xstat

7/24/2019 Xstat

1/58

Analysis of Climate and Weather Data | Extreme Value Analysis An Introduction | HS 2015 | christoph.frei [at] meteoswiss.ch 2

Section 4:

Extreme Value Analysis

An Introduction

7/24/2019 Xstat

2/58


Motivation

What can happen in extremecases?

! Design of weather exposed

constructions (a pylon, wind

power plant, )

! Plan constructions for the

protection against natural

disasters.

Sea dams in the Netherlands

are dimensioned for the one in10000 year event. (?!)

Society is risk-averse !

Storm surge, The Netherlands Feb. 1953

Storm Kyrill, Northern Europe Jan. 2007

7/24/2019 Xstat

3/58


Outline

Theoretical Background

Modelling Block Maxima

Modelling Peaks over Threshold

Additional Remarks

Material based on:

!

Coles 2001 (Chap. 1- 4)

! Hosking 1990, Hosking and Wallis 1993, Zwiers and Kharin 1998,

Martins and Stedinger 2000, Katz et al. 2002, Palutikof et al. 1999.

7/24/2019 Xstat

4/58


Theoretical Background

Section 4: Extreme Value Analysis An Introduction

7/24/2019 Xstat

5/58


Return Value and Return Period

X(T): Return Value Xof Return Period T

! If Tis measured in years: Xis the threshold that is exceeded in one year with

a probability of 1/T. (One or more exceedances!)

! If Tis very large (T>> 1 year) this is equivalent to saying that Xis exceeded

on average once in Tyears.

! X(T): Amplitude as function of rareness (the quantile function)

Example: Engelberg (daily rainfall)

! The T=5-year return value rainfall is X=70 mm. A rainfall of 70 mm isexceeded in one year with a probability of 20%.

! The rainfall that occurred on August 21, 1954 (X=89.6 mm) has a return

period of T=15 years. Such an event (or larger) is expected on averageevery 15 years.

7/24/2019 Xstat

6/58


What if wed use data only?

Extreme 2-day precipitation in Entlebuch (1901-2004/5)

177 mm, 2005.08.21-22

150 mm, 1984.08.09-10

133 mm, 1946.07.05-06

132 mm, 1974.08.22-23

100

180

200

160

140

120

Empirical return period

2004 2005

T=? T=105

T=104 T=53

T=52 T=35

T=35 T=27

7/24/2019 Xstat

7/58


Extreme Value Analysis

Purpose

! Find reliable estimates of X(T)for large T(i.e. rare events),

! even for Tlarger than the period of observation,

!

including estimates of the uncertainty of X(T).

Procedure! Choose an appropriate parametric distribution function

! Calibrate it such that it describes available data well

! Extrapolate distribution function

?

7/24/2019 Xstat

8/58


Distribution of Maxima

Independent identically distributed random vars:

! E.g. representative for daily precipitations in a year (n=365)

! F(x):= prob(Xk!x), theparent distribution (CDF)

The Maximum

! E.g. the largest 24-hour total in a year

! A random variable with:

! Because:

X1, X2, X3, ..., Xn Xk ~ F(x) iid

Mn = max X

1, X

2, X

3, ...,X

n( )

prob Mn ! x( ) = prob X1 ! x,..., Xn ! x( )

= prob X1 ! x( ) " ... "prob Xn ! x( ) =Fnx( )

Mn~ F

n(x)

7/24/2019 Xstat

9/58


Illustration

Distributions of Maxima (Mn)of the exponential distribution

Distributions of Maxima (Mn)

of the normal distribution

For large nthe distribution of Mn

converges to one shape.

Asymptotic shapes for the normal and

the exponential parent distributionsare the same. (The Gumbel Distr.)

Convergence is fast for exponential

but slow for normal parent

distribution.

7/24/2019 Xstat

10/58


Extremal Types Theorem

If distribution of Mnconverges with large n

! I.e. if there exist an> 0, bn, and

a non-degenerate distribution G(x)such that:

then the limit distribution G(x)is one of ! The Gumbel distribution

!

The Frchet distribution

! The Weibull distribution

independent of the parent distribution F(x).

Fn x ! bn

an

"

#$

%

&'

n()* (** G x( )

7/24/2019 Xstat

11/58


Extremal Types Distributions

The Gumbel distribution (CDF)

The Frchet distribution (CDF)

The Weibull distribution (CDF)

G x( ) = exp !exp !x( )( )

G x( ) =0 x!0

exp "x"!( ) x > 0,!> 0#$%

&%

G x( ) = exp ! !x( )

!

( ) x 0

1 x"0

#

$%

&% Ernst HjalmarWaloddi Weibull

1887-1979

Maurice RenFrchet

1878-1973

Emil Julius Gumbel1891-1966

7/24/2019 Xstat

12/58


Asymptotic Distributions Laws

The Central Limit Theorem:! The mean of a large number of iid random variables is distributed like the

Normal Distributionindependently of the parent distribution.

The Extremal Types Theorem:

! The maximum of a large number of iid random variables is distributed like the

Gumbelor Frchetor Weibull Distributionsindependently of the parentdistribution ( if there is convergence at all).

7/24/2019 Xstat

13/58


Convergence?

In theory! No general assurance of convergence but

! convergence is warranted for continuous parent distributions under fairly

general regularity assumptions (well-behaviour of the tail).

In practice! It is justified to presume that data from natural phenomena have a parent

distribution the maxima of which are converging.

! Much more care is required with the assumption that the asymptotic limit is

applicable:

slow convergence with some parent distributions

threshold processes may influence the tail

7/24/2019 Xstat

14/58


Sketch of Proof

Max - Stability

! A distribution functionG(x)is max-stableif there exist ak(>0), and bkso that:

If G(x)is a limit distribution then it must be max-stable:

Gkx( ) =G

x ! bk

ak

"

#$

%

&', k =1, 2, 3, ...

X1,1

X1,2

! X1,n

X2,1

X1,2

! X2,n

" "

Xk,1

Xk,2

! Xk,n

n!"# !## M

(n),1$ G

n!"# !## M

(n),2 $ G

!

n!"# !## M

(n),k $ G

M(n!k)

"GkM

(n!k) "G

n!"

7/24/2019 Xstat

15/58


Sketch of Proof

Gumbel, Frchet and Weibull are max-stable.! Can be verified with simple algebra.

! E.g. Gumbel

There are no other max-stable distributions than Gumbel, Frchet

and Weibull.

!

Proof is complex (function theory).

Gkx( ) = exp !exp !x( )( )

k

= exp !k"exp !x( )( )

= exp !exp !x+ log k( )( )( )= exp !exp ! x ! log k( )( )( )( ) =G x ! log k( )( )

7/24/2019 Xstat

16/58


Generalized Extreme Value Distribution

The GEV Distribution (CDF)

! A combined parametrisation of

all three limit distributions

! Three parameters:

Location (), Scale (!),

Shape (")

! "= 0: Gumbel, unbounded

"> 0: Frchet, lower bound"< 0: Weibull, upper bound

GEV(x;,!,") = exp ! 1+" x !

!

"

#$

%

&'

(

)*

+

,-

!1 "./0

10

230

40

where: 1+"5x!

!> 0

=0; !=1

7/24/2019 Xstat

17/58


Which one is appropriate?

The GEV ! takes a special role in statistics.

! is theoretically appropriate to modelling

extremes (minima, maxima).

Independent of the nature of original data! Wind gusts,

!

Precipitation,

! Stock market changes,

! etc.

7/24/2019 Xstat

18/58


History of Extreme Value Statistics

Boris V. Gnedenko1912-1995

Unification / extension of Extreme Value Theory

Ronald Aylmer Fisher& L.H.C. Tippett

First statement of extremal types theorem

Emil Julius Gumbel1891-1966

Statistical application of theory to estimate extremes

7/24/2019 Xstat

19/58


Modelling of Block Maxima


7/24/2019 Xstat

20/58


The Block Maxima Approach

EstimateX(T)(for rare extremes)

by parametric modelling of Maxima

taken from large blocksof

independent data.

Jenkinson 1955, Gumbel 1958

7/24/2019 Xstat

21/58


Procedure

Build Blocks! Divide full dataset into equal sized chunks of data

! E.g. yearly blocks of 365/366 daily precipitation measurements

Extract Block Maxima

!

Determine the Max for each block

Fit GEV to the Max and estimate X(T)

! Estimate parameters of a GEV fitted to the block maxima.

! Calculate the return value function X(T)and its uncertainty.

7/24/2019 Xstat

22/58


The Gumbel Diagramm

Y x( )=

!log

!log F x( )( )( ) Gumbel Variate

T x( ) =1 1!F(x)( ) Return period

xk k =1,.., N Block Maxima

F x( ) =GEV x;,!,"( ) estimated CDF

Horizontal axis is linear in Y.

The Gumbel DiagrammX(T)

T-axis is transformed such thatGumbel-Distribution is a straight line.

!Tk =

N+1

N+1! rank(xk)

plotting points of

block maxima xk

Y

rank =108

!Tk =

111

3=37a

7/24/2019 Xstat

23/58


The Gumbel Diagram

Extrapolation:200-year return

value: 145 mm

Return period:

Amounts fallen on2005.08.22 have areturn period of 46

years.Return value:10-year return

value: 82 mm

7/24/2019 Xstat

24/58


Parameter Estimation

Maximum Likelihood (ML) Estimation! Numerically maximize the Likelihood Function

! Find , !, "such that the probability of drawing {xk, k=1,.., N}as random sample from the GEV would be largest.

! Pro: Obtain approximate confidence intervals (e.g. Coles 2001).

! Cons: Robustness for small N, numerical procedure

L-Moments Estimation

! Set , !, " so that the first 3 sample L-Moments are equal to thefirst 3 L-Moments of the GEV.

! Formuli for GEV L-moments from Hosking 1990

! Pros: Robustness, analytical solution.

!

Cons: Best estimate only. Uncertainties (CIs) to be inferred separately.

Hosking 1990, Coles 2001

7/24/2019 Xstat

25/58


Parameter Estimation (Example)

Max-Likelihood

= 53.5 mm

!= 12.5 mm" = 0.038

x(T=200) = 127 mm

L-Moments

= 53.2 mm

!= 12.6 mm" = 0.055

x(T=200) = 131 mm

7/24/2019 Xstat

26/58

7/24/2019 Xstat

27/58


Assessing Goodness of Fit

QQ-plot

! Check if GEV can adequately describe the data

! Too small block size and non-stationarities may manifest in inconsistencies

Engelberg

1-day totals

1901-2010

7/24/2019 Xstat

28/58


Uncertainty?

How accurate are parameter estimates ?

How accurate are return values ?

How far is extrapolation justified ?

7/24/2019 Xstat

29/58


X100= 183 km/h

Yearly maximum

wind gusts in

Zurich 1981-1999

Lets roll storms

T(Lothar) = 16 a

7/24/2019 Xstat

30/58


X100= 183 km/h

X100=165 km/h

X100=205 km/h

X100=165 km/h

X100=205 km/h

Lets roll storms

7/24/2019 Xstat

31/58


Lets roll storms

95%

60% Parametric Resampling

Confidence Intervals

Large samplinguncertainty.

X100: 145 - 250 km/h

Uncertainty of X(T)

increases with T.

7/24/2019 Xstat

32/58



Resampling Confidence Intervals! Simulate random samples (same size) and fit GEV

Parametric: random samples from estimated distribution.

Non-parametric: Draw samples (with replacement) from the original data.

! Pros: Generally accurate, applicable to any estimation method

Cons: Computationally demanding.

Asymptotic Maximum Likelihood Confidence (Delta-Method)

!

Likelihood-Theory: For large samples the sampling distribution of parameters

is multivariate Normal. Variances/covariances are inferred from curvature of

Likelihood surface.

!

Pros: fast, analytic

Cons: applicable to MLE only, not very accurate for small samples

7/24/2019 Xstat

33/58



asymmetricsymmetric

90% MLconfidence 90% resamplingconfidence

7/24/2019 Xstat

34/58


What goes wrong here?

ML-CI (Delta Method)

Index for Storminessin Europe.

ERA40, 1958-2002

Vivian,Feb 1990

7/24/2019 Xstat

35/58



Likelihood-Profile Confidence! Similar to asymptotic ML-CI. Exploits higher moments in the shape of the

likelihood function.

! Converges to asymptotic ML-CI for large samples.

! Pros: more accurate for small samples, strong shapes, large Tmakes better use of information in available data,

also accurate for negative shape.

!

Cons: only for ML estimates, computationally demanding.

Coles 2001

7/24/2019 Xstat

36/58


ML vs. Likelihood Profile CI

Likelihood-Profile

ML (Delta Method)

Index for Storminessin Europe.

ERA40, 1958-2002

Vivian,Feb 1990

7/24/2019 Xstat

37/58


Block Size: What is large enough?

A trade-off between biases (too small block size) and large samplingerrors (large blocks, small number of blocks).

Desirable block size depends on parent distribution

! Smaller for precipitation (close to exponential distribution)

!

Larger for temperature (close to Normal distribution)

For the Exponential (Normal) parent distributions GEVs from blocksizes >20 (>50) provide a reasonably bias free approximation for

return periods between 5 and 100 times the block size.

7/24/2019 Xstat

38/58


Other Issues in Practice

Trends violate iid assumption!! Solution: Parametrization of trends. GEV parameters vary

smoothly with time (Katz et al. 2002).

GEV does not make sense for seasonal means (e.g. summer 2003).

Not a Maximum of a large block!

In a network with many stations you will find events with very largelocal return periods every now and then. Citing those results only

is misleading. (Dont fall for sensation journalism!)

7/24/2019 Xstat

39/58


Example: August 2005

MeteoSwiss 2006

7/24/2019 Xstat

40/58


Modelling of Peaks over Threshold


7/24/2019 Xstat

41/58


The Peak-over-Threshold Approach

Estimate X(T)(for rare extremes)

by parametric modeling of independentexceedances above a large threshold.

Davison & Smith 1990

Note: Hydrologists tend to call this the method of Partial Duration Series

7/24/2019 Xstat

42/58


Distribution of Exceedances

Independent identically distributed random variables:

! E.g. daily maximum of wind speed over 30 years of observation

! F(x):= prob(Xk!x), the parent distribution

The Distribution of Exceedances

! ua threshold (e.g. u=10 m/s, about Beaufort 6 or larger)

!

Exceedances:y=xu

! CDF of exceedances Eu(y):

X1, X2, X3, ..., XN Xk~ F(x) iid

Eu y( ) := prob X< u+y X> u( ) Eu y( ) =

F x = u+y( )! F x = u( )1!

F x = u( )

7/24/2019 Xstat

43/58


Clarification

prob X> u+y X> u( ) =1!Eu y( )

prob X> u+y X> u( ) =1!F x = u+y( )1!F(x = u)

" Eu y( ) =

F x = u+y( )!F x = u( )

1!

F x = u( )

7/24/2019 Xstat

44/58


Asymptotic Theorem for Exceedances

If Block Maxima of F(x)asymptote to GEV

then the distribution of exceedancesEu(y)asymptotes (for large u)

to a limit distribution:

! GPD the Generalized Pareto Distribution

! GPD and GEV shape parameters are identical

! GPD and GEV scale parameters are related.

prob Mn < z( ) !GEV z;,!,"( ) forn "#

Eu y( )! GPD y;!!,"( ) for u " #

GPD y;!!,"( ) =1$ 1+! y

!!

%

&'

(

)*

$1"

with !! =! +"+ u $ ( ) > 0

See outline of proof in Coles 2001, p.76

7/24/2019 Xstat

45/58


Generalized Pareto Distribution

The GPD (CDF)

!

Two parameters:Scale (!), Shape (")

! "= 0: Exponential Distribution

!

"< 0: upper bound at !/"

! "> 0: no upper bound

GPD y;!,"( ) =1! 1+! y

"

"

#$

%

&'

!1!

y (0, 1+!y "(0

!=1

Vilfredo Federico Damaso Pareto1848-1923

7/24/2019 Xstat

46/58


Procedure

Select a threshold u

! Should be large enough to be in asymptotic limit (see later)

Extract the exceedances from the dataset

! nvalues out of the total $data values

! Exceedances need to be mutually independent (see later)

Fit GPD to exceedances, yields conditional distribution:

Estimate unconditional distribution and return values

!

with prob(X>u)estimated as n/N(the third model parameter)

! Return values X(T)from the unconditional distribution

prob X> x X> u( ) =1!GPD x !u;!,"( )

prob X> x( ) = prob X> u( ) ! 1"GPD x "u;!,"( )( )

7/24/2019 Xstat

47/58


Estimation and CIs

Exactly as with GEV

Parameter Estimation

! Maximum Likelihood Method (find maximum of Likelihood Function numerically)

! Method of L-Moments (analytical formuli, see Hosking 1990)


!

Resampling

! Asymptotic ML confidence intervals (Delta Method)

! Likelihood profile

Hosking 1990, Coles 2001

7/24/2019 Xstat

48/58


The Exceedance Diagram

Max-Likelihood

!= 11.3 mm" = 0.07

x(T=200) = 133 mm

Threshold:

30 mm

640 independentexceedances

Axis in log(T):Exponential Distribution

("= 0) is a straight line.

90% confidence

7/24/2019 Xstat

49/58


Assumptions

Exceedances are identically distributed! May be violated e.g. by seasonality, by trends

! May be resolved by stratification, explicit modeling

Exceedances are independent

!

May be violated by serial correlation

!

Much more critical than for block maximum approach

! In general resolved by declusteringof original data

! E.g.: Exceedances should be separated by at least mdays (runs-declustering).

Sufficiently large threshold (asymptotic limit)

! Depends on the parent distribution (fast/slow convergence)

! Use additional diagnostics for appropriate choice (see later)

7/24/2019 Xstat

50/58


Threshold Selection

Parameter Dependence on Threshold! If exceedances of threshold u are GPD then exceedances of threshold v(>u)

are also GPD with:

!

Can only be satisfied if:

GPD(y,!v,"v ) =GPD((v!u)+y,!u,"u)

GPD((v!u),!u,"u)for ally > 0

!v =!

u, "

v !! "v ="

u !! "u

At sufficiently large thresholds u:

(a) shape "is independent of threshold(b) modified scale (! "u) is independent of threshold

7/24/2019 Xstat

51/58


Threshold Selection

Diagnostics forthreshold selection

Engelberg,

daily precipitation (mm)

1901-2010

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Be aware:

Stability within

confidence limitsdoes not warrant

truestability.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

approx. stablewithin limits

approx. stablewithin limits

7/24/2019 Xstat

52/58


Threshold Selection

! If exceedances are GPD

! Then expected value of

exceedances is:

Mean Residual Life Plot:

linear

within limits

Mean of exceedance (beyond threshold)of daily precipitation peaks

Engelberg, 1901-2010

prob(Yu

7/24/2019 Xstat

53/58


Block-Max vs. Peak-over-Thresh

Block Maximum:

GEV, 110 maxima

Peak-over-Threshold:

GPD, 640 peaks > 30 mm

127 mm

101 mm

152 mm

120 mm

105 mm

138 mm

X(100)X(100)

7/24/2019 Xstat

54/58


Block-Max vs. Peak-over-Thresh

Block Maximum Approach

Pros

! Theoretical assumptions are

less critical in practice.

! Independence of maxima can

be achieved by selecting largeblock size.

!

More easy to apply.

Cons

! Estimation uncertainties can be

large because sample size issmall.

Peak-over-Threshold Approach

Pros

! Smaller CIs if a small

threshold is justified. (Moreindependent exceedances than

block maxima.)

Cons! Independence assumption is

critical in practice. Need

declustering techniques.

!

Needs diagnostics for threshold

selection. Choice somewhat

ambiguous in practice.

!

Less easy to apply in practice.

7/24/2019 Xstat

55/58


Additional Remarks


7/24/2019 Xstat

56/58


Additional Remarks

Precipitation Extremes! Dependence of return levels on duration is commonly described by an

Intensity-Duration-FrequencyCurves (also Depth-Duration-Frequency) using

scaling relationships between durations.

Koutsoyannis et al. 1998; Geiger, Zeller, Rthlisberger, 1990

Bern

LogPrecipitationRate(In

tensity)

Log Measurement Interval (Duration)

Return Period (Frequency)

7/24/2019 Xstat

57/58


Additional Remarks

Hydrological Extremes! Hydrological processes involve threshold processes. Only very few events

may have been observed from the extreme tail. Very large block size needed.

! Statistics alone may be inadequate. Physical modelling of scenarios instead.

Hourly runoff extremes, Langeten

Liechti, 2008

7/24/2019 Xstat

58/58

Analysis of Climate and Weather Data | Extreme Value Analysis An Introduction | HS 2015 | christoph frei [at] meteoswiss ch 63

Additional Remarks

Spatial Extremes! Utilize concepts of spatial dependence or regional pooling of data to reduce

uncertainties in estimation.

! Combine extreme values analysis and spatial statistics to estimate extremes

for unobserved locations.

Cooley et al. 2007, Alila 2007; Davison et al. 2012

xstat

Documents