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**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)

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**The “Problem” Warmer climate Is there a trend in the extreme field?**

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

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Our extreme events

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**Introduction of non-stationarity**

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

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**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 t2 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

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**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

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**“Varying” threshold Basic method Keep in mind the whole data**

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

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**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

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**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

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**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

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**Empirical results Correct estimation σ = α + β * t**

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

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**Application Data NCEP Reanalyses Daily extreme data Temperature MAX**

Temperature MAX Summer (JJA) North-Atlantic Lat: 30N to 70N Lon: 80W to 40E Covariate Time

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**Trends of Tmax JJA – Pareto**

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

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**Trends of Tmax JJA – Poisson**

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

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**Non-stationary Return Levels**

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

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**Non-stationary Return Levels (2)**

Disputed Description of past evolution Prediction of future evolution Metamathematical problem ! Well-known trade off between fit and prediction

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**Final Quizz Climatological question Statistical 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

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**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

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**Thank You! R project: http://www.r-project.com**

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, , 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,

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**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

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**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

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**CPLM vs. Polynomials Model choice CPLM vs. polynomials**

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

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**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

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**Example Unbounded non-stationarity Question Possible heuristic**

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

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**General method validation - GPD**

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**Empirical estimation of λ**

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 Seasons of Extreme events

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Return levels

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**Piecewise linear Alternative to polynomial fitting**

Linear fragments connection Less risky than polynomial interpolation with high degree for extrapolation Nodes

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**T max JJA Threshold & Xi High temperatures not gaussian**

-0.2 -0.4 High temperatures not gaussian Threshold u is an upper percentile of the series

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Manfred Mudelsee Department of Earth Sciences Boston University, USA

Manfred Mudelsee Department of Earth Sciences Boston University, USA

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