Extreme value statistics

Problems of extrapolating to values we have no data about

Question:Question: Can this be done at all?

unusually large or small

)( ith

it

?)(max ith

~100 years (data)

~500 years (design)

winds

)(v it

How long will it stand?

Extreme value paradigm

is measured:Question: Question: What is the distribution of the largest number?

)( y

y

)(zPN

z

NN yyyz ,...,,max 21

Y Nyyy ,...,, 21

LogicsLogics::

Assume something about iy

Use limit argument: )( N

E.g. independent, identically distributed

Family of limit distributions (models) is obtained

Calibrate the family of models by the measured values of Nz

parent distribution

An example of extreme value statistics

)( y

y

)(xP

x

The 1841 sea level benchmark (centre) on the `Isle of the Dead', Tasmania.  According to Antarctic explorer, Capt. Sir James Clark Ross, it marked mean sea level in 1841. 

Data plots here and below are from Stuart Coles: An Introduction to Statistical Modeling of Extreme Values

Recurrence time: If then the maximum will exceed in T years.

0

/1)(x

TdxxP

0x

F

F

1.5cm

63 fibers

The weakest link problem

F

Problem of trends I

Variables may be non-identically distributed.

Sea level seems to grow.

Problem of trends II

Athletes run now faster than 30 years ago.

Problem of correlations I

Maximum sea level depends, or at least iscorrelated to other variables.

Problem of correlations II

Multivariate extremes

Problem of second-, third-, …, largest values

Problem of exceeding a threshold

Problem of deterministic background processes

Problem of the right choice of variables

1

ln

i

i

M

M

Problem of spatial correlations

is measured:Y Nyyy ,...,, 21

Fisher-Tippett-Gumbel distribution I

Assumption: Independent, identically distributed random variables with

yey )(

y NN yyyz ,...,,max 21

parent distribution

11stst q question:uestion: Can we estimate ?Nz

lnlnNzN

1)( NzN

Note:

NNz

22ndnd q question:uestion: Can we estimate ?

)(zPN

z

Nz22 )( Nzz

eNzN /1)( 1

Homework: Carry out the above estimates for a Gaussian parent

distribution ! 2

)( yey

is measured:Y Nyyy ,...,, 21

Fisher-Tippett-Gumbel distribution II

Assumption: Independent, identically distributed random variables with

yey )(

y NN yyyz ,...,,max 21

parent distribution

)(zPN

z

NzNzx ln

QQuestion:uestion: Can we calculate ?)( NN zP

Probability of : zzN

Nzz

NNNN dyydzzPzM

0

)()()(

)()/1(

)/1()1()( )ln(

xMeNe

NeezMxeNx

NNzNzN

xexexMxP )(')(

Expected that this result does not depend on small detailsof .)( y

y

FTG density function

Fisher-Tippett-Gumbel distribution III

)( y

y

NN yyyz ,...,,max 21

)(zPN

z

Nz

QQuestion:uestion: What is the „fitting to FTG” procedure?

b

ax

bax eebxP

1)(

We do not know the parent distribution!

is measured.

Nbxz The shift in is not known!

The scale of can be chosen at will.Nz

Fitting to:

Asymptotes:

bxe

bx

e

exP /||

/

)(x

x-1 largest smallest

ImportantImportant:: In the simplest EVS paradigm only linear change of variables is allowed. Without this restriction any distribution could be obtained!

FTG function and fittingb

ax

bax e

b exP

1)(

1 0 ba

xee xe

FTG function and fitting: Logscale b

ax

bax e

b exP

1)(

1 0 ba

xee

xe

See example on fitting.

Fisher-Tippett-Fréchet distribution I

NN yyyz ,...,,max 21

)(zPN

z

Nz

Parent distribution: Power decay

is measured.

11stst q question:uestion: Can we estimate the typical maximum?

)1/(1 NzN

22 )( Nzz

,1)( NzN

Nz

Nz

22ndnd q question:uestion: Can we estimate the deviation?

If it exists!

The maximum is on the same scale as the deviation.

)1/(1 NN

)1()( yAy

y

01 A

Fisher-Tippett-Fréchet distribution II

)1/(1 xNz

QQuestion:uestion: Can we calculate ?)( NN zP

Probability of : zzN

Nzz

NNNN dyydzzPzM

0

)()()(

M N ( z )≈[1− 1

z β−1 ]N

≈ e− N

z β−1

1

1

)1()(')(

xexxMxP FTF density function

is measured:Y Nyyy ,...,, 21

Assumption: Independent, identically distributed random variables with parent

NN yyyz ,...,,max 21

1

1

)(

xexM

)1()( yAy

y

)(zPN

z

Nz

01 A

For large : z

Fisher-Tippett-Fréchet distribution III

y

)(zPN

z

Nz

1)(1 )()(

b

ax

exP bax

b

NN yyyz ,...,,max 21

The origin and the scale of x can be chosen at will:

ba,

The function to fit for x>a is

Note that for 2 there is no average!

The kth moment does not exist for 1k

)1()( yAy

y

01 A)1/(1 NzN in is not known!

FTF density function for

2,1,0 ba

xexxP /12)(

is measured:Y Nyyy ,...,, 21

Finite cutoff: Weibull distribution I

Assumption: Independent, identically distributed random variables with

1)()( 1

1

,yaya

y NN yyyz ,...,,max 21

parent distribution

11stst q question:uestion: Can we estimate ?Nza

)1/(1 Nza N

aN

22ndnd q question:uestion: Can we estimate ?22 )( Nzz

0)( a

)(zPN

z

Nz

a

1)( NzN Nza

Nza )1/(1 N

Weibull distribution II

y

parent distribution

a

)1( )1/(1 xNaz

QQuestion:uestion: Can we calculate ?)( NN zP

Probability of : zzN

Nzz

NNNN dyydzzPzM

0

)()()(

)()/)(1()1(1)(1)(11 xMeNxzM xNN

az

N

1)())(1()(')(

xexxMxP

Weibull density function

is measured:Y Nyyy ,...,, 21

Assumption: Independent, identically distributed random variables with

NN yyyz ,...,,max 21

)1/(1 Nza N

0x0)( xP 0x

)(zPN

z

Nz

if

if

1)()( 1

1

,yaya

Weibull distribution III

y

parent distribution

)(zPN

z

Nz

a

ax

axexPbxa

bxa

b

,0

,)()(1)(1

)1/(1 Nza N

NN yyyz ,...,,max 21 is measured.

are not known!,a

,a in is not known!

Fitting to

1)()( 1

1

,yaya

Nand possibly

bThe scale of Nx can be chosen at will.

Weibull function and fitting

1 0 ba

ax

axexP

bxa

bxa

b

0

)()(

1)(1

Notes about the Tmax homework)(,),(,),2(),1( )1(

max)1(

max)1(

max)1(

max NTnTTT

)(,),(,),2(),1( )2(max

)2(max

)2(max

)2(max NTnTTT

)(max

T 2)(max

)(max )( TT

Introduce scaled variables common to all data sets

2)(max

)(max

)(max

)(max)(

)(

)(

TT

TnTxn

0)( nx 1( )()(

nn xx

Find

Average and width of distribution

so all data can be analyzed together.

?

?

?

?

?

?

?

?

What kind of conclusions can be drawn?