efficient extreme value prediction for nonlinear beam vibrations using measured random response...

Nonlinear Dynamics24: 71–101, 2001.© 2001Kluwer Academic Publishers. Printed in the Netherlands.

Efficient Extreme Value Prediction for Nonlinear Beam VibrationsUsing Measured Random Response Histories

J. F. DUNNESchool of Engineering, University of Sussex, Falmer, Brighton BN1 9QT, U.K.

M. GHANBARIGroup Lotus, Hethel, Norwich, Norfolk NR14 8EZ, U.K.

(Received: 21 March 1999; accepted: 28 October 1999)

Abstract. Predicted extreme exceedance probabilities associated with experimental measurements of highly non-linear clamped-clamped beam vibrations driven by band-limited white-noise, are compared using two differentapproaches for application to short data sets. The first approach uses response history measurements to calibratea discrete dynamic model using a Markov moment method appropriately matched to extreme value prediction viafinite element solution of the Fokker–Planck (FPK) equation. The dynamic model is obtained via the Woinowsky–Krieger equation with added empirical damping. Stationary FPK solutions are used to obtain mean crossing rates,and for the purpose of extreme value prediction, crossings are assumed to be independent. The second approachuses a Weissman type I asymptotic estimator, justified by use of the Hasofer–Wang hypothesis test. Both methodsare compared with exceedance probabilities obtained using data from ‘long’ experiments in which dependencebetween extreme values is excluded. The paper shows that by exploiting the Weissman estimator in a ‘forward’predictive mode, very accurate exceedance probabilities can be obtained from relatively small amounts of meas-ured data. The calibrated model based predictions are consistently in error as a result of non-linear coupling effectsnot included in the model – this coupling is implicitly accounted for in the Weissman predictions.

Keywords: Clamped beams, displacements, probabilities, Fokker–Planck, asymptotic theory.

1. Introduction

Forced random vibration of beam structures is of interest in several areas of dynamics, forexample in stress level prediction in aircraft panels exposed to high intensity acoustic loadingabove 120 dB. Large amplitude clamped beams, with immovable ends, vibrate in a highlynonlinear way as a result of several different mechanisms including hardening type geometriceffects on stiffness [1] and nonlinear damping [2]. Much theoretical and experimental work onfree- and forced beam vibration [3–13] has identified the relative importance of these sourcesof nonlinearity, initially for isothermal vibrations, but more recently for thermal-acousticloading [8] – of increasing importance in many practical applications. Experimental studies[6] have shown in particular, that with random type loading, the distribution of axial strain ishighly non-normal, resulting in much reduced fatigue life predictions compared with linearanalysis. Furthermore, a consequence of nonlinearity for very large amplitude studies, is thatdiscrete vibration models may be highly coupled and even if attempts are made to excite one(linear) mode of vibration, more degrees-of-freedom may be needed in the model. Theoret-ical predictions for harmonically forced beam vibrations [10] have confirmed the importanceof coupling, but with random loading, recent findings [12] suggest considerable amplitudesensitivity, where nonlinear coupling is unimportant below a certain amplitude level, but very

72 J. F. Dunne and M. Ghanbari

important above that level. Accurate prediction of large amplitude vibrations of beams withimmovable ends, therefore requires at least a well calibrated discrete dynamic model, whichultimately must be of sufficient order to include any effects of nonlinear coupling. Calibrationnecessarily requires use of measured data, especially since a good damping model is needed,and this alone cannot at present be wholly synthesised. Increasing emphasis is also beingplaced on the use of measured data for condition monitoring purposes, in particular, for on-linedetection of material damage and defects in structures. And although certain types of responsemeasures, such as power spectra, are appropriate for detection of faults in machinery, extremevalue statistics have traditionally been used for the purpose of monitoring the behaviour ofentire structures with random-type loading. There is in fact a relatively long history in theuse of extreme value statistics in design of aircraft, ship, and offshore structures [14–20]. Lessemphasis however has been placed on the use of extreme values for monitoring sub-structures,such as panels and beams, even though it is now possible to monitor many such members inparallel.

In general, the tails of the extreme value distribution function are of greatest interest[21] since much important structural dynamic information can be found there. But designexceedance probability levels are likely to be very low in the range 10−6–10−8 per-service-lifecorresponding to conservative target performance categories used for example in design ofall critical components in nuclear installations, and for safe-life design of principal structuralelements in aerospace applications. Exceedance probability levels studied in practice howeverare typically restricted to a higher range for the following reasons:

(i) Collection of experimental data at realistic failure probability levels would pose majorproblems.

(ii) Prediction in the range 10−2–10−4 is quite acceptable for condition monitoring purposes.

For structural monitoring purposes, there are typically two objectives to meet, both of whichfocus on the tails of the distribution: the first is to confidently extrapolate extreme exceedanceprobability estimates using limited amounts of measured data corresponding to some specifiedduration. It might be required for example, to extrapolate to an exceedance probability levelof say 10−3, using a few hundred samples or less, when in fact for direct measurement andconfident estimation, many thousands of samples would be needed. The second objective is topredict extreme exceedance probabilities corresponding to some relatively long duration, fromdata collected at significantly shorter duration. For example, in an aircraft application, if veryaccurate extreme value statistics associated with an entire flight duration were needed, thendirect exceedance estimation to a level around 10−3, would require about 10000 flights. Sincethis might exceed the design life of the aircraft, it would rarely be useful. In practice thereare three ways in which measured data can be used to meet these objectives: first, measuredexcitation data alone could be used to predict responses via a known dynamic model (per-haps calibrated elsewhere). In the second approach, both measured excitationand responsehistories are used to calibrate the parameters in a dynamic model, which is then used forprediction (via methods discussed in Section 2 for example). The third approach predicts usingan appropriately fitted asymptotic extreme value (probability) model but does not involve useof a dynamic model at all. The main problem with the first approach stems from practicaldifficulties arising in accurately measuring excitation, which is a serious limitation even whena good dynamic model is available. These same practical difficulties also extend to the secondapproach, unless the (dynamic) model calibration process can proceed by using response data

Extreme Value Prediction for Nonlinear Beam Vibrations73

alone. The obvious advantage of having a well-calibrated dynamic model, is that responsestatistics can be predicted under totally different excitation conditions from those prevailingat the time of measurement. The third approach is the most direct, and has the advantage thatno detailed dynamic model information is needed. But the success of the asymptotic approachdepends largely on three factors: (i) on appropriate choice of the ‘domain of attraction’ (or ex-treme value distribution type); (ii) on application to a sufficiently long duration for the sampleextreme value distribution function to have sufficiently converged to the asymptotic model,and (iii) that the data available allows a good fit to the asymptotic model for extrapolationpurposes.

In this paper, for the purpose of predicting extreme exceedance probabilities, two of theseapproaches are compared, namely a calibrated dynamic model based method versus an asymp-totic approach (see Figure 1). These approaches are applied in a novel way to real dataobtained from an experimental clamped-clamped beam rig to establish which is better. Thedetails of various methods available to predict extreme values using dynamic models andasymptotic theory are discussed respectively in [22] and [23] and appropriately covered inSections 2 and 3 and Appendix 1. The dynamic model approach involves parameter estim-ation using measured response histories to calibrate a discrete beam model obtained fromthe Woinowsky-Krieger equation. Extreme exceedance probabilities are then obtained us-ing finite-element solutions of the associated stationary Fokker–Planck equation [22]. Theasymptotic approach uses a non-parametric Weissman estimator [23] to make extreme valuepredictions, which is first justified by an appropriate hypothesis test to confirm the domainof attraction. Both approaches are compared with direct exceedance probability estimatesobtained from data captured in a set of long experimental measurements in which considerablecare has been taken to exclude the possibility of dependence between extreme value samples.

2. Extreme Exceedance Probabilities via a Calibrated Dynamic Model

Focusing initially on the calibrated dynamic model approach, an outline is given here, startingwith approximate extreme value theory, the Fokker–Planck equation and its numerical solu-tion, construction of a large amplitude discrete dynamic beam model, and finally calibrationusing stationary response moments. A calibrated dynamic model, excited by band-limitedwhite noise, can in fact be used to predict extreme value statistics via Markov process theoryusing the stationary (FPK) Fokker–Planck equation [22]. This approach is based on levelcrossing statistics, making the Poisson assumption of independent high level crossings. ThePoisson assumption has been tested extensively using simulated data and shown to give veryaccurate extreme value predictions for SDOF equations [22]. In this section, the role ofcrossing statistics in extreme value theory is briefly outlined.

In principle, threshold crossing statistics offer a general approach to extremes which can beused on any dynamic system whose joint probability density function is available for specificdisplacement and velocity variables. The stationary FPK equation can in fact provide a veryeffective route to the required joint density function, but the method is sensitive to the accuracyof the solution and the Poisson assumption [22]. Accurate solutions of the FPK equation arenot easy to obtain even for two-dimensional problems, restricting the approach to systems withjust a few degrees of freedom. Moreover, even if the FPK equation could be solved easily for aknown dynamic system with large dimension, there remains a question as to how such modelscan be calibrated using measured response data alone. Here the method of moments (based


on Markov process theory) provides an ideally matched parameter estimation technique. Thismethod is however also restricted to low dimensional systems. These restrictions dictate thatattention here focuses on the use of a discrete SDOF model for beam vibration, which asfully explained in [12], is approached part-way via the Woinowsky-Krieger equation. The laststage in the modelling process is totally ad hoc, guided largely by a combination of publishednonlinear damping models, suggested over the past three decades. Only when thead hoctermsare included in the discrete dynamic model, is it then appropriate to describe the detailed formof the matched moment parameter estimation method.

2.1. THRESHOLD CROSSINGAPPROACHUSING THE POISSONASSUMPTION

The mean threshold up-crossing rateν+(uT ), of some high levelUT , is one well-knownroute to extreme value statistics for large durationT [24]. For stationary responses,ν+(uT ) isobtained fromp(z, z) the joint density function of displacement and velocity, namely:

ν+(uT ) =∞∫

0

zp(uT , z) dz. (1)

An approximate extreme value distribution function can be obtained by making the Poissonassumption [15, 24]

FM(uT ) = prob{M ≤ uT } = e−ν+(uT )T , (2)

whereM is an extreme value, which can alternatively be expressed in terms of the exceedanceprobability

PM{M(T ) > uT = q(ε)} = 1− F(uT ) = ε. (3)

Here we introduce the terminology used later for both the threshold level and the correspond-ing exceedance probability, namely the quantileq(ε) (which is identical to variableUT ), andε (the exceedance probability). The reason for defining this dual terminology, stems from thepotentially confusing use of terminologies in the general areas of statistical analysis, differentfrom those used in the analysis of random vibration.

2.2. THE FPK EQUATION – STATIONARY SOLUTION USING THE FINITE ELEMENT

METHOD

To obtain the required joint pdfp(z, z) in Equation (1), Markov process theory (in the formof the FPK equation) gives the most direct approach – this is now summarised and used later.If one assumes that an appropriately calibrated discrete vibration model is expressed in statespace form, then the probabilistic properties of trajectories associated with the response ofsuch systems may be modelled assuming broad-band excitation, via the FPK equation. Whenthe discrete vibration model is put into an appropriate system of first-order equations an Itoequation [25] follows in the form

dz = Q(z, t) dt +G(z, t) db(t), (4)

in which the vectorz(t) represents ann-dimensional vector Markov solution process,Q(z, t)

is ann-dimensional vector of (drift) system functions, andG(z, t) is a deterministic functionand db(t) are increments of a Wiener process with the following properties:

E[b(t)] = 0 (5)


and

E(1bi(t)1bj (t)) = 2Dij1t, (6)

where matrixD = πI (I being the unit matrix). A general form of Fokker–Planck (diffusion)equation associated with Equation (7) can be constructed i.e.

∂

∂tp(z, t | z0, t0) =

n∑i=1

n∑j=1

∂2

∂zi∂zj[(GDGT )ijp] −

n∑i=1

∂

∂zi(Qi(z, t)p), (7)

where the joint transition probability density functionp(z, t | z0, t0), is subject to a varietyof boundary conditions [26]. For autonomous discrete models (which are not explicitly de-pendent on time) the Ito equation reduces to

dz = g(z) dt +G db(t) (8)

and correspondingly the stationary FPK equation is

1

2

n∑i=1

n∑j=1

Bij∂2

∂zi∂zj[p(z)] −

n∑i=1

∂

∂zi[gi(z)p(z)] = 0, (9)

in which B = 2πGGT , i.e. a constantn by n matrix of white noise intensities, with Equa-tion (9) subject to far-field and normalisation boundary conditions imposed over an unrestric-ted region� in the form

p(z)→ 0, z→ ±∞ (10)

and∫�

p(z) dz = 1. (11)

If it possible to generate a solutionp(z) under general conditions, then to obtain the bi-variateprobability density function needed to compute the threshold crossing rate given by Equa-tion (1), it is first necessary to computen− 2 multiple integrations. With the above boundaryconditions imposed on Equation (9), there are very few known explicit solutions, consequentlya considerable number of numerical solution methods have been developed [27, 28]. Resultsare shown later using an FEM-FPK (finite element solution method) which is in fact a formof weighted residual technique in which shape functions are defined over finite regions, ratherthan over the entire variate space. For solving nonstationary problems, a thorough coverage ofthe FEM-FPK methods can be found in [28]. Stationary FPK solutions can be obtained veryefficiently using the particular method developed in [27] in which the same shape functionsare used for approximating, within each element, the nonlinear functionsgi(z) in Equation (9).The unknown jpdfp(z) is approximated, within the interior of a generalised rectangular finiteelement, by a series function

pe(z) =2n∑i=1

piNi(z), (12)


wherepi is the local value of the jpdf at each of the corresponding nodal ‘corners’ of eachelement. Lagrange-type shape functionsNi(z) (constructed from functions withC0 continu-ity) are often used to give a value of unity at node points and zero elsewhere. Using theseshape functions, a numerical solution can only be obtained using the ‘weak’ form of the FPKequation, because the FPK Equation (9) includes a second derivative and therefore shapefunctions need at leastC1 continuity. This obstacle is overcome by integrating the residualequation [27], to obtain the general ‘weak’ form of the FPK equation

1

2

n∑i=1

n∑j=1

Bij

∫R

∂

∂zi[w(z)] ∂

∂zi[p(z)] −

n∑i=1

∫R

gi(z)p(z)∂

∂zi[w(z)] dz = 0, (13)

in whichgi(z) are drift functions,Bij are the elements in the diffusion matrix in Equation (9),andw(z) is the weight function [27] according to Galerkin’s method to be the same as theshape function in Equation (12). The integrals in (13) can replaced by finite summations overeach element by use of (12). This leads to a computationally efficient scheme for the unknownvalues ofpi via a system of linear equations. This system however is indeterminate, and to ob-tain a unique solution in practice using the method described in [27], it is necessary to fix oneof the nodal values (as a constraint) to some arbitrary constant value. Then, by imposing thenormalisation condition, all of the nodal values (including the constrained value) are adjustedto their correct level. Furthermore for practical implementation, a finite region is assumed,extending to 4 or 5 standard deviations for all corresponding variables. The unknown standarddeviations associated with these variables are initially estimated using statistical linearisationapplied to Equation (8). In addition, where possible, use of symmetry about the origin gives acorresponding reduction in the computational burden, by appropriately eliminating unknownnodal values. Construction of the FPK equation associated with an appropriate single coordin-ate discrete dynamic model will shortly be demonstrated and the accuracy of the extreme valueprediction approach of Section 2.1 will then be confirmed using Monte Carlo simulation.

2.3. A DISCRETEDYNAMIC BEAM MODEL

A discrete large amplitude vibration model for a beam with immovable ends, can be con-structed directly via structural FEM [9], or indirectly via the Woinowsky-Kreiger equation[1]. Note whichever approach is used, the normalised central displacement ratiow/r [1],gives an indication of the strength of nonlinearity (r = √I/AB ; AB is the beam sectionalarea, andI the second moment of area). Beam vibration is deemed to be strongly nonlinearwhenw/r � 1. A discrete SDOF vibration model can be extracted from the Woinowsky-Kreiger equation by application of Galerkin’s method [2] using just one term of a conventionalgeneralised displacement [12] giving an undamped hardening-type Duffing model

z(t)+ ω2nz(t)(1+ γ z2(t)) = f (t), (14)

wheref (t) is a generalised excitation term. Parametersω2n andγω2

n in Equation (14) dependon the unknown displacement shape which for realistic forced vibration studies, requiresdamping to be included. The model for damping has to be chosen somewhat ad hoc, since thisis made up from several different mechanisms each being difficult to quantify. A three-termnonlinear damping model has been examined in [12] and shown to be appropriate for largeamplitude SDOF motion. When combined with Equation (14) the resulting discrete dynamicbeam model is

z + C1z + α1zz2+ α2z|z| + ω2

nz(1+ γ z2) = f (t). (15)


Figure 1. Two approaches to stationary extreme value prediction using measured beam response histories.

The stationary FPK Equation (9) associated with Equation (15) can be constructed (initiallysettingC1 = 2ξωn, k3 = ω2

nγ andf (t) = Aω(t), then putting into Ito form (8) gives[dz1

dz2

]=[

z2

−2ξωnz2− α1z2z21 − α2z2|z2| − ω2

nz1− k3z31

]dt +

[0 00 A

] [0

db(t)

](16)

and the corresponding stationary FPK equation (9) can then be written as

πA2∂2f

∂z22

− ∂

∂z1(z2f (z))

+ ∂

∂z2((2ξωnz2+ α1z2z

21+ α2z2|z2| + ω2

nz1+ k3z31)f (z)) = 0 (17)

with the same boundary conditions (10) and (11) as an appropriate two-state process. Equa-tion (17) can be solved using the FEM outlined in Section 2, but the parameters in (15) musthowever be calibrated using measured data for the reasons explained in [12]. Before givingdetails of the specific form of parameter estimation method, it is timely to confirm the accuracyof the FEM based extreme value prediction approach described in Sections 2.1 and 2.2 usingsimulated data applied to the SDOF model (15). An appropriate set of parameter values areused [22], namelyξ = 0.0, α1 = 0.812,α2 = 0.015,ωn = 144.34,k3 = 3021,A = 200.

Figure 2 shows a comparison between extreme exceedance probability predictions ob-tained by using the approach in Section 2.1 and corresponding Monte Carlo simulationsassociated with model (15), corresponding to two different durations of timeT . Here 31× 31node FPK-FEM based solutions are compared in Figure 2. with converging simulations. TheMonte Carlo simulation approach used here is the same as described in [22]. In outline,excitation sample paths are constructed by assembling normally distributed random numbersat discrete time intervals1t , to simulate time histories of a continuous process with band-width fn = 1/21t . A truncated Whitaker filter is used in the integration to interpolate


Figure 2. Extreme exceedance probabilities showing converging Monte Carlo simulations compared with 31×31node FEM-FPK based predictions for different duration: (a)T = 1 second; (b)T = 100 seconds.


to smaller time steps1τ . This allows rapid truncation error convergence, independently ofthe bandwidth. Here a standard 4th-order Runge–Kutta integration scheme was used. Aftertransient removal, multiple random response sections of durationT were obtained, and foreach section, a single maximum value was captured for use in estimating the extreme ex-ceedance probability. To achieve low statistical variability, a large number of extreme valueswas needed. In fact, the sample size needed for a confident extreme exceedance probabilityestimatep, can be deduced fromσp/p = 1/

√Np, whereσp is the standard deviation inp and

N is the sample size. HereN = 1000 extreme values were used for each durationT, givingreasonably confident exceedance estimates above 10−2.

For durationT = 1, (roughly 28 cycles) Figure 2a, shows the Poission assumption (madein Equation (2)) holds above a normalised levelU/σz = 2.8, and forT = 100 seconds, thePoisson assumption is seen in Figure 2b to hold for virtually all exceedance probability levelsof interest. (Note a seemingly course FE mesh of (31× 31) nodes has for practical purposesbeen shown to be quite adequate for the beam model (see [22, figure 4]), where convergingFEM-FPK solutions were obtained using different mesh sizes ranging from (21×21)–(51×51)nodes.)

2.4. CALIBRATION OF SDOF MODEL VIA MATCHED MOMENT METHOD

To calibrate Equation (15) using real data, a suitable parameter estimation technique is needed.In general, this requires almost perfectly simultaneous measurements of excitation force andcorresponding response histories. But when perfectly simultaneous measurements are notavailable, namely when there is a small degree of phase-shift between input and output, thestandard parameter estimation techniques can be very inaccurate [12]. Alternatively, when theexcitation can be modelled as white noise process, and its intensity can be measured, explicitparameters can be estimated accurately using a Markov moment based method [30].

The stationary form of this moment method can also be used to obtain explicit parametersfor white noise driven oscillators when detailed input information is not known. Dampingestimation tends to be very sensitive to the amount of data available since additional (ac-curate) information is needed, such as energy envelope statistics [30] because, as explainedshortly, the moment method alone does not return explicit damping parameters, rather, ratiosof damping-to-intensity. However when parameters from the moment method are to be useddirectly in the FPK equation, explicit information about the excitation is not needed, sincethere is a form ofmatchingbetween the moment equations and the FPK equation. Predictionof response statistics at the same (unknown) measured data excitation intensity can in fact pro-ceed using the estimated damping-to-intensity ratios in the FPK equation. This is in completecontrast to Monte Carlo simulations, where explicit parameter values are always required,allowing sample path information to be generated. Solutions of the FPK equation on the otherhand, contain no explicit sample path information, only regularity properties.

In principle the moment method [30] allows any order of moment equations to be used.The best specific set of moment equations to obtain estimates of stiffness and damping forthe model equation (15) has been established empirically in [12], namely up to order 6. Aftercertain moments have been deemed to be small or zero [12, 30], stiffness parameters can beobtained from the following simplified set of second, fourth, and sixth order equations

2nd order stiffness equation:

[ω2nE(z

21)+ γω2

nE(z41)] = E(z2

2). (18)


4th order stiffness equations:

[ω2nE(z

21z

22)+ γω2

nE(z41z

22)] = E(z4

2), (19)

[ω2nE(z

41)+ γω2

nE(z61)] = 3E(z2

1z22). (20)

6th order stiffness equations:

[ω2nE(z

21z

42)+ γω2

nE(z41z

42)] = E(z6

2), (21)

[ω2nE(z

41z

22)+ γω2

nE(z61z

22)] = 2E(z2

1z42), (22)

[ω2nE(z

61)+ γω2

nE(z81)] = 5E(z4

1z22). (23)

Linear and non-linear damping parameters can be obtained from a corresponding set simpli-fied set of second, fourth, and sixth-order equations as follows

2nd order damping equation:

[C1E(z22)+ α1E(z

21z

22)+ α2E(z

22|z2|)] = πA2. (24)

4th order damping equations:

[C1E(z42)+ α1E(z

42)+ α2E(z

42|z2|)] = 3πA2E(z2

2), (25)

[C1E(z21z

22)+ α1E(z

41z

22)+ α2E(z

21z

22|z2|)] = 2πA2E(z2

1). (26)

6th order damping equations:

[C1E(z62)+ α1E(z

21z

62)+ α2E(z

62|z2|)] = 5πA2E(z4

2), (27)

[C1E(z21z

42)+ α1E(z

41z

42)+ α2E(z

21z

42|z2|)] = 4πA2E(z2

1z22), (28)

[C1E(z41z

62)+ α1E(z

61z

22)+ α2E(z

41z

22|z2|)] = 3πA2E(z4

1). (29)

To obtain stationary moments, the expectation operatorE(•) (used above) is applied to timehistories in the form of a sample averaging operation. Equations (18–23), and (24–29), forma system of linear equations which can be solved in general using the method of least squares.It is shown in [12] that response statistics obtained from the model (15) using just a singleset of estimated parameters are subject to very noticeable scatter. To reduce this scatter useof averaged parameters from 10 sets does in fact show predicted response statistics haveconverged.

This parameter estimation method, is deemed to be ideally ‘matched’ to the FPK equationbecause it generates (to a good approximation) both necessary and sufficient information forsubsequent FPK based probability prediction. By a reverse process, the hierarchy of mo-ment equations (which can indeed be constructed via the FPK equation), could also be usedto obtain the original Ito and corresponding FPK equations. As can be seen from Equa-tions (18–23) and (24–29), application of the moment method for parameter estimation ina SDOF dynamic model, generates explicit stiffness parameters but only ratios of damping-to-excitation-intensity. But this information is sufficient for probability prediction via the FPK


at the same (unknown) excitation intensity (and at known multiples of it). For this reason themethod is deemed as being ‘matched’ for subsequent FPK prediction.

3. Extrapolation Using an Asymptotic Extreme Value Distribution

An alternative to the model calibration and prediction approach discussed previously, is touse measured data to directly fit an asymptotic extreme value distribution. This hopefullycan then be used to predict extreme exceedance probabilities with relatively small amountsof data. The theory of extreme value statistics is well developed [23, 24, 32] in the form ofparticular distribution types, namely whether of type I (Gumbel), type II (Fréchet), and typeIII (weibull). This asymptotic extreme value theory, along with an outline of a data collectionprocedure for model selection, fitting, and extreme value prediction, is given in summary inAppendix 1. There are in general two stages to fitting one of these asymptotic distributions:(i) the distribution type (domain of attraction) must be established for a particular data set;(ii) parameters in the chosen distribution type must be estimated from the data. There areactually good physical reasons why extreme vibrations should be of type I – this will beexamined later. In the particular strategy chosen it is assumed at the outset that the beam datais of type I. A test of hypothesis is used via Equations (A4–A6) to confirm the domain ofattraction, where fitting and prediction uses ‘top order statistics’ via Equations (A2) and (A3)of Appendix 1. The approach outlined is appropriate for estimating probabilities for somedurationT , using data of the same duration. The second objective stated in the introduction,is to predict probabilities for some significantly longer duration, using data collected overshorter duration (where considerably less samples are generally available). Prediction in this‘forward’ mode is possible by assuming short duration extreme values samples are mutuallyindependent. This assumption is justified when the time intervalT is significantly greater thatthe system response correlation time, as demonstrated in Section 4, using the autocorrelationfunction associated with response measurements. The exceedance probabilities can then beestimated for a longer duration using Equation (A7).

4. The Experimental Rig

A relatively simple beam, suitably clamped under experimental conditions, is capable ofgenerating the required data needed to test the two approaches outlined in Sections 2 and3. The essential dynamic requirements of the rig are that the beam can be excited to suf-ficiently large normalised central displacements i.e.w/r � 1 (see Section 2.3), in such away that the frequency response characteristics allow genuinely nonlinear responses to beidentified using band-limited white-noise excitation. Figure 3 shows, in diagrammatic form,the rig in the horizontal plane plus associated instrumentation. A steel beam is used, of lengthbetween clampsL = 1 m, thicknessd = 3 mm, and widthb = 25 mm. With thesedimensions, and with material properties:ρ = 7850 kg/m3 andE = 190 GN/m2, the firstthree (linear) small amplitude natural frequencies of lateral vibration are 15, 42, and 82 Hz.Excitation was achieved via a standard electro-magnetic shaker with bandwidth: 1.5–9 kHz,and maximum force rating of 100 N, positioned near to one of the clamps, giving a max-imum central beam peak-to-peak displacement greater than 16 mm. Force measurement wasachieved via a force transducer (in series with the shaker), and displacements were obtainedvia an accelerometer. Both signals were sampled at 500 Hz using a commercial data acquisi-


Figure 3. Experimental clamped-clamped beam rig in diagrammatic form (horizontal plane view).

Figure 4. Probability density estimate of measured shaker force compared with normal distribution with the samemean and variance.


Figure 5. Measured history of central beam displacement sampled at 500 Hz.

tion system. To obtain accurate displacements from acceleration measurements, appropriatelyfiltered (doubly-integrated) signals from a single accelerometer were used (Brul & Kjaer type4332). The accelerometer was located at the beam mid-span position. Displacements above10 Hz, were obtained from sampled acceleration signals at 500 Hz by feeding these into avery high input impedance charge amplifier (i.e. a type 2625 vibration pick-up pre-amplifier).DC offset and drift, introduced by the integration network, were removed by feeding displace-ment signals into a null-adjuster voltage amplifier. The accelerometer was calibrated with acapacitive type displacement transducer (Wayne Kerr). This had a working range of 0–8 mm,with an accuracy of 1%, and linearity better than 2%. Dynamic calibration of the capacitivetransducer was achieved using a moving table. Measured displacements obtained from boththe capacitive transducer and the accelerometer (positioned on the shaker head and driven atknown frequency) were then compared. Experimentally measured displacements were thenobtained using the accelerometer signals, since the extreme values exceeded the range of thecapacitive transducer. Figure 4 shows an histogram of the measured force compared with anormal distribution with the same mean and variance. Figure 5 shows a 2-second sample ofthe measured (normalised) central beam displacement, showing the quality of the noise-freemeasured response histories. Figure 6a (from [12]) compares typical power spectral densityestimates for the (band-limited) excitation and the central beam response, showing the firstthree linear natural frequencies. The dominant response frequency on Figure 6a is markedwith the symbol①. The second harmonic, denoted by 2×①, indicates a degree of asymmetry.Note the response in the frequency domain at the (shifted) second and third (linear) resonantfrequencies, denoted by symbols② and③, are clearly insignificant. Response measurementscorrespond to normalised displacements with(w/r)rms= 3.3 and the largest measured value(w/r)max ≈ 10 confirms these beam vibrations as being highly nonlinear. Figure 6b showsthe measured response autocorrelation function up to the first one second of time-lag. Thisfigure adds clear justification to the independence assumption made in Section 4, which isneeded for ‘forward’ prediction of exceedance probabilities based on contiguous extremevalues associated with a sufficiently large durationT (for exampleT = 1 second duration).


Figure 6a. Power spectral density estimates of measured force and central beam displacement.

Figure 6b. Autocorrelation function estimate for central beam displacement.


Figure 7a. ‘Short data’ collection strategy for model calibration and for obtaining extreme values by sectioningcontiguous lengths of durationT = 1 second (N = 2500),T = 20 seconds, (N = 120) andT = 40 seconds(N = 60).

Figure 7b. ‘Long data’ collection strategy: direct collection of 1150 ‘independent’ extreme values forT = 1 second,T = 20 seconds andT = 40 seconds with a 10 second interrupt. Total data collection time= 1170 minutes (excluding duration of repeatability tests).

5. Measured Data Collection Strategies for Model Calibration and Extreme ValueExtrapolation

The objectives of the paper involve: (i) accurate prediction of extreme exceedance prob-abilities associated with nominal durationT , and (ii) ‘forward’ prediction of exceedanceprobabilities for durationT , using shorter lengths of data. To meet these requirements, threedifferent types of data are in general needed. One type should be made-up ofhistories, neededto calibrate the dynamic model approach described in Section 2. A second type should com-prise (possibly dependent) extreme values for use with the asymptotic approach of Section 3.And a third type should include a large number of independent extreme values, to providedata for benchmark exceedance estimation against which the dynamic model and asymptoticapproaches can be compared. An essential requirement for appropriately comparing thesetwo methods is that the same data should be used. This means that there are really onlytwodata sources needed, namelyhistories, and extreme values for benchmarking. These typesof data are loosely referred to here as ‘short’ and ‘long’ data respectively. Figures 7a and 7bshow random vibration histories and corresponding data collection strategies for obtaining the‘short’ and ‘long’ data sets (the histories shown have actually been simulated for illustrationpurposes, rather than being real measurements). Three selected nominal durations were tested,namelyT = 1 second,T = 20 seconds, andT = 40 seconds. In testing the dynamicmodel based approach, the same quantity of data can be used throughout. But in testing theasymptotic method, as demonstrated shortly, different sample sizes arise when attempting touse the same data source as for dynamic calibration.

Two batches of ‘short’ data were captured, each comprising 10 sections of measured dis-placement history of 250 seconds duration, sampled at a rate of 500 Hz. Short-data-set-1 wasobtained in one continuous run (taking about 40 minutes in real-time). Short-data-set-2 was


obtained intermittently at various times (to confirm constant mechanical beam properties)during the long data collection periods (which lasted for periods up to 24 hours real time).The numerical choice of 10 sets ofhistoriesper data set, was made on the basis that this num-ber proved sufficient to achieve convergence of predicted extreme exceedance probabilitiesusing the calibrated discrete model FEM–FPK solution method [12] described in Section 2.To obtain extreme values for use in the asymptotic model,historieswere sub-sectioned intocontiguous lengths of selected nominal duration. By sectioning histories of 250 second dur-ation and obtaining extreme values, a different numberN of values is produced. Namely fornominal durationT = 1 second,N = 2500; whereas forT = 20 seconds,N = 120; andfor T = 40 seconds,N = 60. In generating the long data, independent extreme values wereobtained from continuously running beam vibration experiments. For each nominal duration,a total of 1150 extreme values were obtained. Independence was achieved by interruptingthe data capture (for a period of 10 seconds) between each full section of durationT . Theimportant question remaining to be answered is whether exceedances predicted using theapproaches of Sections 2 and 3 (using short data), compare well with exceedance probabilitiesobtained using the (benchmark) estimates (based on long data). The results of this comparisonare given in Section 6.

5.1. CALIBRATED DYNAMIC BEAM MODEL PARAM ETERS VIA MATCHED MOMENT

METHOD

Before showing extreme exceedance probability comparisons, it is appropriate to give someoverall evidence of the suitability of the discrete SDOF beam model, its calibration via thematched moment method, and the accuracy the FEM solution of the FPK equation, by compar-ing predicted marginal density estimates with measurement. In this comparison, the Poissonassumption of Section 2.1 is not involved, thus giving an intermediate check on the ap-proach. The moment method (Equations (18–29)) is applied to the measured beam responsesin short-data-set-1 and the parameters are used in the stationary FPK equation, which issolved numerically with the FEM (Section 2.2). Scatter in predicted responses, as a resultof scatter in the parameter estimates, is reduced by averaging parameters obtained from all10 sections of the data (described in Section 2.4). Note, appropriate simulation studies re-ported in [12], showed that errors in raw parameter estimates (compared with their targetvalues) can be as high as 40%, even though corresponding predictions of marginal densityfunctions and extreme values via the FPK equation, agree very well with their respectivetarget functions. As explained in Section 2.4, stiffness parameters are estimated explicitly,whereas ratios of damping to excitation intensity are obtained, not explicit values, as shown inTable 1. Damping-to-excitation-intensity ratios can be used for direct prediction with the FPKequation, but only at the same (unknown) excitation intensity as the experimental measure-ments (or a known multiple of it). Figure 8 shows the predicted marginal probability densityfunction from the calibrated SDOF model via the FPK equation compared with an estimateobtained from measured central beam displacement. This shows very good agreement over theentire range of the density function. Note application of the equivalent linear damping methodto calibrated model (15) shows that the overall equivalent linear damping is around 1.38%whereas the contribution from the linear part of the damping model is only around 0.18%, andas confirmed in [12], the linear part of the damping can be omitted, leaving a wholly nonlinearmodel.


Figure 8. Marginal probability density function from calibrated SDOF model and FPK equation, compared withhistogram from measured central beam displacement.

Table 1. Calibration parameterestimates via moment method using‘short’-data-set-1.

Stiffness Damping

ω2n = 17573 C1/A

2 = 13.5× 10−6

γω2n = 2322 α1/A

2 = 13.2× 10−6

α2/A2 = 0.47× 10−6

6. A Comparison of Predictions Using the Dynamic Model and AsymptoticExtrapolation

The calibrated model based extreme value prediction approach of Section 2, and the non-parametric asymptotic approach of Section 3 are now compared. Calibrated model predictions(based on all 10 sections of short-data-set-1) are compared with corresponding asymptotic es-timates based on maximum values obtained from contiguous sectioning, first using short-data-set-1, then using short-data-set-2. Both approaches compare extreme exceedance probabilityestimates forT = 1 second,T = 20 seconds, andT = 40 seconds, with those based onlong-data, in which the possibility of dependence existing in the samples has been removedas explained in Section 5. First, attention is focused on the top order statistics and the HasoferWang hypothesis test applied to both short data sets, which is needed to justify subsequent useof the Weissman type I asymptotic estimator. Comparisons are then made for moderately low-level exceedance predictions using data corresponding to the same durationsT = 1, 20, and40 seconds. The final set of results show ‘forward’ extreme exceedance probability predictionusing contiguous section sampling. By sectioning short data into durations ofT = 1 second,(using the relatively abundant)N = 2500 extreme values generated to ‘forward’ predictexceedance probabilities toT = 20 and 40 seconds, respectively.


Figure 9. Maximum values obtained by continuous sectioning of short data for durationT = 1 second:(a) short-data-set-1, (b) short-data-set-2.

Figure 9a shows normalised largest maximum values ofw/r (r = √A/I = 0.866)obtained withinT = 1 second by contiguous sectioning short-data-set-1. Figure 9b showscorresponding maximum values for short-data-set-2. . The normalised mean value estim-ates and standard deviations associated with these maximum values are:µ1 = 5.07 andµ2 = 5.10, σ1 = 1.45 andσ2 = 1.44, respectively – this information shows that thereare no obvious differences in the short data sets and therefore no changes in the mechanicalproperties of the beam during ‘long data’ collection. Figure 10a and 10b show corresponding(normalised) top order statistics obtained by contiguous sectioning short-data-set-1 and short-data-set-2 for durationsT = 1, 20 and 40 seconds, respectively. Figures 11a, 11b, and 11cshow Hasofer–Wang hypothesis test results (Equations (A4) and (A5)) applied to correspond-ing contiguous sections of short-data-set-1 for durationsT = 1 second,T = 20 seconds andT = 40 seconds respectively. Figures 12a, 12b, and 12c show similar hypothesis test resultsapplied to short-data-set-2.

Turning now to the extreme exceedance probability comparisons, Figures 13a, 13b, and13c show results with short-data-set-1 at the three durations, using i) the calibrated modelapproach (Section 2), (ii) asymptotic predictions (Section 3), and (iii) benchmark estimatesfrom long data. The number of estimates used in the Weissman estimator is based on thesemi-empirical optimal top order statistic, Equation (A6), whereN = 2500, 120 and 60,respectively, givingK1 = 75,K20 = 17, andK40 = 12. Figures 14a, 14b, and 14c show a sim-ilar set of exceedance probability predictions using short-data-set-2, where again predictionsto moderately low probability are from data collected at the same durationT . The final set ofpredictions shown in Figures 15 and 16, use short data sets with the Weissman estimator, inthe role of ‘forward’ predictor, i.e. using contiguously sectionedT = 1 second data to predictextreme exceedance probabilities forT = 20 seconds andT = 40 seconds. Figure 15a and15b shows the comparison using short-data-set-1, similar to the information in Figures 13b


Figure 10a. Top order statistics for normalisedw/r using contiguous sectioning of short-data-set-1.

Figure 10b. Top order statistics for normalisedw/r using contiguous sectioning of short-data-set-2.

and 13c, but now additionally using Equation (A7) in the extrapolation. Figures 16a and 16bshow corresponding results applied to short-data-set-2.

7. Discussion of Results

We begin our discussion by focusing on the results of the Hasofer Wang test applied to short-data-set-1. Figures 11a, 11b, and 11c show unequivocally that for the extreme value top orderstatistics associated with this set, the null hypothesis has to be accepted, suggesting that thishigh-level data belongs to a type I domain of attraction. Clearly, below a certain level, for allthree durationsT = 1–40 seconds, the extreme values appear to be of type III rather thantype I. In particular forT = 1 second in Figure 11a, the transition threshold from type III to


Figure 11a. Hasofer–Wang hypothesis test results using short-data-set-1.T = 1 second.

Figure 11b. Hasofer–Wang hypothesis test results using short-data-set-1.T = 20 seconds.

type I is in the interval 100< K1 < 200, whereas for Figures 11b and 11c, correspondingto T = 20 seconds andT = 40 seconds, extreme values are of type III forK20 > 35andK40 > 45, respectively. Figures 10a and 10b can be used to show the magnitude of thenormalised beam displacement at which this transition in domain of attraction (from type IIIto type I) occurs. For all three durations, this transition can clearly be seen to occur in the veryprecise range 7.5 < w/r < 8. Turning attention to short-data-set-2, Figures 12a, 12b, and12c again point to a null hypothesis for the top order statistics, perhaps less obviously than for


Figure 11c. Hasofer–Wang hypothesis test results using short-data-set-1.T = 40 seconds.

Figure 12a. Hasofer–Wang hypothesis test results using short-data-set-2.T = 1 second.

short-data-set-1. Figure 10b shows that the same precise transition range applies for all threedurations.

Since the null hypothesis, is accepted for the top order statistic for both short data sets,predictions using the type I Weissman estimator (Equation (A2)) are justified. Figure 13afor T = 1 second clearly shows excellent agreement between the Weissman estimator ascompared with exceedance probability estimates from the 1150 samples of long data. Thislevel of agreement in Figure 13a also shows that there is no evident dependence in the extremevalues obtained by contiguously sectioning short-data-set-1. This statement can be made withconfidence since, with a sample sizeN = 2500, the effects of statistical scatter should be


Figure 12b. Hasofer–Wang hypothesis test results using short-data-set-2.T = 20 seconds.

Figure 12c. Hasofer–Wang hypothesis test results using short-data-set-2.T = 40 seconds.

very small and can therefore be eliminated. Note the dynamic model based predictions shownin Figure 13a are markedly in error. The reason for this difference can almost certainly beattributed to nonlinear coupling involving more degrees of freedom, which for the reasonsoutlined in [12], cannot be seen in the measured response spectrum Figure 6a. Figures 13b and13c show similar results for Weissman predictions (which are again obtained by contiguouslysectioning short-data-set-1 withT = 20 seconds andT = 40 seconds, respectively, producingN = 120 andN = 60 extreme values, and correspondingly top order statisticsK20 = 17,andK40 = 12). For these durations, the reduced sample size would be expected to increase


Figure 13a. Extreme exceedance probabilities using short-data-set-1 forT = 1 second. FEM–FPK predictionsvia calibrated model - - -; Weissman predictor•; ‘long data’ —.

Figure 13b. Extreme exceedance probabilities using short-data-set-1 forT = 20 seconds. FEM–FPK predictionsvia calibrated model - - -; Weissman predictor•; ‘long data’ —.

statistical scatter, but the results shown are still significantly better than those obtained via thecalibrated dynamic model.

Figure 14a shows, for the similar set of predictions using short-data-set-2, excellent agree-ment with the long data. But the results in Figure 14b forT = 20 seconds, and to a greaterextent in Figure 14c forT = 40 seconds, show that there are very noticeable differencescompared with the ‘long data’ results. This difference stems from statistical scatter, and indeedfor T = 40 seconds, the error is of the same order of magnitude as the calibrated dynamicmodel.

Very much improved results are however shown in Figures 15a and 15b, using short-data-set-1 and Equation (A7) in a ‘forward’ predictive mode forT = 20 seconds, andT =


Figure 13c. Extreme exceedance probabilities using short-data-set-1 forT = 40 seconds. FEM–FPK predictionsvia calibrated model - - -; Weissman predictor•; ‘long data’ —.

Figure 14a. Extreme exceedance probabilities using short-data-set-2 forT = 1 second. FEM–FPK predictionsvia calibrated model - - -; Weissman predictor•; ‘long data’ —.

40 seconds, since they are much less prone to scatter, because the base sample size is nowN = 2500. This improvement is particular evident when the predictions in Figure 15a arecompared with those in Figure 13b, and when Figure 15b is compared with Figure 13c. Thereis obvious improvement for short-data-set-1, but moreover for short-data-set-2, there is veryconsiderable improvement as shown when Figures 16a and 16b are compared respectivelywith Figures 14b and 14c. The results in Figures 15 and 16 show that when independencebetween the extreme values of contiguously sectioned data can be confirmed (as for examplein Figures 13a and 14a) an appropriately justified Weissman type I estimator, proves consider-ably more accurate predictions in a forward predictive mode using Equation (A7), than direct


Figure 14b. Extreme exceedance probabilities using short-data-set-2 forT = 20 seconds. FEM–FPK predictionsvia calibrated model - - -; Weissman predictor•; ‘long data’ —.

Figure 14c. Extreme exceedance probabilities using short-data-set-2 forT = 40 seconds. FEM–FPK predictionsvia calibrated model - - -; Weissman predictor•; ‘long data’ —.

prediction via Equation (A2) alone, i.e. using extreme values at the same duration. Althoughthis method has been tested on measured clamped-clamped beam data, these findings areclearly important for more general areas of random vibration where extreme value statisticsof nonlinearly coupled motions are of interest.


Figure 15a. Forward prediction of extreme exceedance probabilities using short-data-set-1.T = 20 seconds.FEM–FPK calibrated model - - -; Weissman predictor•; ‘long data’ —.

Figure 15b. Forward prediction of extreme exceedance probabilities using short-data-set-1.T = 40 seconds.FEM–FPK calibrated model - - -; Weissman predictor•; ‘long data’ —.

8. Conclusions

Two extreme value prediction approaches, for use with relatively short duration measuredresponse histories have been tested on experimentally generated clamped-clamped beam data.One approach uses a calibrated discrete SDOF dynamic model, from which extreme valuesare obtained via threshold crossing statistics and finite element solution of the FPK equation.The other approach uses a Weissman type I asymptotic extreme value estimator, which is firstjustified by the Hasofer–Wang hypothesis test. Predictions using both methods are compared


Figure 16a. Forward prediction of extreme exceedance probabilities using short-data-set-2.T = 20 seconds.FEM–FPK calibrated model - - -; Weissman predictor•; ‘long data’ —.

Figure 16b. Forward prediction of extreme exceedance probabilities using short-data-set-2.T = 40 seconds.FEM–FPK calibrated model - - -; Weissman predictor•; ‘long data’ —.

with exceedance probability estimates obtained from data measured in long experiments inwhich dependence between adjacent extreme value samples has been virtually eliminated.

The paper shows that the extreme values associated with large amplitude random vibrationsof a clamped-clamped beam, consistently belong to the type I (Gumbel) domain of attraction.It is also shown that Weissman based predictions are significantly better than the exceedanceprobabilities obtained via a calibrated SDOF model using the FPK equation. Moreover, whenthe extreme values of contiguously sectioned response histories are independent, the Weiss-man estimator used in a ‘forward’ predictive role, gives very accurate results since much more


data can be used. These conclusions could be of considerable importance in more general areasof nonlinear random vibration where extreme value statistics are of interest.

Appendix 1

THE ASYMPTOTIC EXTREME VALUE THEORY AND ITS APPLICATION

The theory of extreme value statistics is well developed [23, 24, 32] in the form of particulardistribution types, namely whether of type I (Gumbel), type II (Fréchet), or type III (Weibull).These three distribution types are compactly described by the Pareto family of distributions[10]. The distribution function

F(z) = exp[−(1+ αz)−1/2] (A1)

is classed by parameterα taking zero, positive, or negative values, to determine whether thedistribution is of type I, II or III, respectively [23]. In general there are two stages needed toconfidently use one of these distribution types. First, the distribution type must be establishedfor a particular data set, i.e. the domain of attraction for the (physical) mechanism generatingthe data must be established. Second, the data must then be used to estimate parameters inthe chosen distribution type. Predictions can then be made, but only relating to the calibrationdata. In practice, considerably more effort is focused on fitting the asymptotic model than ondeciding the distribution type. There are in fact several schools of thought on how the datashould be used in the fitting process, namely whether to use all of the data, or a select sample.One approach is to use all of the data, perhaps initially estimating local maxima statistics, fromwhich the extreme-value distribution can in many cases be constructed [33]. Another approachis to transform all of the data using an appropriate functional mapping, prior to fitting the data,in the hope that the transformed data will converge more rapidly to an (asymptotic) extreme-value distribution function [34]. A third approach recognises that lower-level data may begenerated by a fundamentally different mechanism from the upper-most extreme values. Thissuggests lower-level data may actually distort the fitting process and should therefore notbe used. Indeed there is evidence to confirm that using only high level samples i.e. a smallselection of the entire set, offers the best chance of getting a good fit (see, for example, [18]).The precise threshold above which data should be used has led to the development of twomethods of selection, namely the ‘peaks over threshold’ method, and the other via ‘top orderstatistics’. The latter method, which has been rigorously tested in [35], is explained below.This is used in Section 5 on measured beam vibration data in the form recommended in [23].

Top Order StatisticsIn focusing on the use of top order statistics some particular definitions are needed whichare given using the terminology defined in Section 2.1. By collectingN extreme valuesZ1, Z2, . . . ZN , associated with a vibration response historyZ(t) corresponding to some par-ticular durationT , the so calledorder statisticscan be identified, namelyZ1N,Z2N, . . . , ZNNsatisfying inequalitiesZ1N ≥ Z2N ≥ . . . ≥ ZNN . Focusing only on the topK order statistics,the truncated setZ1N,Z2N, . . . , ZKN is then referred to as the ‘K top order statistics’. Whenthe domain of attraction can be established as being of type I, the Weissman (type I) estimator

q(ε) = a ln(K/(Nε))+ ZKN, (A2)


where

a =(

K∑i=1

ZiN/K

)− ZKN (A3)

can be used to predict the extreme quantilesq(ε), (amplitude leveluT ) associated with aparticular target exceedance probabilityε [23]. There are actually good physical reasons whyextreme vibrations should be of type I. But before Equation (A2) can be used with confidence,a test of its suitability is needed to confirm thenull hypothesis, namely that the domain ofattraction is of type I. The Hasofer–Wang hypothesis test [23] can be used to do this, based ontest statisticW , using upper and lower percentage pointsWU andWL of W (defined in [23]).These determine the critical regions to enable thealternative hypotheses(that the data is eithertype II or III) to be accepted or rejected. If the outcome of the test suggests that the domain ofattraction is not of type I, then an appropriate (but more complicated) type II or III estimatoris then needed instead of Equation (A2). In circumstances where the alternative hypothesesare rejected, then the null hypothesis is accepted, and predictions proceed assuming the databelongs to a type I domain of attraction. To be specific, the hypothesis test statisticW is givenby

W = K(z − zkn)2(k − 1)

[∑Ki=1(Z − ZiN)2

] , (A4)

where

Z =(

K∑i=1

ZiN/K

)(A5)

and a semi-empirical optimal value of top order valueK is suggested in [23] as

K = 1.5√N. (A6)

The following conditions are then tested

(i) If W < WL, then the hypothesis that data belongs to a type II domain of attraction isaccepted.

(ii) If W > Wu, then the hypothesis that the data belongs to a type III domain of attraction isaccepted.

(iii) If (i) and (ii) are rejected, the null hypothesis is accepted.

If indeed the null hypothesis is accepted, then the data is deemed to belong to type I and theextreme exceedance probability estimator, Equation (A2), can be used with confidence.

This approach is appropriate for estimating probabilities for some durationT , using dataof the same duration. The second objective stated in the introduction, is to predict probab-ilities for some significantly long duration, using data collected for a short duration (whereconsiderably less samples are generally available). Predicting in a ‘forward’ mode is possibleby assuming extreme value samples for some short durationT are mutually independent. Theexceedance probabilities can then be estimated for a longer durationmT , wherem indexestime, in whichm � 1. The basis for ‘forward’ extrapolation is therefore obtained from the


distribution functionF(z) (corresponding to intervalT ), being used to construct the distri-bution formT , whose distribution is[F(z)]m. The exceedance probability is therefore givenby

εmT = 1− [1− εT (z)]m, (A7)

whereεmT is the exceedance probability for durationmT , andεT (z) corresponds to durationT .

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