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Maximum Likelihood Estimation of Power-Law Exponents for Testing Universality in Complex Systems

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Multidisciplinary Mathematical Modelling

Part of the book series: SEMA SIMAI Springer Series ((ICIAM2019SSSS,volume 11))

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Abstract

Power-law-type distributions are extensively found when studying the behavior of many complex systems. However, due to limitations in data acquisition, empirical datasets often only cover a narrow range of observations, making it difficult to establish power-law behavior unambiguously. In this work, we present a statistical procedure to merge different datasets, with two different aims. First, we obtain a broader fitting range for the statistics of different experiments or observations of the same system. Second, we establish whether two or more different systems may belong to the same universality class. By means of maximum likelihood estimation, this methodology provides rigorous statistical information to discern whether power-law exponents characterizing different datasets can be considered equal to each other or not. This procedure is applied to the Gutenberg–Richter law for earthquakes and for synthetic earthquakes (acoustic emission events) generated in the laboratory: labquakes (Navas-Portella et al. Phys Rev E 100:062106, 2019).

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References

  1. Christensen, K., Moloney, N.R.: Complexity and Criticality. Imperial College Press London (2005)

    Google Scholar 

  2. Bak, P.: How Nature Works: The Science of Self-organized Criticality. Oxford University Press, Oxford (1997)

    MATH  Google Scholar 

  3. Sethna, J.P., Dahmen, K.A., Myers, C.R.: Crackling noise. Nature 410(6825), 242–250 (2001)

    Article  Google Scholar 

  4. Salje, E.K.H., Dahmen, K.A.: Crackling noise in disordered materials. Annu. Rev. Cond. Matter Phys. 5(1), 233–254 (2014)

    Article  Google Scholar 

  5. Gabaix, X.: Power laws in economics: An introduction. J. Econ. Pers. 30(1), 185–206 (2016)

    Article  Google Scholar 

  6. Moreno-Sánchez, I., Font-Clos, F., Corral, A.: Large-scale analysis of Zipf’s law in English texts. PLOS One 11(1), 1–19 (2016)

    Article  Google Scholar 

  7. Corral, Á., González, Á.: Power-law distributions in geoscience revisited. Earth Space Sci. 6, 673–697 (2019)

    Google Scholar 

  8. Kagan, Y.Y.: Earthquakes: Models, Statistics, Testable Forecasts. Wiley, London (2013)

    Google Scholar 

  9. Kagan, Y.Y.: Earthquake size distribution: power-law with exponent \(\beta \equiv \frac {1}{2}\)? Tectonophysics 490(1), 103–114 (2010)

    Google Scholar 

  10. Stanley, H.E.: Scaling, universality, and renormalization: three pillars of modern critical phenomena. Rev. Mod. Phys. 71, S358–S366 (1999)

    Article  Google Scholar 

  11. Stanley, H.E., Amaral, L.A.N., Gopikrishnan, P., Ivanov, P.Ch., Keitt, T.H., Plerou, V.: Scale invariance and universality: organizing principles in complex systems. Phys. A Stat. Mech. App. 281(1), 60–68 (2000)

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. Hanks, T.C., Kanamori, H.: A moment magnitude scale. J. Geophys. Res. Solid Earth 84(B5), 2348–2350 (1979)

    Article  Google Scholar 

  14. Bormann, P.: Are new data suggesting a revision of the current M w and M e scaling formulas? J. Seismol. 19(4), 989–1002 (2015)

    Article  Google Scholar 

  15. Woessner, J., Wiemer, S.: Assessing the quality of earthquake catalogues: estimating the magnitude of completeness and its uncertainty. Bull. Seismol. Soc. Am. 95(2), 684–694 (2005)

    Article  Google Scholar 

  16. González, Á.: The Spanish national earthquake catalogue: evolution, precision and completeness. J. Seism. 21(3), 435–471 (2017)

    Article  Google Scholar 

  17. Schorlemmer, D., Woessner, J.: Probability of detecting an earthquake. Bull. Seismol. Soc. Am. 98(2), 2103–2117 (2008)

    Article  Google Scholar 

  18. Davidsen, J., Green, A.: Are earthquake magnitudes clustered? Phys. Rev. Lett. 106, 108502 (2011)

    Article  Google Scholar 

  19. Hainzl, S.: Rate-dependent incompleteness of earthquake catalogs. Seism. Res. Lett. 87(2A), 337–344 (2016)

    Article  Google Scholar 

  20. Helmstetter, A., Kagan, Y.Y., Jackson, D.D.: Comparison of short-term and time-independent earthquake forecast models for Southern California. Bull. Seismol. Soc. Am. 96(1), 90–106, 02 (2006)

    Google Scholar 

  21. Corral, Á., García-Millán, R., Moloney, N.R., Font-Clos, F.: Phase transition, scaling of moments, and order-parameter distributions in Brownian particles and branching processes with finite-size effects. Phys. Rev. E 97, 062156 (2018)

    Article  Google Scholar 

  22. Serra, I., Corral, Á.: Deviation from power law of the global seismic moment distribution. Sci. Rep. 7(1), 40045 (2017)

    Article  Google Scholar 

  23. Baró, J., Corral, Á., Illa, X., Planes, A., Salje, E.K.H. Schranz, W., Soto-Parra, D.E., Vives, E.: Statistical similarity between the compression of a porous material and earthquakes. Phys. Rev. Lett. 110(8), 088702 (2013)

    Article  Google Scholar 

  24. Nataf, G.F., Castillo-Villa, P.O., Baró, J., Illa, X., Vives, E., Planes, A., Salje, E.K.H.: Avalanches in compressed porous SiO2-based materials. Phys. Rev. E 90(2), 022405 (2014)

    Article  Google Scholar 

  25. Navas-Portella, V., Corral, Á., Vives , E.: Avalanches and force drops in displacement-driven compression of porous glasses. Phys. Rev. E 94(3), 033005 (2016)

    Article  Google Scholar 

  26. Yoshimitsu, N., Kawakata, H., Takahashi, N.: Magnitude −7 level earthquakes: a new lower limit of self-similarity in seismic scaling relationships. Geophys. Res. Lett. 41(13), 4495–4502 (2014)

    Article  Google Scholar 

  27. Main, I.G.: Little earthquakes in the lab. Physics 6:20 (2013)

    Article  Google Scholar 

  28. Uhl, J.T., Pathak, S., Schorlemmer, D., Liu, X., Swindeman, R., Brinkman, B.A.W., LeBlanc, M., Tsekenis, G., Friedman, N., Behringer, R., Denisov, D., Schall, P., Gu, X., Wright, W.J., Hufnagel, T., Jennings, A., Greer, J.R., Liaw, P.K., Becker, T., Dresen, G., Dahmen, K.A.: Universal quake statistics: from compressed nanocrystals to earthquakes. Sci. Rep. 5, 16493 (2015)

    Article  Google Scholar 

  29. Mäkinen, T., Miksic, A., Ovaska, M., Alava, M.J.: Avalanches in wood compression. Phys. Rev. Lett. 115(5), 055501 (2015)

    Article  Google Scholar 

  30. Deluca, A., Corral, Á.: Fitting and goodness-of-fit test of non-truncated and truncated power-law distributions. Acta Geophys. 61(6), 1351–1394 (2013)

    Article  Google Scholar 

  31. Clauset, A., Shalizi, C.R., Newman, M.E.J.: Power-law distributions in empirical data. SIAM Rev. 51, 661–703 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Navas-Portella, V., Serra, I., González, Á., Vives, E., Corral, Á.: Universality of power-law exponents by means of maximum likelihood estimation. Phys. Rev. E 100, 062106 (2019)

    Article  Google Scholar 

  33. Pawitan, Y.: In All Likelihood: Statistical Modelling and Inference Using Likelihood. OUP Oxford, Oxford (2013)

    MATH  Google Scholar 

  34. Navas-Portella, V., Serra, I., Corral, Á., Vives, E.: Increasing power-law range in avalanche amplitude and energy distributions. Phys. Rev. E 97, 022134 (2018)

    Article  Google Scholar 

  35. Baró, J., Vives, E.: Analysis of power-law exponents by maximum-likelihood maps. Phys. Rev. E 85(6), 066121 (2012)

    Article  Google Scholar 

  36. Chou, T.-A., Dziewonski, A.M., Woodhouse, J.H.: Determination of earthquake source parameters from waveform data for studies of global and regional seismicity. J. Geophys. Res 86, 2825–2852 (1981)

    Article  Google Scholar 

  37. Nettles, M., Ekström, G., Dziewonski, A.M.: The Global CMT project 2004–2010: centroid-moment tensors for 13,017 earthquakes. Phys. Earth Planet. Inter. 200–201, 1–9 (2012)

    Google Scholar 

  38. Yang, W., Hauksson, E., Shearer, P.M.: Computing a large refined catalog of focal mechanisms for Southern California (1981–2010): temporal stability of the style of faulting. Bull. Seismol. Soc. Am. 102, 1179–1194 (2012)

    Article  Google Scholar 

  39. Wessel, P., Luis, J., Uieda, L., Scharroo, R., Wobbe, F., Smith, W.H.F., Tian, D.: The generic mapping tools version 6. Geochem. Geophys. Geosyst. 6(11), 5556–5564 (2019)

    Article  Google Scholar 

  40. Hutton, K., Woessner, J., Hauksson, E.: Earthquake monitoring in Southern California for seventy-seven years (1932–2008). Bull. Seismol. Soc. Am. 100(2), 423–446 (2010)

    Article  Google Scholar 

  41. PCI-2—PCI-Based Two-Channel AE Board & System, by Physical Acoustics

    Google Scholar 

  42. Bender, B.: Maximum likelihood estimation of b values for magnitude grouped data. Bull. Seismol. Soc. Am. 73(3), 831–851 (1983)

    Article  Google Scholar 

  43. Pacheco, J.F.E., Scholz, C.H., Sykes, L.R.: Changes in frequency-size relationship from small to large earthquakes. Nature 355, 71–73 (1992)

    Article  Google Scholar 

  44. Yoder, M.R., Holliday, J.R., Turcotte, D.L., Rundle, J.B.: A geometric frequency-magnitude scaling transition: measuring b = 1.5 for large earthquakes. Tectonophysics 532–535, 167–174 (2012)

    Article  Google Scholar 

  45. Kamer, Y., Hiemer, S.: Data-driven spatial b value estimation with applications to California seismicity: to b or not to b. J. Geophys. Res.: Solid Earth 120(7), 5191–5214 (2015)

    Google Scholar 

  46. Plenkers, K., Schorlemmer, D., Kwiatek, G., JAGUARS Research Group: On the probability of detecting picoseismicity. Bull. Seismol. Soc. Am. 101(6), 2579–2591, 12 (2011)

    Google Scholar 

  47. Kwiatek, G., Plenkers, K., Nakatani, M., Yabe, Y., Dresen, G., JAGUARS-Group: Frequency-magnitude characteristics down to magnitude − 4.4 for induced seismicity recorded at Mponeng gold mine. Bull. Seismol. Soc. Am. 100(3), 1165–1173, 06 (2010)

    Google Scholar 

  48. Kagan, Y.: Universality of the seismic moment-frequency relation. Pure Appl. Geophys. 155, 537–573 (1999)

    Article  Google Scholar 

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Acknowledgements

The research leading to these results has received funding from “La Caixa” Foundation. V. N. acknowledges financial support from the Spanish Ministry of Economy and Competitiveness (MINECO, Spain), through the “María de Maeztu” Programme for Units of Excellence in R & D (grant no. MDM-2014-0445) and the Juan de la Cierva research contract FJCI-2016-29307 hold by Á.G.. We also acknowledge financial support from the MINECO under grant nos. FIS2015-71851-P, FIS-PGC2018-099629-B-100, and MAT2016-75823-R and from the Agència de Gestió d’Ajuts Universitaris i de Recerca (AGAUR) under grant no. 2014SGR-1307.

Author information

Authors and Affiliations

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

    Víctor Navas-Portella

  2. Barcelona Graduate School of Mathematics, Barcelona, Spain

    Víctor Navas-Portella

  3. Facultat de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Spain

    Víctor Navas-Portella

  4. Centre de Recerca Matemàtica, Bellaterra, Barcelona, Spain

    Álvaro González

  5. GFZ German Research Centre for Geosciences, Potsdam, Germany

    Álvaro González

  6. Centre de Recerca Matemàtica, Bellaterra, Barcelona, Spain

    Isabel Serra

  7. Barcelona Supercomputing Center, Barcelona, Spain

    Isabel Serra

  8. Departament de Matèria Condensada, Facultat de Física, Universitat de Barcelona, Barcelona, Catalonia, Spain

    Eduard Vives

  9. Universitat de Barcelona Institute of Complex Systems (UBICS), Facultat de Física, Universitat de Barcelona, Barcelona, Catalonia, Spain

    Eduard Vives

  10. Barcelona Graduate School of Mathematics, Barcelona, Spain

    Álvaro Corral

  11. Complexity Science Hub Vienna, Vienna, Austria

    Álvaro Corral

  12. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Barcelona, Spain

    Álvaro Corral

  13. Centre de Recerca Matemàtica, Bellaterra, Barcelona, Spain

    Álvaro Corral

Authors
  1. Víctor Navas-Portella
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  2. Álvaro González
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  3. Isabel Serra
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  4. Eduard Vives
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  5. Álvaro Corral
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Corresponding author

Correspondence to Víctor Navas-Portella .

Editor information

Editors and Affiliations

  1. Centre de Recerca Matemàtica, Barcelona, Spain

    Francesc Font

  2. Centre de Recerca Matemàtica, Barcelona, Spain

    Tim G. Myers

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Navas-Portella, V., González, Á., Serra, I., Vives, E., Corral, Á. (2021). Maximum Likelihood Estimation of Power-Law Exponents for Testing Universality in Complex Systems. In: Font, F., Myers, T.G. (eds) Multidisciplinary Mathematical Modelling. SEMA SIMAI Springer Series(), vol 11. Springer, Cham. https://doi.org/10.1007/978-3-030-64272-3_5

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