t i Extreme value statistics Problems of extrapolating to values we have no data about unusually large or small ~100 years (data) ~500 years (design)
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1 Extrm valu statistics Problms of xtrapolatig to valus w hav o data about uusually larg or small t i ~00 yars (data h( t i { h( }? max t i wids v( t i ~500 yars (dsig Qustio: Ca this b do at all? How log will it stad?
2 Extrm valu paradigm Y Logics: is masurd: y, y,..., { y, y } x max,..., y Assum somthig about yi y E.g. idpdt, idtically distributd Us limit argumt: ( Qustio: What is th distributio of th largst umbr? P ( y 0 y Family of limit distributios (modls is obtaid Calibrat th family of modls by th masurd valus of x P(x Slightly suspicious but o altrativs xist at prst. x
3 A xampl of xtrm valu statistics Th 84 sa lvl bchmark (ctr o th `Isl of th Dad', Tasmaia. Accordig to Atarctic xplorr, Capt. Sir Jams Clark Ross, it markd ma sa lvl i 84. P 0 ( y y P ( x x Figurs ar from Stuart Cols: A Itroductio to Statistical Modlig of Extrm Valus
4 Th wakst lik problm.5cm F 63 fibrs F F
5 Problm of trds I Figurs ar from Stuart Cols: A Itroductio to Statistical Modlig of Extrm Valus
6 Problm of trds II
7 Problm of corrlatios
8 Problm of scod-, third-,, largst valus
9 Problm of iformatio loss
10 Problm of dtrmiistic backgroud procsss
11 Problm of trds ad variabls l M M i i
12 Problm of spatial corrlatios
13 Problm of trds
14 Fishr-Tipptt-Gumbl distributio I Y is masurd: y, y,..., y P 0 ( y y part distributio Assumptio: Idpdt, idtically distributd radom variabls with { y, y } x max,..., y st qustio: Ca w stimat x? y P0 ( x x l + l ot: x d qustio: Ca w stimat? δ ( x x P ( x +δ / δ 0 P(x δ x Homwork: Carry out th abov stimats for a Gaussia part y distributio! P 0 ( x y
15 Fishr-Tipptt-Gumbl distributio II Y is masurd: y, y,..., y P 0 ( y y part distributio Assumptio: Idpdt, idtically distributd radom variabls with { y, y } x max,..., y y Qustio: Ca w calculat P ( x? Probability of x < z : m( z m z z ( z P ( x dx P0 ( y dy 0 z ( zl x ( ( / ( / z x l x P(x δ x x P( x lim d m dx ( z x + l FTG distributio x x Expctd that this rsult dos ot dpd o small y dtails of P 0 ( y but thr is mor grality to this rsult.
16 Fishr-Tipptt-Gumbl distributio III { y, y } x max,..., y P 0 ( y is masurd. part distributio y is ot kow! W do ot kow it! Qustio: What is th fittig to FTG procdur? Th shift z Th scal of x l x is ot kow! ca b chos at will. y Fittig to: P( x b x b a xa b P(x δ Asymptotics: b b x/ b P ( x x / b x + x x x - largst smallst
17 FTG fuctio ad fittig P ( x b x b a xa b a 0 b x x
18 FTG fuctio ad fittig: Logscal P ( x b x b a xa b a 0 b x x S xampl o fittig.
19 Fiit cutoff: Wibull distributio Y is masurd: y, y,..., y Assumptio: Idpdt, idtically distributd radom variabls with { y, y } x max,..., y st qustio: Ca w stimat a x? a x /( β + a P ( x a x 0 P β + 0( y ( a y a + P(x β β part distributio δ a y d qustio: Ca w stimat? δ ( x x x x P0 ( a 0 δ a x δ /( β +
20 Wibull distributio II Y is masurd: y, y,..., y Assumptio: Idpdt, idtically distributd radom variabls with { y, y } x max,..., y Qustio: Ca w calculat P ( x? β + ( β P0 y β + ( a y β a a y 0 part distributio Probability of x < z : m( z m z z ( z P ( x dx P0 ( y dy 0 z β + [ ( ] a ( ( x β + / ( x β+ P(x δ a x /( β + z a + x /( β + x x P( x lim d dx m ( z a + x /( β + ( β + ( x β ( x β+ x 0 Wibull distributio P( x 0 x 0
21 Wibull distributio III a x { y, y } x max,..., is masurd. y Th scal of a i x is ot kow! ca b chos at will. /( β + is ot kow! β + ( β P0 y β + ( a y β a a y 0 part distributio Fittig to P( x β 0 + b ( ax b β ( a b x β + x a x > 0 P(x δ x x
22 Wibull fuctio ad fittig P( x β + b 0 ( ax b β ( a b x β + x a x > a a 0 b
23 ? ots about th Tmax homwork?? T T ( max ( max Fid (, T (, T ( max ( max (, K, T (, K, T ( max ( max (, K, T (, K, T ( max ( max ( ( ( T ( T T max M max max Itroduc scald variabls commo to all data sts ( (??? ( x T ( max ( T ( T ( max T ( max ( max Avrag ad width of distributio? ( ( ( x 0 ( x x so all data ca b aalyzd togthr.? What kid of coclusios ca b draw?
24 Györgyi Géza loadásai
25 Critical ordr-paramtr fluctuatios i th d3 dimsioal Isig modl Dlamott, idrmayr, Tissir T ξ or how dos a iformativ figur look lik T ξ L L ξ / ν a
26 Exampl of EVS i actio: P(v of th rightmost atom i a xpadig gas 3 0 D idal, lastic gas i quilibrium at T: P(v ~ mv /kt ~ (v/ v 0 Box is opd, wait for a log tim. Qustios: ( What is th xpctd vlocity of th rightmost particl? ( What is P(v of th rightmost particl? Elastic collisios: v v + v + v v v' + v' v' + v' v' v' v v vlocitis ar xchagd always icrass (3 Estimat th xpctd vlocity. v max v 0 / (l 6000 m s t v v max 300 m s 3 0
27 otrivial EVS distributios Qustios: ( What ar th rasos for ay othr xtrm satistics to mrg? Aalogous qustio i statistical physics: What ar th rasos for ogaussia statistics of macroscopic (additiv quatitis? ( What ar th simplst calculabl xampls? Idpdt, oidtically distributd variabls. (3 What ca w say about itractig systms? Qustio of wak or strog corrlatios (fiit- or ifiit corrlatio lgth. FTG, Wibull, FTF : High-tmpratur fixd poit. Ar thr critical EVS distributios with uivrsality classs?
28 Gaussia ad ogaussia distributios I P(M d/ L L d M Extsiv quatity i a ocritical systm ( ξ < L Exampl: Isig modl abov ad blow Tc. ctral limit thorm P(M d/ L d/ L Small Gaussia fluctuatios aroud th ma Isig modl at Tc: M ~ L d + γ ν L d M
29 Gaussia ad ogaussia distributios II Extsiv quatity i a critical systm (ξ ~ L o ctral limit thorm ogaussia fluctuatios aroud th ma Exampl: Isig modl at Tc: Emrgc of uivrsal scalig fuctios M ~ L d + γ ν Scalig variabl: x M / M Scalig fuc.: Φ( x M P( M d d3 x M / M x M / M
30 Edwards-Wilkiso (EW itrfac Surfac tsio driv growth Vrtical growth vlocity ca dpd oly o th spatial drivativs of. h h( y, t y 0 L Log wavlgth (gradit xpasio: v h f ( h, h,... +η f ( t y h, y y h,... v y 0 ois i th arrival (Gaussia, whit is assumd + σ y h +... I th systm movig with v 0 : t h σ h +η avrag growth vlocity surfac tsio Stady stat distributio fuctio: Fourir mods: h ( y L [ ] ( h dy 0 P h ~ σ P[ h ] k σ L k h k k σ Lk h k ~ Π k k iky hk Idpdt mods
31 ogaussia distributios - Edwards-Wilkiso (EW itrfac Surfac tsio driv dyamics Φ(x w t h σ h +η x w / w L dy h L 0 h( y, t 0 L? [ ( y, t h] h L k divrgig fluctuatios k P [ h ] k y Statioary stat: P L h [ ] ( h 0 ~ σ Idpdt Fourir mods: dy σ L k h k k σ Lk h k ~ Π h k ~ k k oidtical, sigular fluctuatios sum of idpdt variabls Raso for th failig of th ctral limit thorm
32 Drivatio of th width distributio for EW itrfac ( t h σ h +η Width distributio: I trms of Fourir mods: ormalizatio: Statioary stat: P Gratig fuctio (Laplac trasform: L h ~ σ [ ] ( 0 h Π P( w D( h P( h δ[ w L dy( h h ] σ ws 0 G( s dw P( w D( h Π dh dh k k σ k G( 0 G( s k σ k + k L π 0 k L s L ( h L 0 dy s dy L s ( σk + hk L ~ L 0 G( s ( hh Π k σ k Lk h k Idpdt Fourir mods dy [ ] σ k + L s [ + ] Ls π σ Path itgral of harmoic oscillator
33 Drivatio of th width distributio for EW itrfac ( Width distributio i scalig form: ( ( ( ( Φ w w w w s w w i i i w ds w s w i i i ds w s F s G w P π π Gratig fuctio: ( ( 0 w P dw s G s w [ ] + Ls σ π Avrag width: σ σ π 0 L L ds dg w s [ ] ( 6 + w s F w s π Φ(x w / w x Scalig fuctio: xy y i i i dy x + Φ 6 ( ( π π Simpl pols at 6 / π y Φ 6 3 ( ( x x π π ( ( l l
34 ogaussiaity - width distributio of itrfacs Ag o glass Statioary distributio: w w L P(w Φ Φ( x w L Pictur gallry h h( y, t 0 L w L y w width ~ χ L [ h( y, t h ] L ( L, t dy 0 EW MH KPZ MH PRE50, 639, 3589 (994 PRE65, 0636 (00 PRE50, 3530 (994
35 Exampl of otrivial EVS: oidtically distributd idpdt variabls A giv cofiguratio h(t provids a st of h h(t t 0 T Part distributio Actio: S ~ h ~ h P (lookig for th max of variabls [ ] h π Probability of h max z big lss tha z: M ( z h 0 z π h dh dh ( z
36 Exampl of otrivial EVS: oidtically distributd idpdt variabls c S ~ Probability of z c max big btw z ad z+dz z m z m z P ( (
37 Distributio of xtrmal itsity fluctuatios Math: Drawig from idpdt, oidtically distributd umbrs ( h k h k max Ρ( h k ~ σ k k hk h k PRE68, 0566 (003 with sigular h k ~ k k k π L Fid P ( z dz: th probability of Rsult: z hk z + dz max L L z ( z ( z P
38 /f ois - voltag fluctuatios i rsistors Exampl:. V (t Rhutat film IrO t T S( f ~ V f ~ / f V f πift V ( t T x M.B. Wissma, Rv.Mod.Phys.60, 537(988
39 Turbulc ad th d EW modl Exprimt Critical ω ω c Distributio of rgy dissipatio S. Bramwll t al. atur 396, 55 (998 Distributio of d XY magtizatio blow Tc. (fiit-siz Aji & Goldfld, PRL86, 007 (00 (p- p / σ p (M- M / σ M dissipatio is maily o th fluctuatios of th shar pacak d EW XY T < T c Possibl coctio to xtrm statistics?
40 Width distributios for / f sigals PRE65, (00 h(t 0 T Qustio: Is thr a for which xtrm statistics distributio mrgs? t Statioary distributio for Fourir mods P [ ] h ~ σ w ( hh ~ h h Itgratd powr spctrum w T P(w Φ x w w T Φ ( x δ ( x Dos it hav a itral structur?
41 Extrm statistics for. PRL87, 4060 (00 w scalig variabl σ T P( w σ T y Φ w w σ w ( y T T T w T h(t 0 T Φ w h itgratd powr spctrum ( y y y t Fishr-Tipptt-Gumbl xtrm valu distributio / Φ ( y ~ y ~ h Ctral limit thorm is rstord for /
42 Homworks. S slid Fishr-Tipptt-Gumbl distributio I.. Show that FTG distributio bcoms th Wibull distributio i th β limit. 3. Dtrmi th Tmax distributio for tmpraturs masurd at th Amistad Dam. Try fits with both th FTG ad th Wibull fuctios. 4. What is th vlocity distributio of th rightmost particl i a o-dimsioal, frly xpadig gas?
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