Extreme Value Theory in Civil Engineering

Transcription

1 Extreme Value Theory i Civil Egieerig Baidurya Bhattacharya Dept of Civil Egieerig IIT Kharagpur December 2016 Homepage:

2 Prelimiaries: Retur period IID radom variables {X 1, X 2, X 3, } with CDF F X Occurrece (or success) = {X i > x p } i i th trial p = P{success} = P {X i > x p } = 1 F X (x p ) x p = level correspodig to exceedace probability p Sequece of idepedet ad idetical Beroulli trials: Geometric radom variable Time betwee successive occureces is radom Mea Retur Period associated with x p is 1/p (i uits of trial time iterval)

3 Prelimiaries: Characteristic value IID radom variables {X 1, X 2, X 3, } with CDF F X Success = {X i > x p } i i th trial p = P{success} = P {X i > x p } = 1 F X (x p ) x p = level correspodig to exceedace probability p repeated trials Biomial radom variable Mea umber of occurreces = p x = Characteristic value of X i if mea umber of occurreces, p(x ) =1 that is, 1 F X (x ) = 1/

4 Motivatio: Time-depedet reliability First Passage Time: T = if [ t : C( t, x) < D( t, x), t 0, x Ω] ( tl ) = P T tl Probability of failure: [ ] P f Capacity (C) demad (D) time t L

5 Desig issues Maximum load Miimum capacity Associated ucertaities Data drive

6 Reliability problem statemet Time-depedet failure probability P f ( t) = P[ R( τ ) D L( τ ) 0 for ay τ [0, t]] Simplificatio P ( t) P[ R D Lmax, 0] Need to estimate L max,t f = e t

7 Maximum live load estimatio Maximum live load L = max{ L1, L2,..., ( t) } max, t L N Distributio fuctio Simplificatios idepedece statioarity F max, ( l) = P[ L1 l, L2 l,..., L ( ) L t N t F max, () [ ()] N t L t l = FL l () l]

8 Extreme value theory: problem statemet Sequece of radom variables {X i } What are the limitig forms of Z ad W? Issues Z = max( X, X,..., X ) ~ H 1 2 W = mi( X, X,..., X ) ~ L 1 2 ukow Degeeracy of limit distributios: ifiite Nature of populatio distributios, F i Depedece i the sequece No-statioarity of the sequece

9 IID (classical) case {X i } is a IID sequece X i ad X j are idepedet for i j F i = F are same for all i H (x) ad L (x) are degeerate distributios PZ [ x] = H( x) = F ( x) PW [ x] = L( x) = 1 (1 F( x)) lim H ( x) 0, x< ω( F) = 1, otherwise ω( F) = upper ed poit of F lim L ( x) ( ) 0, x α( F) = 1, otherwise a( F) = lower ed poit of F

10 Code for CDF of Xmax vs. %This program geerates a sequece of IID expoetial RVs ad stores it maximum, xmax %It the repeats the process mct times %The distributio of xmax is plotted %The plot is repeated for a differet clear all; mct=1000; =iput('give \'); for mcti=1:mct, for i=1:, x(i)=-log(rad); ed xmax(mcti)=max(x); freq(mcti)=mcti/(mct+1); ed xmaxsorted=sort(xmax); loglog(xmaxsorted,freq); sizes=['=' um2str()]; hold o; text(xmaxsorted(mct/2),.5,[sizes],'backgroudcolor',[1 1 1],'EdgeColor','black'); axis([0,100,0,1]);

11 Example: degeeracy for large X i ~ Expoetial(1) iid X = max{ X, X,..., X } max 1 2 i= 1,..., cdf of X max x

12 Normalizatio Ca we ormalize Z ad help matters? Z a lim P x = lim H( a + bx ) = H( x) b Issues How may possible forms for H? How does H deped o F? How to fid a ad b What is the speed of covergece?

13 Normalizatio Ca we ormalize W () ad help matters? W( ) a lim P z = lim L( ) ( a + bz ) = Lz ( ) b Issues How may possible forms for L? How does L deped o F? How to fid a ad b What is the speed of covergece?

14 Normalizatio As, Istead of F ( x) = F ( x) max Look at F ( x) where x = f( x, ) Simplest form for x : X X x = a + bx

15 Normalizatio example X i max ~ Expoetial(1) F ( x) = F ( x) = (1 exp( x)) X As, Replace x α( x u) + l( ) exp( α ( x u)) Obtai: Fmax ( x) = lim 1 = exp( exp( α ( x u)))

16 Code for CDF of ormalized Xmax vs. %This program geerates IID Expoetials ad stores the maximum xmax. %xmax is the recetered ad scaled as a fuctio of %The process is repeated times clear all; mct=1000; =iput('give \'); scale=iput('give scale\'); shift=iput('give shift\'); if ==1,scale=1;shift=0;ed for mcti=1:mct, for i=1:, x(i)=-log(rad); ed xmax(mcti)=(max(x)-log()+shift)/scale; freq(mcti)=mcti/(mct+1); ed xmaxsorted=sort(xmax); loglog(xmaxsorted,freq); sizes=['=' um2str()]; hold o; text(xmaxsorted(mct/2),.5,[sizes],'backgroudcolor',[1 1 1],'EdgeColor','black'); axis([0,100,0,1]);

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18 Limit distributios i IID case There are oly three types of o-degeerate distributios H(x) for maxima There are oly three types of o-degeerate distributios L(x) for miima Necessary ad sufficiet coditios exist for F(x) to yield above max or mi distributios Note: Two distributios F ad G are of the same type, if F(x) = G(ax+b) where a, b are costats

19 Geeralized EV distributio for maxima I IID case, H( z) must be of the same type as: H z cz cz 1/ c c( ) = exp (1 + ), 1+ > 0 e c= 0 Gumbel (Type I) distributio: H ( z) = e, < z< γ z e, z > 0 c> 0 Frechet (Type II) distributio: HF ( z) = 0, z 0 c < 0 Weibull (Type III) distributio: H ( z) where γ = 1/ c G W z 1, z > 0 = γ ( z) e, z 0

20 Geeralized EV distributio for miima I IID case, Lz ( ) must be of the same type as: L z = cz cz > 1/ c c( ) 1 exp (1 ), 1 0 e c= 0 Gumbel (Type I) distributio: L ( z) = 1 e, < z< γ e z c> 0 Frechet (Type II) distributio: LF ( z) = 1, z > 0 c < 0 Weibull (Type III) distributio: L ( z) where γ = 1/ c G W z ( z) 1, 0 0, z < 0 = γ z 1 e, z 0

21 ω ( F ) Domais of attractio for maxima 1 F( tx) γ (A) lim = x, γ > 0 t 1 Ft () (B) (1 F( x)) dx <, a 1 F( t + xr( t)) x (C) lim = e, t ω ( F) 1 Ft () * F x F ω F x ay fiite a, α( F), ω( F) = lower ad upper ed poits of F Rt ( ) = E( X t X> t), t> α ( F) F DH ( )if ad oly if ω( F) = ad (A) holds for F F a = 0, b = if( x : 1 F( x) 1 / ) F DH ( )if ad oly if ω( F) < ad (A) holds for W ( ) = ( ( ) 1/ ) a = ω( F), b = ω( F) if( x : 1 F( x) 1 / ) F DH ( )if ad oly if ω( F) = ad (B), (C) hold G a = if( x: 1 F( x) 1 / ), b = Ra ( )

22 α ( F ) Domais of attractio for miima F( tx) γ (A) lim = x, γ > 0 t Ft () a (B) F( x) dx <, F( t + xr( t)) x (C) lim = e, t α ( F) Ft () * F x F F x x ay fiite a, α( F), ω( F) = lower ad upper ed poits of F rt ( ) = Et ( X X< t), t> α ( F) F DL ( )if ad oly if α( F) = ad (A) holds for F F c = 0, d = sup( x : F( x) 1 / ) F DL ( )if ad oly if α( F) > ad (A) holds for W ( ) = ( α( ) 1/ ), < 0 G c = α( F), d = sup( x : F( x) 1 / ) α( F) F DL ( )if ad oly if (B), (C) hold c = sup( x: F( x) 1 / ), d = rc ( )

23 Examples Cauchy Uiform Expoetial Rayleigh

24 Examples Limit distributio for miima from Cauchy paret 1 arcta( x) F( x) = + ; < x< 2 π Check Eq (A) for miima: 1 arcta( tx) + F( tx) lim = lim 2 π t f() t t 1 arcta( t ) + 2 π x ( tx) = lim t t 2 x(1 + t ) = lim = x t tx 1 That is, γ = 1, ad Cauchy lies i the domai of attractio of Frechet for miima. The ormalizig costats are: c d = = ta π 2

25 Examples Limit distributio for maxima from uiform paret The complemetary CDF of the uiform is: 1 F*( x) = 1, x 1 x 1 1 F *( tx) 1 lim lim tx = = x t 1 F*( t) t 1 t Sice γ = 1, uiform gives rise to Weibull maxima. The ormalizig costats are: a = ω( F) = b = ω( F) F = 1 1+ =

26 Examples Limit distributio for miima from expoetial paret 1 F*( x) = 1 exp x exp exp 2 F *( tx) tx tx t x lim = lim = lim = t F*( t) t 1 t exp exp 2 t t t Sice γ = 1, expoetial gives rise to Weibull miima. c d = α( F) = = F α( F) = log 1 x 1

27 Domais of attractio Domai of Attractio Type Distributio For maximum For miimum Normal Gumbel Gumbel Expoetial Gumbel Weibull Log-ormal Gumbel Gumbel Gamma Gumbel Weibull Gumbel M Gumbel Gumbel Gumbel m Gumbel Gumbel Rayleigh Gumbel Weibull Uiform Weibull Weibull Weibull M Weibull Gumbel Weibull m Gumbel Weibull Cauchy Frechet Frechet Pareto Frechet Weibull Frechet M Frechet Gumbel Frechet m Gumbel Frechet M=for maximum m=for miimum

28 Estimatio of EV distributio parameters Block maxima probability plot Retur period plot Problem: ot all data are utilized

29 Geeralized Pareto Distributio Exceedaces of X over high threshold u Defie: Y = X - u G( y) = P[ Y y Y > 0] = 1 [1 + ( cy / a)] 1/ c a > 0, 1 + ( cy / a) > 0 same c as i GEV distributio