The Annals of Applied Statistics

Extremes on river networks

Peiman Asadi, Anthony C. Davison, and Sebastian Engelke

Full-text: Open access


Max-stable processes are the natural extension of the classical extreme-value distributions to the functional setting, and they are increasingly widely used to estimate probabilities of complex extreme events. In this paper we broaden them from the usual situation in which dependence varies according to functions of Euclidean distance to situations in which extreme river discharges at two locations on a river network may be dependent because the locations are flow-connected or because of common meteorological events. In the former case dependence depends on river distance, and in the second it depends on the hydrological distance between the locations, either of which may be very different from their Euclidean distance. Inference for the model parameters is performed using a multivariate threshold likelihood, which is shown by simulation to work well. The ideas are illustrated with data from the upper Danube basin.

Article information

Ann. Appl. Stat., Volume 9, Number 4 (2015), 2023-2050.

Received: February 2015
Revised: July 2015
First available in Project Euclid: 28 January 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Extremal coefficient hydrological distance max-stable process network dependence threshold-based inference upper Danube basin


Asadi, Peiman; Davison, Anthony C.; Engelke, Sebastian. Extremes on river networks. Ann. Appl. Stat. 9 (2015), no. 4, 2023--2050. doi:10.1214/15-AOAS863.

Export citation


  • Asadi, P., Davison, A. C. and Engelke, S. (2015). Supplement to “Extremes on river networks.” DOI:10.1214/15-AOAS863SUPP.
  • Bienvenüe, A. and Robert, C. (2014). Likelihood based inference for high-dimensional extreme value distributions. Available at
  • Blanchet, J. and Davison, A. C. (2011). Spatial modeling of extreme snow depth. Ann. Appl. Stat. 5 1699–1725.
  • Böhm, O. and Wetzel, K.-F. (2006). Flood history of the Danube tributaries Lech and Isar in the Alpine foreland of Germany. Hydrological Sciences Journal 51 784–798.
  • Brown, B. M. and Resnick, S. I. (1977). Extreme values of independent stochastic processes. J. Appl. Probab. 14 732–739.
  • Chandler, R. E. and Bate, S. (2007). Inference for clustered data using the independence loglikelihood. Biometrika 94 167–183.
  • Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer, London.
  • Coles, S., Heffernan, J. and Tawn, J. (1999). Dependence measures for extreme value analyses. Extremes 2 339–365.
  • Coles, S. G. and Tawn, J. A. (1991). Modelling extreme multivariate events. J. R. Stat. Soc. Ser. B. Stat. Methodol. 53 377–392.
  • Cooley, D., Naveau, P. and Poncet, P. (2006). Variograms for spatial max-stable random fields. In Dependence in Probability and Statistics (P. Bertail, P. Soulier and P. Doukhan, eds.). Lecture Notes in Statist. 187 373–390. Springer, New York.
  • Cressie, N., Frey, J., Harch, B. and Smith, M. (2006). Spatial prediction on a river network. J. Agric. Biol. Environ. Stat. 11 127–150.
  • Davison, A. C. and Gholamrezaee, M. M. (2012). Geostatistics of extremes. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468 581–608.
  • de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.
  • Dieker, A. B. and Mikosch, T. (2015). Exact simulation of Brown–Resnick random fields at a finite number of locations. Extremes 18 301–314.
  • Dombry, C., Engelke, S. and Oesting, M. (2016). Exact simulation of max-stable processes. Biometrika 103. To appear.
  • Einmahl, J., Kiriliouk, A., Krajina, A. and Segers, J. (2015). An M-estimator of spatial tail dependence. J. R. Stat. Soc. Ser. B. Stat. Methodol. 77. To appear.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Springer, Berlin.
  • Engelke, S., Kabluchko, Z. and Schlather, M. (2011). An equivalent representation of the Brown–Resnick process. Statist. Probab. Lett. 81 1150–1154.
  • Engelke, S., Malinowski, A., Kabluchko, Z. and Schlather, M. (2015). Estimation of Hüsler–Reiss distributions and Brown–Resnick processes. J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 239–265.
  • Heffernan, J. E. and Tawn, J. A. (2004). A conditional approach for multivariate extreme values. J. R. Stat. Soc. Ser. B. Stat. Methodol. 66 497–546.
  • Huser, R. and Davison, A. C. (2014). Space–time modelling of extreme events. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 439–461.
  • Huser, R., Davison, A. C. and Genton, M. G. (2014). A comparative study of parametric estimators for multivariate extremes. Extremes. Under review.
  • Hüsler, J. and Reiss, R.-D. (1989). Maxima of normal random vectors: Between independence and complete dependence. Statist. Probab. Lett. 7 283–286.
  • Kabluchko, Z. (2011). Extremes of independent Gaussian processes. Extremes 14 285–310.
  • Kabluchko, Z., Schlather, M. and de Haan, L. (2009). Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37 2042–2065.
  • Kallache, M., Rust, H. W., Lange, H. and Kropp, J. P. (2010). Extreme value analysis considering trends: Application to discharge data of the Danube river basin. In Extremis: Disruptive Events and Trends in Climate and Hydrology (J. Kropp and H. Schellnhuber, eds.) 167–184. Springer, Berlin.
  • Katz, R. W., Parlange, M. B. and Naveau, P. (2002). Statistics of extremes in hydrology. Advances in Water Resources 25 1287–1304.
  • Keef, C., Svensson, C. and Tawn, J. A. (2009). Spatial dependence in extreme river flows and precipitation for Great Britain. Journal of Hydrology 378 240–252.
  • Keef, C., Tawn, J. A. and Lamb, R. (2013). Estimating the probability of widespread flood events. Environmetrics 24 13–21.
  • Keef, C., Tawn, J. and Svensson, C. (2009). Spatial risk assessment for extreme river flows. J. R. Stat. Soc. Ser. C. Appl. Stat. 58 601–618.
  • Kundzewicz, Z. W., Ulbrich, U., Brücher, T., Graczyk, D., Krüger, A., Leckebusch, G. C., Menzel, L., Pińskwar, I., Radziejewski, M. and Szwed, M. (2005). Summer floods in central Europe–Climate change track? Natural Hazards 36 165–189.
  • Merz, R. and Blöschl, G. (2005). Flood frequency regionalisation—Spatial proximity vs. catchment attributes. Journal of Hydrology 302 283–306.
  • Oesting, M., Kabluchko, Z. and Schlather, M. (2012). Simulation of Brown–Resnick processes. Extremes 15 89–107.
  • Opitz, T. (2013). Extremal $t$ processes: Elliptical domain of attraction and a spectral representation. J. Multivariate Anal. 122 409–413.
  • Padoan, S. A., Ribatet, M. and Sisson, S. A. (2010). Likelihood-based inference for max-stable processes. J. Amer. Statist. Assoc. 105 263–277.
  • Palutikof, J. P., Brabson, B. B., Lister, D. H. and Adcock, S. T. (1999). A review of methods to calculate extreme wind speeds. Meteorol. Appl. 6 119–132.
  • Renard, B. and Lang, M. (2007). Use of a Gaussian copula for multivariate extreme value analysis: Some case studies in hydrology. Advances in Water Resources 30 897–912.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
  • Rootzén, H. and Tajvidi, N. (2006). Multivariate generalized Pareto distributions. Bernoulli 12 917–930.
  • Salvadori, G. and De Michele, C. (2010). Multivariate multiparameter extreme value models and return periods: A copula approach. Water Resources Research 46 W10501.
  • Schlather, M. (2002). Models for stationary max-stable random fields. Extremes 5 33–44.
  • Schlather, M. and Tawn, J. A. (2003). A dependence measure for multivariate and spatial extreme values: Properties and inference. Biometrika 90 139–156.
  • Skøien, J., Merz, R. and Blöschl, G. (2006). Top-kriging-geostatistics on stream networks. Hydrol. Earth Syst. Sci. 10 277–287.
  • Tawn, J. A. (1988). An extreme-value theory model for dependent observations. Journal of Hydrology 101 227–250.
  • Thibaud, E. and Opitz, T. (2015). Efficient inference and simulation for elliptical Pareto processes. Biometrika 102 855–870.
  • Ver Hoef, J. M. and Peterson, E. E. (2010). A moving average approach for spatial statistical models of stream networks. J. Amer. Statist. Assoc. 105 6–18.
  • Ver Hoef, J. M., Peterson, E. and Theobald, D. (2006). Spatial statistical models that use flow and stream distance. Environ. Ecol. Stat. 13 449–464.
  • Wadsworth, J. L. and Tawn, J. A. (2014). Efficient inference for spatial extreme value processes associated to log-Gaussian random functions. Biometrika 101 1–15.
  • Wang, Y. and Stoev, S. A. (2010). On the structure and representations of max-stable processes. Adv. in Appl. Probab. 42 855–877.

Supplemental materials

  • Supplement to “Extremes on river networks”. The supplementary material contains the following: a PDF document containing the derivation of the new likelihood representation mentioned in Section 4.3.2, results of the simulation study mentioned in Section 4.3.3, and additional details germane to Section 5.3; and R code and data files to reproduce the data analysis and figures.