The Annals of Applied Statistics

A Markov-switching model for heat waves

Benjamin A. Shaby, Brian J. Reich, Daniel Cooley, and Cari G. Kaufman

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Heat waves merit careful study because they inflict severe economic and societal damage. We use an intuitive, informal working definition of a heat wave—a persistent event in the tail of the temperature distribution—to motivate an interpretable latent state extreme value model. A latent variable with dependence in time indicates membership in the heat wave state. The strength of the temporal dependence of the latent variable controls the frequency and persistence of heat waves. Within each heat wave, temperatures are modeled using extreme value distributions, with extremal dependence across time accomplished through an extreme value Markov model. One important virtue of interpretability is that model parameters directly translate into quantities of interest for risk management, so that questions like whether heat waves are becoming longer, more severe or more frequent are easily answered by querying an appropriate fitted model. We demonstrate the latent state model on two recent, calamitous, examples: the European heat wave of 2003 and the Russian heat wave of 2010.

Article information

Ann. Appl. Stat., Volume 10, Number 1 (2016), 74-93.

Received: February 2015
Revised: August 2015
First available in Project Euclid: 25 March 2016

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Zentralblatt MATH identifier

Latent state extremes generalized Pareto distribution extremal dependence


Shaby, Benjamin A.; Reich, Brian J.; Cooley, Daniel; Kaufman, Cari G. A Markov-switching model for heat waves. Ann. Appl. Stat. 10 (2016), no. 1, 74--93. doi:10.1214/15-AOAS873.

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  • Amengual, A., Homar, V., Romero, R., Brooks, H. E., Ramis, C., Gordaliza, M. and Alonso, S. (2014). Projections of heat waves with high impact on human health in Europe. Glob. Planet. Change 119 71–84.
  • Barriopedro, D., Fischer, E. M., Luterbacher, J., Trigo, R. M. and García-Herrera, R. (2011). The hot summer of 2010: Redrawing the temperature record map of Europe. Science 332 220.
  • Clark, R. T., Brown, S. J. and Murphy, J. M. (2006). Modeling northern hemisphere summer heat extreme changes and their uncertainties using a physics ensemble of climate sensitivity experiments. J. Climate 19 4418–4435.
  • Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer London, Ltd., London.
  • Coles, S., Heffernan, J. and Tawn, J. (1999). Dependence measures for extreme value analyses. Extremes 2 339–365.
  • Easterling, D. R., Evans, J. L., Groisman, P. Y., Karl, T. R., Kunkel, K. E. and Ambenje, P. (2000). Observed variability and trends in extreme climate events: A brief review. Bull. Am. Meteorol. Soc. 81 417–425.
  • Ferro, C. A. T. and Segers, J. (2003). Inference for clusters of extreme values. J. R. Stat. Soc. Ser. B. Stat. Methodol. 65 545–556.
  • Fischer, E. M. and Schär, C. (2010). Consistent geographical patterns of changes in high-impact European heatwaves. Nat. Geosci. 3 398–403.
  • Frich, P., Alexander, L. V., Della-Marta, P., Gleason, B., Haylock, M., Klein Tank, A. M. G. and Peterson, T. (2002). Observed coherent changes in climatic extremes during the second half of the twentieth century. Clim. Res. 19 193–212.
  • Frühwirth-Schnatter, S. (2006). Finite Mixture and Markov Switching Models. Springer, New York.
  • Furrer, E. M., Katz, R. W., Walter, M. D. and Furrer, R. (2010). Statistical modeling of hot spells and heat waves. Clim. Res. 43 191–205.
  • Gelman, A., Meng, X.-L. and Stern, H. (1996). Posterior predictive assessment of model fitness via realized discrepancies. Statist. Sinica 6 733–807.
  • Gneiting, T. and Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. J. Amer. Statist. Assoc. 102 359–378.
  • Hanlon, H. M., Morak, S. and Hegerl, G. C. (2013). Detection and prediction of mean and extreme European summer temperatures with a multimodel ensemble. J. Geophys. Res. 118 9631–9641.
  • Huth, R., Kyselỳ, J. and Pokorná, L. (2000). A GCM simulation of heat waves, dry spells, and their relationships to circulation. Clim. Change 46 29–60.
  • Karl, T. R. and Knight, R. W. (1997). The 1995 Chicago heat wave: How likely is a recurrence? Bull. Am. Meteorol. Soc. 78 1107–1120.
  • Khaliq, M. N., St-Hilaire, A., Ouarda, T. and Bobee, B. (2005). Frequency analysis and temporal pattern of occurrences of southern quebec heatwaves. Int. J. Climatol. 25 485–504.
  • Meehl, G. A. and Tebaldi, C. (2004). More intense, more frequent, and longer lasting heat waves in the 21st century. Science 305 994–997.
  • Munich Re (2003). Annual review: Natural catastrophes 2003. Available at
  • Nychka, D., Furrer, R. and Sain, S. (2014). fields: Tools for spatial data. R package version 7.1.
  • Otto, F. E. L., Massey, N., Oldenborgh, G. J., Jones, R. G. and Allen, M. R. (2012). Reconciling two approaches to attribution of the 2010 Russian heat wave. Geophys. Res. Lett. 39 L04702–L04707.
  • Palecki, M. A., Changnon, S. A. and Kunkel, K. E. (2001). The nature and impacts of the July 1999 heat wave in the midwestern United States: Learning from the lessons of 1995. Bull. Am. Meteorol. Soc. 82 1353–1367.
  • Parry, M. L., Canziani, O. F., Palutikof, J. P., van der Linden, P. J. and Hanson, C. E., eds. (2007). Climate Change 2007: Impacts, Adaptation and Vulnerability. Contribution of Working Group II to the Fourth Assessment Report of the IPCC. Cambridge Univ. Press, Cambridge.
  • Peng, R. D., Bobb, J. F., Tebaldi, C., McDaniel, L., Bell, M. L. and Dominici, F. (2011). Toward a quantitative estimate of future heat wave mortality under global climate change. Environ. Health Perspect. 119 701–706.
  • Reich, B. J., Shaby, B. A. and Cooley, D. (2014). A hierarchical model for serially-dependent extremes: A study of heat waves in the western US. J. Agric. Biol. Environ. Stat. 19 119–135.
  • Robine, J. M., Cheung, S. L. K., Le Roy, S., Van Oyen, H., Griffiths, C., Michel, J. P. and Herrmann, F. R. (2008). Death toll exceeded 70,000 in Europe during the summer of 2003. C. R., Biol. 331 171–178.
  • Schär, C., Vidale, P. L., Lüthi, D., Frei, C., Häberli, C., Liniger, M. A. and Appenzeller, C. (2004). The role of increasing temperature variability in European summer heatwaves. Nature 427 332–336.
  • Shaby, B. A., Reich, B. J., Cooley, D. and Kaufman, C. G. (2015). Supplement to “A Markov-switching model for heat waves.” DOI:10.1214/15-AOAS873SUPP.
  • Sibuya, M. (1959). Bivariate extreme statistics. Ann. Inst. Statist. Math. 11 195–210.
  • Smith, R. L. (1985). Maximum likelihood estimation in a class of nonregular cases. Biometrika 72 67–90.
  • Smith, R. L., Tawn, J. A. and Coles, S. G. (1997). Markov chain models for threshold exceedances. Biometrika 84 249–268.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A. (2002). Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B. Stat. Methodol. 64 583–639.

Supplemental materials

  • Additional analysis. The Supplement contains additional exploratory analysis related to the case study, a sensitivity analysis for the prior distribution of the GPD threshold $u$, and results of the model run on 2003 temperatures at several additional sites throughout Western Europe.