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.

c.a.t.ferro@reading.ac.ukwww.met.rdg.ac.uk/~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