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?

)(0 yP

y

)(xP

x

NN yyyx ,...,,max 21

Y Nyyy ,...,, 21

LogicsLogics::

Assume something about iy

Use limit argument: )( N

Slightly suspicious but no alternatives exist at present.

E.g. independent, identically distributed

Family of limit distributions (models) is obtained

Calibrate the family of models by the measured values of Nx

An example of extreme value statistics

)(0 yP

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. 

Figures are from Stuart Coles: An Introduction to Statistical Modeling of Extreme Values

F

F

1.5cm

63 fibers

The weakest link problem

F

Problem of trends IFigures are from Stuart Coles: An Introduction to Statistical Modeling of Extreme Values

Problem of trends II

Problem of correlations

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

Problem of information loss

Problem of deterministic background processes

Problem of trends and variables

1

ln

i

i

M

M

Problem of spatial correlations

Problem of trends

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

Fisher-Tippett-Gumbel distribution I

Assumption: Independent, identically distributed random variables with

yeyP )(0

y NN yyyx ,...,,max 21

parent distribution

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

lnln NxN

1)(0 NxP N

Note:

NNx

22ndnd q question:uestion: Can we estimate ?

)(xP

x

Nx22 )( Nxx

eNxP N /1)(0 1

Homework: Carry out the above estimates for a Gaussian parent

distribution ! 2

)(0yeyP

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

Fisher-Tippett-Gumbel distribution II

Assumption: Independent, identically distributed random variables with

yeyP )(0

y NN yyyx ,...,,max 21

parent distribution

)(xP

x

NxNxz ln

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

Probability of : zxN

Nzz

NNNN dyyPdxxPzm

0

0 )()()(

xeNxNNzNz eNeNeezm )/1()/1()1()( )ln(

xexNdx

dN eNxzmxP

)ln(lim)(

Expected that this result does not depend on small detailsof but there is more generality to this result.

)(0 yPy

FTG distribution

Fisher-Tippett-Gumbel distribution III

yeyP )(0

y

NN yyyx ,...,,max 21 parent distribution

)(xP

x

Nx

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

b

ax

bax ebexP

)(

We do not know it!

is measured.

is not known!N

Nxz lnThe shift is not known!

The scale of can be chosen at will.Nx

Fitting to:

Asymptotics:

bxe

bx

be

bexP |/|

/

)(x

x

-1 largest smallest

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.

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

Finite cutoff: Weibull distribution

Assumption: Independent, identically distributed random variables with

)()( 1

10 yayP

a

y NN yyyx ,...,,max 21

parent distribution

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

)1/(1 Nxa N

aN

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

0)(0 aP

)(xP

x

Nx

a

1)(0 NxP N Nxa

Nxa )1/(1 N

Weibull distribution II

0 )()( 1

10

yayPa

y

parent distribution

)(xP

x

Nx

a

)1/(1 xNaz

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

Probability of : zxN

Nzz

NNNN dyyPdxxPzm

0

0 )()()(

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

az eNxzm

1)()1/(1 ))(1()(lim)(

x

Ndxd

N exxNazmxP

Weibull distribution

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

Assumption: Independent, identically distributed random variables with

NN yyyx ,...,,max 21

)1/(1 Nxa N

0x

0)( xP 0x

Weibull distribution III

0 )()( 1

10

yayPa

y

parent distribution

)(xP

x

Nx

a

0 0

)()(

1)(1

x

axexP

bxa

bxa

b

)1/(1 Nxa N

NN yyyx ,...,,max 21 is measured.

is not known!N

a in is not known!

The scale of can be chosen at will.Nx

Fitting to

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

)(maxT 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?

Györgyi Géza előadásai

Critical order-parameter fluctuations in the d=3 dimensional Ising model

L

2T

1T 1

2

aL

/1

Delamotte, Niedermayer, Tissier

or how does an informative figure look like

Example of EVS in action: P(v) of the rightmost atom in an expanding gas

D=1 ideal, elastic gas in equilibrium at T:

20

2 )v/v(2/v ~~)v( eeP kTm

Box is opened, wait for a long time.

2310N

Questions: (1) What is the expected velocity of the rightmost particle?(2) What is P(v) of the rightmost particle?

(3) Estimate the expected velocity.

Elastic collisions:

22

21

22

21

2121

v'v'vv

v'v'vv

12

21

vv'

vv'

velocities are exchanged

always increases

t

v

maxvvsmN 16000)(lnvv 2/1

0max

sm300 2310

Nontrivial EVS distributions

(1) What are the reasons for any other extreme satistics to emerge?

Analogous question in statistical physics: What are the reasons for nongaussian statistics of macroscopic (additive) quantities?

(2) What are the simplest calculable examples?

Independent, nonidentically distributed variables.

(3) What can we say about interacting systems?

Question of weak or strong correlations (finite- or infinite correlation length).

FTG, Weibull, FTF : High-temperature fixed point.

Are there critical EVS distributions with universality classes?

Questions:

Gaussian and nongaussian distributions I

Extensive quantity in a noncritical system

Ld

d/2L

P(M)

M

Small Gaussian fluctuationsaround the mean

central limit theorem

L) (

Example: Ising modelabove and below Tc.

P(M)

MLd

d/2L d/2

L

Ising model at Tc:

22~2 d

LM

Gaussian and nongaussian distributions II

Extensive quantity in a critical system

nongaussian fluctuationsaround the mean

no central limit theorem

L) ~(

Example: Ising model at Tc:

Scaling variable: )( )( 2 MPMx Scaling func.: 2/ MMx

2/ MMx

d=2

2/ MMx

d=3

Emergence of universal scaling functions

22~2 d

LM

Edwards-Wilkinson (EW) interface

hht

Surface tension driven growth

L

dyheh 0

22 )(

~

P 22

2

222~ kk k hLk

k

hkL

k eeh

P

Noise in the arrival(Gaussian, white is assumed)

average growth velocity

Long wavelength (gradient) expansion:

y),( tyh

0 L

,...),(v 2hhfh yyt

Vertical growth velocity can dependonly on the spatial derivatives of .h

...v,...),( 20

2 hhhf yyy

In the system moving with : 0vsurface tension

Steady state distribution function: Fourier modes: k k

ikyheyh )(

Independent modes

Nongaussian distributions - Edwards-Wilkinson (EW) interface

LhhtyhdyL k

k

L

22

02 ),(1

w

y),( tyh

0 L

hht

Surface tension driven dynamics

Stationary state:

L

dyheh 0

22 )(

~

P

diverging fluctuations

22

2

222~ kk k hLk

k

hkL

k eeh

P

Independent Fourier modes:

sum of independent variables

22 w/wx

(x)

2

2 1~k

hk

Reason for the failing of the central limit theorem

nonidentical,singular fluctuations

?

Derivation of the width distribution for EW interface (1)

hht Stationary state:22

2 khLkke

Independent Fourier modes

Width distribution:

L

L hhdywhhDwP0

2122 ])([ )( )()( P

L

dyheh 0

22 )(

~

P

122||)

2(

~22

Ls

k

hLs

k

kkk

kNedhdhNk

In terms of Fourier modes:

Normalization:

1)0( G

k Lsk

ksG

22

2

)(

1

1

2 221)(n

nLssG

nk L2

Generating function (Laplace transform):

L L

dyhhLs

dyhsw ehDwPedwsG 0

2

0

22

2

)()(

2

0

2 )()()(

Path integral ofharmonic oscillator

Derivation of the width distribution for EW interface (2)

Width distribution in scaling form:

)()()()(2

2

2

22

2

2

2

2 122

122

ww

w

wswwi

i

iwds

wsw

i

i

ids ewsFesGwP

Generating function: )()( 2

0

22 wPedwsG sw

1

1

2 221n

nLs

Average width:

121

2 122

02

Ln

LdsdG

wns

)(1 2

1

16

222

wsF

nn

ws

(x)

22 w/wx

Scaling function:

xy

nn

yi

i

idy ex

162 )1()( 22

Simple poles at

6/22ny

1

6213

22

2)1()(

n

xnn enx

2)1()1( 11

2

2

n

n

n

Nongaussianity - width distribution of interfaces

Picture gallery

KPZ

EW

MHMH

PRE50, 639, 3589 (1994) PRE50, 3530 (1994) PRE65, 026136 (2002)

)(w

w)w( w

2

222 xP

LL

Stationary distribution:

Ag on glass

y),( tyh

0 L

L~w2

202 ),(

1),(w htyhdyL

tLL

width

h

Example of nontrivial EVS: Nonidentically distributed independent variables

N

nnhnS

1

2~

Probability of zhn 2

maxbeing less than z:

N

n

znN

n

zh

nnhnn edhdhezM

n

n

11 0

)1()(

2

2

2

1

~nhn

N

n

nn ehP

A given configuration

t)(th

0 T Parent distribution

provides a set of nh)(th

Action:

(looking for the max of N variables)

Example of nontrivial EVS: Nonidentically distributed independent variables

N

nncnS

1

2~

Probability of zcn 2

maxbeing between z and z+dz

N

nzn

N

m

zm

e

nezP

11 1)1()(

Distribution of extremal intensity fluctuations

dzzP )( dzzhz k 2

maxFind : the probability of

2

~)(

k khk

k eh

Drawing from N independent, nonidentically distributed numbers ( ) with singular 2

kh

2

kh

2Lnk

k

2

maxkh

Math:

PRE68, 056116 (2003)

khk ~2

ResultResult::

L

n

L

nzn

zn

e

nezP

1 1 1)1()(

1/f noise - voltage fluctuations in resistors

Example:Example:

)(21 tVeVift

xTf

fVfS f /1~|| ~)( 2 fVfS f /1~|| ~)( 2

T

)(tV

t

2IrOfilmRhutenate

2.1

1

M.B. Weissman, Rev.Mod.Phys.60, 537(1988)

Turbulence and the d=2 EW model

S. Bramwell et al. Nature 396, 552 (1998)

c

Experiment

Critical Distribution of energy dissipation

Distribution of d=2 XY magnetization below Tc. (finite-size)

M)/M-(M p)/p-(p

cT T XY EW 2d

Aji & Goldenfeld, PRL86, 1007 (2001)

dissipation is mainly on the fluctuations of the shear pancake

Possible connection to extreme statistics?

Width distributions for signals

n nhhh 22

2 ||~)(w

t)(th

0 T

2

~ n nhn

n eh

P

f/1

Stationary distribution for Fourier modes

Integrated power spectrum

Question:Question: Is there anfor which extreme statistics distribution emerges?

PRE65, 046140 (2002)

T

T xP2

222 w

w)w( w

Does it have an internal structure?

1 )()( xx

Extreme statistics for .

n

nh2

2w

t)(th

0 T

1

new scaling variable

integrated power spectrum

Fisher-Tippett-Gumbel extreme value distribution

Central limit theorem is restored for

PRL87, 240601 (2001)

1

yeyey1

2

~ yey 2/1

n~h2

n

T

Ty 22 ww

22

22T ww TT

yPT )w( 2

1

2/1

Homeworks

1. See slide Fisher-Tippett-Gumbel distribution I.

2. Show that FTG distribution becomes the Weibull distribution in the limit.

3. Determine the Tmax distribution for temperatures measured at the Amistad Dam. Try fits with both the FTG and the Weibull functions.

4. What is the velocity distribution of the rightmost particle in a one-dimensional, freely expanding gas?