Slides lyon-2011
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This document summarizes Arthur Charpentier's presentation on modeling heat waves under climate change. It discusses how heat waves are becoming more frequent and intense due to global warming. It also examines different approaches to modeling temperature data, including quantile regression to analyze extremes, modeling the trend, seasonal, and residual components of temperature, and the potential for spatial modeling of residuals between locations. Tail dependence indices are also introduced as a way to quantify dependence in temperature extremes between locations.
![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 and
regional challenge, Dennis Tirpak.
source : Meehklet al. (2009).
3](https://image.slidesharecdn.com/slides-lyon-3-111114191033-phpapp02/85/Slides-lyon-2011-3-320.jpg)
![Arthur CHARPENTIER, modeling heat waves
Quantile regression to describe ’extremal’ temperature
See 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=1
where R(x) = x2 .
Quantile regression can be used, let p ∈ (0, 1), and consider
T
min R(Yt − [β0 + β1 t])
β0 ,β1
t=1
where
x · (p − 1) if x < 0
Rp (x) = x · (p − 1(x < 0)) =
x · p if x > 0
12](https://image.slidesharecdn.com/slides-lyon-3-111114191033-phpapp02/85/Slides-lyon-2011-12-320.jpg)
![Arthur CHARPENTIER, modeling heat waves
Remark :
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) .
−∞ γ
5%, 25%, 75%, and 95% quantile regressions,
13](https://image.slidesharecdn.com/slides-lyon-3-111114191033-phpapp02/85/Slides-lyon-2011-13-320.jpg)
![Arthur CHARPENTIER, modeling heat waves
Tail dependence indices
Ledford & 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 a
tail 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 appelés
indices de queue supérieure et inférieure, respectivement.
29](https://image.slidesharecdn.com/slides-lyon-3-111114191033-phpapp02/85/Slides-lyon-2011-29-320.jpg)
![Arthur CHARPENTIER, modeling heat waves
Long range dependence and regression
Gray, Zhang & Woodward (1989) proposed an extension to model persistent
seasonal series, using Gegenbauer’s polynomial Gd (·), defined as
n
(−2)n Γ(n + d)Γ(n + 2d) 2 −α+1/2 d
n
Gd (x)
n = (1 − x ) (1 − x2 )n+d−1/2 .
n! Γ(d)Γ(2n + 2d) dxn
and such that Gd (·) is the unique polynomial of degree n such that
n
∞
(1 − 2uZ + Z 2 )d = Gd (u) · Z n .
n
n=0
If d ∈ (0, 1/2), and |u| < 1 then
ρ(h) ∼ h2d−1 sin([π − arccos(u)]h) as h → ∞.
43](https://image.slidesharecdn.com/slides-lyon-3-111114191033-phpapp02/85/Slides-lyon-2011-43-320.jpg)
Slides lyon-2011
- 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 plus fréquentes.” Jean-Marc Lamère, délégué général de la Fédération Française des Sociétés 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 and regional challenge, Dennis Tirpak. source : Meehklet al. (2009). 3
- 4. Arthur CHARPENTIER, modeling heat waves The European heatwave of 2003 Third IPCC Assessment, 2001 : treatment of extremes (e.g. trends in extreme high temperature) is “clearly inadequate”. Karl & Trenberth (2003) noticed that “the likely outcome is more frequent heat waves”, “more intense and longer lasting” 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 over 35◦ C for 10 consecutive days, on 4-13 August. Previous records were 4 days in 1998 (8 to 11 of August), and 5 days in 1911 (8 to 12 of August). Similar conditions were found in London, where maximum temperatures peaked above 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 : minimum temperature 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 : minimum temperature 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 temperature Consider the following decomposition Yt = µt + St + Xt where • 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 trend Consider a spline and lowess regression 9
- 10. Arthur CHARPENTIER, modeling heat waves Nonstationarity and linear trend or a polynomial regression,and compare local slopes, 10
- 11. Arthur CHARPENTIER, modeling heat waves Nonstationarity and linear trend Several authors (from Laneet al. (1994) to Blacket al. (2004)) have tried to explain global warming, and to find explanatory factors. As pointed out in Queredaet al. (2000), the “analysis of the trend is difficult and 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 warming can be understood as one of the consequence of “global pollution” (see also Houghton (1997) or Braunet al. (2004) for a detailed study of the impact of transportation). 11
- 12. Arthur CHARPENTIER, modeling heat waves Quantile regression to describe ’extremal’ temperature See 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=1 where R(x) = x2 . Quantile regression can be used, let p ∈ (0, 1), and consider T min R(Yt − [β0 + β1 t]) β0 ,β1 t=1 where x · (p − 1) if x < 0 Rp (x) = x · (p − 1(x < 0)) = x · p if x > 0 12
- 13. Arthur CHARPENTIER, modeling heat waves Remark : 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) . −∞ γ 5%, 25%, 75%, and 95% quantile regressions, 13
- 14. Arthur CHARPENTIER, modeling heat waves Quantile regression to describe ’extremal’ temperature slopes (β1 ) of quantile regressions, as functions of probability q ∈ (0, 1), Remark : constant rate means that scenario 1 is realistic in the Third IPCC Agreement, 2001 : 14
- 15. Arthur CHARPENTIER, modeling heat waves Quantile regression to describe ’extremal’ temperature slopes (β1 ) of quantile regressions, as functions of probability q ∈ (0.9, 1), 15
- 16. Arthur CHARPENTIER, modeling heat waves Regression for yearly maxima Let Mi denote the highest minimal daily temperature observed during year i, 16
- 17. Arthur CHARPENTIER, modeling heat waves Regression for yearly maxima Let Mi denote the highest minimal daily temperature observed during year i, 17
- 18. Arthur CHARPENTIER, modeling heat waves Linear trend, and Gaussian noise Benestad (2003) or Redner & Petersen (2006) temperature for a given (calendar) day is an “independent Gaussian random variable with constant standard deviation σ and a mean that increases at constant speed ν” In the US, ν = 0.03◦ C per year, and σ = 3.5◦ C In Paris, ν = 0.027◦ C per year, and σ = 3.23◦ C Assuming independence it is possible to estimate return periods 18
- 19. Arthur CHARPENTIER, modeling heat waves Linear trend, and Gaussian noise Return period of k consecutive days with temperature exceeding s, 19
- 20. Arthur CHARPENTIER, modeling heat waves The seasonal component Seasonal pattern during the yearly did not use a cosine function to model St but a spline regression (on circular data), 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 their stationary residuals 27
- 28. Arthur CHARPENTIER, modeling heat waves Tail dependence indices For 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→0 such that −1 −1 λL = lim P X ≤ FX (u) |Y ≤ FY (u) , u→0 and −1 −1 λU = lim P X > FX (u) |Y > FY (u) . u→1 28
- 29. Arthur CHARPENTIER, modeling heat waves Tail dependence indices Ledford & 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 a tail 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 appelés indices de queue supérieure et inférieure, respectivement. 29
- 30. Arthur CHARPENTIER, modeling heat waves Tail functions L(·) and R(·) Gaussian copula L and R concentration functions Student t copula L and R concentration functions 1.0 1.0 1.0 1.0 q q q q q q qqqq qq qq q q q q q q q q q q q q qqqq q q q q q q qq qq qq q q qq q q q q q q q q qq q qqq q q q q q q q q q q qq qq q q q q q qq q q q q qqq q q qq q q q q q q q q q q q q q q q q qqq q q qq q q q q qq q q q q qq q qq q qq qqq q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qqqq qq q q q q qq q qq q q qq q qq qq q qq q q q q q q q q q q q q qq q qq q q qq q qq q q qqq q qq qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq qq q q q qq q qq q q q q q q q qq q q q q qq qq q q q q q qqq q q qq q q q qqq q q q q q q q q q q q q qq q q qq qq qq q q q q q q q q q q q q q qq q q q q q q q qq q q q q q q qq qq q qq q q q qq q q q q q q q q q q q q q qq q q q q q qq qq q q q q qq qq qq q qq q q q qq q q qq q qq qq q q q q q q q q q q q qq q q q q q q qqq q q q q qq qq q q qq q qq q qq q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q qq q q q q q q q q q q q q q qq q q qq q q q q q q q q q qq qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q qq q q q q q q q q qq q qq q qq q qq q qq q qq qq q q q q qqq q q q q q qq q q q qq qq q qq q q q qq q q q q qq q q q q q q q q q qq q q q qq q q qq q q q q q q qq q 0.8 q 0.8 0.8 0.8 q q q q qq q q q q q q q q qqq qq q q q q qq q q q q qq qq q qq q q q qq q q q q qq q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qq qq q qq q q q q q q q q q q qq qq qq qq q q q q q q q q q q q qqq q q q q q q qq q q q q q q qq q q q qq q q q q q qq q q qq q qq q q qqq q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q qq q q q q q q q qq q q qqq qq q q q q q q q q q q qq q q q qq q q q qq q q q q q q q q qq q q qq q qq q q qq q qq qq qq q qq q q q q q q q q q q q q q q q q qq q q q q q q q qqq q qq qq q q q q q q q q q q q STUDENT (df=3) q qq q q qq q qq q q q q q q q q qq q q q q q q q q qq q q q qqq q q qq q q q q q q q q q q qq q q qq q q q qq q q q q q q q q q q q GAUSSIAN q q q qq q qq q q q q q q q qq q q q q q q q q qq q q q qq q q q q q q q q q q q q qqq q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q qq qq q q q q q q q q q q q q qq q q q q q q q qq q q q qq qq q qq q qq q q q q q q qq qq q q q q qq q q q q q q q q q qq q qq qqq q qq qqq qqq q qq q qq q q q q q q q q q q qq q q qq q q q q q q q q qq q q q q q qq q q q qq qq q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q qq q q q q q q q q q qq q q qq qq q q q q q qqq q q qq q q q q q q qq q q q q q q q q q q q q q qq q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q 0.6 0.6 0.6 0.6 q q q qq q q q q q qq q qq q q q q q q q q q q q qq q q q qq q q q qq qq q q q q q q q q q q q q q qq q q q q q q q q qq q q q qq q q q qq qq q q q q q q qq q qq q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q qqq q q q q q q q q q q q q qq q q q q q qq q q q qq q q q qq q q q q q q q q q q q qq q q q qq q q q qq qq q q q q qq q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q qq q qq q q q q q q qq qq q q q q qq q q q q q qq q q q q q q q q q q q q q qq q q q qq q q q q q q qq q q q q q q q q q q qq q q q q q qq q q q q q q q qq q q q q q q q qq q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q qq q q q q q q q q qq q qq q qq q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q qq q qq q q q q q q q q q qq q q q q q q q qq qq qq q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q qq q qq q q q qq q q q q qq q q q q q q qq q qq q q q q q q qq q qq qq qq q q q q q q q q q q qq q q qq qq q q q qq q qqq q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q qq q q q q q q q qq q q qq q qq q q q qq q q q q q qq q q q q qq q q q q q q q q qq q q qq q q q q q qqq q qq q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qqq q q q 0.4 0.4 0.4 0.4 qq q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q qq q q q q q qq q q qq q q q q q q q q q q q q q q q q q qq q q qq q q q q q q qq q q q q q q q q q q q q q q q qq q q q qq q q q q qq q q qq q qq q q q q q q qq q qq q q q q q q q q q q q q qq q qq q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q qq q q q qq qq q qq q q q q q qq q q q q q q q q q q q q qq q q q q q qq qq q q q q q q q q q qq q q qq q q q q q q qqqq q q q qq q q q q q qq q q q q qq q qq q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q qq q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q qq q q q q qq q q q qq q q q q q q q q q q q q q q q qq q q q q q qq qq q q q q q q qq qq q q q q q q q q qqq q q q q q q qq q qq q q q q qq q q q qq q q qq qq qq q q q q q q q q q q q qqq q q qq qq qq q q q q q q qqq q q q q qq q q q q qq q q q q q q q q q q q q q q q q q qq q q qq q qq q q q q q q q qq q q q q q q q qqq q q q q q q q q q q qq q q q q qq q q qq q q q q q q q qq qq qqq q q q q q q q q q q q qq q q qq q q q q q q q q q qq q q qq q q q q q q qq q qqq q qqq q q q qq qq q qq qq qq q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q qq q q qq q qq q q qq q q q q q q qq q q q q q q q q q q q q q 0.2 0.2 0.2 0.2 q q q q q q qq q q q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q qq q qq q q q q qq q qq q q q q q q q q q q q q q q q q qq q qq q q qq q qq q q q qq q q q q qq q qqq q q q q q q q qq q q q qq q q q q q q q q q q q q qq q q q q q q q q qq q q q q q qq q q qq q q q qq q qq q q qq q q q q q q q q q q q q q q q q qq q q qq q q q q qq q q q q q q qq q qq q q q q q q q q q q q q qq q q qq q qq q q q q q qq q q q qqq q q q q q q q q q q qq q q q qq q q q q qq q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q qq qq q q q q q q qq qq q q q q q q qq q q qqq qq q q q qq q q q q q q qq q q q q q qq q qq q q q qq q q q q q q q q q q q q qq q q q q q q qq qq q q q q qqq q qq q q qq q qqq q q q q q qq q q q q q q q q qq qq qq q q q q q q q q q q q q qq q q q q q q qq q q q qq q q qq q q qq q q q q q qq q q q q q q qq q q q qqq q q q q q q q q q q q q q q q q q q q q q q qq qq q q qq q q q qqq q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q q q q qq q qq q q q q q q q q qq q q q q q q q q q qqq q q q q q q q q q q qq q q q q q qq q q q q qq q q q q q q q q q q q qq q q q q qq q q qqq q q q q qq q q q q q q q q q qq q q qq q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q qq q qq q q qq q q q qq qqq q qq q q q q q q q q q q q q q qqq qq q qq q q q q q qqq q q q qq q q q q q qq q qq qq q q q qq q qq q q q q qq qq q q qqq q q q qqqq q q q 0.0 0.0 0.0 0.0 q q q q q qqq qq q q q q q q q qq q qq q q qq q q q q qq q qq q q qq q qq q q q qq q q qqq q q qq q q q q q q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 L function (lower tails) R function (upper tails) L function (lower tails) R function (upper tails) Clayton copula L and R concentration functions Gumbel copula L and R concentration functions 1.0 1.0 1.0 1.0 q q q q q q qq q q qq q q q q q q q q q q q qq qqqqq qq q qq q qq q qq q q q q q q q q q q qq q qq q qq q qq q q q q q qq q q q qq qq q 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q qq q q q q q q qq q q qq q q q q q q q q q q qqqq q qq q q q q q 0.0 0.0 0.0 q 0.0 qq q q q q q q q qqq q qq qq qq qq q qq q q q q q qq q qq q qq q q q q qq q q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 L function (lower tails) R function (upper tails) L function (lower tails) R function (upper tails) 30
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Arthur CHARPENTIER, modeling heat waves Tail functions χL (·) and χU (·) Gaussian copula Chi dependence functions Student t copula Chi dependence functions 1.0 1.0 1.0 1.0 qq q q qq q q q q q q q q q q q qq q q qqq qq q q q q q q qq qqq qq qqq q q q q q qq q q q q q q q q q q q qq qqq q q q q q qq qq q q qq q qq q q q q qq qq q q q q q qq q q qq q q q q q q qq q q qq q q q qq q q q q q q q q q q q qq q qqqqq q q qq q q q q q q qq qq q q q q q q q q q qq qq qq q q q q qq q qq q q qq q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q qq qq q q q q qq q q qq q q q q qq q qq q q q q q qq q q q q q q q q q q qq q q q qq q q q q q qq qq q q q qqq q q q q q q q qq q q q q q qq q qq q q q q q q q q q q q q q qq qq q q q q q q q qq qq qq q q q q q q q q q q q q q q q q qq qq q q q q q q q qq q q qq q q q qq q q q q q q q q q q q q qq q q q q qq q q q qq q q q q q qq q q q q q qqq q q qq q q q q q qq q q q q q q q qq q q q q qq q q q q qq qq qq q q q qq q q q q q q qq q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q qq qq q q q q qq q q q qq q q q q q qq q q q q q q q q q q q q q q qq q q q q q qqq q qq q q q q q q q qq q q qq q q q q qq qq q qq q qq q q q qq q qq q q q q q q q q q q q q q q q qq qq q q q q q q q q qq qq q q q q q qq qq q q q q q qq q q qq q qqq qqq q q q q q q q q q qq q q q q q q q q q q qq q q q q q q qq q q q 0.8 0.8 0.8 0.8 qq q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q qq q q qq q q q q q qq q q q q qq q q q q q qq q qq q qq qq q q qq q q qq qq q q q q q q q q q q q qq q q q q q qq q q q q q q q qq qqq q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q qq q q q q q qq qq q q q qq qqq q q qqq qq qq q qq q q q q q q q qq q q q q q q q q qq q qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q qq qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q qq qq q q qq q q q q q qq q qq q q q qq q q q qq q qq qq q q q q q q qq q q q q q q q qq q q q q q q q q qq q q q q q q q qqq q q q q q q qq q qq q q q q qq q q q q q q q qq q q qq q qq q q q qq q q q q q q q q q q q qq q q q q q q q q qq qq q q q q qq q q q q q qq q q q q qq q q qq q q q q qq q q q qqq q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q qq q q q q qq q q q qq q q q q q q q qq qq q q qq q q q qq q q q q q q q qq q q q q q q q q q q q q qq q q q qq qq qq q q q q q q q q q q q qq q q q q q q q q q q q qq q q q q qq q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q STUDENT (df=3) q q q qq q q q q q q q q q q qq qqq qq q q qq q q q q q qq q q q q qq q q q q q q q q q q qq q q q q q qq qq qqq qq qq qq q q qq q qq q qq q q q q q q GAUSSIAN q q q q qq qq q q q q q q q q qq q q q q qq q q q q q q q q qq q q q q q q qq q q q q q q 0.6 0.6 0.6 0.6 q q q q q q q q q q q q q qq q q q q q q q q qq q qq q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q q q qq q q q qq q qqq q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q qq qq q q q q q q q q qq q qq q q q q q q q qq q q q qq q q q qqq q q qq q q q q qq q q q q q qq qq q q q q q q q q qq qq q q q qq q q q q q q q qq q q q q qq q q q q qq q q qq q q q q q q q q qq q q q q q q q q q qq q q qq q q q q q qq q q q qq q q q qq q q q q qq qq q q q q q q q qq q qq q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q qq q q q qq q q q q q q q qq q q q q qq q q qq qq q q q q q q q q q qq q q qq q q qq q q qq q qq q q q qq qq q q q q q q q qq q q q q q q q q q q qq q q q qq q q qq q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q qq q q q q qq qq q q q q qqqq q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq qqq q qq q q q q q q qqq q qq q q q q qq q q q qq qq q q q q q q q qq q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q qqq q q q q q q q qqq q q q q qq q q q qq q q qq q q q qqqqq qq q q qq q q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q qq q q qq q q q q q q q q q q q q q q qq q q q q q q q q qq q q q q q q q q q q q qq qq q q q q q q qq q q q qq q q q q qq q 0.4 0.4 0.4 0.4 qq q q q q q q q qqq q q q q q qq qq q q q q q q q q q q q qq q q q qq qq q q q q q qqq qq q q q q q q q q q q qq q q q q q q q q q qq q q qq q q qq q qq q q q qq q q q q q q q q q q qq q q q q q qq q q q q q q qq q q q q qq q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q qq q qq q q q q q q q qq q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q qqq q q q q qq q q q q q q q qq q q q q q q q q q q q qq qq qq q q q qq q q q q q q q q qq q qq q q q qq qq qq q qq q qq q q qq q q q q q q q qq qq q qq q qq q q q q q q q q qq q q qq q q q q q qq q q q qq q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q qq qq q q q q q q qq qq q q q q q qq q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q qq q q q q q qq q q q q q q q q q qq q q qq q q qq q q q q q q q q q q q qqq q q q q q q q qq q q q q q q qq q q qq q q q q q q q q q q qq q q q q q qq qq q q q q q q q q q q q q q qq q q q q q qq q qqq q q q q q q q q q q q q q q q q q q q qq q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q qq q qq q qq q q q q qq q q qq q q q q q 0.2 0.2 0.2 0.2 qq q qq q q q q q q q q q q q q q q q qq qq q q q q q q q q q q qqq q q q q q q q q q q q qq qq q q q qq q q q q q q qq q q q q q q qq qq qq q q q q q q q q q q q qq q q q q q q q q q qq q qq q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q qq qq q q q q q qq qq q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q qqqq qq q q q q q q q q q q q q q q q q q q q qq q q qq q qq q q q qq qq q q q qq qqqq q qq q qq q qq q q q q q q q q q q qqq q q q qq q q qq q qq q qq q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q q q q qq q q q qqq q qq q q qq q q qqq qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q qq qq q q q q q qq q q q qqq q qq q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q qqq q q qq q q q q q q q q q qq q q q q q q q q q q qq qq qq q q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q qq q q qq q q q q q q qq q q q q q q q q qqq q q qq q q q q q q q q q q qq q qq q q q qq q q q q q q q q q q q q qq q q q q q q q qq q q q q q qq q q q q q q q q q q q q qqq q q q q q qq q q q q qq q qq q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q qq q q qq qq q q q q q q qq q q q q q q q q q q q qq q qq q q q q q q q qq q q qq qq q q q q q q q q q q qq q q q q qq q qq qq q q qq qq q q q q q 0.0 0.0 0.0 0.0 q q q q qqq qq q qq q q qq qq q q q qq q q q q qq q q q q q q qq q q q q qq qq q qq q qq q q q q qq q q q 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 lower tails upper tails lower tails upper tails Clayton copula Chi dependence functions Gumbel copula Chi dependence functions 1.0 1.0 1.0 1.0 q q qq q q q q qq q q q qq qq q q q qq q q q q qqqq qqqqq q q q q q qq q q q qq q q q q q q q q q q q q q qqqq q q qq qq qq q q q q q q qq q q q q q q q q q q q qqq qqqq q q q q q qq q q q q q q q q q qq q q q q qqq q q q q qq q qq q q q qq q q qqq q q q q qq qq q q qqq q q q q qq q q q q q q q q q q q qq q q qq q qq q q q q q qqqq q q q q q q q q q q qq q q q q q q q q q q qq qq q q q qqqqq q qq q qq q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q qq q q q qqq q q q q q q q q q q q q q q q q qqqq q q q q q q q q q qq qq q qq q q q qq q q q q q q q q q qq q qq q q q qq q q q q qq q qq q q q qq q q q q q qqq qqq q q q q q q q q q q q q qq qq q q q q qq q q qq qq q q q q qq q q q q q q q q q q q q qq q q qq q qq q qq q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q qqqq q q qq q qq q q qq q q q q qq q q qq q q q q qq q q q qq q q q q q q q q q q q q qq q q q qqq q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q qq q q q q qq qq q q q q qq q q qq q q q q q q qq q q q q q q q q q q q q q q q qq q q q q qq q q q q q qq q q q q qq q q qq q q q q q qq qqq q q q q q q q qq q q q qqqqq qq q qq q q q q q qq q q qq q q q qq q q q q qq q q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q qq qq q q q q q q q qq q q q q q q q q qq q q 0.8 0.8 0.8 q qq 0.8 q q q q q q q q q q q q q q q q q q q qq q q q qq q q q qq q q q q q q q qq q qq q q qq q q qq q q q qq q qq qq q q q q q q qq q qq q q qq q CLAYTON q q q q q q q q q q q q q qq q q q q q q q qq qq q q q q qq q qq q q qqq q q q q q qqq q qq qq q qq q q q q q q q q qq q q q q q q q q q qqq qq q q qq q q q q q q q q qq q q q q qq q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q qq q qq q qq q qq q q q q q qq q q qq q qq q q q q q q q q q q q qq q q q q qq q q qqqq q q q q q q qq q q q qq q qq q qq q q q q q q q qq q q q qq q q q q q q q qq q q q q qq q q q q q q q qq q q qq q q q qq q q q q qq q q q q q q q q q q q q q q q q q qq q q q qq qq q q q q q q q q q q q qq qq q q q q qq q q q q q q q q qq q q q q q q q q q q q qq q qqq q q q q q q q qq q q q q qq q qq qq q q q q q q q q q q q qq q q q q q qq q q q q qq q q qq q q q qq q q q q qq q q q q q qq q qq q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q qq q q q q q q qq q q qq q q q q qq q q q q q q qq q q q q q q qq q q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q qq q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q qq qq q q q qq q q q q q q q q q q q qq q q q q q q q q q q q q q q q q qq q q q q q q q q q q q qq q q q qq q q q q q q q q q q q qq q q q q q qqq q q q q q q q q q q q qq q q qqq q q q q q qq q q qqq q q q q q qq q q qq q q q q q q q q q q q q q q q q q q 0.6 0.6 0.6 0.6 q q q q q q q q q q qq q q q q q q qq q q q q q q qq q qq q q q q qq qq qq q q q q q q q q q q q q qq q q qq qq q q q q q q qq q qq q q q q qq q q q q q q q qq q q q q q qq q q q q q q q q q q qqq q q q q q qq q q qq q q qq q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q qq q q q q q qq q q q q q q q q q q q q qq q q q qq q q q q q q q q qq qq q qq q q q q qq q q q q q q q q q qqq qq q qq q qq q q q q q q q qq q q q q q qq q qq q q q q q q qq q q q qqq q q q q q q q q qq q GUMBEL q q q q q q q q q qq q q q q q q q q q q q q q q qq qq q q qq q q q qq q q q q q q qq q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q qq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q qq qq qq q q q q q q qq qq q q q q q q q qqq q q q q q qq q qq q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q qq qq q q q q q q q q q q q qq q q q q q q q q qq qq qq q q q q q q q q qq q q q qq q q q qq q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q q q q qq q q qq q qq qq q qq q q q qq q q q q q qq q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q qq q q q q q q qq q q q qq qq qq q q qq q q q q q q qq q q qq qq q qqq q q q q q q q q qq q q q q qqq q q q q q q q q q q q q q q q q q q q qqq qq q q qqq q q q q qq q qq qq q q q qq q q q 0.4 0.4 0.4 0.4 q q q q q q q q q qq q q q q q q q q q qq qq q q q q q q q q q q qq q q q q q q q q q q q qq q q q q q q q qq q q q q q q q qq q q q q q q q qq qq q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q qq q q q q q q q q qqq q q q q q q qq q q q q q q q q q q q q q q qqq q q qqq q q q q q q q qq q q qq q q q q qq q q q qq q q q qq q q q q q qq q q q qq q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q qq q q q qq q qq q qqq q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q qq q q qq q q q q q q q q q q q q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q qq q q q qq q q q q qq q q q q q q q q qq qq q q q q q qq q q q q q q q qq qq q qq q q q qq qqq q q q q qq q q q q q qq q q qq q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q qq q q qq qq q qq qq q q q qq q q q q q q q q q q q q q q q qq q q q q q q q q qq qq q q q qq q q qq qq q q q q q q q q q q q q qqq q qq q q q q qq q q qq q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q qq q q q q q q q qq q q q q q q q qqq q q q q q qq q qq q q qq q q q q qq q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q qq qq qq q q q q q q qq q q q q q q q q q q qq q qq q q q q q q q q q qq q q q q q qq q q q q qq q q qq q q q q qq q qq q q q q q qq q qq q q q q q q q q q qq q q q q q q q q q q q q q q q qq qq q q q qqq qq q q q q q q qq q q q q q q q q q q q q q q q qq q qq qqq q q q q q qq q q q q q q qq q q q q q qq 0.2 0.2 0.2 0.2 q q q q q q qq q q q q q q q q q q qq q q q q q q qq q q q q q q qq q q q q q q q q q qq q q q q q q q q q q qqq q q q q q qq q q q q q q q q q q q q q q qqq q qq q q qqq q q q qq q q q q q q qq q q q q q q q q q q q q q qq q q q q q qq q q q q q q q qq q q q q qq q q q q q q q q qq q qq q qq q qq qq qq q q q q q q qq q qq q q q qq q q q q q q q q q q q q qq q qq q q q q q q q q q q q q q q q qq q q q q qq q q q qq q qq q q qq q q q q q qq q q q qq q q q q q qq q q qq qq q qq q q qq q qq q q q q q qq q q q q q q qq q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qqq q q q q q q qq q q q qq qqq q q q q q q q q q q q q q q qq q q q qqq qq q q q q q q q q q q qq q q q q q q q q qq q q qq q q q q q q q qq qq qq q q q q qq q q q q q qq q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q qq q q q q q q q qq qq q qq q q q q q q q q q qq q q q q q q q qq q qq q q q q q q q q q qq q q q q q q q qq qq qqq q q q q q q q q q q q q q q q q q qq qq q q q q q qq q q q q qq q q q q q qq q q q qq qqqq q qq q qq qq q q q q q q q q q q q q q q q q q q q q q q qqqq q qq q q q q q q qqq q q q q q q q qq qq q q q q q q q q qqq q qq q q qq q q q q q qqqq q qq q q q q q q qq q q q q q qq q q q q qqq q q q q q q q qq q q q q q q q q q qq q qq qqq q q qq q qq q q qq q q q q qq qqq qq q qqq q q q q q q q q qq q q q qq q qq q qq q qqq q q q q 0.0 0.0 0.0 q 0.0 q qq qq qq qq qq q q q q qq q qq q q qq qq qq qq q q qqqq q q q q q q qq q qq q q 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 lower tails upper tails lower tails upper tails 31
- 32. Arthur CHARPENTIER, modeling heat waves Tail functions L(·) and R(·) 32
- 33. Arthur CHARPENTIER, modeling heat waves Tail functions χL (·) and χU (·) 33
- 34. Arthur CHARPENTIER, modeling heat waves Dynamics of the residual part consider an ARMA model to model residuals dynamics φ1 φ2 θ1 θ2 σ2 ARMA(2,2) 1.4196 −0.4733 −0.6581 −0.1032 5.023 (0.0419) (0.0322) (0.0419) (0.00752) It is possible to fit a Gaussian or a Student ARMA process, 34
- 35. Arthur CHARPENTIER, modeling heat waves QQ plot of residuals (Gaussian) QQ plot of residuals (Student) 10 10 5 5 Sample Quantiles Sample quantiles 0 0 −5 −5 −10 −10 −15 −15 −4 −2 0 2 4 −4 −2 0 2 4 Theoretical Quantiles Theoritical Quantiles 35
- 36. Arthur CHARPENTIER, modeling heat waves Extremes and discrete data Recall Fisher-Tippett theorem : let X1 , · · · , Xn , · · · be i.i.d. with cdf F , with right point xF = sup{x; F (x) < 1}, and define Xn:n = sup{X1 , · · · , Xn }. If 1 − F (x) lim −) = 1, (1) x→xF 1 − F (x then there exists 3 possible limits for Xn:n − bn lim P ≤x = G(x), n→∞ an (Fréchet, Gumbel and Weibull), see Leadbetteret al.(1983), and we shall say that F ∈ MDA(G). But Assumption (1) is usually not satisfied for discrete data. Approximations can be derived for standard discrete distributions (Poisson, geometric, etc). Nevertheless, if X is an absolutely continuous random variable, with xF = ∞, with hazard rate satisfying h(x) = d log x → 0 as x → ∞, if X is a discretized version of X, with cdf F , then (1) holds, and further, F ∈ MDA(G) if and only if F ∈ MDA(G). 36
- 37. Arthur CHARPENTIER, modeling heat waves Generalized Pareto and modeling exceedances Instead of looking at maximas, use the threshold approch and Pickamds, Balkema, de Haan theorem : −1/ξ ξx P (X > x + u|X > u) ≈ 1+ βu + with x+ = max{x, 0}, βu > 0 and ξ are respectively scale and shape parameters, in the context of a stationary process (Xt ). For non-stationary processes, Smith (1989) and Davison & Smith (1990) suggested a GPD model with covariate, −1/ξ(z) ξ(z)x P (X > x + u|X > u, Z = z) ≈ 1+ βu (z) + See e.g. Chavez-Desmoulins & Davison (2005) for non-parametric (GAM) models. 37
- 38. Arthur CHARPENTIER, modeling heat waves Generalized Pareto and modeling exceedances But as point out in Eastoe & Tawn (2008) a strong constraint is that the scale parameter has to satisfy, for any v > u, βv (z) = βu (z) + (v − u)ξ(z). Coelho et al. (2008) and Kyslýet al. (2010) suggested to use a time varying threshold u(z), while Eastoe & Tawn (2008) suggested preprocessing ˜ ˜ techniques, and to assume that Xt = µt + σt Xt , with Xt stationary. 38
- 39. Arthur CHARPENTIER, modeling heat waves Generalized Pareto and modeling exceedances What about tails of |Xt | ? 39
- 40. Arthur CHARPENTIER, modeling heat waves Generalized Pareto and modeling exceedances What about tails of |Xt | ? different thresholds u. 40
- 41. Arthur CHARPENTIER, modeling heat waves Long range dependence ? Smith (1993) or Dempster & Liu (1995) suggested that, on a long period, the average annual temperature should be decomposed as follows • an increasing linear trend, • a random component, with long range dependence. E.g.(Yt ) such that Φ(L)(1 − L)d Yt = Θ(L)εt where d ∈ (−1/2, 1/2), and where ∞ d d(d − 1) · · · (d − j + 1) (1 − L) = 1 + (1−)j Lj . j=1 j! i.e. ARFIMA(p, d, q), fractional processes (see e.g. Hurst (1951) or Mandelbrot (1965)) E.g.(Yt ) such that Φ(L)(1 − 2uL + L2 )d Yt = Θ(L)εt , i.e. GARMA(p, d, q) from Hosking (1981) Gegenbauer’s frequency, defined as ω = cos−1 (u), is closely related to the seasonality of the series. Here, u = 2π/365 (because of the annual cycle of temperature). 41
- 42. Arthur CHARPENTIER, modeling heat waves A stationary process (Yt )t∈Z is said to have long range dependence if ∞ |ρX (h)| = ∞, h=1 and short range dependence if not. Smith (1993) “we do not believe that the autoregressive model provides an acceptable method for assessing theses uncertainties” (on temperature series) 42
- 43. Arthur CHARPENTIER, modeling heat waves Long range dependence and regression Gray, Zhang & Woodward (1989) proposed an extension to model persistent seasonal series, using Gegenbauer’s polynomial Gd (·), defined as n (−2)n Γ(n + d)Γ(n + 2d) 2 −α+1/2 d n Gd (x) n = (1 − x ) (1 − x2 )n+d−1/2 . n! Γ(d)Γ(2n + 2d) dxn and such that Gd (·) is the unique polynomial of degree n such that n ∞ (1 − 2uZ + Z 2 )d = Gd (u) · Z n . n n=0 If d ∈ (0, 1/2), and |u| < 1 then ρ(h) ∼ h2d−1 sin([π − arccos(u)]h) as h → ∞. 43
- 44. Arthur CHARPENTIER, modeling heat waves Long range dependence and regression If γX (·) denotes the autocovariance function of a stationary process (Xt )t∈Z , n−1 γX (0) 2 k V ar(X n ) = + 1− γ(k), n n n k=1 where X n is the standard empirical mean of a sample {X1 , ..., Xn } (see Brockwell & Davis (1991), or Smith (1993)). Furthermore, if autocovariance function satisfies γ(h) ∼ a · h2d−1 as h → ∞, then a V ar(X n ) ∼ · n2d−2 , d(2d − 1) as derived in Samarov & Taqqu (1988). And further, the ordinary least squares estimator of the slope β (in the case where the Xi ’s are regressed on some covariate Y ) is still Xi (Yi − Y n ) β= 2 . (Yi − Y n ) 44
- 45. Arthur CHARPENTIER, modeling heat waves As shown in Yajima (1988), and more generally in Yajima (1991) in the case of general regressors, 36a(1 − d) V ar(β) ∼ · n2d−4 . d(1 + d)(2d + 1) 45
- 46. Arthur CHARPENTIER, modeling heat waves Long range dependence and extremes From Leadbetter (1974), let X1 , · · · , Xn , · · · be a stationnary process, and denote Xn:n = max{X1 , · · · , Xn } ; if there exists sequences (an ) and (bn ) such that Xn:n − bn lim P ≤ x = G(x), n→∞ an and if so called D(an z + bn ) holds for all z ∈ R (i.e. not long range dependence), then G is a GEV distribution. Remark but parameters are not necessarily the same as an i.i.d. sequence with the same margins (see extremal index concept. Condition D(un ) from Leadbetter et al. (1983) : for all p, q and n, and indices 1 ≤ i1 < i2 < · · · < ip < j1 < j2 < · · · < jq ≤ n, I J 46
- 47. Arthur CHARPENTIER, modeling heat waves such that j1 − ip ≥ m, P max {Xk } ≤ un − P max{Xi } ≤ un · P max{Xj } ≤ un ≤ α(n, m), k∈I∪J i∈I j∈J where α(n, m) → 0 as n → ∞, for some m = o(n). Remark Note that P max {Xk } ≤ un − P max{Xi } ≤ un · P max{Xj } ≤ un =0 k∈I∪J i∈I j∈J means independence. Remark For a Gaussian sequence, with autocorrelation function ρ(h), D(un ) is satisfied if ρ(n) · log(n) → 0 as n → ∞. I.e. weaker than geometric decay for AR processes. 47
- 48. Arthur CHARPENTIER, modeling heat waves Stability of the dynamics First order autocorrelation of the noise (ρ(1) = corr(Xt , Xt−1 )) and estimation of the fractional index d. Correlation of the noise, between t and t+1 Minimal temperature, fractional index 0.85 0.25 0.80 First order autocorrelation 0.20 Fractional index d 0.75 0.15 0.70 0.10 0.65 0.05 0.60 1920 1940 1960 1980 2000 1920 1940 1960 1980 2000 Time Time 48
- 49. Arthur CHARPENTIER, modeling heat waves On return periods Depends on scenarios for the increasing trend, • an optimistic scenario, where we assume that there will be no more increasing trend in the future, • a pessimistic scenario, where we assume that the trend will remain, with the same slope. Depends on the definition of the heat wave • during 11 consecutive days, the temperature was higher than 19◦ C (type (A)), • during 3 consecutive days, the temperature was higher than 24◦ C (type (B)). 49
- 50. Arthur CHARPENTIER, modeling heat waves short memory short memory long memory short tail noise heavy tail noise short tail noise optimistic 88 years 69 years 53 years pessimistic 79 years 54 years 37 years Table 1 – Periods of return (expected value, in years) before the next heat wave similar with August 2003 (type (A)). 50
- 51. Arthur CHARPENTIER, modeling heat waves short memory short memory long memory short tail noise heavy tail noise short tail noise optimistic 115 years 59 years 76 years pessimistic 102 years 51 years 64 years Table 2 – Periods of return (expected value, in years) before the next heat wave similar with August 2003 (type (B)). 51
- 52. Arthur CHARPENTIER, modeling heat waves On return periods, optimistic scenario Distribution function of the period of return Distribution function of the period of return 1.0 1.0 11 consecutive days exceeding 19 degrees 4 consecutive days exceeding 24 degrees 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 GARMA + Gaussian noise GARMA + Gaussian noise ARMA + t noise ARMA + t noise 0.0 0.0 ARMA + Gaussian noise ARMA + Gaussian noise 0 50 100 150 200 0 50 100 150 200 Years before next heat wave Years before next heat wave 52
- 53. Arthur CHARPENTIER, modeling heat waves On return periods, optimistic scenario Density of the period of return Density of the period of return GARMA + Gaussian noise GARMA + Gaussian noise 0.030 ARMA + t noise ARMA + t noise 0.030 ARMA + Gaussian noise ARMA + Gaussian noise 11 consecutive days exceeding 19 degrees 4 consecutive days exceeding 24 degrees 0.025 0.025 0.020 0.020 0.015 0.015 0.010 0.010 0.005 0.005 0.000 0 50 100 150 200 0.000 0 50 100 150 200 Years before next heat wave Years before next heat wave 53
- 54. Arthur CHARPENTIER, modeling heat waves On return periods, optimistic scenario Distribution function of the period of return Distribution function of the period of return 1.0 1.0 11 consecutive days exceeding 19 degrees 4 consecutive days exceeding 24 degrees 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 GARMA + Gaussian noise GARMA + Gaussian noise ARMA + t noise ARMA + t noise 0.0 0.0 ARMA + Gaussian noise ARMA + Gaussian noise 0 50 100 150 200 0 50 100 150 200 Years before next heat wave Years before next heat wave 54
- 55. Arthur CHARPENTIER, modeling heat waves On return periods, optimistic scenario Density of the period of return Density of the period of return GARMA + Gaussian noise GARMA + Gaussian noise 0.030 0.030 ARMA + t noise ARMA + t noise ARMA + Gaussian noise ARMA + Gaussian noise 11 consecutive days exceeding 19 degrees 4 consecutive days exceeding 24 degrees 0.025 0.025 0.020 0.020 0.015 0.015 0.010 0.010 0.005 0.005 0.000 0.000 0 50 100 150 200 0 50 100 150 200 Years before next heat wave Years before next heat wave 55
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- 58. Arthur CHARPENTIER, modeling heat waves Burt, S. (2004). The August 2003 heatwave in the United Kingdom. Part III : Minimum temperatures. Weather, 59, 272-273. Charpentier, A. (2010). On the return period of the 2003 heat wave. Climatic Change, DOI 10.1007/s10584-010-9944-0 Coelho, C.A.S., Ferro, C.A.T., Stephenson, D.B. & Steinskog, D.J. (2008). Methods for exploring spatial and temporal variability of extreme events in climate data. Journal of Climate, 21, 2072-2092. Davison, A.C. & Smith, R.L. (1990). Models for exceedances over high thresholds. Journal of the Royal Statistical Society. Series B., 52, 393-442. Dempster, A.P. & Liu, C. (1995).Trend and drift in climatological time series. Proceedings of the 6th International Meeting on Statistical Climatology, Galway, Ireland, 21-24. Eastoe, E. & Tawn, J.A. (2009). Modelling non-stationary extremes with application to surface level ozone. Applied Statistician, 58, 25-45. 58
- 59. Arthur CHARPENTIER, modeling heat waves Fink A. H., Brucher, T., Kruger, A., Leckebusch, G.C., Pinto, J.G. & Ulbrich, U. (2004). The 2003 European summer heatwaves and droughtSynoptic diagnosis and impacts. Weather, 59, 209-215. Gray, H.L., Zhang, N.F., & Woodward, W. (1989). On Generalized Fractional Processes. Journal of Time Series Analysis, 10, 233-57. Gumbel E. (1958). Statistics of Extreme. Columbia University Press. Hosking, J. R. M. (1981). Fractional Differencing. Biometrika, 68, 165-76. Houghton, J.T. (1997). Global Warming. Cambridge University Press. Hurst, H.E. (1951). Long-term storage capacity of reservoirs, Trans. Am. Soc. Civil Engineers, 116, 770–799. Karl, T.R. & Trenberth, K.E. (2003). Modern Global Climate Change. Science, 302, 1719. Karl, T.R., & Knight, R.W. (1997). The 1995 Chicago heat wave : How likely is a recurrence ? Bulletin of the American Meteorological Society, 78, 1107-1119. 59
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- 63. Arthur CHARPENTIER, modeling heat waves Quereda Sala, J., Gil Olcina, A., Perez Cuevas, A., Olcina Cantos, J., Rico Amoros, A. & Montón Chiva, E. (2000). Climatic Warming in the Spanish Mediterranean : Natural Trend or Urban Effect. Climatic Change, 46, 473-483. Redner, S. & Petersen, M.R. (2006). Role of global warming on the statistics of record-breaking temperatures. Physical Review E. Rocha Souza, L. & Soares, L.J. (2007). Electricity rationing and public response. Energy Economics, 29, 296-311. Samarov, A. & Taqqu, M. (1988). On the efficiency of the sample mean in long-memory noise. Journal of Time Series Analysis, 9, 191-200. Schar, C., Luigi Vidale, P., Luthi, D., Frei, C., Haberli, C., Liniger, M.A. & Appenzeller, C. (2004). The role of increasing temperature variability in European summer heatwaves. Nature, 427, 332-336. Smith, R. (1989). Extreme value analysis of environmental time series : an application to trend detection in ground-level ozone. Statistical Sciences, 4, 367-393. 63
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