Extreme value statistics
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Transcript of Extreme value statistics
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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?
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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
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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
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F
F
1.5cm
63 fibers
The weakest link problem
F
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Problem of trends I
Variables may be non-identically distributed.
Sea level seems to grow.
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Problem of trends II
Athletes run now faster than 30 years ago.
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Problem of correlations I
Maximum sea level depends, or at least iscorrelated to other variables.
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Problem of correlations II
Multivariate extremes
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Problem of second-, third-, …, largest values
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Problem of exceeding a threshold
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Problem of deterministic background processes
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Problem of the right choice of variables
1
ln
i
i
M
M
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Problem of spatial correlations
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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
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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
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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!
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FTG function and fittingb
ax
bax e
b exP
1)(
1 0 ba
xee xe
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FTG function and fitting: Logscale b
ax
bax e
b exP
1)(
1 0 ba
xee
xe
See example on fitting.
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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
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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
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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!
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FTF density function for
2,1,0 ba
xexxP /12)(
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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
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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
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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.
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Weibull function and fitting
1 0 ba
ax
axexP
bxa
bxa
b
0
)()(
1)(1
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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.
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What kind of conclusions can be drawn?