen, B.R., C.J. Neumann, and M.A.S. Davis (1988), A tropical cyclone data tape for theNorthAtlantic basin, 1886–1983: contents, limitations, and uses, NOAA Technical Memorandum NWS NHC 22, National Hurricane Center, Miami, USA, http://www.nhc.noaa.gov/pdf/NWS-NHC-1988-22.pdf.

    Google Scholar 

  • Jensen, H.J. (1998), Self-Organized Criticality. Emergent Complex Behavior in Physical and Biological Systems, Cambridge University Press, Cambridge.

    Book  Google Scholar 

  • Johnson, N.L., S. Kotz, and N. Balakrishnan (1994), Continuous Univariate Distributions. Vol. 1, 2nd ed., John Wiley & Sons Inc., New York.

    Google Scholar 

  • Johnson, N.L., A.W. Kemp, and S. Kotz (2005), Univariate Discrete Distributions, 3rd ed., John Wiley & Sons Inc., Hoboken.

    Book  Google Scholar 

  • JTWC (2012), Annual tropical cyclone report, Joint Typhoon Warning Center, http://www.usno.navy.mil/NOOC/nmfc-ph/RSS/jtwc/best_tracks.

    Google Scholar 

  • Kagan, Y.Y. (2002), Seismic moment distribution revisited: I. Statistical results, Geophys. J. Int. 148,3, 520–541, DOI: 10.1046/j.1365-246x.2002.01594.x.

    Article  Google Scholar 

  • Kalbfleisch, J.D., and R.L. Prentice (2002), The Statistical Analysis of Failure Time Data, 2nd ed., John Wiley & Sons Inc., Hoboken.

    Book  Google Scholar 

  • Kanamori, H., and E.E. Brodsky (2004), The physics of earthquakes, Rep. Prog. Phys. 67,8, 1429–1496, DOI: 10.1088/0034-4885/67/8/R03.

    Article  Google Scholar 

  • Klafter, J., M.F. Shlesinger, and G. Zumofen (1996), Beyond Brownian motion, Phys. Today 49,2, 33–39, DOI: 10.1063/1.881487.

    Article  Google Scholar 

  • Kolmogorov, A.N. (1956), Foundations of the Theory of Probability, 2nd ed., Chelsea Pub. Co., New York.

    Google Scholar 

  • Krane, K.S. (1988), Introductory Nuclear Physics, JohnWiley & Sons Inc., New York.

    Google Scholar 

  • Lahaie, F., and J.R. Grasso (1998), A fluid-rock interaction cellular automaton of volcano mechanics: Application to the Piton de la Fournaise, J. Geophys. Res. 103,B5, 9637–9650, DOI: 10.1029/98JB00202.

    Article  Google Scholar 

  • Main, I.G., L. Li, J. McCloskey, and M. Naylor (2008), Effect of the Sumatran megaearthquake on the global magnitude cut-off and event rate, Nature Geosci. 1,3, 142, DOI: 10.1038/ngeo141.

    Article  Google Scholar 

  • Malamud, B.D. (2004), Tails of natural hazards, Phys. World 17,8, 31–35.

    Google Scholar 

  • Malamud, B.D., G. Morein, and D.L. Turcotte (1998), Forest fires: An example of self-organized critical behavior, Science 281,5384, 1840–1842, DOI: 10.1126/science.281.5384.1840.

    Article  Google Scholar 

  • Malamud, B.D., J.D.A. Millington, and G.L.W. Perry (2005), Characterizing wildfire regimes in the United States, Proc. Natl. Acad. Sci. USA 102,13, 4694–4699, DOI: 10.1073/pnas.0500880102.

    Article  Google Scholar 

  • Malmgren, R.D., D.B. Stouffer, A.E. Motter, and L.A.N. Amaral (2008), A Poissonian explanation for heavy tails in e-mail communication, Proc. Natl. Acad. Sci. USA 105,47, 18153–18158, DOI: 10.1073/pnas.0800332105.

    Article  Google Scholar 

  • McClelland, L., T. Simkin, M. Summers, E. Nielsen, and T.C. Stein (eds.) (1989), Global Volcanism 1975–1985, Prentice Hall, Englewood Cliffs.

    Google Scholar 

  • Mitzenmacher, M. (2004), A brief history of generative models for power law and lognormal distributions, Internet Math. 1,2, 226–251, DOI: 10.1080/15427951.2004.10129088.

    Article  Google Scholar 

  • Newman, M.E.J. (2005), Power laws, Pareto distributions and Zipf’s law, Contemp. Phys. 46,5, 323–351, DOI: 10.1080/00107510500052444.

    Article  Google Scholar 

  • NHC (2012), National Hurricane Center, http://www.nhc.noaa.gov/pastall.shtml#hurdat.

    Google Scholar 

  • Paczuski, M., S. Boettcher, and M. Baiesi (2005), Interoccurrence times in the Bak-Tang-Wiesenfeld sandpile model: A comparison with the observed statistics of solar flares, Phys. Rev. Lett. 95,18, 181102, DOI: 10.1103/Phys-RevLett.95.181102.

    Article  Google Scholar 

  • Peters, O., and K. Christensen (2006), Rain viewed as relaxational events, J. Hydrol. 328,1–2, 46–55, DOI: 10.1016/j.hydrol.2005.11.045.

    Article  Google Scholar 

  • Peters, O., and J.D. Neelin (2006), Critical phenomena in atmospheric precipitation, Nat. Phys. 2, 393–396, DOI: 10.1038/nphys314.

    Article  Google Scholar 

  • Peters, O., C. Hertlein, and K. Christensen (2001), A complexity view of rainfall, Phys. Rev. Lett. 88,1, 018701, DOI: 10.1103/PhysRevLett.88.018701.

    Article  Google Scholar 

  • Peters, O., A. Deluca, A. Corral, J.D. Neelin, and C.E. Holloway (2010), Universality of rain event size distributions, J. Stat. Mech. 2010, P11030, DOI: 10.1088/1742-5468/2010/11/P11030.

    Article  Google Scholar 

  • Press, W.H., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery (1992), Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed., Cambridge University Press, Cambridge.

    Google Scholar 

  • Pruessner, G. (2012), Self-Organised Criticality: Theory, Models and Characterisation, Cambridge University Press, Cambridge.

    Book  Google Scholar 

  • Pueyo, S., and R. Jovani (2006), Comment on “A keystone mutualism drives pattern in a power function”, Science 313,5794, 1739c–1740c, DOI: 10.1126/science.1129595.

    Article  Google Scholar 

  • Ross, S. (2002), A First Course in Probability, 6th ed., Pearson Education, 528 pp.

    Google Scholar 

  • Saichev, A., and D. Sornette (2006), “Universal” distribution of interearthquake times explained, Phys. Rev. Lett. 97,7, 078501, DOI: 10.1103/Phys-RevLett.97.078501.

    Article  Google Scholar 

  • Sethna, J.P., K.A. Dahmen, and C.R. Myers (2001), Crackling noise, Nature 410,6825, 242–250, DOI: 10.1038/35065675.

    Article  Google Scholar 

  • Shearer, P., E. Hauksson, and G. Lin (2005), Southern California hypocenter relocation with waveform cross-correlation. Part 2: Results using source-specific station terms and cluster analysis, Bull. Seismol. Soc. Am. 95,3, 904–915, DOI: 10.1785/0120040168.

    Article  Google Scholar 

  • Shiryaev, A.N. (1996), Probability, 2nd ed., Graduate Texts in Mathematics, Springer, New York.

    Book  Google Scholar 

  • Silverman, B.W. (1986), Density Estimation for Statistics and Data Analysis, Chapman and Hall, New York.

    Google Scholar 

  • Sornette, D. (2004), Critical Phenomena in Natural Sciences. Chaos, Fractals, Selforganization and Disorder: Concepts and Tools, 2nd ed., Springer, Berlin.

    Google Scholar 

  • Takayasu, H. (1990), Fractals in the Physical Sciences, Manchester University Press, Manchester.

    Google Scholar 

  • Utsu, T. (1999), Representation and analysis of the earthquake size distribution: a historical review and some new approaches, Pure Appl. Geophys. 155,2–4, 509–535, DOI: 10.1007/s000240050276.

    Article  Google Scholar 

  • Utsu, T. (2002), Statistical features of seismicity. In: W.H.K. Lee, H. Kanamori, P.C. Jennings, and C. Kisslinger (eds.), International Handbook of Earthquake and Engineering Seismology, Part A, Vol. 81, Academic Press, Amsterdam, 719–732, DOI: 10.1016/S0074-6142(02)80246-7.

    Google Scholar 

  • Utsu, T., Y. Ogata, and R. Matsu’ura (1995), The centenary of the Omori formula for a decay law of aftershock activity, J. Phys. Earth 43,1, 1–33, DOI: 10.4294/jpe1952.43.1.

    Article  Google Scholar 

  • Wanliss, J.A., and J.M. Weygand (2007), Power law burst lifetime distribution of the SYM-H index, Geophys. Res. Lett. 34,4, L04107, DOI: 10.1029/2006GL028235.

    Article  Google Scholar 

  • White, E.P., B.J. Enquist, and J.L. Green (2008), On estimating the exponent of powerlaw frequency distributions, Ecology 89,4, 905–912, DOI: 10.1890/07-1288.1.

    Article  Google Scholar 

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  1. Centre de Recerca Matemàtica, Edifici C, Bellaterra, Barcelona, Spain

    Anna Deluca & Álvaro Corral

  2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Cerdanyola, Spain

    Anna Deluca & Álvaro Corral

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  1. Anna Deluca
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Deluca, A., Corral, Á. Fitting and goodness-of-fit test of non-truncated and truncated power-law distributions. Acta Geophys. 61, 1351–1394 (2013). https://doi.org/10.2478/s11600-013-0154-9

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