A Monte Carlo Simulation of Multivariate General


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1 A Monte Carlo Simulation of Multivariate General Pareto Distribution an its Application Luo Yao 1, Sui Danan, Wang Dongxiao,*, Zhou Zhenwei, He Weihong 1, Shi Hui 1 South China Sea Institute of Oceanology, Chinese Acaemy of Sciences, Guangzhou, China State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Acaemy of Sciences, Guangzhou, China School of Civil Engineering an Transportation, South China University of Technology, China China Water Resources Pearl River Planning Surveying & Designing CO., LTD, China Corresponence to: (Luo Yao), *Corresponing author: (Wang Dongxiao) Abstract: The paper presents a MGPD (Multivariate General Pareto Distribution) metho an buils the solving metho of MGPD by a Monte Carlo simulation for the marine environmental extreme value parameters. The simulation metho is prove to be feasible by analyzing the joint probability of wave weight an its concomitant win from a hyrological station in SCS (South China Sea). The MGPD is the natural istribution of the MPOT (Multivariate Peaks Over Threshol) sampling metho, an has the extreme value theory backgroun. The existing epenence function can be use in the MGPD, so it may escribe more variables which have ifferent epenence relationships. The MGPD metho improves the efficiency of the extremes in raw ata. For the win an the concomitant wave in years (101), the number of the win an wave selecte is average 1 each year. Finally, by using the CP (Conitional Probability), a Monte Carlo Simulation metho base on the MGPD is aopte in case of the return perio of base shear. Key Wors: MPOT, extreme wave, extreme value theory, Monte Carlo 1. Introuction Statistical moeling of extreme values (EV) plays a crucial role in the esign an
2 risk evaluation of ocean engineering. Problems concerning ocean environmental extremes are often multivariate in character. An example is the ocean environments (incluing waves, win an currents) are all contributing to the forces experience by offshore systems uring the typhoon. So the severity of such a typhoon event may be escribe by a function of win spee peak an concomitant wave height, currents, etc. (or switch their orer). When the force of a system is ominate by both win an concomitant wave it may be sufficient to employ the 0year return wave an 0year return win as a esign criterion. However, the 0year return win an 0year return wave are frequently not occurre at the same time. Therefore, any simple analysis assuming a perfect correlation between the win an waves is likely to overestimate the esign value (Morton an Bowers, 1). So, analyzing the encounter probability among the ocean environments by the multivariate istribution can offer useful reference to evaluate project s safety an cost. In Multivariate EV theory, two sampling methos, the Block Maxima metho an the POT (Peaks Over Threshols) metho have been evelope. They correspon respectively to two natural istributions, MGEVD (Multivariate Generalize Extreme Value Distribution) an MGPD. MGEVD is the natural istribution of the block maxima of all components. A typical example is that block is a year an the block maxima are the Annual Maxima Series (AMS). An MGEVD have evelope extensively uring the last ecaes too. This is witnesse by several literatures (Morton an Bowers, 1; Sheng, 001; YANG an ZHANG, 01); MGPD shoul be the theoretical istribution of MPOT metho, in which the sample inclues all extreme values which are larger than a suitable threshol. MPOT improves the efficiency of the extremes in raw ata, an is superior to other methos (such as the annual maximum, Luo an Zhu, 01). Rootzén an Tajvii (00) suggest, base on the research by Tajvii (1), that MGPD shoul be characterize by the following couple properties: (i) exceeances (of suitably coorinate levels) asymptotically have a MGPD if an only if componentwise maxima asymptotically are EVD, (ii) the MGPD is the only one
3 which is preserve uner (a suitably coorinate) change of exceeance levels. MPOT metho has a high utilization rate of raw ata. Besies, Luo et al. (01) selecte 0 or 0year samples from about 0year wave raw ata arbitrarily. The samples were analyze by POT an AMS respectively, an the result shows that the return levels with the POT metho are closer to the return levels from 0year wave raw ata an have smaller fluctuations than AMS. Because the POT metho can get as much extreme information as possible from raw ata, its result may be more stable. MGPD is wiely use recently, Morton an Bowers (1) are base on the response function with wave an win spee of anchoring semisubmersible platforms enabling to analyze extreme anchorage force an corresponing wave height an win spee by using logical extreme value istribution. They in t use the natural istribution of the MPOT metho  MGPD, to fitting samples but bivariate extreme value istribution to fitting the MPOT samples. Coles an Tawn (1) using the same min. MGPD theory has improve greatly in recent years, but the efinition of MGPD still nees further research. Bivariate threshol methos were evelope by Joe et al. (1) an Smith (1) base on point process theory. MGPD has been the focus of research an the etail about MGPD can be foun in Rootzén an Tajvii (00), Tajvii (1), Beirlant et al. (00) an Falk et al. (00). However, ue to ifficulties of MGPD in solving proceure, (in general, with the imension increase, the calculate quantity an complexity rapily o), the application of MGPD in ocean engineering has been restricte. Monte Carlo simulation is feasible to solve these problems because it only changes inner prouct operation an the complexity of algorithm oesn't increase with imension ecreases. Liu et al. (10) use the Monte Carlo simulation for the esign of offshore platforms, an practical examples prove its fast calculation spee an high precision in Compoun Extreme Value Distribution. Philippe (000) presente a new parameter estimation metho of bivariate extreme value istribution by using Monte Carlo simulation. Shi (1) presents a Monte Carlo metho from a simple trivariate neste logistic moel.
4 Stephenson (00) gives methos for simulating from symmetric an asymmetric versions of the multivariate logistic istribution, an compares many the Monte Carlo simulation methos of multiimensional extreme istribution. We evelop a proceure to hanle the application of MGPD in marine engineering esign. The paper uses the Monte Carlo simulation to solve the MGPD equation. The Monte Carlo metho is introuce in section. Funamental to the application of MGPD is the choice of the optimal joint threshol an the estimation of the joint ensity. These aspects inclue in an example are iscusse in section. Finally, the avantage of MGPD an its Monte Carlo simulation are outline.. MONTE CARLO SIMULATION OF MGPD MGPD THEORY It is well known that MGEVD (Multivariate Generalize Extreme Value Distributions) arise, like in the univariate case, as the limiting istributions of suitably scale componentwise maxima of inepenent an ientically istribute ranom vectors. If for inepenent X,..., 1 X n following F, there exist vectors, a n, b n R, a 0, such that n 1 max i1,..., n( Xi bn ) n n P F ( anx bn) G( x) an (1) where Gx ( ) is a MGEVD, an F is in the omain of attraction of G. We note this by F D( G). The MGPD is base on the extreme value theory an has been wiely use in many fiels. In one imension, Generalize Extreme Value istribution (GEVD) is the theoretical istribution of all variation block maxima. GPD escribes the properties of extreme of all variation over threshol after eclustering, so calle POT istribution. Base on the relationship of GPD an GEVD: H( x) 1log( G( x)),log( G( x)) 1, the istribution function of MGPD can be euce:
5 W X G x1 x ( ) 1 log( (,..., )) x x xid,...,, log Gx1,..., x 1 i1 x i 1 i x i1 i () where ( x1,..., x ) x U, U is a neighborhoo of zero in the negative quarant (,0), D is the Pickans epenence function in the unit simplex R 1 on the omain of efinition, R { x[0, ) x 1} i ; 1 i1 G( x,, x n ) is a MGEVD function which marginal istribution is negative exponential istribution (etail in René Michel, 00). The MGPD can simply use existing multivariate extreme epenence function ue to it is euce from MGEV, greatly enriche the expression of correlation of MGPD. MGPD has a variety of ifferent types of istribution functions (Coles et al., ) varying from Pickans epenence function eq. (). Logistic epenence function is ease to use an has the favorable statistical properties, an wiely use to hyrology, financial an other fiels. 1 1 r Dr t1,..., t 1 ti 1 ti i1 i1 r 1 r () 1 r r r i r i1 W ( x) 1 ( ( x ) ) 1 x, () 1 where r is the correlation parameter of epenence function an r>1. x i in the interval (1, 0), are variables of stanarization. The Bivariate Logistic GPD ensity function is W w x y r xy x y x y xy r1 r r 1/ r (, ) ( 1)( ) [( ) ( ) ] 0, 0 The correlation parameter r can be evaluate by step by step metho: evaluate by using bivariate extreme value istribution firstly then introuce to istribution function; or be evaluate by global metho, estimate the parameter by using the maximum likelihoo for the ensity function w. The global metho evaluate results more reliable ue to the final function form are to be concerne, but the processes of evaluate are more complex. The maximumlikelihoo function is ()
6 n L( r) ln( w ( x, y )) () i1 r i i. SIMULATION METHOD The Monte Carlo Simulation metho of multivariate istribution is relatively complex, because of generating multivariate ranom an relevant vectors involve. By a transformation metho, the variables become inepenence. An then, every variable is generate a ranom vector. Finally by the inverse transformation, the ranom vectors of the multivariate istribution are obtaine. The simulation metho was suggeste by René(00). Using polar coorinate to emonstrate the simulate metho of MGPD better: x x T ( x,..., x ) (,...,, x... x ) ( z,..., z, c), () 1 1 p x1... x x1... x where Tp change vector ( x1,..., x ) into polar coorinate. C x1 x an Z ( x / C,, / C ) are raial component an angular component, respectively. 1 x 1 They calle Pickans polar coorinate. In the Pickans polar coorinate, W( X) presents ifferent properties. Presume that ( X1,..., X ) follow multivariate generalize Pareto istribution W( X ) an its Pickans epenence function D exists orer ifferential, efine the Pickans ensity of H(X) x1,, x 1 z, c c H T 1 z, c Presume R 1 p z z 0 an constant c0 0 existing in a neighborhoo of zero, then the simulation metho of MGPD is: 1) generate uniform ranom numbers on unit simplex 1 R ; ) generate ranom vector z z 1, () base on the ensity function () z f() z of Z ( z1,, z 1) in the Pickans polar coorinate combine with AcceptanceRejection Metho; ) generate uniform ranom numbers on c,0 0 ; an )
7 calculate vector cz1,, cz 1, c c z i1 i which is ranom vector of satisfy the multivariate over threshol istribution. The c 0 above is the joint threshol in MGPD metho. This paper etermines the threshol by using the principle on Coles an Tawn (1).. JOINT PROBABILITY DISTRIBUTION With the evelopment of offshore engineering, joint probability stuy for extreme sea environments such as win, waves, ties an streams is beginning to receive much more attention. But for the selection of esign criteria for marine structures, a clear stanar about the joint probability has not been foun. API (American Petroleum Institute), DNV (Det Norske Veritas) an so on in t propose an explicit metho for joint variables although they mae some relevant rules. API RPALRFD (1) suggests three options, one of which is Any reasonable combination of win spee, wave height, an current spee that results in the 0year return perio combine platform loa, e.g. base shear or base overturning moment, but oes not provie etails of how to etermine the appropriate extreme conitions in a multivariate environment. The joint the return perio of two variables nee to consier the probability of encounter between variables. In other wor, 0years return wave may not counter 0year return win spee. CP can represent the probability of encounter between extreme value of main marine environmental elements an extreme value of its accompanie marine environmental elements. So, it is critical to use CP to escribe the probability of their joint together an analyze the effect of all kins of marine environmental elements to the engineering. The joint istribution of bivariate Pareto istribution function W(x, y) is W( x, y) Pr( X x, Y y). An Wx ( x ) an Wy ( y) are marginal istribution of x an y respectively. Conitional extreme value istribution can be
8 CP 1: CP : CP : CP : Pr( X x, Y y) 1 WX( x) WY( y) W( x, y) Pr( X x Y y) Pr( Y y) 1 WY ( y) Pr( X x, Y y) WX ( x) W ( x, y) Pr( X x Y y) Pr( Y y) 1 WY ( y) Pr( X x, Y y) WY ( y) W( x, y) Pr( X x Y y) Pr( Y y) WY ( y) Pr( X x, Y y) W ( x, y) Pr( X x Y y) Pr( Y y) W ( y) Other four CP istributions can be euce by swapping two variables. Y () () () (1). CASE The ocean environment is multivariate with waves, win an currents all contributing to the forces experience by marine structures. Any simple assumes that there is either perfect epenence or inepenence among them are unreasonable for a esign criterion. The section combine with CP for analysis of the esign criteria (taking the 0year return value as an example) about base shear of a plat. Assume the function of base shear an win velocity an wave height for a certain type of structure is Z(x, y)= 0.x 0.1y (1) where x an y are win spee an wave height, with their units m/s an m respectively. Z represents base shear an the unit is KN..1 THE DATA The ata are a sequence of (101) years of wave height recors an synchronous win spee recors, which was recore four times a ay from an ocean hyrological station (ZL) near. o N,. o E (fig. 1) in SCS, which is a typical typhoon sea area. In the raw ata, the maximum wins reache 0m/s an the maximum wave height is.0m. Due to restrictions with observation technologies at the time, the win spee an the wave height were kept only the integer an one ecimal place respectively. This will influence the level of precision of extreme value, so in fig. (a), the all the win values in the range 0. an 0. (stanar unit) show the
9 same probability.. DECLUSTERING The sample of MPOT metho is from the extreme value of blocks, so the first stage in a multivariate extreme value analysis is to ientify a set of eclustere events. For satisfying inepenent ranom istribution assumption, the principle of eclustering is keeping samples inepenence. In SCS, typhoon occurs frequently an causes almost all extreme win spee an wave height. Generally the influence of typhoon for one point may last several ays or one week. Exploratory analysis suggeste that a winow corresponing to five ays is appropriate. But, if the interval between the ajacent extremes is less than ays, then we nee to elete smaller values from the ajacent extremes in orer to keep inepenence. After eclustering, the block maxima shoul be extracte as extreme values. Generally, in a bivariate analysis of wave heights an win spees, there are four efinitions of maximum ata set in a certain block: (1) maximum wave height an maximum win spee in a block; () maximum wave height an its concomitant win spee in a block; () maximum win velocity an its concomitant wave height in a block; Liu et al. (00) suggests that efinition (1) is the simplest one, but it may lea to overconservative results for they are not observe at the same time; for efinitions () an (), only one of the extreme observations is selecte, so we have to consier which has a major influence for offshore structures. For example, if wave loa has a major influence on a certain structure, efinition () woul be a goo choice. In case that we o not know which loa has a ominating influence on a structure, we usually make extreme value analysis on efinitions () an () respectively, an finally choose the more angerous one. In the present stuy, the case is use solely for testing the valiity of the simulation metho of MGPD an giving an example of its application. So efinition () is an arbitrary choice. That is win spee is the ominating variable. After eclustering accoring to the requirements of inepenence, 1 groups of extreme win spee
10 an corresponing wave height are selecte.. MARGINAL TRANSFORMATION AND JOINT THRESHOLD In section.1, the variables of MGPD must be in a neighborhoo of zero in the negative quarant. By a suitable marginal transformation, we can transfer a margin into a uniform margin in a neighborhoo of zero. After many experiments, it is foun that marginal istributions of extreme win spees an wave heights in 1 ata sets can be escribe by GEVD: x 1 F(x)= P(X < x)= exp [1 ( )], 0 (1), where an, are three variables of GEVD. They are estimate by using maximum likelihoo estimate. This is an approach of estimation suggeste in Section.1 of Beirlant et al. (00) an Section. of Coles (001). Fig. shows the probability plot an probability plots (incluing the % confience intervals) of marginal istribution before fitting MGPD. To stanarize the margins, the marginal istribution of MGPD is negative exponential istribution. Accoring to Taylor expansion, we get logf ( x ) log(1 F( x ) 1) F( x ) 1 (1), i i i where Fx ( i ) is GEVD of variable i ( i 1,),which represent win an wave respectively (etail in RenéMichel, 00). The epenence moels between extreme variables have been suggeste: Logistic, Bilogistic an Dirichlet. However, it appears that the choice of epenence moel is not usually critical to the accuracy of the final moel (Morton an Bowers, 1). So the simple bivariate Logistic GPD was selecte. The MGPD moel of the paper is base on multivariate extreme value istribution, the joint threshol can be calculate by the metho in section.. The joint threshol is c0 0., an there are 0 groups of combination of win spee an wave height over c 0. Fig. (a) shows that the samples
11 of over threshol value. In the left subfigure, c0 0. is a curve, an the right sie of the curve are over threshol value. In the right subfigure c0 0. is a line, the topright sie of the line are over threshol value of the ata converte. The correlation parameter r of epenence function is estimate by the maximumlikelihoo metho, an the joint istribution is showe in Fig. (b).. COMPARISON OF STOCHASTIC SIMULATION RESULTS For the annual maximum of win spee an wave height, Pearson type Ⅲ istribution is use to obtain return perio values of win spee an wave height in oneimensional (Fig. ). The Pearson type Ⅲ istribution is ( ) (1) x 1 F(x)= P(X < x)= ( x ) exp[ ( x )] x Fig. shows the ata of stochastic simulation by N=0000 an N=0000. Due to the accuracy of the raw ata, the 0m/s win event is so much that they can t be fitte by MGPD well. So the simulation results an the measurement points are not overlap completely for the 0m/s win event in Fig.. But the simulation results are in basic conformity with the actual situation, an this represents that the MGPD simulation metho is worke. The scatter iagrams show the results irectly, they nee further quantitative analysis to show the ifferences of them objectively. A couple of CP are to be use to the paper: 1) P( H h V v) an ) P( H h V v) which means 1) the probability of the wave height over a staner h uner the win spee over the staner v an ) the probability of the wave height less than a staner h uner the win spee less than the staner v, respectively. Both of them are actually responing the probability of extreme value wave height an its corresponing win spee joint occur. Fig. represent calculating the CP P( H h V v) by group h.m v m/ s. Using the Monte Carlo metho calculate its CP by the result of simulation base on the efinition of conitional istribution. As it s showe in Fig., the ifference value of the simulation an the moel are relate to the simulation times N. an
12 Relative ifference value has been reuce with the increase of simulation times. When the simulation times up to, the relative error value of simulation results an calculation results is 0.1%, shows that the simulation results error is acceptable. Base on the results by simulation times e, Tab. an Tab. show the calculation results of two ifferent CP. The tables represent groups calculation an stochastic simulation results of CP 1 an on ifferent combination of wave height an win spee. The two results are closely. For instance, the calculate result of the probability of wave height of the esign frequency more than % encounter win spee of the esign frequency more than % is.% where stochastic simulation result is.%, the relate error only 0.%. The calculate result of the probability of wave height of the esign frequency more than % encounter win spee of the esign frequency more than % is 0.0% where stochastic simulation result is.%, the relate error only.%. Synthesizing the appearing probability of extreme sea environment is to the benefit of fin a balance between the engineering investment an risk, an can provie the scientific basis for the risk preestimates.. ANALYSIS OF RETURN VALUE For reasonable combination of win spee an wave height, the metho of analyzing the multivariate extreme environment is offere by CP. Because win spee is the ominating variable in the paper, we use CP 1: P( H h V v) as a stanar. The return value of the base shear follows the couple principles: (1) if P( H h V v ) >%, the 0year base shear is the combination of 0year win 0 0 spee an 0year wave height; () if P( H h0 V v0 ) <%, the 0year base shear is the combination of 0year win spee an the wave height corresponing to P( H h V v ) =%. The value % was selecte only to show how to 0 0 etermine esign criterion by the CP, an the paper o not analyze whether the value is appropriate for the marine structure. In the metho, it can be avoie that the encounter probability of win spee an wave height is smaller.
13 The result of CP 1: P( H h V v) is in tab.. We can know that the estimates of 0year win spee an wave height are.1m/s an.1m respectively, an their CP1 is 1.. Accoring to the above principles, this is meeting the principle (). Using the Monte Carlo metho, we can get P( H.01 V v0) =%. So the 0year base shear is Z(.1,.01 )=.KN.. Discussion an Conclusions NEW POSSIBILITIES BASED ON MONTE CARLO SIMULATION The esign sea states for a particular location are often use in esign an assessment of coastal an ocean engineering. These esign sea states might be jointly ecie by multivariate ocean environment factors, such as combinations of wave conitions an water levels with given joint return perios. A potentially better approach is possible base on Monte Carlo simulation of a wie range of ocean environment factors, from which the structure variables in ifferent return perios can be etermine irectly. Once the longterm (such as thousans of years) sea state ata has been simulate, several structure variables or ocean environment factors can be assesse quite quickly by the law of large numbers. If the raw ata is enough, the MGPD can be use in extreme analysis base on or morevariables. In this case, Monte Carlo simulation becomes a musthave tool, because it is ifficult to solve the highimension MGPD an get the structure variables in ifferent return perios.. CONCLUSIONS The MGPD is the nature istribution of MPOT metho, which can ig up more extreme information from the raw ata. The moel base on the extreme value theory which is wellfoune an the intrinsic properties of all extreme variables are into consieration. The MGPD is a useful basis for extreme value analysis of the offshore environment. The Monte Carlo simulation of MGPD provies estimates of the 0year base shear, taking into account the conitional probability, which represents
14 the encounter probability between variables such as win spee an wave height. CP1 inclues the encounter probability of extreme events an provies the theory basis for fining the best balance point between engineering cost an risk. The analysis of the return mooring forces at SCS illustrates the practicalities of Monte Carlo simulation of MGPD. Acknowlegments This work was supporte by the 'Strategic Priority Research Program' of the Chinese Acaemy of Sciences (XDA10); References API RP ALRFD Planning, Designing an Constructing for Fixe Offshore Platforms Loa an Resistance Factor Design, ISO : 1(E), 1. Beirlant J, Segers J, Teugels J. Statistics of extremes, theory an applications. Wiley, Chichester 00. Coles S G, Tawn J A. Moelling extreme multivariate events. Journal of the Royal Statistical Society ; B, . Coles S G, Tawn J A. Statistical methos for multivariate extremes: an application to structural esign. Applie Statistics 1;, 1. Falk M, Hüsler J, Reiss, R D. Laws of Small Numbers: Extremes an Rare events. n en. Birkhäuser, Basel 00. Joe H, Smith R L, Weissman I. Bivariate threshol methos for extremes. Journal of the Royal Statistical Society 1; B, 1. Liu Defu, Shi Jiangang, Zhou Zhigang. The application of Montecarlo metho in the Marine engineering esign, Acta Oceanologica Sinica 10; 1() . (in chinese) Liu, D. F., Wen, S. Q. an Wang, L. P.. PoissonGumbel mixe compoun istribution an its application, Chin. Sci. Bull. 00; ()
15 Luo Yao, Zhu Liangsheng. A New Moel of Peaks over Threshol for Multivariate Extremes [J], 01. China Ocean Eng., ():1~ Luo Yao, Zhu Liangsheng, Hu Jinpeng. A new metho for etermining threshol in using PGCEVD to calculate return values of typhoon wave height [J]. China Ocean Eng., 01, (): 1 0. Morton I D, Moul G. Extreme value analysis in a multivariate offshore environment. Applie Ocean Research 1; 1, 01. Philippe Caperaa. Estimation of a Bivariate Extreme Value Distribution. Extreme 000; (),  RenéM. Simulation of certain multivariate generalize Pareto istributions. Extremes 00;,. Rootzen, Tajvii, Multivariate Generalize Pareto Distributions, Preprint 00:, Chalmers Tekniska Häogskola Gäoteborg, Rootzén H, Tajvii N. Multivariate generalize Pareto istributions. Bernoulli 00; 1, 10. Shi D.. Moment estimation for multivariate extreme value istribution in a neste logistic moel. Ann. Inst. Statist. Math. 1; 1, . Sheng Yue. The Gumbel logistic moel for representing a multivariate storm event. Avances in Water Resources 001; Smith R L. Multivariate threshol methos. In Galambos, J., Lechner, J., Simiu, E., eitors, Extreme Value Theory an Applications 1; pages . Kluwer, Dorrecht. Stephenson A. Simulating multivariate extreme value istributions of logistic type. Extremes 00;,. Tajvii N, Characterisation an Some Statistical Aspects of Univariate an Multivariate Generalise Pareto Distributions, Department of Mathematics, Chalmers Tekniska Hogskola Goteborg, 1
16 Xiaochen YANG, Qinghe ZHANG, Joint probability istribution of wins an waves from wave simulation of 0 years (100) in Bohai Bay, Water Science an Engineering 01; (): 0. Tab.1 parameters of marginal istribution Win spee Wave height Tab. comparision of the results of CP 1 RP (year) V(m/s) RP H(m) c s c s c s c s c s RP: return perios, a: calculation results by analytic solution, s: simulation results Tab. comparision of the results of CP RP (year) V(m/s) RP H(m) c s c s c s c s c s RP: return perios, a: calculation results by analytic solution, s: simulation results 1
17 Fig. 1 Location of ZL ocean hyrological station Fig. fitting testing of marginal istribution, (a) win spee (b) wave height
18 Fig. (a) over threshol value of wave height an win spee (b) joint istribution of extreme value of wave height an win spee Fig. calculation of win spee samples an wave height return value N=0000 N=0000 Fig. value of over threshol an ata of stochastic simulation
19 Conition probability simulation calculate The number of simulation Fig. The change of simulation accuracy
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