By Didier Sornette

Ideas, equipment and strategies of statistical physics within the examine of correlated, in addition to uncorrelated, phenomena are being utilized ever more and more within the ordinary sciences, biology and economics in an try and comprehend and version the big variability and hazards of phenomena. this is often the 1st textbook written by way of a well known specialist that gives a contemporary up to date advent for employees open air statistical physics. The emphasis of the booklet is on a transparent realizing of innovations and strategies, whereas it additionally presents the instruments that may be of speedy use in purposes. even though this e-book advanced out of a path for graduate scholars, will probably be of serious curiosity to researchers and engineers, in addition to to post-docs in geophysics and meteorology.

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**Extra resources for Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools **

**Example text**

This evolution of the apparent average as a function of the sampling size has obvious important consequences and should be kept in mind. 5 Measure of Variations from Central Tendency When repeating a measurement or an observation several times, one expects them to be within an interval anchored at the central tendency (when welldeﬁned) of a certain width. This width is a measure of the variations. There are several ways to measure the width of the variations. A ﬁrst measure is the average of the absolute value of the spread (Dabs ) deﬁned by ∞ Dabs = −∞ |x − x1/2 |P (x) dx .

The GPBH theorem yields an approximation to the tail P¯< (x) by a GPD as a tail estimator: nu ¯ ˆa G(x/ξ, ˆ) . 93) P¯< (x) = N ˆa The estimates of the two parameters ξ, ˆ can be obtained through the Maximum Likelihood estimation (ML). The log-likelihood L is given by L = −nu ln a − 1 + nu 1 ξ ln 1 + i=1 ξyi a . 94) Maximization of the log-likelihood ln L can be done numerically. The limit standard deviations of the ML-estimates as nu → +∞ can be easily obtained [274]: 1+ξ σξ = √ ; nu σa = 2(1 + ξ)/nu .

In general, the rate of convergence depends very much on the tail of the distribution of the random variables. For instance, Gaussian random variables have their EV distribution converging to the Gumbel law only as O(1/ ln N ) as above. For random variables distributed according to the exponential pdf, the convergence rate is much faster, as O(1/N 2 ). In addition, the situation is complicated by the fact that the convergence rate depends on the precise choice of normalising constants. This is much more complex and distribution dependent than the situation for the convergence to the Gaussian for the sum of random numbers as given by the central limit theorem discussed in Chap.