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Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Analysis of non-stationary climatic extreme events Didier Dacunha-Castelle.

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Presentation on theme: "Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Analysis of non-stationary climatic extreme events Didier Dacunha-Castelle."— Presentation transcript:

1 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Analysis of non-stationary climatic extreme events Didier Dacunha-Castelle (U Orsay) Farida Malek (U Orsay) Sylvie Parey (R&D EDF) Pascal Yiou (LSCE) MARTA NOGAJ

2 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement 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 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Our extreme events

4 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Introduction of non-stationarity Amplitude of Extremes –Generalized Pareto Distribution Dates of Extremes –Poisson Distribution 1 )( 1)( t ux uXxXP Scale parameter depends on covariate t Intensity parameter depends on covariate t

5 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement 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 + β t »Quadratic - α + β t + γ t 2 –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

6 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Non-stationary caveats Non-stationarity depends on a covariate t –Nature Time Other (GHG, NAO) 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 tests performed – non-stationarity rejected in > 90% STATIONARY ξ Varying threshold in the GPD? = GEV model with varying μ parameter –Attempt with elimination of mean trend

7 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Varying threshold 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

8 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Method description for non-stationary GPD/Poisson Parameter estimation –Maximum likelihood Model choice for σ(t) & I(t) –Likelihood ratio test Best degree choice - polynomial Best number of nodes – piecewise linear Checking the adequacy of the models –Classical Goodness of fit tests Uncertainty estimation –Confidence Intervals

9 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement 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

10 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Bypassing the lack of asymptotics Analysis of previous procedure through simulation –N simulations 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 Order/parameters

11 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement 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) Percentage of correct estimations of the order of the models depending on the initial values and the length of the observations

12 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Application Data –NCEP Reanalyses –Daily extreme data –Temperature MAX –Summer (JJA) –North-Atlantic Lat: 30N to 70N Lon: 80W to 40E –Covariate Time

13 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Trends of Tmax JJA – Pareto σ increasing σ decreasing σ(t) = σ σ (t) = σ 0 + σ 1 t σ (t) = σ 0 + σ 1 t + σ 2 t 2 Non-stationary σ (Amplitudes) Varying threshold Mean variation has been eliminated σ increasing σ decreasing Sigma degree Tmax JJA

14 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Trends of Tmax JJA – Poisson λ increasing λ decreasing λ(t) = λ λ (t) = α + β t λ (t) = α + β t + γ t 2 Non-stationary λ (Frequencies) λ increasing λ decreasing Varying threshold Mean variation has been eliminated Intensity degree Tmax JJA

15 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Non-stationary Return Levels Return Level: –N RP (z): number of exceedances of z in RP (return period) –z : Return Level for RP EN RP (z)=1 Different concept from the usual stationary case: –Assumption of correctness of extrapolation in the future –Depends highly on position in time

16 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Non-stationary Return Levels (2) Disputed –Description of past evolution –Prediction of future evolution Metamathematical problem ! Well-known trade off between fit and prediction

17 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement 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

18 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement But is it final ? Climatological perspectives –Other covariates –Analyses of model simulations –Other physical domains (E2C2 program) Statistical perspectives –Introduction of a spatial context –Analysis of clusters Length of extremes + droughts

19 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Thank You! R project: 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, , D. Dacunha-Castelle and E. Gassiat, Testing in locally conic models and application to mixture models ESAIM P et S, 1, 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,

20 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement

21 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement The Menu POT model Introduction of non-stationarity –GPD/Poisson model –Descriptive analysis –Varying threshold Trend detection – method description Method Analysis –Problems of lack of asymptotic convergence –Empirical results –Statistical considerations about CPLMs Application –Climatological maps Return Levels –Prediction

22 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement 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) 1,2,3… parts

23 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement 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

24 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement 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 σ 2 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

25 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Example Unbounded non-stationarity Classical asymptotic fails if: –E.g. m(t)=α 0 + α 1 t + α 2 t 2 (α 1 α 2 0) »In fine, the deterministic mean makes the extremes –Possible heuristic Usage justified if –α 0 (T) << logT –α 1 (T) logT / T –α 2 (T) logT/T 2 Question Cf. later in my presentation

26 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement General method validation - GPD

27 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Tmin DJF - Poisson Seasons of Extreme events Empirical estimation of λ Lat: 32N Lon: 5W Empirical estimation - the histogram of Poisson with fitted Poisson λ covariate for GP

28 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Return levels

29 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement Piecewise linear Alternative to polynomial fitting –Linear fragments connection Less risky than polynomial interpolation with high degree for extrapolation Nodes

30 Marta Nogaj Laboratoire des Sciences du Climat et de lEnvironnement T max JJA Threshold & Xi High temperatures not gaussian Threshold u is an upper percentile of the series


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