Marta Nogaj ([email protected]) Laboratoire des Sciences du Climat et de l’Environnement Analysis...

Marta Nogaj ([email protected])

Laboratoire des Sciences du Climat et de l’Environnement

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



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



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



Our extreme events



Introduction of non-stationarity

• Amplitude of Extremes– Generalized Pareto Distribution

• Dates of Extremes– Poisson Distribution

1

)(1)(

tuxuXxXP

Scale parameter depends on covariate t

Intensity parameter depends on covariate t

!

)(exp)())((

n

ttntN

n



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

• xClassification of grid points based on the dynamical evolution

of extremes and not their absolute values



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?



“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

• 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)

1947 1975 2004

YearS

easo

nal

mea

n te

mpe

ratu

re

Spline adjustment for seasonal mean temperature

for 8E-50N (Germany)



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



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



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



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 depending on initial values and observation length ratio

0

20

40

60

80

100

0.0

1

0.0

5

0.5

0.2

5

0.1 1

2.5 5 10

50

(α/β) / length(t)

Perc

en

tag

e

200 observations

2000 observations

Percentage of correct estimations of the order of the models depending on the initial values and the length of the

observations



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

dtt ii 1



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



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



Trends of Tmax JJA – Pareto

σ in

crea

sing

σ d

ecre

asin

g

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

σ in

crea

sing

σ d

ecre

asin

g

Sigma degree Tmax JJA Sigma degree Tmax JJA



Trends of Tmax JJA – Poisson

λ in

crea

sing

λ de

crea

sing

λ(t) = λ

λ (t) = α + β t

λ (t) = α + β t + γ t2

Non-stationary λ

(Frequencies)

λ in

crea

sing

λ de

crea

sing

“ Varying threshold ”

Mean variation has been eliminated

Intensity degree Tmax JJAIntensity degree Tmax JJA



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 arbitraryIdea of a varying threshold has been proved useful

These joint models improve the quality of climatological interpretation



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



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



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



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



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.



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



General methodvalidation - GPD



Tmin DJF - Poisson

Seasons of Extreme events

Em

piric

al e

stim

atio

n o

f λ

Lat: 32N

Lon: 5W

Empirical estimation - the histogram of Poisson with fitted Poisson λ covariate for GP 512

1958 1972 1985 1999 2003



Poisson validation



GPD validation linear



Return levels



Piecewise linear

• Alternative to polynomial fitting– Linear fragments connection

• Less risky than polynomial interpolation with high degree for extrapolation

Nodes



General methodvalidation - GPD



T max JJAThreshold & Xi

-0.2

-0.4

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

Marta Nogaj ([email protected]) Laboratoire des Sciences du Climat et de l’Environnement Analysis...

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