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 ~100 years (data) ~500 years (design) winds How long will it stand?

Extreme value paradigm is measured: Question: Question: What is the distribution of the largest number? Logics: Assume something about Use limit argument: Slightly suspicious but no alternatives exist at present. E.g. independent, identically distributed Family of limit distributions (models) is obtained Calibrate the family of models by the measured values of

An example of extreme value statistics 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 Figures are from Stuart Coles: An Introduction to Statistical Modeling of Extreme Values

F F 1.5cm 63 fibers The weakest link problem F

Problem of trends I Figures are from Stuart Coles: An Introduction to Statistical Modeling of Extreme Values

Problem of trends II

Problem of correlations

Problem of second-, third-, …, largest values

Problem of information loss

Problem of deterministic background processes

Problem of trends and variables

Problem of spatial correlations

Problem of trends

is measured: Fisher-Tippett-Gumbel distribution I Assumption: Independent, identically distributed random variables with parent distribution 1 st question: 1 st question: Can we estimate ? Note: 2 nd question: 2 nd question: Can we estimate ? Homework: Carry out the above estimates for a Gaussian parent distribution !

is measured: Fisher-Tippett-Gumbel distribution II Assumption: Independent, identically distributed random variables with parent distribution Question: Question: Can we calculate ? Probability of : Expected that this result does not depend on small details of but there is more generality to this result. FTG distribution

Fisher-Tippett-Gumbel distribution III parent distribution Question: Question: What is the fitting to FTG procedure? We do not know it! is measured. is not known! The shift is not known! The scale of can be chosen at will. Fitting to: Asymptotics: -1 largest smallest

FTG function and fitting

FTG function and fitting: Logscale See example on fitting.

is measured: Finite cutoff: Weibull distribution Assumption: Independent, identically distributed random variables with parent distribution 1 st question: 1 st question: Can we estimate ? 2 nd question: 2 nd question: Can we estimate ?

Weibull distribution II parent distribution Question: Question: Can we calculate ? Probability of : Weibull distribution is measured: Assumption: Independent, identically distributed random variables with

Weibull distribution III parent distribution is measured. is not known! in is not known! The scale of can be chosen at will. Fitting to

Weibull function and fitting

Notes about the Tmax homework Introduce scaled variables common to all data sets Find Average and width of distribution so all data can be analyzed together. ? ? ? ? ? ? ? ? What kind of conclusions can be drawn?

Györgyi Géza előadásai

Critical order-parameter fluctuations in the d=3 dimensional Ising model L Delamotte, Niedermayer, Tissier or how does an informative figure look like

Example of EVS in action: P(v) of the rightmost atom in an expanding gas D=1 ideal, elastic gas in equilibrium at T: Box is opened, wait for a long time. Questions: (1)What is the expected velocity of the rightmost particle? (2) What is P(v) of the rightmost particle? (3) Estimate the expected velocity. Elastic collisions: velocities are exchanged always increases

Nontrivial EVS distributions (1) What are the reasons for any other extreme satistics to emerge? Analogous question in statistical physics: What are the reasons for nongaussian statistics of macroscopic (additive) quantities? (2) What are the simplest calculable examples? Independent, nonidentically distributed variables. (3) What can we say about interacting systems? Question of weak or strong correlations (finite- or infinite correlation length). FTG, Weibull, FTF : High-temperature fixed point. Are there critical EVS distributions with universality classes? Questions:

Gaussian and nongaussian distributions I Extensive quantity in a noncritical system L d d/2 L P(M) M Small Gaussian fluctuations around the mean central limit theorem Example: Ising model above and below Tc. P(M) M L d d/2 L L Ising model at Tc:

Gaussian and nongaussian distributions II Extensive quantity in a critical system nongaussian fluctuations around the mean no central limit theorem Example: Ising model at Tc: Scaling variable: Scaling func.: d=2 d=3 Emergence of universal scaling functions

Edwards-Wilkinson (EW) interface Surface tension driven growth Noise in the arrival (Gaussian, white is assumed) average growth velocity Long wavelength (gradient) expansion: Vertical growth velocity can depend only on the spatial derivatives of. In the system moving with : surface tension Steady state distribution function: Fourier modes: Independent modes

Nongaussian distributions - Edwards-Wilkinson (EW) interface Surface tension driven dynamics Stationary state: diverging fluctuations Independent Fourier modes: sum of independent variables Reason for the failing of the central limit theorem nonidentical, singular fluctuations ?

Derivation of the width distribution for EW interface (1) Stationary state: Independent Fourier modes Width distribution: In terms of Fourier modes: Normalization: Generating function (Laplace transform): Path integral of harmonic oscillator

Derivation of the width distribution for EW interface (2) Width distribution in scaling form: Generating function: Average width: Scaling function: Simple poles at

Nongaussianity - width distribution of interfaces Picture gallery KPZ EW MH PRE50, 639, 3589 (1994)PRE50, 3530 (1994)PRE65, (2002) Stationary distribution: Ag on glass width

Example of nontrivial EVS: Nonidentically distributed independent variables Probability of being less than z: A given configuration Parent distribution provides a set of Action: (looking for the max of N variables)

Example of nontrivial EVS: Nonidentically distributed independent variables Probability of being between z and z+dz

Distribution of extremal intensity fluctuations Find : the probability of Drawing from N independent, nonidentically distributed numbers ( ) with singular Math: PRE68, (2003) Result:

1/f noise - voltage fluctuations in resistors Example: M.B. Weissman, Rev.Mod.Phys.60, 537(1988)

Turbulence and the d=2 EW model S. Bramwell et al. Nature 396, 552 (1998 ) Experiment Critical Distribution of energy dissipation Distribution of d=2 XY magnetization below Tc. (finite-size) Aji & Goldenfeld, PRL86, 1007 (2001) dissipation is mainly on the fluctuations of the shear pancake Possible connection to extreme statistics?

Width distributions for signals Stationary distribution for Fourier modes Integrated power spectrum Question: Is there an for which extreme statistics distribution emerges? PRE65, (2002) Does it have an internal structure?

 Extreme statistics for. new scaling variable integrated power spectrum Fisher-Tippett-Gumbel extreme value distribution Central limit theorem is restored for PRL87, (2001) 1   y ey ey    1  2 ~ y ey    2/1    n~h 2 n T T y    22 ww T ww TT   yP T   )w( 2 1  2/1 

Homeworks 1.See slide Fisher-Tippett-Gumbel distribution I. 2.Show that FTG distribution becomes the Weibull distribution in the limit. 3.Determine the Tmax distribution for temperatures measured at the Amistad Dam. Try fits with both the FTG and the Weibull functions. 4.What is the velocity distribution of the rightmost particle in a one-dimensional, freely expanding gas?