Post on 28-Mar-2015
Marta Nogaj (marta.nogaj@cea.fr)
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
Our extreme events
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
“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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
Method descriptionfor 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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
Application
• Data– NCEP Reanalyses– Daily extreme data
• 1947-2004
– Temperature MAX– Summer (JJA)– North-Atlantic
• Lat: 30N to 70N• Lon: 80W to 40E
– Covariate• Time
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
Non-stationary Return Levels (2)
• Disputed– Description of past evolution– Prediction of future evolution
• Metamathematical problem !
Well-known trade off between fit and prediction
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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.
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
General methodvalidation - GPD
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
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
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
Return levels
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
Piecewise linear
• Alternative to polynomial fitting– Linear fragments connection
• Less risky than polynomial interpolation with high degree for extrapolation
Nodes
Marta Nogaj (marta.nogaj@cea.fr)
Laboratoire des Sciences du Climat et de l’Environnement
T max JJAThreshold & Xi
-0.2
-0.4
High temperatures not gaussian Threshold u is an upper percentile of the series