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Quantiles x p is the p-quantile of the control if An estimate of x p is whereare the order statistics.
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Chris Ferro
Climate Analysis GroupDepartment of Meteorology
University of Reading
Extremes in a Varied Climate
1. Significance of distributional changes2. Extreme-value analysis of gridded data
Significance of Changes
Compare daily data at single grid point in
Assume X1, …, Xn have same distributionAssume Y1, …, Yn have same distribution
21002070}{scenario19901960}{control
1
1
n
n
, Y, Y, X, X
Quantiles
xp is the p-quantile of the control if
An estimate of xp is
where are the order statistics.
.Pr pxX p
)()1( nXX
,ˆ ])5.0([ pnp Xx
Daily Tmin at Wengen (DJF)
pp xy ˆagainstˆ pxy pp againstˆˆ
Dots mark the 1, 5, 10, 25, 50, 75, 90, 95 and 99% quantiles
Confidence Intervals
Quantify the uncertainty due to finite samples.
A (1 – α)-confidence interval for
is (Lp, Up) if
ppp xyd
.1)Pr( ppp UdL
Resampling
Replicating the experiment would reveal sampling variation of the estimate.
Mimic replication by resampling from data.
Must preserve any dependence– time (e.g. pre-whitening, blocking)– space (e.g. pairing)
Bootstrapping 1
For b = 1, …, B (large)resample {X1
*, …, Xn*} and {Y1
*, …, Yn*}
computeThen
*,
ˆbpd
]5.0)2/1[(,ˆ]5.0)2/[(,ˆ
*)(
*)(
BbdUBbdL
Ubpp
Lbpp
U
L
Daily Tmin at Wengen (DJF)
90% confidence intervals
pd
p
Descriptive Hypotheses
1. No change: Y Xdp = 0 for all preject unless
2. Location change: Y X + m for some mdp = m for some m and all preject unless
3. Location-scale change: Y sX + m
4. Scale change (if natural origin): Y sX
pUL pp allfor 0
pmUmL pp all and somefor
Simultaneous Intervals
Simultaneous (1 – α)-confidence intervals for dp1, …, dpm are (Lp1, Up1), …, (Lpm, Upm) if
Wider than pointwise intervals, e.g.
.1),...,1for Pr( miUdLiii ppp
.1)1(
)Pr()Pr(1
m
m
ippp iii
UdL
Bootstrapping 2
For each pbootstrapset
with k chosen to give correct confidence level:estimate level by proportion of the B sets
with at least one point outside the intervals.
*)(
*)1(
ˆˆBpp dd
*)1(
*)(
ˆandˆkBppkpp dUdL
*,
*,
ˆ,,ˆ1 bpbp m
dd
Daily Tmin at Wengen (DJF)
90% pointwise intervals90% simultaneous intervals
pd
p
Simplest hypothesis not rejectedTmin (DJF) Tmax (JJA)
Other issues
• Interpretation tricky when distributions change within samples: adjust for trends
• Bootstrapping quantiles is difficult: more sophisticated bootstrap methods
• Field significance• Computational cost
Conclusions
• Quantiles describe entire distribution• Confidence intervals quantify uncertainty• Bootstrapping can account for dependence
Extreme-value Analysis
• Summarise extremes at each grid point• Estimate return levels and other quantities• Framework for quantifying uncertainty
• Summarise model output• Validate and compare models• Downscale model output
Annual Maxima
Let Ys,t be the largest daily precipitation value at grid point s in year t.
s = 1, …, 11766 grid pointst = 1, …, 31 years
Assume Ys,1, …, Ys,31 are independent and have the same distribution.
GEV Distribution
Probability theory suggests the generalised extreme-value distribution for annual maxima:
with parameters
s
s
ssts
yyY
/1
, 1expPr
.,, ssss
Return Levels
zs(m) is the m-year return level at s if
i.e. exceeded once every m years on average.
,/1)(Pr , mmzY sts
111log)(s
mmz
s
sss
Fitting the GEV
Find to maximise likelihood:
Estimate has varianceEstimate is obtained from
),,( ssss
31
1
31
1,,,
31,31,1,1,
Prlog
,,Prlog
t tstststs
ssssss
lyY
yYyYl
.)}ˆ({)ˆvar( 12 sss l s.s)(ˆ mzs
Results 1 – Parameters
μ σ γ
Results 1 – Return Levels
z(100) % standard error
Pooling Grid Points
Often if r is close to s.Assume if r is a neighbour (r ~ s).Maximise
Using 9 × 31 observations increases precision.
sr
sr
31
1 ~,
*
t srstrss
ll
Standard Errors
Taylor expansion yields
Independence of grid points implies H = V,leaving
For dependence, estimate V using variance of
)}.(var{and)}({ **2ssss lVlEH
,)ˆvar( 11 HVHs
.)}ˆ({)ˆvar( 1*21 sss lH
.31,,1for)ˆ(~
, tlsr
str
Results 2 – Parameters
μ σ γ
Ori
gina
lPo
oled
Results 2 – Return Levels
z(100) % standard error
Ori
gina
lPo
oled
Local Variations
Potential bias if for r ~ s.Reduce bias by modelling, e.g.
Should exploit physical knowledge.
sr
.),ALTALT(),ALTALT(
sr
srssr
srssr
Results 3 – Parameters
μ
Ori
gina
lPo
oled
II
σ γ
Results 2 – Parameters
μ σ γ
Ori
gina
lPo
oled
Results 3 – Return LevelsO
rigi
nal
z(100) % standard error
Pool
ed II
Results 2 – Return Levels
z(100) % standard error
Ori
gina
lPo
oled
Conclusions
• Pooling canclarify extremal behaviourincrease precisionintroduce bias
• Careful modelling can reduce bias
Future Directions
• Diagnostics for adequacy of GEV model• Threshold exceedances; k-largest maxima• Size and shape of neighbourhoods• Less weight on more distant grid points• More accurate standard error estimates• Model any changes through time• Clustering of extremes in space and time
References• Davison & Hinkley (1997) Bootstrap Methods and their
Application. Cambridge University Press.• Davison & Ramesh (2000) Local likelihood smoothing of
sample extremes. J. Royal Statistical Soc. B, 62, 191–208.• Smith (1990) Regional estimation from spatially dependent
data. www.unc.edu/depts/statistics/faculty/rsmith.html• Wilks (1997) Resampling hypothesis tests for autocorrelated
fields. J. Climate, 10, 65 – 82.
[email protected]/~sws02caf
Commentary
Annual maximum at grid point s is GEV.
1. Estimate using data at s.
2. Estimate using data in neighbourhood of s.
Bias can occur if
3. Allow for local variation in parameters..~neighboursfor srsr
),,( ssss
