Extreme Value Theory in Civil Engineeringbaidurya/dissemination/ev.pdfExtreme Value Theory in Civil...
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Extreme Value Theory in Civil Engineering
Baidurya Bhattacharya
Dept of Civil Engineering IIT Kharagpur
December 2016 Homepage: www.facweb.iitkgp.ernet.in/~baidurya/
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Preliminaries: Return period
• IID random variables X1 , X2 , X3,… with CDF FX – Occurrence (or success) = Xi > xp in ith trial – p = Psuccess = P Xi > xp = 1 – FX(xp) – xp = level corresponding to exceedance probability p
• Sequence of independent and identical Bernoulli trials: – Geometric random variable – Time between successive occurences is random – “Mean Return Period” associated with xp is 1/p (in units of trial
time interval)
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Preliminaries: Characteristic value
• IID random variables X1 , X2 , X3,… with CDF FX – Success = Xi > xp in ith trial – p = Psuccess = P Xi > xp = 1 – FX(xp) – xp = level corresponding to exceedance probability p
• n repeated trials – Binomial random variable – Mean number of occurrences = np
• xn = Characteristic value of Xi – if mean number of occurrences, np(xn) =1 – that is, 1 – FX(xn) = 1/n
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Motivation: Time-dependent reliability
],0),,(),(:[inf Ω∈≥<= xtxtDxtCtTFirst Passage Time:
Probability of failure: ][)( LLf tTPtP ≤=
Capacity (C
) dem
and (D)
time tL
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Design issues
• Maximum load • Minimum capacity • Associated uncertainties • Data driven
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Reliability problem statement
• Time-dependent failure probability
• Simplification
• Need to estimate Lmax,t
]],0[anyfor0)()([)( tLDRPtPf ∈≤−−= τττ
]0[)( max, ≤−−= tef LDRPtP
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Maximum live load estimation
• Maximum live load
• Distribution function
• Simplifications – independence – stationarity
,...,,max )(21max, tNt LLLL =
],...,,[)( )(21max,lLlLlLPlF tNL t
≤≤≤=
( )max, ( ) [ ( )]N t
L t LF l F l=
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Extreme value theory: problem statement
• Sequence of random variables Xi • What are the limiting forms of Zn and Wn ?
• Issues – n unknown – Degeneracy of limit distributions: n infinite – Nature of population distributions, Fi
– Dependence in the sequence – Non-stationarity of the sequence
1 2
1 2
max( , ,..., ) ~min( , ,..., ) ~
n n n
n n n
Z X X X HW X X X L
==
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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IID (classical) case
• Xi is an IID sequence – Xi and Xj are independent for i ≠ j – Fi = F are same for all i
• Hn(x) and Ln(x) are
degenerate distributions
[ ] ( ) ( )
[ ] ( ) 1 (1 ( ))
nn n
nn n
P Z x H x F x
P W x L x F x
≤ = =
≤ = = − −
( ) upper end point of
0, ( )lim ( )
1,otherwisenn
F F
x FH x
ω
ω→∞
=
<=
( )
( ) lower end point of
0, ( )lim ( )
1, otherwisenn
a F F
x FL x
α→∞
=
≤=
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Code for CDF of Xmax vs. n
• %This program generates a sequence of n IID exponential RVs and stores it maximum, xmax
• %It then repeats the process mct times
• %The distribution of xmax is plotted
• %The plot is repeated for a different n
• clear all;
• mct=1000;
• n=input('give n\n');
• for mcti=1:mct,
• for i=1:n,
• x(i)=-log(rand);
• end
• xmax(mcti)=max(x);
• freq(mcti)=mcti/(mct+1);
• end
• xmaxsorted=sort(xmax);
• loglog(xmaxsorted,freq);
• sizes=['n=' num2str(n)];
• hold on;
• text(xmaxsorted(mct/2),.5,[sizes],'BackgroundColor',[1 1 1],'EdgeColor','black');
• axis([0,100,0,1]);
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Example: degeneracy for large n
max 1 21,...,
~ Exponential(1) iidmax , ,...,
i
ni n
XX X X X
==
cdf o
f Xm
ax
x Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Normalization
• Can we normalize Zn and help matters?
• Issues – How many possible forms for H ? – How does H depend on F ? – How to find an and bn – What is the speed of convergence?
lim lim ( ) ( )n nn n nn n
n
Z aP x H a b x H xb→∞ →∞
−≤ = + =
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Normalization
• Can we normalize W(n) and help matters?
• Issues – How many possible forms for L ? – How does L depend on F ? – How to find an and bn – What is the speed of convergence?
( )( )lim lim ( ) ( )n nn n nn n
n
W aP z L a b z L z
b→∞ →∞
− ≤ = + =
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Normalization
max
As ,Instead of ( ) ( )
Look at ( ) where ( , )Simplest form for :
nX
nX n n
n
n n n
nF x F x
F x x f x nx
x a b x
→∞
=
=
= +
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Normalization example
max
max
~ Exponential(1)
( ) ( ) (1 exp( ))As ,Replace ( ) ln( )
exp( ( ))Obtain: ( ) lim 1
exp( exp( ( )))
in n
X
n
n
X
F x F x xn
x x u n
x uF xnx u
α
α
α
→∞
= = − −→∞
← − +
− − = −
= − − −
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Code for CDF of normalized Xmax vs. n
• %This program generates n IID Exponentials and stores the maximum xmax.
• %xmax is then recentered and scaled as a function of n
• %The process is repeated n times
• clear all;
• mct=1000;
• n=input('give n\n');
• scale=input('give scale\n');
• shift=input('give shift\n');
• if n==1,scale=1;shift=0;end
• for mcti=1:mct,
• for i=1:n,
• x(i)=-log(rand);
• end
• xmax(mcti)=(max(x)-log(n)+shift)/scale;
• freq(mcti)=mcti/(mct+1);
• end
• xmaxsorted=sort(xmax);
• loglog(xmaxsorted,freq);
• sizes=['n=' num2str(n)];
• hold on;
• text(xmaxsorted(mct/2),.5,[sizes],'BackgroundColor',[1 1 1],'EdgeColor','black');
• axis([0,100,0,1]);
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Limit distributions in IID case
• There are only three types of non-degenerate distributions H(x) for maxima
• There are only three types of non-degenerate distributions L(x) for minima
• Necessary and sufficient conditions exist for F(x) to yield above max or min distributions – Note: Two distributions F and G are of the same “type”, if
F(x) = G(ax+b) where a, b are constants
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Generalized EV distribution for maxima
1/
In IID case, ( ) must be of the same type as:
( ) exp (1 ) , 1 0
0 Gumbel (Type I) distribution: ( ) ,
, 00 Frechet (Type II) distribution: ( )0 , 0
0 Weibull
z
cc
eG
z
F
H z
H z cz cz
c H z e z
e zc H zz
c
γ
−
−
−
−
−
= − + + >
= ⇒ = −∞ < < ∞
>> ⇒ = ≤
< ⇒( )
| 1 / |where
1, 0(Type III) distribution: ( )
, 0W z
c
zH z
e zγ
γ
−− −
=
>= ≤
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Generalized EV distribution for minima
1/
( )
In IID case, ( ) must be of the same type as:
( ) 1 exp (1 ) , 1 0
0 Gumbel (Type I) distribution: ( ) 1 ,
1 , 00 Frechet (Type II) distribution: ( )1 , 0
0
z
cc
eG
z
F
L z
L z cz cz
c L z e z
e zc L zz
c
γ−
−
−
− −
= − − − − >
= ⇒ = − −∞ < < ∞
− ≤> ⇒ = >
< ⇒
| 1 / |where
0, 0Weibull (Type III) distribution: ( )
1 , 0W z
c
zL z
e zγ
γ
−
=
<= − ≥
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Domains of attraction for maxima
( )
( )
any finite , ( ), ( ) lower and upper end points of
( ) ( | ), ( )
1 ( )(A) lim , 01 ( )
(B) (1 ( )) ,
1 ( ( ))(C) lim ,1 ( )
t
F
a
x
t F
a F F F
R t E X t X t t F
F tx xF t
F x dx
F t xR t eF t
γ
ω
ω
α ω
α
γ−
→∞
−
→
=
= − > >
−= >
−
− < ∞
− +=
−
∫
*
0, inf( : 1 ( ) 1 / )
( ), ( ) inf( : 1 ( ) 1 / )
( ) if and only if ( ) and (A) holds for
( ) if and only if ( ) and (A) holds for
( ) ( ( ) 1/ )
( ) if and only if ( ) a
n n
n n
F
W
G
a b x F x n
a F b F x F x n
F D H F F
F D H F
F x F F x
F D H Fω ω
ω
ω
ω
ω
= = − ≤
= = − − ≤
• ∈ = ∞
• ∈ < ∞
= −
• ∈ = ∞inf( : 1 ( ) 1 / ), ( )
nd (B), (C) hold
n n na x F x n b R a= − ≤ =
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Domains of attraction for minima
( )
( )
any finite , ( ), ( ) lower and upper end points of
( ) ( | ), ( )
( )(A) lim , 0( )
(B) ( ) ,
( ( ))(C) lim ,( )
t
a
F
x
t F
a F F F
r t E t X X t t F
F tx xF t
F x dx
F t xr t eF t
γ
α
α
α ω
α
γ−
→∞
→
=
= − < >
= >
< ∞
+=
∫
*
0, sup( : ( ) 1 / )
( ), sup( : ( ) 1 / ) ( )
( ) if and only if ( ) and (A) holds for
( ) if and only if ( ) and (A) holds for
( ) ( ( ) 1/ ), 0
( ) if and only if (B),
n n
n n
F
W
G
c d x F x n
c F d x F x n F
F D L F F
F D L F
F x F F x x
F D Lα α
α
α
α
= = ≤
= = ≤ −
• ∈ = −∞
• ∈ > −∞
= − <
• ∈sup( : ( ) 1 / ), ( )
(C) hold
n n nc x F x n d r c= ≤ =
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Examples
• Cauchy • Uniform • Exponential • Rayleigh
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Examples
• Limit distribution for minima from Cauchy parent
2
2
21
2 2
1 arctan( )( ) ;2
Check Eqn (A) for minima: 1 arctan( )
( ) 2lim lim 1 arctan( )( )2
1 ( )lim 11(1 )lim
1
That is, 1, and Cauchy lies in the domain of attr
t t
t
t
xF x x
txF tx
tf t
xtx
tx t x
t x
π
π
π
γ
→−∞ →−∞
→−∞
−
→−∞
= + −∞ < < ∞
+=
+
+=
++
= =+
= − action of Frechet for minima.The normalizing constants are:
0
1 1tan2
n
n
c
dn
π
=
= −
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Examples
• Limit distribution for maxima from uniform parent
1
1
The complemetary CDF of the uniform is:1*( ) 1 , 1
11 *( )lim lim 11 *( )
Since = 1, uniform gives rise to Weibull maxima. The normalizing constants are:
( ) 11( ) 1
t t
n
n
F x xx
F tx tx xF t
t
a F
b F Fn
γ
ω
ω
−
→∞ →∞
−
= − ≤
−= =
−
= =
= − −
1 11 1n n
= − + =
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Examples
21
2
1*( ) 1 exp
1 1 1 11 exp exp*( )lim lim lim
1 1 1*( ) 1 exp exp
Since = 1, exponential gives rise to Weibull minima. ( ) 0
t t t
n
n
F xx
F tx tx tx t x xF t
t t t
c F
d
γα
−
→−∞ →−∞ →−∞
= −
− = = = −
= =
= 1 1 1 1( ) log 1F Fn n n
α− − = − − ≈
• Limit distribution for minima from exponential parent
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur
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Domains of attraction
Domain of Attraction Type
Distribution For maximum For minimumNormal Gumbel GumbelExponential Gumbel WeibullLog-normal Gumbel GumbelGamma Gumbel WeibullGumbel M Gumbel GumbelGumbel m Gumbel GumbelRayleigh Gumbel WeibullUniform Weibull WeibullWeibull M Weibull GumbelWeibull m Gumbel WeibullCauchy Frechet FrechetPareto Frechet WeibullFrechet M Frechet GumbelFrechet m Gumbel Frechet
M=for maximumm=for minimum
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Estimation of EV distribution parameters
• Block maxima – probability plot • Return period plot • Problem: not all data are utilized
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Generalized Pareto Distribution
1/( ) [ | 0] 1 [1 ( / )] cG y P Y y Y cy a −= ≤ > = − +0, 1 ( / ) 0a cy a> + >
Exceedances of X over high threshold u Define: Y = X - u
same c as in GEV distribution
Baidurya Bhattacharya, Professor of Civil Engineering, Indian Institute of Technology Kharagpur