, Volume 21, Issue 4, pp 601–628 | Cite as

Prediction of catastrophes in space over time

  • Anastassia BaxevaniEmail author
  • Richard Wilson


Predicting rare events, such as high level up-crossings, for spatio-temporal processes plays an important role in the analysis of the occurrence and impact of potential catastrophes in, for example, environmental settings. Designing a system which predicts these events with high probability, but with few false alarms, is clearly desirable. In this paper an optimal alarm system in space over time is introduced and studied in detail. These results generalize those obtained by de Maré (Ann. Probab. 8, 841–850, 1980) and Lindgren (Ann. Probab. 8, 775–792, 1980, Ann. Probab. 13, 804–824, 1985) for stationary stochastic processes evolving in continuous time and are applied here to stationary Gaussian random fields.


Gaussian random fields Spatio-temporal models Upcrossings Palm distribution Alarms Optimal prediction Catastrophes Likelihood ratio Reliability 

AMS 2000 Subject Classifications

Primary—60G10, 60G25, 60G70 


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The first author would like to thank Dr Manuel Scotto for an invitation to the Department of Mathematics at the University of Aveiro, Portugal that lead to the idea of the present paper.


  1. Adler, R.J., Taylor, J.E.: Random fields and geometry springer monographs in mathematics (2007)Google Scholar
  2. Ailliot, P., Baxevani, A., Cuzol, A., Monbet, V., Raillard, N.: Space-time models for moving fields with an application to significant wave height fields. Environmetrics 22, 354–369 (2011)MathSciNetCrossRefGoogle Scholar
  3. Anderson, T.W.: Introduction to Multivariate Statistical Analysis. Wiley, New York (1958)zbMATHGoogle Scholar
  4. Antunes, K.F., Turkman, M., Turkman, M.A.: A Bayesian approach to event prediction. J. Time Ser. Anal. 24(6), 631–646 (2003)MathSciNetCrossRefGoogle Scholar
  5. Baxevani, C., Rychlik, I., Borget, A.: Spatial models for the variability of the significant wave height on the world oceans. Int. J. Offshore Polar Eng. 18, 1–7 (2008)Google Scholar
  6. Baxevani, I., Rychlik, A.: Fatigue life prediction for a vessel sailing the north atlantic route. Probab. Eng. Mech. 22, 159–169 (2007)CrossRefGoogle Scholar
  7. Baxevani, I., Rychlik, A., Wilsson, R.: A new method for modelling the space variability of significant wave height. Extremes 8, 267–294 (2005)MathSciNetCrossRefGoogle Scholar
  8. Baxevani, S., Rychlik, I., Caires, A.: Spatio-temporal statistical modelling of significant wave height. Environmetrics 20, 14–31 (2009)MathSciNetCrossRefGoogle Scholar
  9. Yu, K.B.: On the number of exits across the boundary of a region by a vector stochastic process. Theor. Probability Appl. 13, 320–324 (1968)CrossRefGoogle Scholar
  10. Hopkinson, M., Alber, A.P., Covich, C.S., Lugo, A.E., Van Bloem, S.J.: Forecasting effects of sea-level rise and windstorms on coastal and inland ecosystems. Front. Ecol. Environ. 6, 255–263 (2008)CrossRefGoogle Scholar
  11. Buishand, A., de Haan, L., Zhou, C.: On spatial extremes; with application to a rainfall problem. Ann. Appl. Stat. 2, 624–642 (2008)MathSciNetCrossRefGoogle Scholar
  12. De Maré, J.: Optimal prediction of catastrophes with applications to Gaussian processes. Ann. Probab. 8, 841–850 (1980)MathSciNetCrossRefGoogle Scholar
  13. Friederichs, P.: Statistical downscaling of extreme precipitation events using extreme value theory. Extremes 13, 109–132 (2010)MathSciNetCrossRefGoogle Scholar
  14. Fuentes, I., Izzeldin, A.-M., Kalotychou, E.: On forecasting daily sock volatility: the role of intraday information and market conditions. Int. J. Forecast. 25, 259–281 (2009)CrossRefGoogle Scholar
  15. Koop, G., Tole, L.: Measuring the health effects of air pollution to what extent can we really say that people are dying from bad air?. J. Environ. Econ. Manag. 47, 30–54 (2004)CrossRefGoogle Scholar
  16. Lee, J.M.: Introduction to smooth manifolds springer graduate texts in mathematics (2013)Google Scholar
  17. Lindgren, G.: Model processes in nonlinear prediction with application to detection and alarm. Ann. Probab. 8, 775–792 (1980)MathSciNetCrossRefGoogle Scholar
  18. Lindgren, G.: Optimal prediction of level crossings in gaussian processes and sequences. Ann. Probab. 13, 804–824 (1985)MathSciNetCrossRefGoogle Scholar
  19. Costa, C., Scotto, M.G., Pereira, I.: Optimal alarm systems for FIAParch processes. REVSTAT 8, 37–55 (2010)MathSciNetzbMATHGoogle Scholar
  20. Niedzielski, T., Kosek, W.: Forecasting sea level anomalies from TOPEX/poseidon and Jason-1 satellite altimetry. J. Geod. 83, 469–476 (2009)CrossRefGoogle Scholar
  21. Smith, R.L., Davis, J.M., Sacks, J., Speckman, P., Styer, P.: Regression models for air pollution and daily mortality: analysis of data from Birmingham, Alabama. Environmetrics 11, 719–743 (2000)CrossRefGoogle Scholar
  22. Scotto, A.M., Alonso, M.G., Barbosa, S.M.: Clustering time series of sea levels: extreme value approach. Waterway, Port, Coastal, and Ocean Engrg. 136, 215–225 (2010)CrossRefGoogle Scholar
  23. Svensson, A., Holst, J.: Prediction of high water levels in the Baltic. J. of the Turkish Stat Assoc. 1(3), 33–46 (1998)Google Scholar
  24. Svensson, J., Lindquist, R., Holst, A., Lindgren, G.: Optimal prediction of catastrophes in autoregressive moving-average processes. J Time Ser. Anal. 17(5), 511–531 (1995)MathSciNetCrossRefGoogle Scholar
  25. Thomas, L.C.: A survey of credit and behavioural scoring: forecasting financial risk of lending to consumers. Int. J. Forecast. 16, 149–172 (2000)CrossRefGoogle Scholar
  26. Tobias, A., Scotto, M.G.: Prediction of extreme ozone levels in Barcelona, Spain. Environ. Monit Assess. 100, 23–32 (2005)CrossRefGoogle Scholar
  27. Turkman, K.F., Amaral Turkman, M.A.: Optimal screening methods. J. R. Statist. Soc. B 51(2), 287–295 (1989)MathSciNetzbMATHGoogle Scholar
  28. Weatherford, L.R., Kimes, S.E.: Forecasting methods for hotel revenue management an evaluation. Int. J. Forecast. 19(3), 405–419 (2003)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CyprusNicosiaCyprus
  2. 2.School of Mathematics and PhysicsThe University of QueenslandSt LuciaAustralia

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