Extreme value statistics

Finley, H. F., An Extreme-value Statistical Analysis of Maximum Pit Depths and Time to First Perforation , Corrosion, 23, 83 (1%7)... 

Landel and Fedors (53, 54) have recently explored the usefulness of extreme value statistics applied to the statistical distribution of rupture in various unfilled polymer specimens. Both breaking stress and breaking strain of natural rubber (47) and styrene butadiene elastomers (53, 54) may be described by the double exponential distribution... 

Z. P. Bazant, and S. D. Pang, Activation energy based extreme value statistics and size effect in brittle and quasibrittle fracture. Journal of the Mechanics and Physics of Solids, 55, 91-131 (2007). 

The reason why we chose a distribution of the form given in Eq. (65) is that in d = 1 the probability P E) is known to have that form exactly for the case of delta correlated random potentials (see [27]). For d > 1 Lifehits [19] argued that the form given by Eq. (65) is also valid. Our goal now is to see if the distribution Eq. (64) derived using the 1-step RSB solution is indeed consistent with the distribution Eq. (66) predicted using extreme value statistics. 

Korb el al. proposed a model for dynamics of water molecules at protein interfaces, characterized by the occurrence of variable-strength water binding sites. They used extreme-value statistics of rare events, which led to a Pareto distribution of the reorientational correlation times and a power law in the Larmor frequency for spin-lattice relaxation in D2O at low magnetic fields. The method was applied to the analysis of multiple-field relaxation measurements on D2O in cross-linked protein systems (see section 3.4). The reorientational dynamics of interfacial water molecules next to surfaces of varying hydrophobicity was investigated by Stirnemann and co-workers. Making use of MD simulations and analytical models, they were able to explain non-monotonous variation of water reorientational dynamics with surface hydrophobicity. In a similar study, Laage and Thompson modelled reorientation dynamics of water confined in hydrophilic and hydrophobic nanopores. 

These relationships can be interpreted in terms of extreme value statistics by assuming different types of distribution laws [23],... 

Probability distribution was introduced in the section in this chapter on Terminology, and the components of a normal distribution and how to work with them were discussed. Some distributions are not normal because they are skewed to one side of the mode. Unless the values can be treated by some mathematical function to yield a new set of values that approximates a normal distribution, the use of normal statistical techniques could lead to erroneous conclusions. Sometimes the distribution is skewed in such a way that the logarithms of the values have a normal distribution. Fortunately, although the values are not amenable to standard statistical methods, their logarithms are. In contrast, maximum pit depths have an extreme value distribution and should be treated with extreme value statistics. We wUl discuss these three distributions. 

A number of statistical transformations have been proposed to quantify the distributions in pitting variables. Gumbel is given the credit for the original development of extreme value statistics (EVS) for the characterization of pit depth distribution [13]. The EVS procedure is to measure maximum pit depths on several replicate specimens that have pitted, then arrange the pit depth values in order of increasing rank. The Gumbel distribution expressed in Eq 1, where X and a are the location and scale parameters, respectively, can then be used to characterize the dataset and estimate the extreme pit depth that possibly can affect the system from which the data was initially produced. 

The statistical analysis procedure most useful in evaluating corrosion on pipes is the extreme value statistical method. The referenced publications provide details on extreme value statistics and their application to pipeline corrosion. The extreme value statistical approach analyzes the maximum pit depth (or minimum wall thickness) per unit length to estimate the range of corrosion data for the pipe. The results, of course, assume that the pipe run is in the same condition as the sample tested. The more samples tested, the greater the accuracy of the result. 

Normal probability and extreme value statistics have already been discussed as means of evaluating corrosion smd soil resistivity. Statistical methods have been applied to determine when steel tanks will fail [38]. Individual soil characteristics, such as resistivity, pH, and soluble ion content, are unsatisfactory methods of predicting the corrosion behavior of a structure in a particular soil. The statistical method is based on a regression analj is of all of the soil characteristics that affect corrosion as compared to actual corrosion of structures found in those soils. This method 5delds a mean time to corrosion failure (MTCF) as opposed to an actual corrosion rate. Its validity for structures other than steel tanks has not been demonstrated. 

In most instances, a group of ceramic or glass samples produced under nominally identical conditions will have worst flaws that vary in severity and location. As a consequence, strength values for those samples will vary, often over a rather wide range. The distribution of failure stresses is usually analyzed in terms of the extreme value statistics developed by Weibull. The most common functional form used in these statistical treatments is... 

A. Repko, P. H. A. J. M. van Gelder, H. G. Voortman and J. K. Vrijling, Bivariate description of offshore wave conditions with physics-based extreme value statistics,... 

For alloys showing a good resistance to pitting with t in years and D in mm, K can be typically close to 0.75. Estimation of the life time of material on the basis of controlled laboratory tests is of little use. However, by extreme value statistical treatment of the data quantitatively, the life and time of materials can be satisfactorily predicted [32,33]. 

The extreme value statistics analysis can be estimated according to the following procedure [35] first, all calculated extreme value data are arranged in order from the smallest and then, the probability F Y) is calculated as -[MI N +1)], where Mis the rank in the ordered extreme value and the total number of extreme value data. The reduced variant (F) can be calculated by the formula Y = -In -In [F(Y)]. ... 

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