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

Deluca, A.; Corrales, A.

doi:10.2478/s11600-013-0154-9

Artykuł - szczegóły

Tytuł artykułu

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

Autorzy

Deluca A. , Corrales A.

Wybrane pełne teksty z tego czasopisma

https://www.springer.com/journal/11600

Identyfikatory

DOI

10.2478/s11600-013-0154-9

Warianty tytułu

Języki publikacji

Abstrakty

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 proce dure, 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.

Słowa kluczowe

power-law distribution estimation goodness-of-fit test binning seismic-moment distribution waiting-time distribution tropical-cyclone energy

Wydawca

Instytut Geofizyki PAN
Springer

Czasopismo

Acta Geophysica

Rocznik

2013

Tom

Vol. 61, no. 6

Strony

1351--1394

Opis fizyczny

Bibliogr. 98 poz.

Twórcy

autor

Deluca A.

ADeluca@crm.cat

Centre de Recerca Matemàtica, Edifici C, Bellaterra, Barcelona, Spain
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Cerdanyola, Spain

autor

Corrales A.

ACorral@crm.cat

Centre de Recerca Matemàtica, Edifici C, Bellaterra, Barcelona, Spain
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Cerdanyola, Spain

Bibliografia

1. Aban, I.B., M.M. Meerschaert, and A.K. Panorska (2006), Parameter estimation forthe truncated Pareto distribution, J. Am. Stat. Assoc.101, 473, 270-277, DOI:10.1198/016214505000000411.
2. Aschwanden, M.J. (2013), SOC systems in astrophysics. In: M.J. Aschwanden (ed.),Self-Organized Criticality Systems, Open Academic Press, Berlin, 439-478.
3. 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.
4. Bak, P. (1996),How Nature Works: The Science of Self-Organized Criticality, Copernicus, New York.
5. 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.
6. Barndorff-Nielsen, O. (1978),Information and Exponential Families in Statistical Theory, John Wiley & Sons Inc., New York, 238 pp.
7. Baró, J., and E. Vives (2012), Analysis of power-law exponents by maximum-likelihood maps, Phys. Rev. E85, 6, 066121, DOI: 10.1103/PhysRevE.85.066121.
8. Bauke, H. (2007), Parameter estimation for power-law distributions by maximum likelihood methods, Eur. Phys. J. B58, 2, 167-173, DOI: 10.1140/epjb/e2007-00219-y.
9. 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.
10. 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.
11. 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.
12. 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.
13. 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.
14. Burroughs, S.M., and S.F. Tebbens (2005), Power-law scaling and probabilistic fore-casting of tsunami runup heights, Pure Appl. Geophys.162, 2, 331-342, DOI:10.1007/s00024-004-2603-5.
15. 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.
16. Casella, G., and R.L. Berger (2002),Statistical Inference, 2nd ed., Duxbury Advanced Series, Duxbury Thomson Learning, Pacific Grove, 660 pp.
17. Chicheportiche, R., and J.-P. Bouchaud (2012), Weighted Kolmogorov–Smirnov test: Accounting for the tails, Phys. Rev. E86, 4, 041115, DOI: 10.1103/Phys-RevE.86.041115.
18. Christensen, K., and N.R. Moloney (2005),Complexity and Criticality, Imperial College Press Advanced Physics Texts, Vol. 1, Imperial College Press, London.
19. 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.
20. Chu, S.Y.F., L.P. Ekström, and R.B. Firestone (1999), The Lund/LBNL Nuclear DataSearch, Version 2.
21. 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.
22. Corpo Forestale dello Stato (2012), http://www.corpoforestale.it.
23. Corral, A. (2003), Local distributions and rate fluctuations in a unified scaling law forearthquakes, Phys. Rev. E68, 3, 035102, DOI: 10.1103/PhysRevE.68.035102.
24. 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.
25. Corral, A. (2004b), Universal local versus unified global scaling laws in the statistics of seismicity, Physica A340, 4, 590-597, DOI: 10.1016/j.physa.2004.05.010.
26. Corral, A. (2005), Comment on “Do earthquakes exhibit self-organized criticality?”, Phys. Rev. Lett.95, 15, 159801, DOI: 10.1103/PhysRevLett.95.159801.
27. Corral, A. (2006), Universal earthquake-occurrence jumps, correlations with time, and anomalous diffusion, Phys. Rev. Lett.97, 17, 178501, DOI: 10.1103/Phys-Rev Lett.97.178501.
28. 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.
29. 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 .
30. Corral, A. (2009b), Statistical tests for scaling in the inter-event times of earth-quakes in California, Int. J. Mod. Phys. B23, 28-29, 5570-5582, DOI:10.1142/S0217979209063869.
31. 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.
32. 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.
33. 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.
34. 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.
35. Corral, A., L. Telesca, and R. Lasaponara (2008), Scaling and correlations in the dynamics of forest-fire occurrence, Phys. Rev. E77, 1 016101, DOI:10.1103/PhysRevE.77.016101.
36. Corral, A., A. Ossó, and J.E. Llebot (2010), Scaling of tropical-cyclone dissipation, Nature Phys.6, 693-696, DOI: 10.1038/nphys1725.
37. Corral, A., F. Font, and J. Camacho (2011), Noncharacteristic half-lives in radioactive decay, Phys. Rev. E83, 6, 066103, DOI: 10.1103/PhysRevE.83.066103.
38. Corral, A., A. Deluca, and R. Ferrer-i-Cancho (2012), A practical recipe to fit discrete power-law distributions, arXiv:1209.1270.
39. 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.
40. 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.
41. del Castillo, J. (2013), Exponential models, Lecture notes (unpublished).
42. 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 .
43. del Castillo, J., J. Daoudi, and I. Serra (2012), The full-tails gamma distribution applied to model extreme values, arXiv:1211.0130.
44. Devroye, L. (1986),Non-Uniform Random Variate Generation, Springer-Verlag, NewYork.
45. Dickman, R. (2003), Rain, power laws, and advection, Phys. Rev. Lett.90, 10, 108701,DOI: 10.1103/PhysRevLett.90.108701 .
46. Durrett, R. (2010),Probability: Theory and Examples, 4th ed., Cambridge University Press, Cambridge.
47. Emanuel, K. (2005a),Divine Wind: the History and Science of Hurricanes, Oxford University Press, New York.
48. Emanuel, K. (2005b), Increasing destructiveness of tropical cyclones over the past 30years,Nature 436, 7051, 686-688, DOI: 10.1038/nature03906.
49. Evans, M., N. Hastings, and B. Peacock (2000),Statistical Distributions, 3rd ed., John Wiley & Sons Inc., New York.
50. 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 .
51. 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.
52. Geist, E.L., and T. Parsons (2008), Distribution of tsunami inter event times, Geophys. Res. Lett.35, 2, L02612, DOI: 10.1029/2007GL032690.
53. Goldstein, M.L., S.A. Morris, and G.G. Yen (2004), Problems with fitting to the power-law distribution, Eur. Phys. J. B41, 2, 255-258, DOI: 10.1140 /epjb/e2004-00316-5.
54. Gutenberg, B., and C.F. Richter (1944), Frequency of earthquakes in California, Bull. Seismol. Soc. Am.34, 4, 185-188.
55. 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.
56. Hergarten, S. (2002), Self-Organized Criticality in Earth Systems, Springer, Berlin.
57. Jarvinen, B.R., C.J. Neumann, and M.A.S. Davis (1988), A tropical cyclone data tape for the North Atlantic 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.
58. Jensen, H.J. (1998),Self-Organized Criticality. Emergent Complex Behavior in Physical and Biological Systems, Cambridge University Press, Cambridge.
59. Johnson, N.L., S. Kotz, and N. Balakrishnan (1994),Continuous Univariate Distributions. Vol. 1, 2nd ed., John Wiley & Sons Inc., New York.
60. Johnson, N.L., A.W. Kemp, and S. Kotz (2005), Univariate Discrete Distributions, 3rded., John Wiley & Sons Inc., Hoboken.
61. JTWC (2012), Annual tropical cyclone report, Joint Typhoon Warning Center,http://www.usno.navy.mil/NOOC/nmfc-ph/RSS/jtwc/best_tracks.
62. 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.
63. Kalbfleisch, J.D., and R.L. Prentice (2002),The Statistical Analysis of Failure Time Data, 2nd ed., John Wiley & Sons Inc., Hoboken.
64. 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.
65. Klafter, J., M.F. Shlesinger, and G. Zumofen (1996), Beyond Brownian motion,Phys.Today49, 2, 33-39, DOI: 10.1063/1.881487.
66. Kolmogorov, A.N. (1956),Foundations of the Theory of Probability, 2nd ed., Chelsea Pub. Co., New York.
67. Krane, K.S. (1988),Introductory Nuclear Physics, John Wiley & Sons Inc., New York.
68. 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.
69. Main, I.G., L. Li, J. McCloskey, and M. Naylor (2008), Effect of the Sumatran mega-earthquake on the global magnitude cut-off and event rate,Nature Geosci.1,3, 142, DOI: 10.1038/ngeo141.
70. Malamud, B.D. (2004), Tails of natural hazards, Phys. World 17, 8, 31-35.
71. Malamud, B.D., G. Morein, and D.L. Turcotte (1998), Forest fires: An example of self-organized critical behavior, Science281, 5384, 1840-1842, DOI: 10.1126/sci-ence.281.5384.1840.
72. Malamud, B.D., J.D.A. Millington, and G.L.W. Perry (2005), Characterizing wildfire regimes in the United States, Proc. Natl. Acad. Sci. USA102, 13, 4694-4699,DOI: 10.1073/pnas.0500880102.
73. 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.USA105, 47, 18153-18158, DOI: 10.1073/pnas.0800332105.
74. McClelland, L., T. Simkin, M. Summers, E. Nielsen, and T.C. Stein (eds.) (1989), Global Volcanism 1975-1985, Prentice Hall, Englewood Cliffs.
75. 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.
76. Newman, M.E.J. (2005), Power laws, Pareto distributions and Zipf’s law, Contemp.Phys.46, 5, 323 -351, DOI: 10.1080/00107510500052444.
77. NHC (2012), National Hurricane Center, http://www.nhc.noaa.gov/pastall.shtml# hurdat.
78. 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.
79. 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.
80. Peters, O., and J.D. Neelin (2006), Critical phenomena in atmospheric precipitation, Nat. Phys.2, 393-396, DOI: 10.1038/nphys314.
81. 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.
82. 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.
83. 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.
84. Pruessner, G. (2012),Self-Organised Criticality: Theory, Models and Characterisation, Cambridge University Press, Cambridge.
85. Pueyo, S., and R. Jovani (2006), Comment on “A keystone mutualism drives pattern in a power function”, Science313, 5794, 1739c–1740c, DOI: 10.1126/sci-ence.1129595.
86. Ross, S. (2002),A First Course in Probability, 6th ed., Pearson Education, 528 pp.
87. 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.
88. Sethna, J.P., K.A. Dahmen, and C.R. Myers (2001), Crackling noise,Nature410, 6825,242-250, DOI: 10.1038/35065675.
89. 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.
90. Shiryaev, A.N. (1996),Probability, 2nd ed., Graduate Texts in Mathematics, Springer, New York.
91. Silverman, B.W. (1986),Density Estimation for Statistics and Data Analysis, Chapman and Hall, New York.
92. Sornette, D. (2004),Critical Phenomena in Natural Sciences. Chaos, Fractals, Selforganization and Disorder: Concepts and Tools, 2nd ed., Springer, Berlin.
93. Takayasu, H. (1990),Fractals in the Physical Sciences, Manchester University Press, Manchester.
94. 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.
95. 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.
96. Utsu, T., Y. Ogata, and R. Matsu’ura (1995), The centenary of the Omori formula for a decay law of aftershock activity, J. Phys. Earth43, 1, 1-33, DOI:10.4294/jpe1952.43.1.
97. 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.
98. White, E.P., B.J. Enquist, and J.L. Green (2008), On estimating the exponent of power-law frequency distributions,Ecology89, 4, 905-912, DOI: 10.1890/07-1288.1.

Typ dokumentu

Bibliografia

Identyfikator YADDA

bwmeta1.element.baztech-936ef384-7dd1-486d-9706-06c2a8a4951b