Skip to main content
SpringerLink
Log in

Asymmetric Laplace Mixture Modelling of Incomplete Power-Law Distributions: Application to ‘Seismicity Vision’

  • Conference paper
  • First Online:
Book cover Advances in Computer Vision (CVC 2019)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 944))

Included in the following conference series:

Abstract

Data used in statistical analyses are often limited to a narrow range over which the quantity of interest is observed to be reliable. The power-law behavior of many natural processes is only observed above a threshold below which information is discarded due to detection limitations. These incomplete data can however also be described by a power law, and the distribution over the full quantity range reformulated as an asymmetric Laplace (AL) distribution. With the detection process heterogeneous in space and time in realistic conditions, the data can be modelled by a mixture of AL components. Using seismicity as example, we describe an asymmetric Laplace mixture model (ALMM), which considers ambiguous overlapping components - as observed in Nature - based on a semi-supervised hard Expectation-Maximization algorithm. We show that the ALMM fits reasonably well incomplete data and that the number of AL components can be related to the seismic network density. We conclude that the full range of data can be used in statistical analyses, including in computer vision. In the case of seismicity, a ten-fold increase in sample size is possible which provides, for example, a better spatial pattern resolution to improve the correlation between fault features and earthquake labels.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Clauset, A., Shalizi, C.R., Newman, M.E.J.: Power-Law distributions in empirical data. SIAM Rev. 51(4), 661–703 (2009)

    Article  MathSciNet  Google Scholar 

  2. Hartley, M.J., Revankar, N.S.: On the estimation of the Pareto Law from under-reported data. J. Econometrics 2, 327–341 (1974)

    Article  MathSciNet  Google Scholar 

  3. Hinkley, D.V., Revankar, N.S.: Estimation of the Pareto Law from under-reported data. J. Econometrics 5, 1–11 (1977)

    Article  MathSciNet  Google Scholar 

  4. Kotz, S., Kozubowski, T.J., Podgorski, K.: The Laplace Distribution and Generalizations. Birkhäuser, Boston (2001)

    Book  Google Scholar 

  5. Mignan, A.: Functional shape of the earthquake frequency-magnitude distribution and completeness magnitude. J. Geophys. Res. 117, B08302 (2012)

    Article  Google Scholar 

  6. Bishop, C.: Pattern Recognition and Machine Learning. Springer, New York (2006)

    MATH  Google Scholar 

  7. Franczak, B.C., Browne, R.P., McNicholas, P.D.: Mixtures of shifted asymmetric laplace distributions. IEEE Trans. Pattern Anal. Mach. Intell. 36(6), 1149–1157 (2014)

    Article  Google Scholar 

  8. Mignan, A., Werner, M.J., Wiemer, S., Chen, C.-C., Wu, Y.-M.: Bayesian estimation of the spatially varying completeness magnitude of earthquake catalogs. Bull. Seismol. Soc. Am. 101(3), 1371–1385 (2011)

    Article  Google Scholar 

  9. Rouet-Leduc, B., Hulbert, C., Lubbers, N., Barros, K., Humphreys, C.J., Johnson, P.A.: Machine learning predicts laboratory earthquakes. Geophys. Res. Lett. 44, 9276–9282 (2017)

    Article  Google Scholar 

  10. DeVries, P.M.R., Viegas, F., Wattenberg, M., Meade, B.J.: Deep learning of aftershock patterns following large earthquakes. Nature 560, 632–634 (2018)

    Article  Google Scholar 

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

    Article  Google Scholar 

  12. Devroye, L.: Non-uniform Random Variate Generation. Springer, New York (1986)

    Book  Google Scholar 

  13. Ogata, Y., Katsura, K.: Immediate and updated forecasting of aftershock hazard. Geophys. Res. Lett. 33, L10305 (2006)

    Article  Google Scholar 

  14. Kijko, A., Smit, A.: Estimation of the frequency-magnitude Gutenberg-Richter b-value without making assumptions on levels of completeness. Seismol. Res. Lett. 88, 311–318 (2017)

    Article  Google Scholar 

  15. Mignan, A., Chouliaras, G.: Fifty Years of Seismic Network Performance in Greece (1964-2013): Spatiotemporal Evolution of the Completeness Magnitude. Seismol. Res. Lett. 85, 657–667 (2014)

    Article  Google Scholar 

  16. Mignan, A., Chen, C.-C.: The spatial scale of detected seismicity. Pure. Appl. Geophys. 173, 117–124 (2016)

    Article  Google Scholar 

  17. Broido, A.D., Clauset, A.: Scale-free networks are rare. https://arxiv.org/abs/1801.03400. Accessed 28 Oct 2018

  18. Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B (Methodological) 39(1), 1–38 (1977)

    MathSciNet  MATH  Google Scholar 

  19. Jain, A.K.: Data clustering: 50 years beyond K-means. Pattern Recogn. Lett. 31, 651–666 (2010)

    Article  Google Scholar 

  20. Aki, K.: Maximum likelihood estimate of b in the formula log N=a-bM and its confidence limits. Bull. Earthq. Res. Inst. 43, 237–239 (1965)

    Google Scholar 

  21. Celeux, G., Hurn, M., Robert, C.P.: Computational and inferential difficulties with mixture posterior distributions. J. Am. Stat. Assoc. 94(451), 957–970 (2000)

    Article  MathSciNet  Google Scholar 

  22. Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6(2), 461–464 (1978)

    Article  MathSciNet  Google Scholar 

  23. Voronoi, G.F.: Nouvelles applications des paramètres continus à la théorie des formes quadratiques, 2ème Mémoire: Recherches sur les parallélloèdres primitifs. J. fûr die. Reine und Angewandte Mathematik 134, 198–287 (1908)

    MathSciNet  MATH  Google Scholar 

  24. Lee, D.T., Schachter, B.J.: Two algorithms for constructing a Delaunay triangulation. Int. J. Comput. Inf. Sci. 9, 219–242 (1980)

    Article  MathSciNet  Google Scholar 

  25. Wu, Y.M., Chang, C.-H., Zhao, L., Teng, T.L., Nakamura, M.: A comprehensive relocation of earthquakes in Taiwan from 1991 to 2005. Bull. Seismol. Soc. Am. 98(3), 1471–1481 (2008)

    Article  Google Scholar 

  26. Efron, B.: Second thoughts on the bootstrap. Stat. Sci. 18(2), 135–140 (2003)

    Article  MathSciNet  Google Scholar 

  27. Mignan, A.: The debate on the prognostic value of earthquake foreshocks: a meta-analysis. Sci. Rep. 4, 4099 (2014)

    Article  Google Scholar 

  28. Bannister, S., Fry, B., Reyners, M., Ristau, J., Zhang, H.: Fine-scale relocation of aftershocks of the 22 February Mw 6.2 Christchurch earthquake using double-difference tomography. Seismol. Res. Lett. 82(6), 839–845 (2011)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Swiss Federal Institute of Technology, Sonneggstr. 5, 8092, Zurich, Switzerland

    Arnaud Mignan

Authors
  1. Arnaud Mignan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arnaud Mignan .

Editor information

Editors and Affiliations

  1. Saga University, Saga, Saga, Japan

    Kohei Arai

  2. The Science and Information (SAI) Organization, Bradford, West Yorkshire, UK

    Supriya Kapoor

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Mignan, A. (2020). Asymmetric Laplace Mixture Modelling of Incomplete Power-Law Distributions: Application to ‘Seismicity Vision’. In: Arai, K., Kapoor, S. (eds) Advances in Computer Vision. CVC 2019. Advances in Intelligent Systems and Computing, vol 944. Springer, Cham. https://doi.org/10.1007/978-3-030-17798-0_4

Download citation

Publish with us

Policies and ethics

Search

Navigation