Mathematical Geosciences

, Volume 45, Issue 7, pp 851–872 | Cite as

Robust Estimation for the Weibull Process Applied to Eruption Records

  • Ting Wang
  • Mark Bebbington


The Weibull process is a parsimoniously parameterized nonhomogeneous Poisson process with monotonic trend, which has been widely used in reliability applications. It has also been used in volcanology to model the process of eruption onsets for a volcano with waning or waxing activity, and thus produce hazard forecasts. However, particularly in the latter application, problems with missing or spurious data can strongly influence the parameter estimates, which are usually obtained by maximizing the log likelihood function, and hence the future hazard. We show how theory developed for robust estimation of a nonhomogeneous Poisson process can be implemented for the Weibull process. The flank eruptions of Mt. Etna, in Sicily, is one of the most complete and best studied records of volcanism. Nevertheless, a number of different catalogs exist. We show how these can be at least partially reconciled by robust estimation, and how the more dubious regions of the catalogs can be identified.


Flank eruptions M-estimator Mt. Etna Nonhomogeneous Poisson process 



This work was carried out while the first author was supported as a Massey University postdoctoral fellow by the Natural Hazards Research Platform and the Earthquake Commission. We thank an anonymous reviewer and the editor for helpful suggestions on the original manuscript.


  1. Andersen PK, Borgan Ø, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer, New York CrossRefGoogle Scholar
  2. Andronico D, Lodato L (2005) Effusive activity at Mount Etna volcano (Italy) during the 20th century: a contribution to volcanic hazard assessment. Nat Hazards 36:407–443 CrossRefGoogle Scholar
  3. Assunção R, Guttorp P (1999) Robustness for inhomogeneous Poisson point processes. Ann Inst Stat Math 51:657–678 CrossRefGoogle Scholar
  4. Bain LJ, Englehardt M, Wright FT (1985) Tests for an increasing trend in the intensity of a Poisson process: a power study. J Am Stat Assoc 80:419–422 CrossRefGoogle Scholar
  5. Bebbington MS (2007) Identifying volcanic regimes using hidden Markov models. Geophys J Int 171:921–942 CrossRefGoogle Scholar
  6. Bebbington MS (2008) Incorporating the eruptive history in a stochastic model for volcanic eruptions. J Volcanol Geotherm Res 175:325–333 CrossRefGoogle Scholar
  7. Bebbington MS (2010) Trends and clustering in the onsets of volcanic eruptions. J Geophys Res 115:B01203 CrossRefGoogle Scholar
  8. Bebbington MS, Harte D (2003) The linked stress release model for spatio-temporal seismicity: formulations, procedures and applications. Geophys J Int 154:925–946 CrossRefGoogle Scholar
  9. Bebbington MS, Lai CD (1996) On nonhomogeneous models for volcanic eruptions. Math Geol 28:585–600 CrossRefGoogle Scholar
  10. Behncke B, Neri M, Nagay A (2005) Lava flow hazard at Mount Etna (Italy): new data from a GIS-based study. Special Paper 396, Geological Society of America Google Scholar
  11. Branca S, Del Carlo P (2004) Eruptions of Mt. Etna during the past 3,200 years: a revised compilation integrating the historical and stratigraphic records. In: Bonaccorso A, Calvari S, Coltelli M, Del Negro C, Falsaperla S (eds) Mt. Etna: volcano laboratory. Geophysical monograph series, vol 143. Am Geophys Union, Washington, pp 1–27 CrossRefGoogle Scholar
  12. Branca S, Del Carlo P (2005) Types of eruptions of Etna volcano AD 1670–2003: implications for short-term eruptive behaviour. Bull Volcanol 67:732–742 CrossRefGoogle Scholar
  13. Crow TH (1974) Reliability analysis for complex repairable systems. In: Proschan F, Serfling RJ (eds) Reliability and biometry. SIAM, Philadelphia, pp 379–410 Google Scholar
  14. Daley DJ, Vere-Jones D (1988) An introduction to the theory of point processes. Springer, New York Google Scholar
  15. Deligne NI, Coles SG, Sparks RSJ (2010) Recurrence rates of large explosive volcanic eruptions. J Geophys Res 115:B06203 CrossRefGoogle Scholar
  16. Diggle P (1985) A kernel method for smoothing point process data. Appl Stat 34:138–147 CrossRefGoogle Scholar
  17. Grillenzoni C (2008) Robust nonparametric estimation of the intensity function of point data. AStA Adv Stat Anal 92:117–134 CrossRefGoogle Scholar
  18. Guttorp P, Thompson ML (1991) Estimating second-order parameters of volcanicity from historical data. J Am Stat Assoc 86:578–583 CrossRefGoogle Scholar
  19. Ho C-H (1991) Nonhomogeneous Poisson model for volcanic eruptions. Math Geol 23:167–173 CrossRefGoogle Scholar
  20. Ho C-H (1992) Statistical control chart for regime identification in volcanic time-series. Math Geol 24:775–787 CrossRefGoogle Scholar
  21. Krasker WS, Welsch RF (1982) Efficient bounded-influence regression estimation. J Am Stat Assoc 77:595–604 CrossRefGoogle Scholar
  22. Künsch HR (1984) Infinitesimal robustness for autoregressive processes. Ann Stat 12:843–863 CrossRefGoogle Scholar
  23. Lawless JF, Nadeau C (1995) Some simple robust methods for the analysis of recurrent events. Technometrics 37:158–168 CrossRefGoogle Scholar
  24. Maronna RA, Martin RD, Yohai VJ (2006) Robust statistics: theory and methods. Wiley, London CrossRefGoogle Scholar
  25. Martin RD, Yohai VJ (1986) Influence functionals for time series. Ann Stat 14:781–818 CrossRefGoogle Scholar
  26. Martin RD, Yohai VJ (1991) Bias robust estimation of autoregression parameters. In: Stahel W, Weisberg S (eds) Directions in robust statistics and diagnostics. Springer, New York, pp 233–246 CrossRefGoogle Scholar
  27. Marzocchi W, Bebbington M (2012) Probabilistic eruption forecasting at short and long time scales. Bull Volcanol 74:1777–1805 CrossRefGoogle Scholar
  28. Mulargia F, Tinti S, Boschi E (1985) A statistical analysis of flank eruptions on Etna volcano. J Volcanol Geotherm Res 23:263–272 CrossRefGoogle Scholar
  29. Mulargia F, Marzocchi W, Gasperini P (1992) Statistical identification of physical patterns which accompany eruptive activity on Mount Etna, Sicily. J Volcanol Geotherm Res 53:289–296 CrossRefGoogle Scholar
  30. Ng ETM, Cook RJ (1999) Adjusted score tests of homogeneity for Poisson processes. J Am Stat Assoc 94:308–319 CrossRefGoogle Scholar
  31. Ng ETM, Cook RJ (2000) A comparison of some random effect models for parameter estimation in recurrent events. Math Comput Model 32:11–26 CrossRefGoogle Scholar
  32. Ogata Y (1988) Statistical models for earthquake occurrences and residual analysis for point processes. J Am Stat Assoc 83:9–27 CrossRefGoogle Scholar
  33. Puente CE, Bierkens MFP, Diaz-Granados MA, Dik PE, López MM (1993) Practical use of analytically derived runoff models based on rainfall point processes. Water Resour Res 29:3551–3560 CrossRefGoogle Scholar
  34. Salvi F, Scandone R, Palma C (2006) Statistical analysis of the historical activity of Mount Etna, aimed at the evaluation of volcanic hazard. J Volcanol Geotherm Res 154:159–168 CrossRefGoogle Scholar
  35. Sandri L, Marzocchi W, Gasperini P (2005) Some insights on the occurrence of recent volcanic eruptions of Mount Etna volcano, Sicily, Italy. Geophys J Int 163:1203–1218 CrossRefGoogle Scholar
  36. Siebert L, Simkin T (2002) Volcanoes of the world: an illustrated catalog of Holocene Volcanoes and their eruptions.
  37. Smethurst L, James MR, Pinkerton H, Tawn JA (2009) A statistical analysis of eruptive activity on Mount Etna, Sicily. Geophys J Int 179:655–666 CrossRefGoogle Scholar
  38. Stefanski LA, Carroll RJ, Ruppert D (1986) Optimally bounded score functions for generalized linear-models with applications to logistic regression. Biometrika 73:413–424 Google Scholar
  39. Tanguy J-C, Condomines M, Le Goff M, Chillemi V, La Delfa S, Pantane G (2007) Mount Etna eruptions of the last 2,750 years: revised chronology and location through archeomagnetic and 226Ra–230Th dating. Bull Volcanol 70:55–83 CrossRefGoogle Scholar
  40. Tanguy J-C, Le Goff M, Arrighi S, Principe C, La Delfa S, Pantane G (2009) The history of Italian volcanoes revised by archeomagnetism. Eos 90:349–350 CrossRefGoogle Scholar
  41. Vere-Jones D (1992) Statistical methods for the description and display of earthquake catalogs. In: Walden AT, Guttorp P (eds) Statistics in the environmental and earth sciences. Edward Arnold, London, pp 220–246 Google Scholar
  42. Wadge G (1982) Steady state volcanism: evidence from eruption histories of polygenetic volcanoes. J Geophys Res 87:4035–4049 CrossRefGoogle Scholar
  43. Wang T, Bebbington M (2012) Estimating the likelihood of an eruption from a volcano with missing onsets in its record. J Volcanol Geotherm Res 243–244:14–23 CrossRefGoogle Scholar
  44. Yoshida N, Hayashi T (1990) On the robust estimation in Poisson processes with periodic intensities. Ann Inst Stat Math 42:489–507 CrossRefGoogle Scholar
  45. Yu JW, Tian GL, Tang ML (2008) Statistical inference and prediction for the Weibull process with incomplete observations. Comput Stat Data Anal 52:1587–1603 CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OtagoDunedinNew Zealand
  2. 2.Volcanic Risk SolutionsMassey UniversityPalmerston NorthNew Zealand

Personalised recommendations