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.

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Presentation transcript:

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: E.g. independent, identically distributed Family of limit distributions (models) is obtained Calibrate the family of models by the measured values of parent distribution

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 Data plots here and below are from Stuart Coles: An Introduction to Statistical Modeling of Extreme Values Recurrence time: If then the maximum will exceed in T years.

F F 1.5cm 63 fibers The weakest link problem F

Problem of trends I Variables may be non-identically distributed. Sea level seems to grow.

Problem of trends II Athletes run now faster than 30 years ago.

Problem of correlations I Maximum sea level depends, or at least is correlated to other variables.

Problem of correlations II Multivariate extremes

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

Problem of exceeding a threshold

Problem of deterministic background processes

Problem of the right choice of variables

Problem of spatial correlations

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. FTG density function

Fisher-Tippett-Gumbel distribution III Question: Question: What is the „fitting to FTG” procedure? We do not know the parent distribution! is measured. The shift in is not known! The scale of can be chosen at will. Fitting to: Asymptotes: -1 largest smallest Important: Important: In the simplest EVS paradigm only linear change of variables is allowed. Without this restriction any distribution could be obtained!

FTG function and fitting

FTG function and fitting: Logscale See example on fitting.

Fisher-Tippett-Fréchet distribution I Parent distribution: Power decay is measured. 1 st question: 1 st question: Can we estimate the typical maximum? 2 nd question: 2 nd question: Can we estimate the deviation? If it exists! The maximum is on the same scale as the deviation.

Fisher-Tippett-Fréchet distribution II Question: Question: Can we calculate ? Probability of : FTF density function is measured: Assumption: Independent, identically distributed random variables with parent For large :

Fisher-Tippett-Fréchet distribution III The origin and the scale of x can be chosen at will: The function to fit for x>a is Note that forthere is no average! The kth moment does not exist for in is not known!

FTF density function for

is measured: Finite cutoff: Weibull distribution I 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 density function is measured: Assumption: Independent, identically distributed random variables with if

Weibull distribution III parent distribution is measured. are not known! in is not known! Fitting to and possibly The scaleofcan be chosen at will.

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?