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, 27-29 September 2010
Outline
- The related Extreme Value (EV) models
briefly review
- various parameter estimators and their characteristics
- confidence intervals of return level or risk estimates
- EV models with covariates
- approaches for assessing trends in extremes
& approaches for projecting changes in extremes, with examples
- on-going/future works
The related extreme value models – GEV and GPD
1. Block Maxima (BM) (e.g., annual maxima x) ~ Generalized Extreme Value (GEV) family of distributions
  0 - Fréchet (type II, heavier upper tail)
x ~ GEV (  ,  ,  )
  0 - Gumbel (type I tail, unbounded)
  0 - Weibull (type III, bounded upper tail)
Choice of block size – trade off between bias and efficiency of estimates
Type II
Annual maxima ~ small samples  large uncertainties in estimates
 Has motivated … modeling of more data, e.g., POT/GPD approach
Type III
x
2. POTs - Peak excesses Over Threshold u: y=(x-u) ~ Generalized Pareto Distribution (GPD) family:
 0
y  ( x  u ) ~ GPD (~ ,  | u )
  0 - Exponential (type I)
 0
Choice of threshold
- Pareto (type II)
- Special case of beta (type III)
u – trade off between bias and efficiency of estimates
GPD has the threshold stability property  minimum suitable threshold, maximizing the sample size
► Goodness of fit test - Anderson-Darling statistic A2, also used to choose a suitable threshold
GEV and GPD: same

 , and   ~   and
average No. of
POTs per year
  u  (~   )/ 
GPD (~ˆ , ˆ | u , ˆ )  eGEV ( ˆ , ˆ , ˆ )
Bias and efficiency of estimators for GPD
Zhang and Stephens (2009) propose a New estimator: based on MLE but with a data-based prior  estimates always exist;
least biased &
Efficiency in estimating shape (N=500)
Bias in estimating scale (N=500)
Bias in estimating shape (N=500)
Efficiency in estimating scale (N=500)
most efficient:
MOM
MOM
MLE: Maximum Likelihood
MOM: Method of Moment
…
Bias
PWM: Probability Weighted Moment
PWM
Efficiency
PWM
LME
MLE
NEW – least biased
Shape ξ
would recommend:
LME
NEW - least biased
NEW – most efficient
NEW – most efficient
LME
LME
MLE
PWM
PWM
MLE
MLE
Shape ξ
MOM
MOM
Shape ξ
Shape ξ
not available for GEV
- use the New estimator for fitting GPDs
- use PWM (Hosking et al. 1985) or MLE for fitting GEVs
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
Effects of sample size and estimator used
on return value estimates (stationary EV models) - for data of Type II tail (ξ>0)
95% Confidence Interval (CI)
(heavier upper tail)
95% Confidence Internal (CI) for 20-year return value
95% Confidence Interval (CI)
Sample size N (years)
95% Confidence Internal (CI) for 100-year return value
Sample size N (years)
each: from 115 re-sampled
daily precipitation series
Effects of sample size and estimator used
95% Confidence Interval (CI)
on return value estimates (stationary EV models) - for data of Type III tail (ξ<0):
(bounded upper tail)
95% Confidence Internal (CI) for 20-year return value
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
- The GPD is again most stable
Sample size N (years)
- The GEVpwm is comparable to
GPD for small samples and
for lower return levels (i.e., 20-yr)
- The GPD is most stable, especially for data of a heavy tail
- The GEVmle is least stable
A famous scientist: “Stationarity is dead in the changing climate.”
One way to represent non-stationarity in extremes:
use “non-stationary” EV models,
e.g., GPD or GEV with time-dependent parameters
?
functions   t  f  G t , Pt 
of covariates:  log(  t )  f  G t , Pt 
   f G , P 

t
t
 t
?
functions    f  ( t )
of time itself:  log(  t )  f  ( t )
Where/when to use which model?
t
   f (t )

 t
Non-stationary EV models with parameters being functions of time t:
e.g., 
2
ˆ t  aˆ 0  aˆ1t  aˆ 2 t
ˆ
ˆ
ˆ 2
 log( ˆ t )  b0  b1t  b 2 t
- not good for use to make projections
of change in extremes, because
Hold the estimated pars. = extrapolate the fitted trend,
not desirable
- good for assessing trends in the observed/projected extremes (examples later)
EV models with covariates ( good predictors that are also well simulated by climate models)
- can be used to project changes in extremes that correspond to the projected changes in the predictors
- such projected extremes could still be stationary, with natural fluctuations around the mean climate
(examples later)
- advantage: no extrapolation of trends
- assumption: The predictor-predictand relationship will hold under the projected climate - reasonable
Allowing the EV model parameters to have different types of trends:
No trend
location
: t  o 
linear trends
 t  a o  a 1t 
l
polynomial trends
quadratic (polynomial) trends
 t  a o  a 1t  a 2 t
p
(log-)scal e :  t   o  log(  t )  bo  b1t  log(  t )  bo  b1t  b 2 t
l
shape :  t   o 
p
 t  c o  c1 t 
l
 t  c o  c1 t  c 2 t
p
1 .5

 t  a o  a 1t  a 2 t
2
1 .5
 log(  t )  bo  b1t  b 2 t
2
1 .5
q
q

 t  c o  c1 t  c 2 t
q
 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
Next, a few examples
2
Example 1 – geostrophic wind extremes
Regional average series:
a. summer (DJF)
Alexander, Wang, Wan, & Trewin ( 2010)  A significant decline
in storminess (geo-wind extremes) over southeast Australia:
Trends in annual 99th percentiles
BGD
b. autumn (MAM)
PBM
MBD
PMR
MDR
DGH
c. winter (JJA)
RDC
CDH
d. spring (SON)
How has the distribution of the
geo-wind extremes changed?
Extremes of geo-wind speeds over MDR (MilduraA-DeniliquinA-Robe), Australia
95% CI
Trend in annual 99th percentiles
1914
11-point Gaussian
smoothed
1952
1990
Notable ↓ in the
location & scale:
An early 20C’s 10yr event
a 156yr event in late 20C
(~15 times less frequent)
Stationary EV models
fitted to 3 segments: GPDs
(each of 37-yr data) GEVs
Better to search in the
GEV tree for the best
non-stationary GEV fit:
95% CI
1914
1952
1990
a 10yr event in the 1900’s
↓
a 209yr event in the 2000’s
(~20 times less frequent)
signif.↓
signif. ↓
insignif. trend
Similar results,
but with 37-yr data
GEVmle fits (to AMs)
are much less stable
than GPD fits (to POTs)
Example 2: Trends in annual minimum and annual maximum temperatures (from daily minT and
daily maxT, respectively)
A Canadian site:
Modern Exp Farm
(Manitoba; 1904-2008)
linear ↑: +3.3˚C/century
linear ↓: -1.38˚C/century
signif. ↑
signif. ↓
No change in scale & shape
No change in scale & shape
It has become much less cold, and also less hot!
(quite common across Canada)
Trends in annual minimum and annual maximum temperatures at Australian sites:
Melbourne (086071; 1856-2009):
No change in scale & shape
No signif. change!
quadratic ↑ (+2.5˚C in 1935-2009:75yr)
It has become much less cold but not hotter.
Urbanization effect
Melbourne suburb Laverton RAAF (087031; 1944-2009):
urban
suburb
10-yr event in 1945
↓
3-yr event now
No change in scale & shape
linear ↑ (+1.2˚C/century); 1 - p = 93%
variability ↑; 1 - p = 94.2%
Trends in annual minimum and annual maximum temperatures at Australian sites (cont’d):
Sydney (066062; 1859-2009):
linear ↑: +1.0˚C/century
linear ↑: +1.09˚C/century
both extremes have increased
No change in scale & shape
No change in scale & shape
It has become less cold and hotter!
Darwin (014015; 1941-2009):
Changes mainly in shape 
New
heat
records!
Shrinking lower tail; 1 – p =90.1%
Significantly heavier upper tail
Warming is common to all these sites, but in very different ways!
An example of EV models with covariates for projecting
changes in extremes/risk
GEV (  t ,  t ,  t )
GPD (~t ,  t | u t )
functions of good predictors,
e.g., geostrophic wind energy anomalies Gt and SLP anomalies Pt 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
First, need to use observations to calibrate the predictors-predictand relationship, e.g.,:
GEV ( ˆ t , ˆ t , ˆt )
with
 ˆ t  fˆ G t , Pt 

ˆ
 log( ˆ t )  f  G t , Pt 
 ˆ  ˆ
 t
- important to choose a period of observations as baseline period  define the observed baseline climate
- 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
- involves no extrapolation of trends, but assumes that the observed relationship will hold
e.g., design (T-yr return) value: H T ( t )  fˆ ( G , P )  model projected predictors
t
t
risk: p ( t )  Prob ( H t  H * )  1  FˆGEV ( H * | fˆ ( G , P ))
t
t
Next: examples of using nested GEV models with covariates to project changes in extremes
CGCM2 (Canadian model) projected changes in return values of winter extreme SWH (A2 scenario)
1990’s 20-yr.
return values (m)
12
SWH of ~13 m:
- once every 20 yr.
in the 1990’s climate;
- once every ~10 yr.
in the 2050’s climate
Contour interval: 1.0 m
changes (cm) in 20-yr
return values by 2050
(2050’s – 1990’s)
projected return periods
as of year 2050
10
14
18
14
14
14
Contour interval: 10 cm.
Shading: at least 5% significance
Contour interval: 2 yr.
Shading: at least 5% significance
Changes in probability density function (pdf) – projected by CGCM2 for IS92a scenario
(solid curves ~ average of 3 runs; non-solid curves ~ individual runs)
JFM
Iceland
no sig.
changes
N. CORM
OND
sig. increase
K-13
STATFJORD
p21=0.887
p31=0.986
FRIGG
FORTIES
AUK
EKOFISK
JFM
slight increase
p21=0.803
p31=0.962
Ireland
GULFAKS
U.K.
CGCM2 projections - IS92a scenario
1790jfm
N.
CORM
K-13
STATFJORD
AUK
Canada
GULFAKS
HIBERNIA
COHASSET
USA
Africa
JFM
JFM
sig. increase
p21=0.998
p31=1.000
sig. decrease
p21=0.921
p31=0.999
A2 scenario
2030
A2 scenario
Projected changes are typically
-non-linear
-dependent on scenario
on location
Trends in location par.
of extreme SWH
no climate change projected for the early 21C
- example of stationarity in projected
extremes using GEV with covariates
just natural fluctuations around the mean
2030
IS92a scenario
K-13
N. CORM
IS92a scenario
STATFJORD
FRIGG
FORTIES
2030
2030
AUK
A2 scenario
2030
EKOFISK
Variation is obvious from
one scenario to another,
GULFAKS
one location
to another
IS92a scenario
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?