Extreme value statistics ms of extrapolating to values we have no data about Question: Question: Can this be done at all? unusually large or small ) ( i t h i t ? ) ( max i t h ~100 years (data) ~500 years (design) winds ) ( v i t How long will it stand?
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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.
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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?