Abstract
Rare weather and climate events, such as heat waves and floods, can bring tremendous social costs. Climate data is often limited in duration and spatial coverage, and so climate forecasting has often turned to simulations of climate models to make better predictions of rare weather events. However very long simulations of complex models, in order to obtain accurate probability estimates, may be prohibitively slow. It is an important scientific problem to develop probabilistic and dynamical techniques to estimate the probabilities of rare events accurately from limited data. In this paper we compare four modern methods of estimating the probability of rare events: the generalized extreme value (GEV) method from classical extreme value theory; two importance sampling techniques, geneaological particle analysis (GPA) and the Giardina-Kurchan-Lecomte-Tailleur (GKLT) algorithm; as well as brute force Monte Carlo (MC). With these techniques we estimate the probabilities of rare events in three dynamical models: the Ornstein-Uhlenbeck process, the Lorenz ’96 system and PlaSim (a climate model). We keep the computational effort constant and see how well the rare event probability estimation of each technique compares to a gold standard afforded by a very long run control. Somewhat surprisingly we find that classical extreme value theory methods outperform GPA, GKLT and MC at estimating rare events.
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Carney, M., Azencott, R., Nicol, M.: Non-stationarity of summer temperature extremes in Texas. Int. J. Climatol. 40(1), 620–640 (2020)
Lucarini, V., et al.: Extremes and Recurrence in Dynamical Systems, 312 pp. Wiley, Hoboken (2016)
Bucklew, J.: Introduction to Rare Event Simulation. Springer Series in Statistics. Springer, New York (2004)
Carney, M., Kantz, H.: Robust Regional clustering and modeling of nonstationary summer temperature extremes across Germany (preprint)
Coles, S.: An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics, 4th edn. Springer, New York (2007)
Collet, P.: Statistics of closest return for some non-uniformly hyperbolic systems. Ergodic Theorem Dyn. Syst. 21, 401–420 (2001)
Giardina, C., Kurchan, J., Lecomte, V., Tailleur, J.: Simulating rare events in dynamical processes. J. Stat. Phys. 145, 787–811 (2011)
Gumbel, E.J.: Statistics of Extremes. Columbia University Press, New York (1958)
Hoskins, B., Simons, A.: A multi-layer spectral model and the semi-implicit method. Q. J. R Meteorol. Soc. 101, 637–655 (1975)
Fraedrich, K., Kirk, E., Lunkeit, F.: PUMA Portable University Model of the Atmosphere. World Data Center for Climate (WDCC) at DKRZ (2009)
Freitas, J., Freitas, A., Todd, M.: Hitting times and extreme values. Probab. Theory Relat. Fields 147(3), 675–710 (2010)
Freita, A.C., Freitas, J., Todd, M.: Speed of convergence for laws of rare events and escape rates. Stoch. Proc. App. 125, 1653–1687 (2015)
Galambos, J.: The Asymptotic Theory of Extreme Order Statistics. Wiley, Hoboken (1978)
Galfi, V., Lucarini, V., Wouters, J.: A large deviation theory-based analysis of heat waves and cold spells in a simplified model of the general circulation of the atmosphere. J. Stat. Mech. Theory Exp. 3(3), 033404 (2019). 39 pp
Gupta, C., Holland, M., Nicol, M.: Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps. Ergodic Theory Dyn. Syst. 31(5), 1363–1390 (2011)
Wouters, J., Bouchet, F.: Rare event computation in deterministic chaotic systems using genealogical particle analysis. J. Phys. A: Math. Theor. 49, 374002 (2016)
Del Moral, P., Garnier, J.: Genealogical particle analysis of rare events. Ann. App. Prob. 15(4), 2496–2534 (2005)
Ragone, F., Wouters, J., Bouchet, F.: Computation of extreme heat waves in climate models using a large deviation algorithm. PNAS 115(1), 24–29 (2018)
Giardina, C., Kurchan, J., Peliti, L.: Direct evaluation of large-deviation functions. Phys. Rev. Lett. 96, 120603 (2006)
Tailleur, J., Kurchan, J.: Probing rare physical trajectories with Lyapunov weighted dynamics. Nat. Phys. 3, 203–207 (2007)
Hall, P.: On the rate of convergence of normal extremes. J. Appl. Prob. 16(2), 433–439 (1979)
Holland, M., Nicol, M.: Stochast. Dyn. 15(4), 1550028 (2015). 23 pp
Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, Heidelberg (1980)
Lorenz, E.N.: Predictability–a problem partly solved. In: Seminar on Predictability, vol. I, ECMWF (1996)
Lorenz, E.N.: Designing chaotic models. J. Atmos. Sci. 62(5), 1574–1587 (2005)
Del Moral, P.: Feynman-Kac Formulae. Genealogical and Interacting Particle Systems with Applications. Probability and its Applications. Springer, New York (2004)
Ragone, F., Wouters, J., Bouchet, F.: Computation of extreme heat waves in climate models using a large deviation algorithm. Proc. Natl. Acad. Sci. U.S.A. 115(1), 24–29 (2018)
Rubino, G., Tuffin, B.: Introduction to Rare Event Simulation. Rare Event Simulation Using Monte Carlo Methods, pp. 1-13. Wiley, Chichester (2009)
Wouters, J., Bouchet, F.: Rare event computation in deterministic chaotic systems using genealogical particle analysis. J. Phys. A 49(37), 374002 (2016). 24 pp
Acknowledgements
We warmly thank Frank Lunkeit at Universität Hamburg for very helpful discussions and advice concerning PlaSim. MN was supported in part by NSF Grant DMS 1600780.
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Carney, M., Kantz, H., Nicol, M. (2020). Analysis and Simulation of Extremes and Rare Events in Complex Systems. In: Junge, O., Schütze, O., Froyland, G., Ober-Blöbaum, S., Padberg-Gehle, K. (eds) Advances in Dynamics, Optimization and Computation. SON 2020. Studies in Systems, Decision and Control, vol 304. Springer, Cham. https://doi.org/10.1007/978-3-030-51264-4_7
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DOI: https://doi.org/10.1007/978-3-030-51264-4_7
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