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- 1. Arthur CHARPENTIER, modeling heat waves Modeling heat-waves : return period for non-stationary extremes Arthur Charpentier Université du Québec à Montréal charpentier.arthur@uqam.ca http ://freakonometrics.blog.free.fr/ Changement climatique et gestion des risques, Lyon, Movember 2011 1
- 2. Arthur CHARPENTIER, modeling heat waves Motivation“Tous nos scénarios à dix ans montrent une aggravation des tempêtes,inondations, sécheresses. Les catastrophes centennales vont devenir plusfréquentes.” Jean-Marc Lamère, délégué général de la Fédération Française desSociétés d’Assurances, 2003.Climate change, from Third IPCC Agreement ,2001 : 2
- 3. Arthur CHARPENTIER, modeling heat waves Motivation“there is no longer any doubt that the Earth’s climate is changing [...] globally,nine of the past 10 years have been the warmest since records began in 1861”,February 2005, opening the conference Climate change : a global, national andregional challenge, Dennis Tirpak.source : Meehklet al. (2009). 3
- 4. Arthur CHARPENTIER, modeling heat waves The European heatwave of 2003Third IPCC Assessment, 2001 : treatment of extremes (e.g. trends in extremehigh temperature) is “clearly inadequate”. Karl & Trenberth (2003) noticedthat “the likely outcome is more frequent heat waves”, “more intense and longerlasting” added Meehl & Tebaldi (2004).In Nîmes, there were more than 30 days with temperatures higher than 35◦ C(versus 4 in hot summers, and 12 in the previous heat wave, in 1947).Similarly, the average maximum (minimum) temperature in Paris peaked over35◦ C for 10 consecutive days, on 4-13 August. Previous records were 4 days in1998 (8 to 11 of August), and 5 days in 1911 (8 to 12 of August).Similar conditions were found in London, where maximum temperatures peakedabove 30◦ C during the period 4-13 August(see e.g. Burt (2004), Burt & Eden (2004) and Fink et al. (2004).) 4
- 5. Arthur CHARPENTIER, modeling heat waves q q q q q q q q q q q 20 q q q q q Temperature in °C q q q q q q q q q q q q q q q q q q q q q 15 q q q q q q q q q q q q q q q q q q q q q q q q 10 q juil. 02 juil. 12 juil. 22 août 01 août 11 août 21 août 31 5
- 6. Arthur CHARPENTIER, modeling heat waves Which temperature might be interesting ?Karl & Knight (1997) , modeling of the 1995 heatwave in Chicago : minimumtemperature should be most important for health impact (see also Kovats &Koppe (2005)), several nights with no relief from very warm nighttime 6
- 7. Arthur CHARPENTIER, modeling heat waves Which temperature might be interesting ?Karl & Knight (1997) , modeling of the 1995 heatwave in Chicago : minimumtemperature should be most important for health impact (see also Kovats &Koppe (2005)), several nights with no relief from very warm night-time 7
- 8. Arthur CHARPENTIER, modeling heat waves Modeling temperatureConsider the following decomposition Yt = µt + St + Xtwhere• µt is a (linear) general tendency• St is a seasonal cycle• Xt is the remaining (stationary) noise 8
- 9. Arthur CHARPENTIER, modeling heat waves Nonstationarity and linear trendConsider a spline and lowess regression 9
- 10. Arthur CHARPENTIER, modeling heat waves Nonstationarity and linear trendor a polynomial regression,and compare local slopes, 10
- 11. Arthur CHARPENTIER, modeling heat waves Nonstationarity and linear trendSeveral authors (from Laneet al. (1994) to Blacket al. (2004)) have tried toexplain global warming, and to ﬁnd explanatory factors.As pointed out in Queredaet al. (2000), the “analysis of the trend is diﬃcultand could be biased by non-climatic processes such as the urban eﬀect”. In fact,“most of the temperature rise could be due to an urban eﬀect” : global warmingcan be understood as one of the consequence of “global pollution” (see alsoHoughton (1997) or Braunet al. (2004) for a detailed study of the impact oftransportation). 11
- 12. Arthur CHARPENTIER, modeling heat waves Quantile regression to describe ’extremal’ temperatureSee Yan (2002), Meehl & Tebaldi (2004), or Alexanderet al. (2006).Least square used to estimate a linear model for E(Yt ), T T min (Yt − [β0 + β1 t])2 = min R(Yt − [β0 + β1 t]) β0 ,β1 β0 ,β1 t=1 t=1where R(x) = x2 .Quantile regression can be used, let p ∈ (0, 1), and consider T min R(Yt − [β0 + β1 t]) β0 ,β1 t=1where x · (p − 1) if x < 0 Rp (x) = x · (p − 1(x < 0)) = x · p if x > 0 12
- 13. Arthur CHARPENTIER, modeling heat wavesRemark : E(Y ) = argmin E (Y − γ)2 = argmin (Y − γ)2 dF (y)while γ +∞ Qp (Y ) = argmin [p − 1] · [γ − x]dF (x) + p · [x − γ]dF (x) . −∞ γ5%, 25%, 75%, and 95% quantile regressions, 13
- 14. Arthur CHARPENTIER, modeling heat waves Quantile regression to describe ’extremal’ temperatureslopes (β1 ) of quantile regressions, as functions of probability q ∈ (0, 1),Remark : constant rate means that scenario 1 is realistic in the Third IPCCAgreement, 2001 : 14
- 15. Arthur CHARPENTIER, modeling heat waves Quantile regression to describe ’extremal’ temperatureslopes (β1 ) of quantile regressions, as functions of probability q ∈ (0.9, 1), 15
- 16. Arthur CHARPENTIER, modeling heat waves Regression for yearly maximaLet Mi denote the highest minimal daily temperature observed during year i, 16
- 17. Arthur CHARPENTIER, modeling heat waves Regression for yearly maximaLet Mi denote the highest minimal daily temperature observed during year i, 17
- 18. Arthur CHARPENTIER, modeling heat waves Linear trend, and Gaussian noiseBenestad (2003) or Redner & Petersen (2006)temperature for a given (calendar) day is an “independent Gaussian randomvariable with constant standard deviation σ and a mean that increases at constantspeed ν”In the US, ν = 0.03◦ C per year, and σ = 3.5◦ CIn Paris, ν = 0.027◦ C per year, and σ = 3.23◦ CAssuming independence it is possible to estimate return periods 18
- 19. Arthur CHARPENTIER, modeling heat waves Linear trend, and Gaussian noiseReturn period of k consecutive days with temperature exceeding s, 19
- 20. Arthur CHARPENTIER, modeling heat waves The seasonal componentSeasonal pattern during the yearlydid not use a cosine function to model St but a spline regression (on circulardata), 20
- 21. Arthur CHARPENTIER, modeling heat waves The residual part (or stationary component)Let Xt = Yt − β0 + β1 t + St 21
- 22. Arthur CHARPENTIER, modeling heat waves The residual part (or stationary component)Xt might look stationary, 22
- 23. Arthur CHARPENTIER, modeling heat waves The residual part (or stationary component)but the variance of Xt seems to have a seasonal pattern, 23
- 24. Arthur CHARPENTIER, modeling heat waves The residual part (or stationary component)Dispersion and variance of residuals, graph of |Xt | series 24
- 25. Arthur CHARPENTIER, modeling heat waves A spatial model for residuals ?Consider here two series of temperatures (e.g. Paris and Marseille). 25
- 26. Arthur CHARPENTIER, modeling heat waves A spatial model for residuals ?Consider here two series of temperatures (e.g. Paris and Marseille). 26
- 27. Arthur CHARPENTIER, modeling heat waves A spatial model for residuals ?Consider here two series of temperatures (e.g. Paris and Marseille), and theirstationary residuals 27
- 28. Arthur CHARPENTIER, modeling heat waves Tail dependence indicesFor the lower tail L(z) = P(U < z, V < z)/z = C(z, z)/z = P(U < z|V < z) = P(V < z|U < z),and for the upper tail R(z) = P(U > z, V > z)/(1 − z) = P(U > z|V > z).see Joe (1990). Deﬁne λU = R(1) = lim R(z) and λL = L(0) = lim L(z). z→1 z→0such that −1 −1 λL = lim P X ≤ FX (u) |Y ≤ FY (u) , u→0and −1 −1 λU = lim P X > FX (u) |Y > FY (u) . u→1 28
- 29. Arthur CHARPENTIER, modeling heat waves Tail dependence indicesLedford & Tawn (1996) suggested the following alternative approach,• for independent random variables, P(X > t, Y > t) = P(X > t) × P(Y > t) = P(X > t)2 ,• for comonotonic random variables, P(X > t, Y > t) = P(X > t) = P(X > t)1 ,Assume that P(X > t, Y > t) ∼ P(X > t)1/η as t → ∞, where η ∈ (0, 1] will be atail index.Deﬁne 2 log(1 − z) 2 log(1 − z) χU (z) = − 1 and χL (z) = −1 log C (z, z) log C(z, z)with ηU = (1 + limz→0 χU (z))/2 and ηL = (1 + limz→0 χL (z))/2 sont appelésindices de queue supérieure et inférieure, respectivement. 29

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