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Extreme value analysis and projection in light of the changing climate Xiaolan L. Wang Climate Research Division Science and Technology Branch Environment Canada WCRP-UNESCO (GEWEX/CLIVAR/IHP) Workshop on Extreme Analysis Paris, France, September 2010

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Outline - The related Extreme Value (EV) models briefly review - EV models with covariates - approaches for assessing trends in extremes & approaches for projecting changes in extremes, with examples - various parameter estimators and their characteristics - confidence intervals of return level or risk estimates - on-going/future works

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The related extreme value models – GEV and GPD 1. Block Maxima (BM) (e.g., annual maxima x ) ~ Generalized Extreme Value (GEV) family of distributions - Fréchet (type II, heavier upper tail) - Weibull (type III, bounded upper tail) - Gumbel (type I tail, unbounded) Choice of block size – trade off between bias and efficiency of estimates Annual maxima ~ small samples large uncertainties in estimates Has motivated … modeling of more data, e.g., POT/GPD approach Choice of threshold u – trade off between bias and efficiency of estimates GPD has the threshold stability property minimum suitable threshold, maximizing the sample size 2. POTs - Peak excesses Over Threshold u: y=(x-u) ~ Generalized Pareto Distribution (GPD) family: - Pareto (type II) - Exponential (type I) - Special case of beta (type III) Type II Type III ► Goodness of fit test - Anderson-Darling statistic A 2, also used to choose a suitable threshold average No. of POTs per year GEV and GPD: x

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Bias and efficiency of estimators for GPD Important to give a Confidence Interval (CI) of EV model estimates: - for some estimators: Asymptotic CI asymptotic variance of the par. estimates but not necessarily genuine! - Genuine CI adjusted bootstrap re-sampling (Coles & Simiu, 2003) – used for results shown in this talk would recommend: - use the New estimator for fitting GPDs - use PWM (Hosking et al. 1985) or MLE for fitting GEVs not available for GEV Zhang and Stephens (2009) propose a New estimator: based on MLE but with a data-based prior estimates always exist; least biased & most efficient: Shape ξ MOM PWM MLE NEW - least biased LME Shape ξ MOM PWM MLE NEW – least biased LME Shape ξ MOM PWM MLE NEW – most efficient LME Shape ξ PWM MOM NEW – most efficient MLE LME Bias in estimating scale (N=500)Bias in estimating shape (N=500)Efficiency in estimating scale (N=500) Efficiency in estimating shape (N=500) Bias Efficiency PWM: Probability Weighted Moment MLE: Maximum Likelihood MOM: Method of Moment …

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Effects of sample size and estimator used on return value estimates (stationary EV models) - for data of Type II tail ( ξ>0 ) (heavier upper tail) each: from 115 re-sampled daily precipitation series 95% Confidence Interval (CI) 95% Confidence Internal (CI) for 20-year return value Sample size N (years) 95% Confidence Interval (CI) Sample size N (years) 95% Confidence Internal (CI) for 100-year return value Least stable most stable

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Effects of sample size and estimator used on return value estimates (stationary EV models) - for data of Type III tail ( ξ<0 ): each: from 103 re-sampled daily Tmax series (bounded upper tail) 95% Confidence Interval (CI) Sample size N (years) 95% Confidence Internal (CI) for 100-year return value most stable Least stable - The GPD is most stable, especially for data of a heavy tail - The GEVmle is least stable 95% Confidence Interval (CI) 95% Confidence Internal (CI) for 20-year return value Sample size N (years) - The GPD is again most stable - The GEVpwm is comparable to GPD for small samples and for lower return levels (i.e., 20-yr)

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Where/when to use which model? A famous scientist: “ Stationarity is dead in the changing climate.” e.g., GPD or GEV with time-dependent parameters One way to represent non-stationarity in extremes: use “non-stationary” EV models, functions of covariates: functions of time itself: ??

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e.g., Hold the estimated pars. = extrapolate the fitted trend, not desirable - advantage: no extrapolation of trends - assumption: The predictor-predictand relationship will hold under the projected climate - reasonable Non-stationary EV models with parameters being functions of time t: - not good for use to make projections of change in extremes, because - good for assessing trends in the observed/projected extremes (examples later) - can be used to project changes in extremes that correspond to the projected changes in the predictors EV models with covariates ( good predictors that are also well simulated by climate models) - such projected extremes could still be stationary, with natural fluctuations around the mean climate (examples later)

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a hierarchy of nested GEV models, a GEV tree, for assessing trends in extremes Use a likelihood ratio test to choose the best-fit model, estimating trend type & significance Allowing the EV model parameters to have different types of trends: No trend linear trends polynomial trends quadratic (polynomial) trends Next, a few examples

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a. summer (DJF) b. autumn (MAM) c. winter (JJA) d. spring (SON) Fig. 3: South-east Australian region averages of seasonal P95 and P99 storm indices, along with Gaussian filtered curves and linear trends for the indicated seasons over the period of Alexander, Wang, Wan, & Trewin ( 2010) A significant decline in storminess (geo-wind extremes) over southeast Australia: Trends in annual 99 th percentiles RDC PBM BGD PMR CDH DGH MDR MBD How has the distribution of the geo-wind extremes changed? Regional average series: Example 1 – geostrophic wind extremes

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95% CI Extremes of geo-wind speeds over MDR (MilduraA-DeniliquinA-Robe), Australia Trend in annual 99 th percentiles 11-point Gaussian smoothed significant decline Stationary EV models fitted to 3 segments: GPDs (each of 37-yr data) GEVs Notable ↓ in the location & scale: An early 20C’s 10yr event a 156yr event in late 20C (~15 times less frequent) 95% CI Similar results, but with 37-yr data GEVmle fits (to AMs) are much less stable than GPD fits (to POTs) Better to search in the GEV tree for the best non-stationary GEV fit: signif. ↓ insignif. trend signif. ↓ a 10yr event in the 1900’s ↓ a 209yr event in the 2000’s (~20 times less frequent)

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linear ↓: -1.38˚C/century signif. ↓ No change in scale & shape linear ↑: +3.3˚C/century signif. ↑ No change in scale & shape Modern Exp Farm (Manitoba; ) A Canadian site: Example 2: Trends in annual minimum and annual maximum temperatures (from daily minT and daily maxT, respectively) It has become much less cold, and also less hot! (quite common across Canada)

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Laverton RAAF (087031; ): variability ↑; 1 - p = 94.2% linear ↑ (+1.2˚C/century); 1 - p = 93% No change in scale & shape 10-yr event in 1945 ↓ 3-yr event now Melbourne suburb - Quadratic trend quadratic ↑ (+2.5˚C in :75yr) increasing rate of warming Melbourne (086071; ): Trends in annual minimum and annual maximum temperatures at Australian sites: No signif. change! It has become much less cold but not hotter. No change in scale & shape Urbanization effect urban suburb

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Trends in annual minimum and annual maximum temperatures at Australian sites (cont’d): Sydney (066062; ): both extremes have increased linear ↑: +1.09˚C/century linear ↑: +1.0˚C/century It has become less cold and hotter! Darwin (014015; ): Changes mainly in shape Shrinking lower tail; 1 – p =90.1% Significantly heavier upper tail New heat records! No change in scale & shape Warming is common to all these sites, but in very different ways!

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An example of EV models with covariates for projecting changes in extremes/risk functions of good predictors, e.g., geostrophic wind energy anomalies G t and SLP anomalies P t are good predictors for ocean wave heights A hierarchy of nested EV models with covariates - Use a likelihood ratio test to find the best-fit model, best predictors

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with - use climate model projected possible future changes (future - baseline) in the predictors to project … diminish the effects of any difference between the observed and simulated baseline climates preserve the observed climatological value and pattern of extremes – important for engineering design - important to choose a period of observations as baseline period define the observed baseline climate e.g., design (T-yr return) value: model projected predictors risk: no - involves no extrapolation of trends, but assumes that the observed relationship will hold First, need to use observations to calibrate the predictors-predictand relationship, e.g.,: Next: examples of using nested GEV models with covariates to project changes in extremes

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CGCM2 (Canadian model) projected changes in return values of winter extreme SWH (A2 scenario) 1990’s 20-yr. return values (m) Contour interval: 1.0 m projected return periods as of year SWH of ~13 m: - once every 20 yr. in the 1990’s climate; - once every ~10 yr. in the 2050’s climate Contour interval: 2 yr. Shading: at least 5% significance 12 Contour interval: 10 cm. Shading: at least 5% significance changes (cm) in 20-yr return values by 2050 (2050’s – 1990’s)

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Ireland U.K. Norway Iceland Changes in probability density function (pdf) – projected by CGCM2 for IS92a scenario (solid curves ~ average of 3 runs; non-solid curves ~ individual runs) JFM no sig. changes OND sig. increase p21=0.887 p31=0.986 slight increase p21=0.803 p31=0.962 JFM

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sig. increase p21=0.998 p31=1.000 JFM sig. decrease p21=0.921 p31=0.999 CGCM2 projections - IS92a scenario USA Canada Africa

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Trends in location par. of extreme SWH A2 scenario IS92a scenario 2030 Norway Variation is obvious from one scenario to another, one location to another A2 scenarioIS92a scenario A2 scenarioIS92a scenario 2030 no climate change projected for the early 21C - example of stationarity in projected extremes using GEV with covariates just natural fluctuations around the mean Projected changes are typically - non-linear - dependent on scenario on location

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The results on SWH are from the particular simulations for the particular forcing scenarios - old We will use the CMIP5 simulations to make new projections of global ocean wave extremes Other on-going/future works in this topic - Developing a software package for extreme analysis, including design-value/risk estimation and trend characterization, and for projecting changes in extremes (incl. GEV and GPD models, with and without covariates; in R and FORTRAN) - Apply the methods to analyze trends in various climate extremes (temperature, storm, wind, waves…) Thank you very much for your kind attention! Questions/comments?

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