Analysis of non-stationary climatic extreme events

1. Analysis of non-stationary climatic extreme events MARTA NOGAJ Didier Dacunha-Castelle (U Orsay) Farida Malek (U Orsay) Sylvie Parey (R&D EDF) Pascal Yiou (LSCE)

2. The “Problem” • Warmer climate • Trend in the average field • Is there a trend in the extreme field? • Is it similar to the average? • Economical & Social impact = climatological concern • Analysis and prediction of the temporal evolution of spatial extremes

3. The Menu • Our extremes • Introduction of non-stationarity • GPD/Poisson model • Descriptive analysis • “Varying threshold” • Method Description • Method Analysis • Problems of lack of asymptotic convergence • Empirical results • Statistical considerations about CPLMs • Application • Climatological maps • Return Levels • Prediction

4. Our extreme events

5. é ù x - 1/-x x u > > = + P ( X x X u ) 1 ê ú s t ( ) ë û Scale parameter depends on covariate t Intensity parameter depends on covariate t Introduction of non-stationarity • Amplitude of Extremes • Generalized Pareto Distribution • Dates of Extremes • Poisson Distribution - 1 ( )

6. Descriptive analysis • Preliminary studies • Non-parametric models for σ(t) and I(t) • Cubic Splines  Non-stationarity in extremes is apparent • Hint on form of covariate model • Choice of 2 classes of models • Polynomials • Stationary – constant α • Linear – α + βt • Quadratic - α + βt + γt2 • Continuous piecewise linear models (CPLM) • Consistent with the requirement of a climatic spatial classification • x Classification of grid points based on the dynamical evolution of extremes and not their absolute values

7. Non-stationary caveats • Stationary or non-stationary ξ? • ξ: physical property of a region • Previous analyses on temperature data show little variation of ξ (e.g. Parey et al.) • Difficult to estimate  STATIONARY ξ • Non-stationarity depends on a covariate t • Nature • Time • Other (GHG, NAO) • Varying threshold in the GPD? = GEV model with varying μ parameter • Attempt with elimination of trend of the whole sample • Extreme non-stationarity different from the whole data set trend?

8. “Varying”threshold Spline adjustment for seasonal mean temperature for 8E-50N (Germany) • Basic method • Forget data under the threshold, keep the extremes • Try and check for non-stationarity • Keep in mind the whole data  Varying threshold • Theory complex • Alternative  non-parametric method • Spline adjustment to seasonal mean • Subtraction of this mean variation ≈ equivalent to the variation of the threshold • What remains? • Xt=Yt + m(t) Yt – iid • σ(t) – amplitude • I(t) - frequency • Elimination of m(t) σ(t) & I(t) ?  Black box (Work in progress) Seasonal mean temperature 1947 1975 2004 Year

9. Method descriptionfor non-stationary GPD/Poisson • Parameter estimation • Maximum likelihood • Model choice for σ(t) & I(t) • Likelihood ratio test or an AIC • Best degree choice - polynomial • Best number of nodes – piecewise linear • Checking the adequacy of the models (polynomials) • Change of clock for the Poisson / Renormalization for GP • Classical tests • Uncertainty estimation • Confidence Intervals

10. Asymptotic properties • No obvious extension of the stationary EVT • Classical asymptotic theory does not always work • E.g. Malek & Nogaj 2005 • Linear Poisson Intensity • Convergence speeds to normal law differ for the 2 parameters • Quadratic Poisson Intensity • Non convergent (non trivial) estimator for the constant term • The highest degree is predominant when t  ∞ • Confidence Intervals • Usage, as often proposed, of the observed information matrix is “perhaps” incorrect • Empirical information matrices might not converge • Solution • Analysis through simulations

11. Bypassing the lack of asymptotics • Analysis of previous procedure through simulation • GPD • Simulation of data from a GPD distribution with polynomial σ(t) • Poisson • Simulation of data from a Poisson distribution with polynomial I(t) using change of clock • Estimation from simulation repetitions • order (stationary/linear/quadratic) • parameters of models • Confidence Interval computation • Correction check

12. Percentage of correct estimations of the order of the models depending on the initial values and the length of the observations Empirical results • Correct estimation • Depends on the length of data (length of t) • Depends on initial parameters • σ = α + β * t • α/β < length(t)

13. 1,2,3… parts Continuous Piecewise Linear Models(CPLM) • GPD & Poisson • Difficulty • Non-identifiable • (as mixtures or ARMA processes) • Classical Likelihood tests do not apply • D. Dacunha-Castelle & Gassiat E., ESAIM (´99), Annals of Statistics (´97) • In practice • Artificial separation of nodes • d – distance (non trivial to determine)

14. CPLM vs. Polynomials • Model choice • Polynomial models and piecewise models are not nested • No statistical comparison • CPLM vs. polynomials • Advantages • “Objective” cut of time • Climatic periods • Possible asymptotic theory • Disadvantages • Statistical problems of non-identifiability • Higher number of parameters

15. Application • Data • NCEP Reanalyses • Daily extreme data • 1947-2004 • Temperature MAX/MIN • Summer (JJA)/ Winter (DJF) • North-Atlantic • Lat: 30N to 70N • Lon: 80W to 40E • Covariate • Time

16. Trends of Tmax JJA – Pareto σ(t) = σ σ (t)= σ0 + σ1 t σ (t)= σ0 + σ1 t + σ2 t2 Non-stationary σ (Amplitudes) “ Varying threshold ” Mean variation has been eliminated Sigma degree Tmax JJA Sigma degree Tmax JJA σ increasing σ increasing σ decreasing σ decreasing

17. Trends of Tmax JJA – Poisson λ(t) = λ λ (t)= α + β t λ(t)= α+ β t + γ t2 Non-stationary λ (Frequencies) “ Varying threshold ” Mean variation has been eliminated Intensity degree Tmax JJA Intensity degree Tmax JJA λ increasing λ increasing λ decreasing λ decreasing

18. Climatological model interpretation • GEV – GPD/Poisson comparison • GEV • μ is the mean (a natural trend) • σ is the variance  Interpretation is straight forward • GPD/Poisson • σ is the mean as well as σ2 is the variance • I(t) has a clear interpretation of the frequency of events • The threshold u is somehow arbitrary • Idea of a varying threshold has been proved useful • These joint models improve the quality of climatological interpretation

19. Non-stationary Return Levels • Return Level: • NRP(z): number of exceedances of z in RP (return period) • z : Return Level for RP • ENRP(z)=1 • Different concept from the usual stationary case: • Assumption of correctness of extrapolation in the future • Depends highly on position in time

20. Non-stationary Return Levels (2) • Disputed • Description of past evolution • Prediction of future evolution • Metamathematical problem ! • CPLM – better choice Well-known trade off between fit and prediction

21. Final Quizz • Climatological question • Are extreme events varying? • Is the variation of extreme events similar to the variation of the average and the variance? • Statistical question • Can we estimate extreme values variability? • Can we adapt the theory to a non-stationary context? • Statistical answer • Possible trend detection in extreme events • Connected statistical problems have been identified & analyzed  BE CAREFUL! • Climatological answer • Detected regions of the dynamical variation of extreme events • Amplitude / Occurrence • “Varying threshold” method used to “separate” extreme variability from the average field • Different covariates allowed us to investigate the cause of the trend in extremes • GHG – comparable with monotonic trend (time) • NAO – no major effect on extreme climate

22. But is it “final” ? • Climatological perspectives • Other covariates: • THC, weather regimes, land use percentage, solar constant • Analyses of model simulations • IPSL, other time periods • Other physical domains (E2C2 program) • Geophysics, Hydrology • Statistical perspectives • Introduction of a “spatial” context • Analysis of “clusters” • Length of extremes + droughts

23. Thank You! R project: http://www.r-project.com CLIMSTAT: http://www.ipsl.jussieu.fr/CLIMSTAT/ Nogaj et al., “Intensity and frequency of Temperature Extremes over the North Atlantic Region”, GRL (submitted 2005) Malek F. and Nogaj M., “Asymptotique des Poissons non-stationnaires”, Canadian Statistical Journal (submitted 2005) D. Dacunha-Castelle and E. Gassiat ,”Testing the order of a model using locally conic parameterization: population mixtures and stationary ARMA processes“ Annals of Stat.,  27, 4, 1178-1209, 1999. D. Dacunha-Castelle and E. Gassiat, “Testing in locally conic models and application to mixture models”  ESAIM P et S, 1, 1997. Parey S. et al., “Trends in extreme high temperatures in France: statistical approach and results”, Climate Change (submitted 2005 ) Naveau P. et al. Statistical Analysis of Climate Extremes. ``Comptes rendus Geosciences de l'Academie des Sciences". (2005, in press) Coles S. (2001) An Introduction to Statistical Modeling of Extreme Values, Springer Verlag Davison A and Smith R. (1990) Models for exceedances over high thresholds. Journal of the Royal Statistical Society, 52, 393-442.

25. Example • Unbounded non-stationarity • Classical asymptotic fails if: • E.g. m(t)=α0 + α1t + α2t2 (α1α2 ≠0) • In fine, the deterministic mean “makes” the extremes • Possible heuristic • Usage justified if • α0(T) << logT • α1(T) ≤ logT / T • α2(T) ≤ logT/T2 • Question • Cf. later in my presentation

26. General methodvalidation - GPD

27. Tmin DJF - Poisson Empirical estimation - the histogram of Poisson with fitted Poisson λ covariate for GP 512 Lat: 32N Lon: 5W Empirical estimation of λ 1958 1972 1985 1999 2003 Seasons of Extreme events

28. Poisson validation

29. GPD validation linear

30. GPD validation linear

31. Return levels

32. Nodes Piecewise linear • Alternative to polynomial fitting • Linear fragments connection • Less risky than polynomial interpolation with high degree for extrapolation

33. General methodvalidation - GPD

34. T max JJAThreshold & Xi -0.2 -0.4 High temperatures not gaussian Threshold u is an upper percentile of the series