Trans RINA, Vol 154, Part A2, Intl J Maritime Eng, Apr-Jun 2012 TECHNICAL NOTE


ON THE APPLICATION OF THE EXTREME VALUE THEORY IN SHIP’S STRENGTH CALCULATIONS (DOI No: 10.3940/rina.ijme.2012.a2.223tn)


L D Ivanov, retired (formerly with American Bureau of Shipping), USA


SUMMARY A procedure is proposed for application of the extreme value theory (EVT) approach considering not only the maximal value of the corresponding random variable but also its probability of exceedance. It substantially reduces the probability of exceedance of any given limit value used in the case when traditional EVT is applied. Examples are provided to illustrate its application when records of the random process are available.


NOMENCLATURE


BM = Bending moment CB = Block coefficient CDF = Cumulative Distribution Function EVT = Extreme Value Theory MSW = Still water bending moment Mt = Total bending moment MW = Wave bending moment MW,h = Hogging wave bending moment


MW,s = Sagging wave bending moment PDF = Probability Density Function POE = Probability of Exceedance POO = Probability of Occurrence T = Time


1. INTRODUCTION


When records of a random process (e.g., obtained from full-scale measurement, model tests or numerically generated) are available, statistical analysis may be performed in two ways – either by application of the individual amplitude


statistics or by extreme value


statistics. In the former, all amplitudes are considered in the analysis while in the latter – only the maximal amplitudes in each time window.


When analyzing records of hull girder bending


stresses/moments, the duration of each time window might be e.g. 5, 10, 20, 30 etc. minutes. If extreme value statistics is applied, only the maximal amplitude will be extracted from each time window. The probabilistic distribution type and probability of exceedance (POE) of the maximum amplitude in the


window is not considered as the present EVT stipulates. According to this theory


(Ochi, 1958/2004, Kotz & Nadarajah, 2000),


maximal variable among X1, X2,….,Xn, i.e. 


Mnn (1) (x) max X , X , , X  12


and X1, X2, …, Xn are independent and identically distributed variables, then the Cumulative Distribution Functions (CDF) F are:


©2012: The Royal Institution of Naval Architects


corresponding time 1989; Gumbel,


if Mn is the F (x) F (x) ...F (x) F (x) XX 2 1  X X n The CDF of Mn will be


FMn (m)=P(X m,X m,.X m)= F (m)    (3)


n 12 X n 


where n = number of observations within a given service life (e.g., number of cycles).


The EVT provides formulas (see, e.g., Ochi 1989) for calculation of the parameters of the newly derived extreme value distributions (Gumbel-, Frechet-, Weibull – type) using the information for the corresponding parent probabilistic distributions. This approach is the most frequently used due to its convenience, especially in design stages. As an example, the formulas given below for the CDF illustrates the way it can be done for asymptotic extreme value distribution type one (Ochi, 1989):


Fy exp exp y  ee   


 


e


Where  = location parameter  = scale parameter


ye = corresponding random variant (e.g., MW)


The parameters  and are calculated following the procedure described by (Ochi M, 1989).


The parameter  is the probable extreme value expected to occur in “n” observations (e.g., “n” cycles). It can be evaluated from the parent CDF (e.g., MW) for which the probability of exceedance (POE) of this value is 1/n, i.e.:


F() = 1 – 1/n (5)


Where F() = parent CDF of MW calculated for ye = the design MW in class societies’ rules).


The other parameter  is calculated by the formula (Ochi, 1989):


 = f() / [1 – F()] where f = parent probability density function (PDF); A-89 (6)  (4) (2)


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