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Extreme Value Modelling in Climate Science:Why do it and how it can fail!

Aims:• What the heck do we mean by “extreme”? • Summary of statistical methods used in climate science Statistics for modelling the process rather than for just making indices• Some examples of extreme value modelling:

• Problem 1: Properties of drought indices• Problem 2: Trends in extreme gridded temperatures• Problem 2: Trends in largest annual skew tides;

Professor David B. StephensonU. of Exeter

NCAR summer colloquium, 8 June 2011© 2011 D.B.Stephenson@exeter.ac.uk

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Some wet and windy extremes

Extra-tropical cyclone

Hurricane

Polar low

Extra-tropical cyclone

Convective severe storm

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Some dry and hot extremesDrought

Wild fireDust storm

Dust storm

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All are complex multivariate spatio-temporal events!So to massively simplify, it is helpful to focus in on the time evolution of single variable related to the event e.g. wind speeds of major extratropical cyclones passing by London, losses to an insurers, etc.

MARKED POINT PROCESS: random times, random marks

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What do we mean by “extreme”?Large meteorological values

Maximum value (i.e. a local extremum) Exceedance above a high threshold Record breaker (time-varying threshold

equal to max of previously observed values)

Rare event in the tail of distribution (e.g. less than 1 in 100 years – p=0.01)

Large losses (severe or high-impact)(e.g. $200 billion if hurricane hits Miami) hazard, vulnerability, and exposure

Gare Montparnasse, 22 Oct 1895

Stephenson, D.B. (2008): Chapter 1: Definition, diagnosis, and origin of extreme weather and climate events, In Climate Extremes and Society , R. Murnane and H. Diaz (Eds), Cambridge University Press, pp 348 pp.

),()),(( txetxhVRisk

NOTE! Extremeness is not a binary property of an event but an ordering of a process

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IPCC 2001 definitionsSimple extremes:

“individual local weather variables exceeding critical levels on a continuous scale”

Complex extremes:“severe weather associated with particular climatic phenomena, often requiringa critical combination of variables”

Extreme weather event:“An extreme weather event is an event that is rare within its statistical referencedistribution at a particular place. Definitions of "rare" vary, but an extremeweather event would normally be as rare or rarer than the 10th or 90th percentile.”

Extreme climate event:“an average of a number of weather events over a certain period of time which is itself extreme (e.g.rainfall over a season)”

X~N(0,1) Y~N(0.5,1.5)

px=rank(x)/(n+1)

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How might extreme events change?

Changes in location, scale, and shape all lead to big changes in the tail of thedistribution.

Some physical argumentsexist for changes in locationand scale.

E.g. multiplicative changein precipitation due to increased humidity(change in scale)

Scale change impacts high quantiles!Example: Normal variable1% increase in standard deviation s shifts the 10-year return value (x0.9) by 1.28s and the 200-year return value (x0.995) by 2.58s.

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How can we relate the tails …

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to the bulk of the distribution?

PDF = Probability Density Function Or … Probable Dinosaur Function??

Change in scale Change in shape

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Quantile attributionDescribe the changes in quantiles in terms of changes in the location, the scale, and the shape of the parent distribution:

0.5 0.5( )

shape changes

IQRX X X X

IQR

The quantile shift is the sum of:• a location effect (shift in median)• a scale effect (change in IQR)• a shape effect

Ferro, C.A.T., D.B. Stephenson, and A. Hannachi, 2005: Simple non-parametric techniques for exploring changing probability distributions of weather, J. Climate, 18, 4344 4354.

Beniston, M. and Stephenson, D.B. (2004): Extreme climatic events and their evolution under changing climatic conditions, Global and Planetary Change, 44, pp 1-9

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Example: Regional Model Simulations of daily Tmax

Changes in location, scale and shape all important

T90ΔT90 (2071-2100 minus 1971-2000)

ΔT90-Δm

ΔT90-Δm-(T90-m) Δs/s

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Statistical methods used in climate science Extreme indices – sample statistics Basic extreme value modelling

GEV modelling of block maxima GPD modelling of excesses above high

threshold Point process model of exceedances

More complex EVT models Inclusion of explanatory factors

(e.g. trend, ENSO, etc.) Spatial pooling Max stable processes Bayesian hierarchical models + many more

Other stochastic process models

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Extreme indices are useful and easy but … They don’t always measure extreme

values in the tail of the distribution! They often confound changes in rate

and magnitude They strongly depend on threshold and

so make model comparison difficult They say nothing about extreme

behaviour for rarer extreme events at higher thresholds

They generally don’t involve probability so fail to quantify uncertainty (no inferential model)

More informative approach: model the extremal process using statisticalmodels whose parameters are then sufficient to provide complete summaries of all other possible statistics (and can simulate!)

See: Katz, R.W. (2010) “Statistics of Extremes in Climate Change”, Climatic Change, 100, 71-76

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Furthermore … indices are not METRICS!One should avoid the word “metric” unless the statistic has distance properties! Index, sample/descriptive statistic, or measure is a more sensible name!

Oxford English Dictionary: Metric - A binary function of a topological space which gives, for any two points of the space, a value equal to the distance between them, or a value treated as analogous to distance for analysis.

Properties of a metric:d(x, y) ≥ 0     d(x, y) = 0   if and only if   x = yd(x, y) = d(y, x)  d(x, z) ≤ d(x, y) + d(y, z)

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Universal Poisson process for extremes

1/

2 1

For a large number of

independent and identically

distributed values and a

sufficiently high threshold :

Pr

( ) 1

n Λ

n

z

N ~ Poisson(Λ)

Λ e(N n)

n!

zΛ t t

N=number of pointswith Z>z

t=t1 t=t2

Miraculous limit theorem for tails of i.i.d. variables!

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Probability models for maxima and excesses

1/

2 1

1/

lim ,

Pr ( ) 1

Pr max( ) Pr ( ) 0

Pr | ( ) / ( )

1 ( )

n Λ

n z

Λ e z(N n) Λ t t

n!

Z z N z e

Z z Z u z u

z uu

Generalized Extreme Value (GEV) distribution

Gener

alized Pareto Distribution (GPD)

Note: extremal properties are characterised by only three parameters (for ANY underlying distribution!)

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Why use these probability models? Model parameters are sufficient for providing a

complete threshold-independent description of extremal properties. All other statistics of the extremal process are a function of these three parameters.

The models provide a rigorous probability framework for making inference about extremal behaviour. Their mathematically justifiable parametric form allows more precise inference about tail properties.

Model can be used to smoothly interpolate between empirical quantiles/probabilities. Such interpolation has made efficient use of all the large values;

Model can be used to extrapolate out carefully to rarer less frequently (or never!) observed events AND provide intervals for such predictions!

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Problem 1: Do 2 drought series have similar extremal properties?

Observed indexn=90

Reconstructed indexn=5000

Data example kindlyprovided by Eleanor Burke,Met Office

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Do 2 drought series have similar extremal properties?

Observed indexn=90

+

Reconstructed indexn=5000

Data example kindlyprovided by Eleanor Burke,Met Office

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Return level plots

Outlier in the extended data set? Slight kink at 2.5 in d1

),)1/(1(

periodsreturn empirical versusquantiles Empirical

][1

iyni

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Quantile-Quantile plot

Empirical distributions similar except for the big outlier in d1

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Modelling the excesses using GPD

i

/11

/1

Zof ceIndependen .2

,lim

support Asymptotic 1.

1

)()(

~1~1

)(

)(~

~11|Pr)(

un

uZE

uzzf

u

uzuZzZzF

:sAssumption

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Use of mean excess to find a suitable threshold u

GPD implies linear behaviour in mean excess for u from about 0 to 1 Try fits with u=0.5 as threshold

Observedd0

Simulatedd1

1

u)X|u-E(X

u

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Nested model approach

data extended

data obs

1

0

~~~

~~

~1~1

)(

10

10

0

0

0

/11

i

i

i

a

X

X

X

H

H

uzzf

modelContrast

model Null

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Is there a statistically significant difference at 5% level?

• Difference in deviance 1050.4-1053.6+2*2=0.8 p=0.67• Parameter estimates 0.234/0.321=0.729 p=0.23

0.299/0.313=0.956 p=0.17

Maximum Likelihood EstimatesModel No. of

paramsAkaike Inf. Criterion

Null 2 1050.4 0.631

(0.023)

--- -0.107

(0.024)

---

Contrast 4 1053.6 0.629

(0.023)

0.234

(0.321)

-0.105

(0.024)

-0.299

(0.313)

Contrast with outlier

4 1085.2 0.599

(0.021)

0.264

(0.321)

-0.042

(0.019)

-0.361

(0.313)

pnts ~/ˆ

pnts ~/ˆ

2242 ~ DD

0~

1~ 0 1

No significant difference between the exceedances at 5% level

4.6107.0/631.05.0/~-ulimit upper Predicted 00

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Model checking: do the quantiles match?

No! The null model underestimates the empirical quantiles

1)Pr(~

)Pr())(1(

1

)Pr()|Pr(

1

)Pr(

1

uZTuz

uZzF

uZuZzZ

zZT

T

value Return

period Return

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Model checking: are estimates stable?

No! Constant up to u=1.7 but then trends for larger values?!

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Model checking: uniform in time?

Uniform distribution in time and exponential between events

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Problem 2: Extremes in surface temperatureCoelho, C.A.S., Ferro, C.A.T., Stephenson, D.B. and Steinskog, D.J. (2008): Methods for exploring spatial and temporal variability of extreme events in climate data, Journal of Climate, 21, pp 2072-2092

Observed surface temperatures 1870-2005 Monthly mean gridded surface temperature (HadCRUT2v)

5 degree resolution Summer months only: June July August Grid points with >50% missing values and SH are omitted.

-150 -100 -50 0 50 100 150

02

04

06

08

0

Maximum temperature

0 5 10 15 20 25 30 35 40Celsius

Maximum monthly temperatures

30year

Tem

pera

ture

(Cel

sius)

2001 2002 2003 2004 2005 2006

-50

510

1520 a)

Long term trend in mean

75th quantile (uy,m = 16.2ºC)

2003 exceedance

Excess (Ty,m – uy,m)

Non-stationarity due to seasonality and long term trends

Example: Grid point in Central Europe (12.5ºE, 47.5ºN)

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GPD scale and shape estimates1

0 1 0

Pr( | ) 1

log

z uZ z Z u

x

Shape parameter is mainly negative suggesting finite upper temperature.

Spatial pooling has been used to get more reliable less noisey shape estimates

Scale parameter is large over high-latitude land areas AND shows some dependence on x=ENSO.

0e

1e

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How significant is ENSO on extremes?

Null hypothesis of no effect can only be rejected with confidence over tropical Pacific and Northern Continents

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Use of covariates in models

“with four parameters I can fit an elephant and with five I can make him wiggle his trunk.” - John von Neumann

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Model can be used to estimate return periods

-150 -100 -50 0 50 100 150

020

4060

80

a) August 2003: Excesses above 75% threshold

0 1 2 3 4Celsius

-150 -100 -50 0 50 100 1500

2040

6080

b) August 2003: Return period

1 5 10 50 150 500years

return period of 133 years for August 2003 event over Europe

Excess for August 2003

Return period for the excessfor August 2003

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Spatial poolingPool over local grid points but allow for spatial variation by including local spatial covariates to reduce bias (bias-variance tradeoff).

For each grid point, estimate 5 GPD parameters by maximising the following likelihood over the 8 neighbouring grid points:

0,

,,0,,

,,

1

11

,

,

)()(log

),;(

jijjii

jjjyjiiii

xjijijjii

jjiijjii

ij

jjiiij

yyxx

yfL

No spatial pooling: 2 parameters from n data valuesLocal pooling: 5 parameters from 9n data values

Coelho et al., 2008:Methods for Exploring Spatial and TemporalVariability of Extreme Events in Climate Data, J. Climate

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-150 -100 -50 0 50 100 150

02

04

06

08

0

b) Chi bar (75th quantile) Central Europe

-0.4 -0.1 0.1 0.4 0.7 1

Teleconnections of extremesBivariate measure of extremal dependency:

2 log Pr( )1

log Pr(( ) & ( ))

Coles et al., Extremes, (1999)

Y u

X u Y u

association with extremes in subtropical Atlantic

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Problem 3: Is there a time trend in extreme skew tides?10 largest skew tides for each of n=149 years

Is there a time trend in the extremes?

Dots show largest values

Line is linear fit to the meanof the 10 values

Data example kindlyprovided by Tom Howard,Met Office

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r Largest Order model

0

0

10

1

0

)(

/1

12

!1)(Pr)max(Pr

1)( Pr

,lim

X

es

rzNzZ

zttΛ

n!

eΛn)(N

zn

r

s

sr

Λn

model Trend

ondistributiOrder Largest r

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• Estimates for null model are similar for r=1,5,10• Estimates get more precise for larger r• Null model has slightly better AIC than trend model• Trend model has trouble estimating trend parameter

Could either constrain shape=0 and/or pool over more data

Maximum Likelihood EstimatesModel No. of

paramsAIC

Null

r=10

3 -6882.1 0.661

(0.0096)

--- 0.147

(0.0065)

0.033

(0.025)

Null

r=5

3 -2469.2 0.659

(0.0099)

--- 0.146

(0.0068)

0.050

(0.034)

Null

r=1

3 -91.5 0.658

(0.014)

--- 0.146

(0.0099)

0.031

(0.059)

Trend

r=10

4 -6880.4 0.754

(0.0095)

-4.57E-5

(???)

0.147

(0.0042)

0.032

(0.0025)

01 0 0

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Model checking: null model r=10

Model slightly underestimates largest r=1 and r=2 quantiles

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Model checking: trend model r=10

Including a time trend does not improve the r=1 and 2 fits

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Summary Sufficiently large values of an independent

identically distributed variable can be described by a 3-parameter non-homogenous Poisson process;

This leads to simple parametric forms for the distribution of maxima and r-largest values (GEV) and exceedances above a high threshold (GPD);

MLE can be used to estimate the parameters (but estimates are often sensitive to individual values);

Non-stationarity can be accounted forby making model parameters systematic functions of covariates;

Spatial pooling can be used to obtain more precise estimates but covariates have to be included to avoid bias

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Some outstanding questions … 1. What do extreme indices really tell us about extremes?

2. How best to develop well-specified extreme value models that account for non-stationarity (non-identical distributions) caused by natural and climate change processes?

3. How to deal with large sampling uncertainty due to the rarity of events and shortness of available observational records? Robust estimation in the presence of outlier events?

4. What can imperfect climate models tell us about real world extremes? How to bias correct model errors in extremes?

5. How to develop and test well-specified inferential frameworks for prediction and attribution of real world extremes from multi-model ensembles?

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ReferencesStephenson, D.B. (2008): Chapter 1: Definition, diagnosis, and origin of extreme weather and climate events, In Climate Extremes and Society , Cambridge University Press, pp 348 pp.Definitions of what we mean by extreme, rare, severe and high-impact events

Ferro, C.A.T., D.B. Stephenson, and A. Hannachi, 2005: Simple non-parametric techniques for exploring changing probability distributions of weather, J. Climate, 18, 4344 4354. Attribution of changes in extremes to changes in bulk distribution

Beniston, M. and Stephenson, D.B. (2004): Extreme climatic events and their evolution under changing climatic conditions, Global and Planetary Change, 44, pp 1-9 Time-varying attribution of changes in heat wave extremes to changes in bulk distribution

Coelho, C.A.S., Ferro, C.A.T., Stephenson, D.B. and Steinskog, D.J. (2008): Methods for exploring spatial and temporal variability of extreme events in climate data, Journal of Climate, 21, pp 2072-2092GPD fits to gridded data including covariates. Spatial pooling and teleconnection methods.

Antoniadou, A., Besse, P., Fougeres, A.-L., Le Gall, C. and Stephenson, D.B. (2001): L Oscillation Atlantique Nord NAO: et son influence sur le climat europeen, Revue de Statistique Applique , XLIX (3), pp 39-60 One of the earliest papers to use climate covariates in EVT fits – NAO effect on CET extremes

Stuart Coles, An Introduction to Statistical Modeling of Extreme Values, Springer. Excellent overview of extreme value theory.

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There are worse things than extreme climate …

e.g. extreme ironing!

Thanks for your attentiond.b.stephenson@exeter.ac.uk

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Tubing Boulder Creek on Sunday?

Sunday noonwhitewatertubing.com

See me today if you are

interested.

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Proposed taxonomy of atmospheric extremes

Rareweather/climate

events

Rare and Severe events

Rare and Non-Severe Events

Rare, Severe, Acute events

e.g. hurricane in New England

Rare, SevereChronic events

e.g. European blocking

Rare, Non-severe, Acute events

e.g. hurricane over the South Atlantic ocean

Rare, Non-severe, Chronic events

e.g. Atlantic blocking

Rarity

Severity

Rapidity

Acute: Having a rapid onset and following a short but severe course.Chronic: Lasting for a long period of time or marked by frequent recurrenceStephenson, D.B. (2008): Chapter 1: Definition, diagnosis, and origin of extreme weather and climate events,

In Climate Extremes and Society , R. Murnane and H. Diaz (Eds), Cambridge University Press, pp 348 pp.