Papers About Extreme Value and Crossing Levels by Arvid Naess

Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2013, Article ID 797014, 15 pageshttp://dx.doi.org/10.1155/2013/797014

Research ArticleEstimation of Extreme Values by the Average ConditionalExceedance Rate Method

A. Naess,1 O. Gaidai,2 and O. Karpa3

1 Department of Mathematical Sciences and CeSOS, Norwegian University of Science and Technology, 7491 Trondheim, Norway2Norwegian Marine Technology Research Institute, 7491 Trondheim, Norway3 Centre for Ships and Ocean Structures (CeSOS), Norwegian University of Science and Technology, 7491 Trondheim, Norway

Correspondence should be addressed to A. Naess; [email protected]

Received 18 October 2012; Revised 22 December 2012; Accepted 9 January 2013

Academic Editor: A. Thavaneswaran

Copyright 2013 A. Naess et al.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper details a method for extreme value prediction on the basis of a sampled time series. The method is specifically designedto account for statistical dependence between the sampled data points in a precisemanner. In fact, if properly used, the newmethodwill provide statistical estimates of the exact extreme value distribution provided by the data in most cases of practical interest. Itavoids the problem of having to decluster the data to ensure independence, which is a requisite component in the application of, forexample, the standard peaks-over-threshold method. The proposed method also targets the use of subasymptotic data to improveprediction accuracy. The method will be demonstrated by application to both synthetic and real data. From a practical point ofview, it seems to perform better than the POT and block extremes methods, and, with an appropriate modification, it is directlyapplicable to nonstationary time series.

1. Introduction

Extreme value statistics, even in applications, are generallybased on asymptotic results. This is done either by assumingthat the epochal extremes, for example, yearly extreme windspeeds at a given location, are distributed according to thegeneralized (asymptotic) extreme value distribution withunknown parameters to be estimated on the basis of the ob-served data [1, 2]. Or it is assumed that the exceedances abovehigh thresholds follow a generalized (asymptotic) Paretodistribution with parameters that are estimated from the data[14]. The major problem with both of these approaches isthat the asymptotic extreme value theory itself cannot be usedin practice to decide to what extent it is applicable for theobserved data. And since the statistical tests to decide thisissue are rarely precise enough to completely settle this prob-lem, the assumption that a specific asymptotic extreme valuedistribution is the appropriate distribution for the observeddata is based more or less on faith or convenience.

On the other hand, one can reasonably assume that inmost cases long time series obtained from practical measure-ments do contain values that are large enough to provide

useful information about extreme events that are trulyasymptotic. This cannot be strictly proved in general, ofcourse, but the accumulated experience indicates that asymp-totic extreme value distributions do provide reasonable, if notalways very accurate, predictions when based on measureddata. This is amply documented in the vast literature on thesubject, and good references to this literature are [2, 5, 6]. Inan effort to improve on the current situation, we have tried todevelop an approach to the extreme value prediction problemthat is less restrictive and more flexible than the ones basedon asymptotic theory. The approach is based on two separatecomponentswhich are designed to improve on two importantaspects of extreme value prediction based on observed data.The first component has the capability to accurately captureand display the effect of statistical dependence in the data,which opens for the opportunity of using all the available datain the analysis. The second component is then constructedso as to make it possible to incorporate to a certain extentalso the subasymptotic part of the data into the estimationof extreme values, which is of some importance for accurateestimation. We have used the proposed method on a widevariety of estimation problems, and our experience is that

2 Journal of Probability and Statistics

it represents a very powerful addition to the toolbox ofmethods for extreme value estimation. Needless to say, whatis presented in this paper is by no means considered a closedchapter. It is a novel method, and it is to be expected thatseveral aspects of the proposed approach will see significantimprovements.

2. Cascade of Conditioning Approximations

In this section, a sequence of nonparametric distributionfunctions will be constructed that converges to the exactextreme value distribution for the time series considered.Thisconstitutes the core of the proposed approach.

Consider a stochastic process (), which has beenobserved over a time interval, (0, ) say. Assume that values1, . . . ,

, which have been derived from the observed

process, are allocated to the discrete times 1, . . . ,

in (0, ).

This could be simply the observed values of () at each, = 1, . . . , , or it could be average values or peak

values over smaller time intervals centered at the s. Our

goal in this paper is to accurately determine the distributionfunction of the extreme value

= max{

; = 1, . . . , }.

Specifically, we want to estimate () = Prob( )

accurately for large values of . An underlying premise for thedevelopment in this paper is that a rational approach to thestudy of the extreme values of the sampled time series is toconsider exceedances of the individual random variables

above given thresholds, as in classical extreme value theory.The alternative approach of considering the exceedances byupcrossing of given thresholds by a continuous stochasticprocess has been developed in [7, 8] along lines similar to thatadopted here.The approach taken in the present paper seemsto be the appropriate way to deal with the recorded data timeseries of, for example, the hourly or daily largest wind speedsobserved at a given location.

From the definition of () it follows that

() = Prob ( ) = Prob {

, . . . ,

1 }

= Prob { |

1 , . . . ,

1 }

Prob {1 , . . . ,

1 }

=

=2

Prob { |

1 , . . . ,

1 }

Prob (1 ) .

(1)

In general, the variables are statistically dependent.

Hence, instead of assuming that all the are statistically

independent, which leads to the classical approximation

() 1() :=

=1

Prob ( ) , (2)

where :=means by definition, the following one-step mem-ory approximation will, to a certain extent, account for thedependence between the

s,

Prob { |

1 , . . . ,

1 }

Prob { |

1 } ,

(3)

for 2 . With this approximation, it is obtained that

() 2()

:=

=2

Prob { |

1 }Prob (

1 ) .

(4)

By conditioning on one more data point, the one-step mem-ory approximation is extended to

Prob { |

1 , . . . ,

1 }

Prob { |

1 ,

2 } ,

(5)

where 3 , which leads to the approximation

() 3() :=

=3

Prob { |

1 ,

2 }

Prob {2 |

1 }Prob (

1 ) .

(6)

For a general , 2 , it is obtained that

() ()

:=

=

Prob { |

1 , . . . ,

+1 }

1

=2

Prob { |

1 . . . ,

1 }

Prob (1 ) ,

(7)

where () = ().

It should be noted that the one-step memory approxi-mation adopted above is not a Markov chain approximation[911], nor do the -step memory approximations lead toth-order Markov chains [12, 13]. An effort to relinquish theMarkov chain assumption to obtain an approximate distribu-tion of clusters of extremes is reported in [14].

It is of interest to have a closer look at the values for ()obtained by using (7) as compared to (2). Now, (2) can berewritten in the form

() 1() =

=1

(1 1()) , (8)

Journal of Probability and Statistics 3

where 1() = Prob{

> }, = 1, . . . , . Then the approx-

imation based on assuming independent data can be writtenas

() 1() := exp(

=1

1()) , . (9)

Alternatively, (7) can be expressed as,

() () =

=

(1 ())

1

=1

(1 ()) , (10)

where () = Prob{

> |

1 , . . . ,

+1 }, for

2, denotes the exceedance probability conditional on 1 previous nonexceedances. From (10) it is now obtainedthat

() () := exp(

=

()

1

=1

()) ,

,

(11)

and () () as with

() = () for .

For the cascade of approximations () to have practical

significance, it is implicitly assumed that there is a cut-offvalue

satisfying

such that effectively

() =

(). It may be noted that for -dependent stationary data

sequences, that is, for data whereand

are independent

whenever | | > , then () = +1() exactly, and, under

rather mild conditions on the joint distributions of the data,lim

1() = lim

() [15]. In fact, it can be shown

that lim

1() = lim

() is true for weaker con-

ditions than -dependence [16]. However, for finite values of the picture is much more complex, and purely asymptoticresults should be used with some caution. Cartwright [17]used the notion of -dependence to investigate the effect onextremes of correlation in sea wave data time series.

Returning to (11), extreme value prediction by the con-ditioning approach described above reduces to estimation of(combinations) of the

() functions. In accordance with

the previous assumption about a cut-off value , for all -

values of interest, , so that 1=1() is effectively

negligible compared to=(). Hence, for simplicity, the

following approximation is adopted, which is applicable toboth stationary and nonstationary data,

() = exp(

=

()) , 1. (12)

Going back to the definition of 1(), it follows that

=11() is equal to the expected number of exceedances

of the threshold during the time interval (0, ). Equation(9) therefore expresses the approximation that the stream ofexceedance events constitute a (nonstationary) Poisson pro-cess. This opens for an understanding of (12) by interpretingthe expressions

=() as the expected effective number

of independent exceedance events provided by conditioningon 1 previous observations.

3. Empirical Estimation of the AverageConditional Exceedance Rates

The concept of average conditional exceedance rate (ACER)of order is now introduced as follows:

() =

1

+ 1

=

() , = 1, 2, . . . . (13)

In general, this ACER function also depends on the numberof data points.

In practice, there are typically two scenarios for theunderlying process (). Either we may consider to be astationary process, or, in fact, even an ergodic process. Thealternative is to view() as a process that depends on certainparameters whose variation in time may be modelled as anergodic process in its own right. For each set of values of theparameters, the premise is that () can then be modelled asan ergodic process. This would be the scenario that can beused to model long-term statistics [18, 19].

For both these scenarios, the empirical estimation of theACER function

() proceeds in a completely analogous

way by counting the total number of favourable incidents,that is, exceedances combined with the requisite number ofpreceding nonexceedances, for the total data time series andthen finally dividing by + 1 . This can be shown toapply for the long-term situation.

A few more details on the numerical estimation of ()

for 2 may be appropriate. We start by introducing thefollowing random functions:

() = 1 {

> ,

1 , . . . ,

+1 } ,

= , . . . , , = 2, 3, . . . ,

() = 1 {

1 , . . . ,

+1 } ,

= , . . . , , = 2, . . . ,

(14)

where 1{A} denotes the indicator function of some eventA.Then,

() =

E [()]

E [()]

, = , . . . , , = 2, . . . , (15)

where E[] denotes the expectation operator. Assuming anergodic process, then obviously

() =

() = =

(), and by replacing ensemble means with correspond-

ing time averages, it may be assumed that for the time seriesat hand

() = lim

=()

=()

, (16)

where () and

() are the realized values of

() and

(), respectively, for the observed time series.Clearly, lim

E[()] = 1. Hence, lim

()/

() = 1, where

() =

=E [()]

+ 1. (17)


The advantage of using themodified ACER function () for

2 is that it is easier to use for nonstationary or long-term statistics than

(). Since our focus is on the values of

the ACER functions at the extreme levels, we may use anyfunction that provides correct predictions of the appropriateACER function at these extreme levels.

To see why (17) may be applicable for nonstationary timeseries, it is recognized that

() exp(

=

()) = exp(

=

E [()]

E [()]

)

exp(

=

E [()]) .

(18)

If the time series can be segmented into blocks, such thatE[()] remains approximately constant within each block

and such that

E[()]

() for a sufficient

range of -values, wheredenotes the set of indices for block

no. , = 1, . . . , , then=

E[()]

=(). Hence,

() exp ( ( + 1) ()) , (19)

where

() =

1

+ 1

=

() . (20)

It is of interest to note what events are actually countedfor the estimation of the various

(), 2. Let us

start with 2(). It follows from the definition of

2() that

2() ( 1) can be interpreted as the expected number of

exceedances above the level , satisfying the condition that anexceedance is counted only if it is immediately preceded by anon-exceedance. A reinterpretation of this is that

2() (

1) equals the average number of clumps of exceedancesabove , for the realizations considered, where a clump ofexceedances is defined as a maximum number of consecutiveexceedances above . In general,

() ( + 1) then

equals the average number of clumps of exceedances above separated by at least 1 nonexceedances. If the timeseries analysed is obtained by extracting local peak valuesfrom a narrow band response process, it is interesting tonote the similarity between the ACER approximations andthe envelope approximations for extreme value prediction[7, 20]. For alternative statistical approaches to account forthe effect of clustering on the extreme value distribution, thereader may consult [2126]. In these works, the emphasis ison the notion of an extremal index, which characterizes theclumping or clustering tendency of the data and its effect onthe extreme value distribution. In the ACER functions, theseeffects are automatically accounted for.

Now, let us look at the problem of estimating a confidenceinterval for

(), assuming a stationary time series. If

realizations of the requisite length of the time series isavailable, or, if one long realization can be segmented into

subseries, then the sample standard deviation () can be

estimated by the standard formula

()2

=1

1

=1

(()

()

())2

, (21)

where ()() denotes the ACER function estimate from real-

ization no. , and () =

=1()

()/.

Assuming that realizations are independent, for a suitablenumber , for example, 20, (21) leads to a good approx-imation of the 95% confidence interval CI = ((), +())for the value

(), where

() =

()

1.96 ()

. (22)

Alternatively, and which also applies to the non-station-ary case, it is consistent with the adopted approach to assumethat the stream of conditional exceedances over a threshold constitute a Poisson process, possibly non-homogeneous.Hence, the variance of the estimator

() of

(), where

() =

=()

+ 1

(23)

is Var[()] =

().Therefore, for high levels , the approx-

imate limits of a 95% confidence interval of (), and also

(), can be written as

() =

()(1

1.96

( + 1) ()

) . (24)

4. Estimation of Extremes for the AsymptoticGumbel Case

The second component of the approach to extreme valueestimation presented in this paper was originally derived fora time series with an asymptotic extreme value distributionof the Gumbel type, compared with [27]. We have thereforechosen to highlight this case first, also because the extensionof the asymptotic distribution to a parametric class of extremevalue distribution tails that are capable of capturing to someextent subasymptotic behaviour is more transparent, andperhaps more obvious, for the Gumbel case. The reasonbehind the efforts to extend the extreme value distributions tothe subasymptotic range is the fact that the ACER functionsallow us to use not only asymptotic data, which is clearly anadvantage since proving that observed extremes are trulyasymptotic is really a nontrivial task.

The implication of the asymptotic distribution being ofthe Gumbel type on the possible subasymptotic functionalforms of

() cannot easily be decided in any detail.However,

using the asymptotic form as a guide, it is assumed thatthe behaviour of the mean exceedance rate in the tail isdominated by a function of the form exp{( )} ( 1 ), where , , and are suitable constants, and

1is an


appropriately chosen tail marker. Hence, it will be assumedthat,

() =

() exp {

(

)

} , 1, (25)

where the function () is slowly varying, compared with

the exponential function exp{(

)} and

, , and

are suitable constants, that in general will be dependent on .Note that the value

= () = 1 corresponds to the asymp-

totic Gumbel distribution, which is then a special case of theassumed tail behaviour.

From (25) it follows that

log

log(()

())

= log (

) log (

) .

(26)

Therefore, under the assumptions made, a plot of log | log(

()/

())| versus log(

) will exhibit a

perfectly linear tail behaviour.It is realized that if the function

() could be replaced

by a constant value, say , one would immediately be in a

position to apply a linear extrapolation strategy for deep tailprediction problems. In general,

() is not constant, but its

variation in the tail region is often sufficiently slow to allowfor its replacement by a constant, possibly by adjusting the tailmarker

1.Theproposed statistical approach to the prediction

of extreme values is therefore based on the assumption thatwe can write,

() =

exp {

(

)

} , 1, (27)

where , , , and

are appropriately chosen constants. In

a certain sense, this is aminimal class of parametric functionsthat can be used for this purpose which makes it possible toachieve three important goals. Firstly, the parametric classcontains the asymptotic form given by

= = 1 as a

special case. Secondly, the class is flexible enough to capture,to a certain extent, subasymptotic behaviour of any extremevalue distribution, that is, asymptotically Gumbel. Thirdly,the parametric functions agree with a wide range of knownspecial cases, of which a very important example is theextreme value distribution for a regular stationary Gaussianprocess, which has

= 2.

The viability of this approach has been successfully dem-onstrated by the authors formean up-crossing rate estimationfor extreme value statistics of the response processes relatedto a wide range of different dynamical systems, comparedwith [7, 8].

As to the question of finding the parameters , , , (the subscript , if it applies, is suppressed), the adoptedapproach is to determine these parameters byminimizing thefollowingmean square error function, with respect to all fourarguments,

(, , , ) =

=1

log (

) log + (

)

2

, (28)

where 1< <

denotes the levels where the ACER func-

tion has been estimated, denotes a weight factor that

puts more emphasis on the more reliably estimated ().

The choice of weight factor is to some extent arbitrary. Wehave previously used

= (log+(

) log(

)) with

= 1 and 2, combined with a Levenberg-Marquardt leastsquares optimization method [28]. This has usually workedwell provided reasonable and initial values for the parameterswere chosen. Note that the form of

puts some restriction

on the use of the data. Usually, there is a level beyondwhich

is no longer defined, that is, (

) becomes negative.

Hence, the summation in (28) has to stop before that happens.Also, the data should be preconditioned by establishing thetail marker

1based on inspection of the empirical ACER

functions.In general, to improve robustness of results, it is recom-

mended to apply a nonlinearly constrained optimization [29].The set of constraints is written as

log ( )

0,

0 < < +,

min

<

1,

0 < < +,

0 < < 5.

(29)

Here, the first nonlinear inequality constraint is evident, sinceunder our assumptionwe have

() = exp{(

)}, and

() < 1 by definition.

A Note of Caution. When the parameter is equal to 1.0 orclose to it, that is, the distribution is close to the Gumbeldistribution, the optimization problem becomes ill-definedor close to ill-defined. It is seen that when = 1.0, there isan infinity of (, ) values that gives exactly the same valueof (, , , ). Hence, there is no well-defined optimum inparameter space.There are simply toomany parameters.Thisproblem is alleviated by fixing the -value, and the obviouschoice is = 1.

Although the Levenberg-Marquardt method generallyworks well with four or, when appropriate, three parameters,we have also developed a more direct and transparentoptimization method for the problem at hand. It is realizedby scrutinizing (28) that if and are fixed, the optimizationproblem reduces to a standard weighted linear regressionproblem. That is, with both and fixed, the optimal valuesof and log are found using closed form weighted linearregression formulas in terms of

, = log (

) and

=

( ). In that light, it can also be concluded that the best

linear unbiased estimators (BLUE) are obtained for= 2

,

where 2= Var[

] (empirical) [30, 31]. Unfortunately, this

is not a very practical weight factor for the kind of problemwe have here because the summation in (28) then typicallywould have to stop at undesirably small values of

.


It is obtained that the optimal values of and are givenby the relations

(, ) =

=1( ) (

)

=1( )2,

log (, ) = + (, ) ,

(30)

where = =1/

=1, with a similar definition of .

To calculate the final optimal set of parameters, onemay use the Levenberg-Marquardt method on the function(, ) = (

(, ), , ,

(, )) to find the optimal values

and , and then use (30) to calculate the corresponding and .For a simple construction of a confidence interval for

the predicted, deep tail extreme value given by a particularACER function as provided by the fitted parametric curve, theempirical confidence band is reanchored to the fitted curveby centering the individual confidence intervals CI

0.95for the

point estimates of the ACER function on the fitted curve.Under the premise that the specified class of parametriccurves fully describes the behaviour of the ACER functions inthe tail, parametric curves are fitted as described above to theboundaries of the reanchored confidence band. These curvesare used to determine a first estimate of a 95% confidenceinterval of the predicted extreme value. To obtain a moreprecise estimate of the confidence interval, a bootstrappingmethod would be recommended. A comparison of estimatedconfidence intervals by both these methods will be presentedin the section on extreme value prediction for synthetic data.As a final point, it has been observed that the predicted valueis not very sensitive to the choice of

1, provided it is chosen

with some care. This property is easily recognized by lookingat the way the optimized fitting is done. If the tail marker isin the appropriate domain of the ACER function, the optimalfitted curve does not change appreciably by moving the tailmarker.

5. Estimation of Extremes for the General Case

For independent data in the general case, the ACER function1() can be expressed asymptotically as

1()

[1 + ( ( ))]1/

, (31)

where > 0, , are constants. This follows from theexplicit form of the so-called generalized extreme value(GEV) distribution Coles [1].

Again, the implication of this assumption on the possiblesubasymptotic functional forms of

() in the general case

is not a trivial matter. The approach we have chosen is toassume that the class of parametric functions needed forthe prediction of extreme values for the general case can bemodelled on the relation between the Gumbel distributionand the general extreme value distribution. While the exten-sion of the asymptotic Gumbel case to the proposed class ofsubasymptotic distributions was fairly transparent, this is notequally so for the general case. However, using a similar kind

of approximation, the behaviour of the mean exceedance ratein the subasymptotic part of the tail is assumed to follow afunction largely of the form [1 + (( ))]1/ (

1

), where > 0, , > 0, and > 0 are suitable constants,and

1is an appropriately chosen tail level. Hence, it will be

assumed that [32]

() =

() [1 +

((

)

)]1/

, 1, (32)

where the function () is weakly varying, compared with

the function [1 + ((

))]1/ and

> 0,

, > 0

and > 0 are suitable constants, that in general will be

dependent on . Note that the values = 1 and

() = 1 cor-

responds to the asymptotic limit, which is then a special caseof the general expression given in (25). Another method toaccount for subasymptotic effects has recently been proposedby Eastoe and Tawn [33], building on ideas developed byTawn [34], Ledford and Tawn [35] and Heffernan and Tawn[36]. In this approach, the asymptotic form of the marginaldistribution of exceedances is kept, but it is modified by amultiplicative factor accounting for the dependence structureof exceedances within a cluster.

An alternative form to (32) would be to assume that

() = [1 +

((

)

+ ())]

1/

, 1,

(33)

where the function () is weakly varying compared with

the function (

) . However, for estimation purposes, it

turns out that the form given by (25) is preferable as it leads tosimpler estimation procedures. This aspect will be discussedlater in the paper.

For practical identification of the ACER functions givenby (32), it is expedient to assume that the unknown function() varies sufficiently slowly to be replaced by a constant.

In general, () is not constant, but its variation in the

tail region is assumed to be sufficiently slow to allow for itsreplacement by a constant. Hence, as in the Gumbel case, itis in effect assumed that

() can be replaced by a constant

for 1, for an appropriate choice of tail marker

1. For

simplicity of notation, in the following we will suppress theindex on the ACER functions, which will then be written as

() = [1 + ( )

]

, 1, (34)

where = 1/, = .For the analysis of data, first the tail marker

1is

provisionally identified from visual inspection of the logplot (, ln

()). The value chosen for

1corresponds to the

beginning of regular tail behaviour in a sense to be discussedbelow.

The optimization process to estimate the parameters isdone relative to the log plot, as for theGumbel case.Themeansquare error function to be minimized is in the general casewritten as (, , , , ) =

=1

log (

) log

+ log [1 + ( )

]

2

,

(35)

where is a weight factor as previously defined.


An option for estimating the five parameters , , ,, is again to use the Levenberg-Marquardt least squaresoptimization method, which can be simplified also in thiscase by observing that if , , and are fixed in (28), theoptimization problem reduces to a standard weighted linearregression problem.That is, with , , and fixed, the optimalvalues of and log are found using closed form weightedlinear regression formulas in terms of

, = log (

) and

= 1 + (

).

It is obtained that the optimal values of and log are given by relations similar to (30). To calculate the finaloptimal set of parameters, the Levenberg-Marquardt meth-od may then be used on the function (, , ) =(, , ,

(, , ),

(, , )) to find the optimal values ,

, and , and then the corresponding and can becalculated. The optimal values of the parameters may, forexample, also be found by a sequential quadratic program-ming (SQP) method [37].

6. The Gumbel Method

To offer a comparison of the predictions obtained by themethod proposed in this paper with those obtained by othermethods, we will use the predictions given by the two meth-ods that seem to bemost favored by practitioners, theGumbelmethod and the peaks-over-threshold (POT) method, pro-vided, of course, that the correct asymptotic extreme valuedistribution is of the Gumbel type.

The Gumbel method is based on recording epochalextreme values and fitting these values to a correspondingGumbel distribution [38]. By assuming that the recordedextreme value data are Gumbel distributed, then representingthe obtained data set of extreme values as a Gumbel probabil-ity plot should ideally result in a straight line. In practice, onecannot expect this to happen, but on the premise that the datafollow a Gumbel distribution, a straight line can be fitted tothe data. Due to its simplicity, a popular method for fittingthis straight line is the method of moments, which is alsoreasonably stable for limited sets of data. That is, writing theGumbel distribution of the extreme value

as

Prob ( ) = exp { exp ( ( ))} , (36)

it is known that the parameters and are related to themeanvalue

and standard deviation

of () as follows:

= 0.57722

1 and = 1.28255/[39].The estimates

of and

obtained from the available sample therefore

provides estimates of and , which leads to the fittedGumbel distribution by the moment method.

Typically, a specified quantile value of the fitted Gumbeldistribution is then extracted and used in a design considera-tion. To be specific, let us assume that the requested quantilevalue is the 100(1 )% fractile, where is usually a smallnumber, for example, = 0.1. To quantify the uncertaintyassociated with the obtained 100(1 )% fractile value basedon a sample of size , the 95% confidence interval of thisvalue is often used. A good estimate of this confidenceinterval can be obtained by using a parametric bootstrappingmethod [40, 41]. Note that the assumption that the initial

extreme values are actually generated with good approxima-tion fromaGumbel distribution cannot easily be verifiedwithany accuracy in general, which is a drawback of this method.Comparedwith the POTmethod, the Gumbelmethodwouldalso seem to use much less of the information available inthe data. This may explain why the POT method has becomeincreasingly popular over the past years, but the Gumbelmethod is still widely used in practice.

7. The Peaks-over-Threshold Method

7.1.TheGeneralized ParetoDistribution. ThePOTmethod forindependent data is based on what is called the generalizedPareto (GP) distribution (defined below) in the followingmanner: it has been shown in [42] that asymptotically theexcess values above a high level will follow a GP distributionif and only if the parent distribution belongs to the domainof attraction of one of the extreme value distributions. Theassumption of a Poisson process model for the exceedancetimes combined with GP distributed excesses can be shownto lead to the generalized extreme value (GEV) distributionfor corresponding extremes, see below.The expression for theGP distribution is

() = (; , ) = Prob ( ) = 1 (1 +

)

1/

+

.

(37)

Here > 0 is a scale parameter and ( < < ) deter-mines the shape of the distribution. ()

+= max(0, ).

The asymptotic result referred to above implies that(37) can be used to represent the conditional cumulativedistribution function of the excess = of the observedvariates over the threshold , given that > for issufficiently large [42]. The cases > 0, = 0, and < 0correspond to Frechet (Type II), Gumbel (Type I), andreverseWeibull (Type III) domains of attraction, respectively,compared with section below.

For = 0, which corresponds to the Gumbel extremevalue distribution, the expression between the parenthesesin (37) is understood in a limiting sense as the exponentialdistribution

() = (; , 0) = exp(

) . (38)

Since the recorded data in practice are rarely indepen-dent, a declustering technique is commonly used to filter thedata to achieve approximate independence [1, 2].

7.2. Return Periods. The return period of a given value

of in terms of a specified length of time , for example,a year, is defined as the inverse of the probability that thespecified value will be exceeded in any time interval of length. If denotes the mean exceedance rate of the threshold per length of time (i.e., the average number of data pointsabove the threshold per ), then the return period of thevalue of corresponding to the level

= + is given by

the relation

=1

Prob ( > )=

1

Prob ( > ). (39)


Hence, it follows that

Prob ( ) = 1 1(). (40)

Invoking (1) for = 0 leads to the result

=

[1 ()]

. (41)

Similarly, for = 0, it is found that,

= + ln () , (42)

where is the threshold used in the estimation of and .

8. Extreme Value Prediction for Synthetic Data

In this section, we illustrate the performance of the ACERmethod and also the 95%CI estimation.We consider 20 yearsof synthetic wind speed data, amounting to 2000 data points,which is not much for detailed statistics. However, this casemay represent a real situationwhen nothing but a limited datasample is available. In this case, it is crucial to provide extremevalue estimates utilizing all data available. As we will see, thetail extrapolation technique proposed in this paper performsbetter than asymptotic methods such as POT or Gumbel.

The extreme value statistics will first be analyzed byapplication to synthetic data for which the exact extremevalues can be calculated [43]. In particular, it is assumedthat the underlying (normalized) stochastic process () isstationary and Gaussian with mean value zero and standarddeviation equal to one. It is also assumed that the mean zeroup-crossing rate +(0) is such that the product +(0) = 103,where = 1 year, which seems to be typical for the windspeed process.Using the Poisson assumption, the distributionof the yearly extreme value of () is then calculated by theformula

1 yr() = exp {+ () } = exp{103 exp(

2

2)} ,

(43)

where = 1 year and +() is the mean up-crossing rate peryear, is the scaled wind speed. The 100-year return periodvalue 100 yr is then calculated from the relation1 yr(100 yr) =1 1/100, which gives 100 yr = 4.80.

The Monte Carlo simulated data to be used for thesynthetic example are generated based on the observationthat the peak events extracted from measurements of thewind speed process, are usually separated by 3-4 days. This isdone to obtain approximately independent data, as requiredby the POTmethod. In accordance with this, peak event dataare generated from the extreme value distribution

3 d() = exp{ exp(

2

2)} , (44)

where = +(0) = 10, which corresponds to = 3.65 days,and 1 yr() = (3 d())100.

Since the data points (i.e., = 3.65 days maxima) areindependent,

() is independent of . Therefore, we put =

1. Since we have 100 data from one year, the data amounts to2000 data points. For estimation of a 95% confidence intervalfor each estimated value of the ACER function

1() for the

chosen range of -values, the required standard deviation in(22) was based on 20 estimates of the ACER function usingthe yearly data. This provided a 95% confidence band onthe optimally fitted curve based on 2000 data. From thesedata, the predicted 100-year return level is obtained from1(100 yr) = 10

4. A nonparametric bootstrapping methodwas also used to estimate a 95% confidence interval based on1000 resamples of size 2000.

The POT prediction of the 100-year return level wasbased on using maximum likelihood estimates (MLE) of theparameters in (37) for a specific choice of threshold. The95% confidence interval was obtained from the parametri-cally bootstrapped PDF of the POT estimate for the giventhreshold. A sample of 1000 data sets was used. One of theunfortunate features of the POTmethod is that the estimated100 year value may vary significantly with the choice ofthreshold. So also for the synthetic data.We have followed thestandard recommended procedures for identifying a suitablethreshold [1].

Note that in spite of the fact that the true asymptoticdistribution of exceedances is the exponential distribution in(38), the POT method used here is based on adopting (37).The reason is simply that this is the recommended procedure[1], which is somewhat unfortunate but understandable.The reason being that the GP distribution provides greaterflexibility in terms of curve fitting. If the correct asymptoticdistribution of exceedances had been used on this example,poor results for the estimated return period values would beobtained.The price to pay for using theGP distribution is thatthe estimated parametersmay easily lead to an asymptoticallyinconsistent extreme value distribution.

The 100-year return level predicted by the Gumbel meth-od was based on using themethod of moments for parameterestimation on the sample of 20 yearly extremes. This choiceof estimation method is due to the small sample of extremevalues. The 95% confidence interval was obtained from theparametrically bootstrapped PDF of the Gumbel prediction.This was based on a sample of size 10,000 data sets of 20 yearlyextremes. The results obtained by the method of momentswere compared with the corresponding results obtained byusing the maximum likelihood method. While there wereindividual differences, the overall picture was one of verygood agreement.

In order to get an idea about the performance of theACER, POT, and Gumbel methods, 100 independent 20-year MC simulations as discussed above were done. Table 1compares predicted values and confidence intervals for aselection of 10 cases together with average values over the 100simulated cases. It is seen that the average of the 100 predicted100-year return levels is slightly better for the ACER methodthan for both the POT and the Gumbel methods. But moresignificantly, the range of predicted 100-year return levels bythe ACER method is 4.345.36, while the same for the POTmethod is 4.195.87 and for the Gumbel method is 4.415.71.


Table 1: 100-year return level estimates and 95% CI (BCI = CI by bootstrap) for A = ACER, G = Gumbel, and P = POT. Exact value = 4.80.

Sim. No. A 100 ACI ABCI G 100 GBCI P 100 PBCI1 5.07 (4.67, 5.21) (4.69, 5.42) 4.41 (4.14, 4.73) 4.29 (4.13, 4.52)10 4.65 (4.27, 4.94) (4.37, 5.03) 4.92 (4.40, 5.58) 4.88 (4.42, 5.40)20 4.86 (4.49, 5.06) (4.44, 5.19) 5.04 (4.54, 5.63) 5.04 (4.48, 5.74)30 4.75 (4.22, 5.01) (4.33, 5.02) 4.75 (4.27, 5.32) 4.69 (4.24, 5.26)40 4.54 (4.20, 4.74) (4.27, 4.88) 4.80 (4.31, 5.39) 4.73 (4.19, 5.31)50 4.80 (4.35, 5.05) (4.42, 5.14) 4.91 (4.41, 5.50) 4.79 (4.31, 5.34)60 4.84 (4.36, 5.20) (4.48, 5.19) 4.85 (4.36, 5.43) 4.71 (4.32, 5.23)70 5.02 (4.47, 5.31) (4.62, 5.36) 4.96 (4.47, 5.53) 4.97 (4.47, 5.71)80 4.59 (4.33, 4.81) (4.38, 4.98) 4.76 (4.31, 5.31) 4.68 (4.15, 5.27)90 4.84 (4.49, 5.11) (4.60, 5.30) 4.77 (4.34, 5.32) 4.41 (4.23, 4.64)100 4.62 (4.29, 5.05) (4.45, 5.09) 4.79 (4.31, 5.41) 4.53 (4.05, 4.88)Av. 100 4.82 (4.41, 5.09) (4.48, 5.18) 4.84 (4.37, 5.40) 4.72 (4.27, 5.23)

Hence, in this case the ACER method performs consistentlybetter than both these methods. It is also observed from theestimated 95% confidence intervals that theACERmethod, asimplemented in this paper, provides slightly higher accuracythan the other two methods. Lastly, it is pointed out that theconfidence intervals of the 100-year return level estimatedby the ACER method obtained by either the simplifiedextrapolated confidence band approach or by nonparametricbootstrapping are very similar, except for a slight mean shift.As a final comparison, the 100 bootstrapped confidence inter-vals obtained for the ACER and Gumbel methods missedthe target value three times, while for the POT method thisnumber was 18.

An example of the ACER plot and results obtained forone set of data is presented in Figure 1.The predicted 100-yearvalue is 4.85 with a predicted 95% confidence interval (4.45,5.09). Figure 2 presents POT predictions based on MLE fordifferent thresholds in terms of the number of data pointsabove the threshold. The predicted value is 4.7 at = 204,while the 95% confidence interval is (4.25, 5.28). The samedata set as in Figure 1 was used. This was also used for theGumbel plot shown in Figure 3. In this case the predictedvalue based on themethod ofmoments (MM) is 100 yrMM = 4.75with a parametric bootstrapped 95% confidence interval of(4.34, 5.27). Prediction based on the Gumbel-Lieblein BLUEmethod (GL), compared with for example, Cook [44], is100 yrGL = 4.73with a parametric bootstrapped 95% confidenceinterval equal to (4.35, 5.14).

9. Measured Wind Speed Data

In this section, we analyze real wind speed data, measuredat two weather stations off the coast of Norway: at Nordyanand at Hekkingen, see Figure 4. Extreme wind speed predic-tion is an important issue for design of structures exposed tothe weather variations. Significant efforts have been devotedto the problemof predicting extremewind speeds on the basisof measured data by various authors over several decades,see, for example, [4548] for extensive references to previouswork.

2.5 3 3.5 4 4.5 5 5.5

4.85

101

102

103

104

105

ACER1(

)

Figure 1: Synthetic data ACER 1, Monte Carlo simulation ();

optimized curve fit (); empirical 95% confidence band (- -);optimized confidence band ( ). Tail marker

1= 2.3.

Hourly maximum gust wind was recorded during the13 years 19992012 at Nordyan and the 14 years 19982012 at Hekkingen. The objective is to estimate a 100-yearwind speed. Variation in the wind speed caused by seasonalvariations in the wind climate during the year makes thewind speed a non-stationary process on the scale of months.Moreover, due to global climate change, yearly statistics mayvary on the scale of years. The latter is, however, a slowprocess, and for the purpose of long-term prediction weassume here that within a time span of 100 years a quasi-stationary model of the wind speeds applies. This may not beentirely true, of course.

9.1. Nordyan. Figure 5 highlights the cascade of ACERestimates

1, . . . ,

96, for the case of 13 years of hourly data

recorded at theNordyanweather station.Here, 96is consid-

ered to represent the final converged results. By converged,we mean that

96 for > 96 in the tail, so that there is no


100 130 160 190 204 230 250

4.6

4.62

4.64

4.66

4.68

4.7

4.72

4.74

4.76

4.7

100

yr

Figure 2: The point estimate 100 yr of the 100-year return periodvalue based on 20 years synthetic data as a function of the number of data points above threshold. The return level estimate = 4.7 at = 204.

3.6 3.8 4 4.2 4.4 4.6 4.8

0

1

2

3

4

5

4.751

ln

(ln((+1)/))

Figure 3: The point estimate 100 yr of the 100-year return periodvalue based on 20 years synthetic data. Lines are fitted by themethod of momentssolid line () and the Gumbel-LiebleinBLUE methoddash-dotted lite (- -). The return level estimateby the method of moments is 4.75, by the Gumbel-Lieblein BLUEmethod is 4.73.

need to consider conditioning of an even higher order than96. Figure 5 reveals a rather strong statistical dependencebetween consecutive data, which is clearly reflected in theeffect of conditioning on previous data values. It is also inter-esting to observe that this effect is to some extent capturedalready by

2, that is, by conditioning only on the value of the

previous data point. Subsequent conditioning on more thanone previous data point does not lead to substantial changesin ACER values, especially for tail values. On the other hand,to bring out fully the dependence structure of these data, itwas necessary to carry the conditioning process to (at least)the 96th ACER function, as discussed above.

However, from a practical point of view, the most impor-tant information provided by theACERplot of Figure 5 is that

Hekkingen Fyr 88690

Nordyan Fyr 75410

Figure 4: Wind speed measurement stations.

3.5 4 4.5 5 5.5 6 6.5 7

101

102

103

104

= 1 = 2

= 4 = 24

= 48

= 72

= 96

/

ACER(

)

Figure 5: Nordyan wind speed statistics, 13 years hourly data.Comparison between ACER estimates for different degrees of con-ditioning. = 6.01m/s.

for the prediction of a 100-year value, one may use the firstACER function.The reason for this is that Figure 5 shows thatall the ACER functions coalesce in the far tail. Hence, wemayuse any of the ACER functions for the prediction. Then, theobvious choice is to use the first ACER function, which allowsus to use all the data in its estimation and thereby increaseaccuracy.

In Figure 6 is shown the results of parametric estimationof the return value and its 95% CI for 13 years of hourly


3 4 5 6 7 8 9

8.62

101

102

103

104

106

105

/

ACER1(

)

Figure 6: Nordyanwind speed statistics, 13 years hourly data. 1()

(); optimized curve fit (); empirical 95% confidence band (- -);optimized confidence band ( ). Tail marker

1= 12.5m/s = 2.08

( = 6.01m/s).

100 120 140 161 180 2007.6

7.7

7.8

7.9

8

8.1

7.95

100

yr/

Figure 7:The point estimate 100 yr of the 100-year return level basedon 13 years hourly data as a function of the number of data pointsabove threshold. = 6.01m/s.

maxima. The predicted 100-year return speed is 100 yr =51.85m/s with 95% confidence interval (48.4, 53.1). = 13years of data may not be enough to guarantee (22), since werequired 20. Nevertheless, for simplicity, we use it hereeven with = 13, accepting that it may not be very accurate.

Figure 7 presents POT predictions for different thresholdnumbers based on MLE. The POT prediction is 100 yr =47.8m/s at threshold = 161, while the bootstrapped 95%confidence interval is found to be (44.8, 52.7) m/s basedon 10,000 generated samples. It is interesting to observe theunstable characteristics of the predictions over a range ofthreshold values, while they are quite stable on either side ofthis range giving predictions that are more in line with theresults from the other two methods.

Figure 8 presents a Gumbel plot based on the 13 yearlyextremes extracted from the 13 years of hourly data. The

6 6.5 7 7.5 8 8.5 9 9.5

0

1

2

3

4

5

1

/

ln

(ln((+1)/))

8.56 9.23

Figure 8: Nordyan wind speed statistics, 13 years of hourlydata. Gumbel plot of yearly extremes. Lines are fitted by themethod of momentssolid line () and the Gumbel-LiebleinBLUE methoddash-dotted lite (- -). = 6.01m/s.

Table 2: Predicted 100-year return period levels for NordyanFyr weather station by the ACER method for different degrees ofconditioning, annual maxima, and POT methods, respectively.

Method Spec 100 yr, m/s 95% CI (100 yr), m/s

ACER, various

1 51.85 (48.4, 53.1)2 51.48 (46.1, 54.1)4 52.56 (46.7, 55.7)24 52.90 (47.0, 56.2)48 54.62 (47.7, 57.6)72 53.81 (46.9, 58.3)96 54.97 (47.5, 60.5)

Annual maxima MM 51.5 (45.2, 59.3)GL 55.5 (48.0, 64.9)

POT 47.8 (44.8, 52.7)

Gumbel prediction based on the method of moments (MM)is 100 yrMM = 51.5m/s, with a parametric bootstrapped 95%confidence interval equal to (45.2, 59.3) m/s, while predictionbased on the Gumbel-Lieblein BLUEmethod (GL) is 100 yrGL =55.5m/s, with a parametric bootstrapped 95% confidenceinterval equal to (48.0, 64.9) m/s.

In Table 2 the 100-year return period values for theNordyan station are listed together with the predicted 95%confidence intervals for all methods.

9.2. Hekkingen. Figure 9 shows the cascade of estimatedACER functions

1, . . . ,

96for the case of 14 years of hourly

data. As for Nordyan, 96

is used to represent the finalconverged results. Figure 9 also reveals a rather strong sta-tistical dependence between consecutive data at moderatewind speed levels.This effect is again to some extent capturedalready by

2, so that subsequent conditioning on more than

one previous data point does not lead to substantial changesin ACER values, especially for tail values.


4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

104

103

102

/

= 1

= 2

= 4

= 24

= 48

= 72

= 96

ACER(

)

Figure 9: Hekkingen wind speed statistics, 14 years hourly data.Comparison between ACER estimates for different degrees of con-ditioning. = 5.72m/s.

5 6 7 8 9 10 11 12

10.6

106

105

104

103

102

/

ACER1(

)

Figure 10: Hekkingen wind speed statistics, 14 years hourly data.1() (); optimized curve fit (); empirical 95% confidence band

(- -); optimized confidence band ( ). Tail marker 1= 23m/s =

4.02 ( = 5.72m/s).

Also, for the Hekkingen data, the ACER plot of Figure 9indicates that the ACER functions coalesce in the far tail.Hence, for the practical prediction of a 100-year value, onemay use the first ACER function.

In Figure 10 is shown the results of parametric estimationof the return value and its 95% CI for 14 years of hourlymaxima. The predicted 100-year return speed is 100 yr =60.47m/s with 95% confidence interval (53.1, 64.9). Equation(22) has been used also for this example with = 14.

Figure 11 presents POT predictions for different thresholdnumbers based on MLE. The POT prediction is 100 yr =53.48m/s at threshold = 183, while the bootstrapped

Table 3: Predicted 100-year return period levels for NordyanFyr weather station by the ACER method for different degrees ofconditioning, annual maxima, and POT methods, respectively.

Method Spec 100 yr, m/s 95% CI (100 yr), m/s

ACER, various

1 60.47 (53.1, 64.9)

2 62.23 (53.3, 70.0)

4 63.03 (53.0, 74.5)

24 60.63 (51.3, 70.7)

48 60.44 (51.3, 77.0)

72 58.06 (51.2, 66.4)

96 59.19 (52.0, 68.3)

Annual maxima MM 58.10 (50.8, 67.3)

GL 60.63 (53.0, 70.1)

POT 53.48 (48.9, 57.0)

100 140 185 220 260 300

9.26

9.3

9.34

9.38

9.42

9.35

100

yr/

Figure 11: The point estimate 100 yr of the 100-year return levelbased on 14 years hourly data as a function of the number of datapoints above threshold. = 5.72m/s.

95% confidence interval is found to be (48.9, 57.0) m/s basedon 10,000 generated samples. It is interesting to observethe unstable characteristics of the predictions over a range ofthreshold values, while they are quite stable on either side ofthis range giving predictions that are more in line with theresults from the other two methods.

Figure 12 presents a Gumbel plot based on the 14 yearlyextremes extracted from the 14 years of hourly data. TheGumbel prediction based on the method of moments (MM)is 100 yrMM = 58.10m/s, with a parametric bootstrapped 95%confidence interval equal to (50.8, 67.3)m/s. Prediction basedon the Gumbel-Lieblein BLUE method (GL) is 100 yrGL =60.63m/s, with a parametric bootstrapped 95% confidenceinterval equal to (53.0, 70.1) m/s.

In Table 3, the 100-year return period values for theHekkingen station are listed together with the predicted 95%confidence intervals for all methods.


ln(ln((+1)/))

7 7.5 8 8.5 9 9.5 10 10.5

0

1

2

3

4

5

10.2 10.6

/

1

Figure 12: Hekkingen wind speed statistics, 14 years of hourlydata. Gumbel plot of yearly extremes. Lines are fitted by themethod of momentssolid line () and the Gumbel-LiebleinBLUE methoddash-dotted lite (- -). = 5.72m/s.

10. Extreme Value Prediction for a NarrowBand Process

In engineering mechanics, a classical extreme response pre-diction problem is the case of a lightly damped mechani-cal oscillator subjected to random forces. To illustrate thisprediction problem, we will investigate the response pro-cess of a linear mechanical oscillator driven by a Gaussianwhite noise. Let () denote the displacement response; thedynamic model can then be expressed as, () + 2

() +

2

() = (), where = relative damping,

= undamped

eigenfrequency, and()= a stationaryGaussianwhite noise(of suitable intensity). By choosing a small value for , theresponse time series will exhibit narrow band characteristics,that is, the spectral density of the response process ()will assume significant values only over a narrow rangeof frequencies. This manifests itself by producing a strongbeating of the response time series, which means that the sizeof the response peakswill change slowly in time, see Figure 13.A consequence of this is that neighbouring peaks are stronglycorrelated, and there is a conspicuous clumping of the peakvalues. Hence the problemwith accurate prediction, since theusual assumption of independent peak values is then violated.

Many approximations have been proposed to deal withthis correlation problem, but no completely satisfactorysolution has been presented. In this section, we will showthat the ACER method solves this problem efficiently andelegantly in a statistical sense. In Figure 14 are shown someof the ACER functions for the example time series. It maybe verified from Figure 13 that there are approximately 32sample points between two neighbouring peaks in the timeseries. To illustrate a point, we have chosen to analyze thetime series consisting of all sample points.Usually, in practice,only the time series obtained by extracting the peak valueswould be used for the ACER analysis. In the present case,the first ACER function is then based on assuming that all

60 70 80 90 100 110 120

0

0.5

1

1.5

2

2.5

Time (s)

2.5

2

1.5

1

0.5

(

)

Figure 13: Part of the narrow-band response time series of the linearoscillator with fully sampled and peak values indicated.

the sampled data points are independent, which is obviouslycompletely wrong. The second ACER function, which isbased on counting each exceedance with an immediatelypreceding nonexceedance, is nothing but an upcrossing rate.Using this ACER function is largely equivalent to assumingindependent peak values. It is now interesting to observethat the 25th ACER function can hardly be distinguishedfrom the second ACER function. In fact, the ACER functionsafter the second do not change appreciably until one starts toapproach the 32nd, which corresponds to hitting the previouspeak value in the conditioning process. So, the importantinformation concerning the dependence structure in thepresent time series seems to reside in the peak values, whichmay not be very surprising. It is seen that the ACER functionsshow a significant change in value as a result of accountingfor the correlation effects in the time series. To verify thefull dependence structure in the time series, it is necessaryto continue the conditioning process down to at least the64th ACER function. In the present case, there is virtuallyno difference between the 32nd and the 64th, which showsthat the dependence structure in this particular time series iscaptured almost completely by conditioning on the previouspeak value. It is interesting to contrast the method of dealingwith the effect of sampling frequency discussed here with thatof [49].

To illustrate the results obtained by extracting only thepeak values from the time series, which would be theapproach typically chosen in an engineering analysis, theACER plots for this case is shown in Figure 15. By comparingresults from Figures 14 and 15, it can be verified that theyare in very close agreement by recognizing that the secondACER function in Figure 14 corresponds to the first ACERfunction in Figure 15, and by noting that there is a factor ofapproximately 32 between corresponding ACER functions inthe two figures. This is due to the fact that the time series ofpeak values contains about 32 times less data than the originaltime series.


1.5 2 2.5 3 3.5 4

101

102

103

104

105

= 1 = 2 = 25

= 32 = 64

/

ACER(

)

Figure 14: Comparison between ACER estimates for differentdegrees of conditioning for the narrow-band time series.

1 1.5 2 2.5 3 3.5 4

103

102

101

/

= 1

= 2

= 4

= 5 = 3

ACER(

)

Figure 15: Comparison between ACER estimates for differentdegrees of conditioning based on the time series of the peak values,compared with Figure 13.

11. Concluding Remarks

This paper studies a newmethod for extreme value predictionfor sampled time series.Themethod is based on the introduc-tion of a conditional average exceedance rate (ACER), whichallows dependence in the time series to be properly and easilyaccounted for. Declustering of the data is therefore avoided,and all the data are used in the analysis. Significantly, theproposed method also aims at capturing to some extent thesubasymptotic form of the extreme value distribution.

Results for wind speeds, both synthetic and measured,are used to illustrate the method. An estimation problem

related to applications in mechanics is also presented. Thevalidation of the method is done by comparison with exactresults (when available), or other widely used methods forextreme value statistics, such as the Gumbel and the peaks-over-threshold (POT) methods. Comparison of the variousestimates indicate that the proposed method provides moreaccurate results than the Gumbel and POT methods.

Subject to certain restrictions, the proposed method alsoapplies to nonstationary time series, but it cannot directlypredict for example, the effect of climate change in the formof long-term trends in the average exceedance rates extendingbeyond the data. This must be incorporated into the analysisby explicit modelling techniques.

As a final remark, it may be noted that the ACERmethodas described in this paper has a natural extension to higherdimensional distributions. The implication is that, it is thenpossible to provide estimates of for example, the exact bivari-ate extreme value distribution for a suitable set of data [50].However, as is easily recognized, the extrapolation problem isnot as simply dealt with as for the univariate case studied inthis paper.

Acknowledgment

This work was supported by the Research Council of Norway(NFR) through the Centre for Ships and Ocean Structures(CeSOS) at the Norwegian University of Science and Tech-nology.

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Applied Ocean Research 28 (2006) 18www.elsevier.com/locate/apor

Numerical methods for calculating the crossing rate of high and extremeresponse levels of compliant offshore structures subjected to random waves

A. Naessa,b,, H.C. Karlsenb, P.S. Teigenc

aCentre for Ships and Ocean Structures, Norwegian University of Science and Technology, A. Getz vei 1, NO-7491, Trondheim, NorwaybDepartment of Mathematical Sciences, Norwegian University of Science and Technology, A. Getz vei 1, NO-7491, Trondheim, Norway

c Statoil Research Centre, Rotvoll, Trondheim, Norway

Received 16 December 2005; received in revised form 3 April 2006; accepted 8 April 2006Available online 5 June 2006

Abstract

The focus of the paper is on methods for calculating the mean upcrossing rate of stationary stochastic processes that can be represented assecond order stochastic Volterra series. This is the current state-of-the-art representation of the horizontal motion response of e.g. a tension legplatform in random seas. Until recently, there has been no method available for accurately calculating the mean level upcrossing rate of suchresponse processes. Since the mean upcrossing rate is a key parameter for estimating the large and extreme responses it is clearly of importanceto develop methods for its calculation. The paper describes in some detail a numerical method for calculating the mean level upcrossing rateof a stochastic response process of the type considered. Since no approximations are made, the only source of inaccuracy is in the numericalcalculation, which can be controlled. In addition to this exact method, two approximate methods are also discussed.c 2006 Elsevier Ltd. All rights reserved.Keywords: Second order stochastic Volterra model; Mean crossing rate; Extreme response; Slow drift response; Method of steepest descent

1. Introduction

The problem of calculating the extreme response ofcompliant offshore structures like tension leg platforms ormoored spar buoys in random seas, has been a challenge formany years, and, in fact, it still represents a challenge. Startingwith the state-of-the-art representation of the horizontalexcursions of moored, floating offshore structures in randomseas as a second order stochastic Volterra series, we shall inthis paper develop a general method for estimating the extremeresponse of such structures. Even if the Volterra series modelwas formulated more than 30 years ago, it is not until quiterecently that general numerical methods have become availablethat allow accurate calculation of the probability distributionand, perhaps more importantly, the mean upcrossing rate ofthe total response process. This last quantity is the crucialparameter for estimating extreme responses.

During the 1980s significant efforts were directed towardsdeveloping methods for calculating the response statistics of

Corresponding author. Tel.: +47 73 59 70 53; fax: +47 73 59 35 24.E-mail address: [email protected] (A. Naess).

compliant offshore structures subjected to random waves. Thelist of contributions is long. To mention but a few, whichalso contain references to other work focussing on responsestatistics, see [16]. However, none of these works succeeded indeveloping a general method that made it possible to calculatethe exact statistical distribution of the total response process,not to mention the much harder problem of calculating the meanupcrossing rate.

A general method for solving the first problem of calculatingthe exact statistical distribution was presented in [7]. Then ittook almost a decade before a general method for solving thesecond problem was outlined [8]. This method was developedfurther in [9]. While the method is mathematically sound, initialefforts to carry out the requisite calculations have revealed thatsome care is needed in setting up the numerical algorithms.The work presented in this paper is part of continued effortsto set up a robust and accurate numerical procedure. It shouldbe emphasized that while the discussion in this paper is limitedto the long-crested seas case, the method described also coversthe short-crested seas case, cf. [6]. A recent paper presentinga discussion and comparison of some approximate proceduresfor calculating the mean upcrossing rate is notable [10].

0141-1187/$ - see front matter c 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.apor.2006.04.001

2 A. Naess et al. / Applied Ocean Research 28 (2006) 18

2. The response process

The response process Z(t) that is considered here is assumedto be represented as a second order stochastic Volterra series.This would apply to the state of the art representation ofe.g. the surge response of a large volume, compliant offshorestructure in random waves. This response would consist of acombination of the wave frequency component Z1(t) and theslow-drift component Z2(t), that is, Z(t) = Z1(t) + Z2(t).Naess [6] describes the standard representation of the tworesponse components leading to a second order Volterra seriesmodel for the total response. To alleviate the statistical analysisof the response process, it has been shown [2,6] that the slow-drift response Z2(t) can be expressed as

Z2(t) =Nj=1

j {W2 j1(t)2 +W2 j (t)2}. (1)

Here W j (t), j = 1, . . . , 2N are real stationary GaussianN (0, 1)-processes. The coefficients j are obtained by solvingthe eigenvalue problem (assumed nonsingular)

Qu j = ju j (2)to find the eigenvalues j and orthonormal eigenvectors u j ,j = 1, . . . , N , of the N N -matrix Q = (Qi j ), where

Qi j = H2(i , j ) 12 [SX (i )SX ( j )]1/21. (3)

Here H2(, ) denotes the quadratic transfer function betweenthe waves and the surge response, cf. [2,6], SX () denotesthe one-sided spectral density of the waves, and 0 < 1 < < N is a suitable discretization of the frequency axis.The stochastic processes W j (t) can be represented as follows(i = 1)

W2 j1(t)+ iW2 j (t) =2

Nk=1

u j (k)Bkeik t (4)

where u j (k) denotes the kth component of u j and {Bk}is a set of independent, complex Gaussian N (0, 1)-variableswith independent and identically distributed real and imaginaryparts. The representation can be arranged so that W2 j (t)becomes the Hilbert transform of W2 j1(t), cf. [6]. For eachfixed t , {W j (t)} becomes a set of independent Gaussianvariables.

Having achieved the desired representation of the quadraticresponse Z2(t), it can then be shown that the linear responsecan be expressed as

Z1(t) =2Nj=1

jW j (t). (5)

The (real) parameters j are given by the relations

2 j1 + i2 j =N

k=1H1(k)

SX (k)1 u j (k) (6)

where H1() denotes the linear transfer function between thewaves and the surge response. Based on the representationsgiven by Eqs. (1) and (5), [11] describes how to calculatethe statistical moments of the response process Z(t), whilea general and accurate numerical method for calculating thePDF of Z(t) is given in [7]. However, for important predictionpurposes, like extreme response estimation, the crucial quantityis the mean rate of level upcrossings by the response process.

3. The mean crossing rate

Let N+Z ( ) denote the rate of upcrossings of the level by Z(t), cf. [12], and let +Z ( ) = E[N+Z ( )], that is,+Z ( ) denotes the mean rate of upcrossings of the level . Asdiscussed in [9], under suitable regularity conditions on theresponse process, which can be adopted here, the followingformula can be used

+Z ( ) = 0

s fZ Z (, s)ds (7)

where fZ Z (, ) denotes the joint PDF of Z(0) and Z(0) =dZ(t)/dt |t=0. Eq. (7) is often referred to as the Riceformula [13]. +Z ( ) is assumed throughout to be finite.

Calculating the mean crossing rate of a stochastic processrepresented as a second order stochastic Volterra series directlyfrom Eq. (7) has turned out to be very difficult due to thedifficulties of calculating the joint PDF fZ Z (, ). However, thiscan be circumvented by invoking the concept of characteristicfunction.

Denote the characteristic function of the joint variable(Z , Z) by MZ Z (, ), or, for simplicity of notation, by M(, ).Then

M(u, v) = MZ Z (u, v) = E[exp(iuZ + iv Z)]. (8)Assuming that M(, ) is an integrable function, that is,M(, ) L1(R2), it follows that

fZ Z (, s) =1

(2pi)2

M(u, v)

exp (iu ivs) dudv. (9)By substituting from Eq. (9) back into Eq. (7), the meancrossing rate is formally expressed in terms of the characteristicfunction, but this is not a very practical expression.

The solution to this is obtained by considering thecharacteristic function as a function of two complex variables.It can then often be shown that this new function becomesholomorphic in suitable regions of C2, where C denotes thecomplex plane. As shown in detail in [14], under suitableconditions, the use of complex function theory allows thederivation of two alternative expressions for the crossing rate.Here we shall focus on one of these alternatives, viz.

+Z ( ) = 1

(2pi)2

iaia

ibib

1

w2M(z, w)eizdzdw (10)

where 0 < a < a1 for some positive constant a1, and b0 < b 0.

A. Naess et al. / Applied Ocean Research 28 (2006) 18 3

To actually carry out the calculations, the joint characteristicfunction needs to be known. It has been shown [8] that forthe second order stochastic Volterra series, it can be givenin closed form. To this end, consider the multidimensionalGaussian vectors W = (W1, . . . ,Wn) ( denotes transposition)and W = (W1, . . . , Wn), where n = 2N . It is obtained that thecovariance matrix of (W , W ) is given by

=(11 1221 22

)(11)

where 11 = I = the n n identity matrix, 12 =(ri j ) = (E[Wi W j ]), 21 = (E[WiW j ]) and 22 = (si j ) =(E[Wi W j ]); i, j = 1, . . . , n. ri j = r j i and 12 = 21. Itfollows from Eq. (4), that the entries of the covariance matrix can be expressed in terms of the eigenvectors u j , cf. [2]. Let

Ri j =N

k=1(ik)ui (k)u j (k). (12)

Then it can be shown that

r2i1,2 j1 = r2i,2 j = R(Ri j ) (13)while

r2i1,2 j = r2i,2 j1 = =(Ri j ) (14)whereR(z) denotes the real part and =(z) the imaginary part ofz. Similarly, let

Si j =N

k=12kui (k)u j (k)

. (15)

Then

s2i1,2 j1 = s2i,2 j = R(Si j ) (16)while

s2i1,2 j = s2i,2 j1 = =(Si j ). (17)By this, the covariance matrix is completely specified.

It is convenient to introduce a new set of eigenvalues j ,j = 1, . . . , n defined by 2i1 = 2i = i , i = 1, . . . , N .Let = diag(1, . . . , n) be the diagonal matrix with theparameters j on the diagonal, and let = (1, . . . , n). Itcan now be shown that [8]

M(u, v)

= exp{12ln(det(A)) 1

2v2 V + 1

2t A1 t

}(18)

where

A = I 2 i u 2 i v (21 + 12 )+ 4 v2 V (19)V = 22 21 12 (20)t =

(i u I + i v12 2 v2 V

). (21)

4. Numerical calculation

Previous efforts to carry out numerical calculation of themean crossing rate using Eq. (10) have been reported in [9].These initial investigations indicated that the method had thepotential to provide very accurate numerical results. We shallrewrite Eq. (10) as follows

+Z ( ) = 1

(2pi)2

iaia

1

w2I (, w)dw (22)

where

I = I (, w) = ibib

M(z, w) eizdz

= ibib

exp{iz + lnM(z, w)}dz. (23)

A numerical calculation of the mean crossing rate can start bycalculating the function I (, w) for specified values of andw. However, a direct numerical integration of Eq. (23) is madedifficult by the oscillatory term exp{iR(z) }. This problemcan be avoided by invoking the method of steepest descent, alsocalled the saddle point method. For this purpose, we write

g(z) = g(z;w) = iz + lnM(z, w)= (x, y)+ i(x, y) (24)

where z = x + iy. (x, y) and (x, y) become real harmonicfunctions when g(z) is holomorphic. The idea is to identifythe saddle point of the surface (x, y) (x, y) closest tothe integration line from ib to ib. By shifting thisintegration line to a new integration contour that passes throughthe saddle point, and then follows the path of steepest descentaway from the saddle point, it can be shown that the function(x, y) stays constant, and therefore the oscillatory term in theintegral degenerates to a constant. This is a main advantage ofthe method of steepest descent for numerical calculations. Itcan be shown that the integral does not change its value as longas the function g(z) is a holomorphic function in the regionbounded by the two integration contours and if the integralsvanish along the contour segments required to close the region.

If zs denotes the identified saddle point, where g(zs) =0, the steepest descent path away from the saddle point willfollow the direction given by g(z), for z 6= zs , cf. [15].Typically, the singular points of the function g will be aroundthe imaginary axis, which indicates that the direction of thepaths of steepest descent emanating from the saddle point willtypically not deviate substantially from a direction orthogonalto the imaginary axis. This provides a guide for setting up anumerical integration procedure based on the path of steepestdescent. First the saddle point zs is identified. Then the path ofsteepest descent starting at zs and going right is approximatedby the sequence of points {z j }j=0 calculated as follows:z0 = zs z1 = zs + h (25)

1z j = g(z j )

|g(z j )| h, j = 1, 2, . . . (26)z j+1 = z j +1z j , j = 1, 2, . . . (27)where h is a small positive constant.

4 A. Naess et al. / Applied Ocean Research 28 (2006) 18

Similarly, the path of steepest descent going left isapproximated by the sequence {z j }j=0 calculated byz1 = zs h (28)

1z j = g(z j )

|g(z j )| h, j = 1,2, . . . (29)z j1 = z j +1z j , j = 1,2, . . . . (30)

A numerical estimate I of I can be obtained as follows.

I = I+ + I (31)where

I+ = h2exp{g(zs)} +

Kj=1

1z j exp{g(z j )} (32)

and

I = h2exp{g(zs)}

Kj=1

1z j exp{g(z j )} (33)

for a suitably large integer K .A numerical estimate +Z ( ) of the mean crossing rate can

now be obtained by the sum

+Z ( ) = 1

(2pi)2R

{L

j=L

1

w2j

I (, w j )1w j

}(34)

where the discretization points w j are chosen to follow thenegative real axis from a suitably large negative number up toa point at , where 0 < a, then follow a semi-circle inthe lower half plane to on the positive real axis, and finallyfollow this axis to a suitably large positive number. Since thenumerical estimate does not necessarily have an imaginary partthat is exactly equal to zero, the real part operator has beenapplied.

Generally, the CPU time required to carry out thecomputations above can be quite long, depending on the size ofthe problem, which is related to the number N of eigenvalues.It is therefore of interest to see if approximating formulasare accurate enough. The first such approximation we shallhave a look at is the Laplace approximation for the innerintegral over the saddle point [15]. The simplest version of thisapproximation, adapted to the situation at hand, leads to theresult

I = I (, w) 2pi 2g(zs ;w)

x2

exp{g(zs;w)} (35)

which can be substituted directly into Eq. (34), leading to anapproximation of +Z ( ), which is denoted by

+Z ( ).

This approximation can be exploited in the following way:(1) The full method is used for an inner interval of w-values,which contribute significantly to the integral in Eq. (22). (2) ALaplace approximation is then used in an outer interval of w-values where the contribution is less than significant. Of course,the level of significance is chosen according to some suitable

criterion. By this procedure, the CPU time was reduced bya factor of about 3. This method will be referred to as thehybrid method, and the corresponding approximation of +Z ( )is denoted by +Z ( ).

A simple approximation proposed in [16,17] is worth acloser scrutiny. It is based on the widely adopted simplifyingassumption that the displacement process is independent of thevelocity process. This leads to an alternative approximation of+Z ( ), which we shall denote by

+Z ( ). It is given by the

formula

+Z ( ) = +Z (ref)fZ ( )

fZ (ref)(36)

where fZ denotes the marginal PDF of the surge response,and ref denotes a suitable reference level, typically the meanresponse. Here, ref has been chosen as the point where fZassumes its maximum, which correspond

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Papers About Extreme Value and Crossing Levels by Arvid Naess