extreme value theory in civil engineeringbaidurya/dissemination/ev.pdfextreme value theory in civil...

Extreme Value Theory in Civil Engineering

Baidurya Bhattacharya

Dept of Civil Engineering IIT Kharagpur

December 2016 Homepage: www.facweb.iitkgp.ernet.in/~baidurya/

http://www.facweb.iitkgp.ernet.in/~baidurya/

http://www.facweb.iitkgp.ernet.in/~baidurya/

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

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


Motivation: Time-dependent reliability

],0),,(),(:[inf Ω∈≥<= xtxtDxtCtTFirst Passage Time:

Probability of failure: ][)( LLf tTPtP ≤=

Capacity (C

) dem

and (D)

time tL


Design issues

• Maximum load • Minimum capacity • Associated uncertainties • Data driven


Reliability problem statement

• Time-dependent failure probability

• Simplification

• Need to estimate Lmax,t

]],0[anyfor0)()([)( tLDRPtPf ∈≤−−= τττ

]0[)( max, ≤−−= tef LDRPtP


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=


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

==


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→∞

=

≤=


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]);


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

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→∞ →∞

−≤ = + =


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→∞ →∞

− ≤ = + =


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

→∞

=

=

= +


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

→∞

= = − −→∞

← − +

− − = −

= − − −


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]);


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


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

−− −

=

>= ≤


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

−

=

<= − ≥


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


( ) ( ( ) 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= − ≤ =

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 / ) ( )



( ) ( ( ) 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= ≤ =

Examples

• Cauchy • Uniform • Exponential • Rayleigh


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

=

= −


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

= − + =


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


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

Estimation of EV distribution parameters

• Block maxima – probability plot • Return period plot • Problem: not all data are utilized


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


extreme value theory in civil engineeringbaidurya/dissemination/ev.pdfextreme value theory in civil...

Documents