The Use of L-Moments in the Peak Over Threshold Approach for Estimating Extreme Quantiles of Wind Velocity

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1 The Use of L-Momets i the Pea Over Threshold Approah for Estimatig Extreme Quatiles of Wid Veloity M.D. Padey Uiversity of Waterloo, Otario, Caada P.H.A.J.M. va Gelder & J.K. Vrijlig Delft Uiversity of Tehology, Delft, Netherlads ABSTRACT: A alterative approah to the estimatio of the tail idex of extreme value distributios i the POT approah is desribed. The proposed estimatio method is based o sample L-momets. Quatile estimatio of wid loadig is preseted as a ase study. INTRODUCTION A fudametal theorem of extreme value theory states that suffiietly large values of idepedet idetially distributed radom variables geerally follow oe of the three extreme value distributios, amely, Frehet, Gumbel ad reverse Weibull. The upper tail of Frehet ad Gumbel distributios is ifiite, whereas the reverse Weibull distributio has fiite upper tail. I the estimatio of extreme quatiles of wid loadig, Frehet ad Gumbel distributios are ormally used. However, it has bee show that the Gumbel ad Frehet models results i urealistially high failure probability of wid sesitive strutures due perhaps to the ubouded upper tail of the distributio (Elligwood 98). The POT (peas over threshold) approah for modellig the tail of a distributio has reeived osiderable attetio sie it has bee show that the Pareto distributio (GPD) arises as the limitig distributio of peas (or exesses) X u of a radom variable X over a high threshold u (Piads 975). The etral objetive of this paper is to evaluate the effetiveess of the method of L-momets for estimatig the GPD parameters, ad ompare its performae i quatile preditio agaist that of the de Haa method (994). Mote Carlo simulatios ad atual wid veloity data olleted at various statios i the Uited States have bee utilized for ompariso purposes. The paper is orgaized as follows. The ext setio reviews basi oepts of the POT approah ad the de Haa estimatio method. The L-momet method is briefly outlied i Setio 3. Setio 4 presets umerial results usig Mote Carlo simulatios ad atual wid speed data. The mai olusios ad a list of referees are preseted i Setio 5 ad 6, respetively. 2 PEAKS OVER THRESHOLD AND DE HAAN 2. Simiu ad Heert's studies I a reet study Simiu ad Heert (996) applied the Pea Over Threshold (POT) approah to aalyze the wid speed data ad fit a appropriate extreme value distributio. The POT approah eables the aalyst to use all the data exeedig a suffiietly high threshold, i otrast with the lassial extreme value aalysis whih typially uses aual extreme values. Simiu ad Heert (996) oluded that the reverse Weibull distributio is a more appropriate distributio tha the Gumbel or Frehet. Their olusio is also supported by a physial fat that otoradi extreme wids are expeted to be bouded. 2.2 POT The appliatio of POT approah ivolves two mai steps: seletio of threshold ad estimatio of the tail idex usig order statistis above the threshold. Sie the tail idex is foud to be very sesitive the threshold value, subsequet quatile estimates also beome sesitive to the threshold. This is a major weaess of the POT approah, whih is also ofirmed from the aalysis of Simiu ad Heert (996). I fat, the flutuatio of quatiles of wid loadig with threshold is so severe that o reliable desig estimates ould be reommeded. Let X, X 2,..., X be a series of idepedet radom observatios of a radom variable X with the distributio futio (DF) F(x). To model the upper tail of F(x), osider exeedaes of X over a

2 threshold u ad let Y, Y 2,..., Y deote the exesses (or peas), i.e., Y i = X i - u. Piads (975) showed that, i some asymptoti sese, the oditioal distributio of exesses follow the geeralized Pareto distributio. Thus the DF of Y i = [(X i - u) X i > u], i =,2,...,, is give as y ( h) Gy ( ) = + a / () where h, a ad deote the loatio, sale ad shape parameters, respetively. Geerally, the loatio parameter is tae as zero. The distributio is ubouded, i.e., < y < if ad bouded as < y < a/ if <. The expoetial DF is a speial ase of eq.() whe =. The GPD exhibits a uique threshold-stability property, i.e., if X follows GPD the the oditioal distributio of exesses, i.e., G(y), also follows GPD with the same shape parameter as that of X. It a also be show that the distributio of maximum exesses, i.e., Z = max (Y, Y 2,..., Y ) follows the geeralized extreme value distributio provided that exeedaes over the threshold are geerated from a Poisso proess (Daviso ad Smith 99). These elegat mathematial properties have motivated the use of the GPD model i extreme quatile estimatio. The alulatio of a quatile value, x R, orrespodig to a R-year retur period is based o the quatile of peas, Y, orrespodig to a retur period of λr, where λ is the mea exeedae (or rossig) rate per year of X over u. Thus, xr = G + u λr (2) where G - (p) deotes the Pareto quatile futio. If deotes the umber of samples olleted over m years ad is the umber of exeedaes, the the mea rossig rate is estimated as λ = /m. As the threshold is lowered to ilude more data i the iferee, the rossig rate λ ireases due to ireasig values of. A iterestig trade-off is thus revealed from eq.(2), i.e., the more the data osidered, the farther i the tail regio oe has to go for quatile estimatio due to ireasig values of the effetive retur period λr. Coversely, by ilusio of additioal data the auray of tail modellig must irease at a higher rate tha the movemet farther i the tail regio. Otherwise, the auray of POT estimates would deteriorate. I this sese, it is expeted that a optimal threshold might exists that would results i the miimum error (Caer ad Maes 998, Dougherty et al. 999). 2.3 Parameter estimatio by De Haa De Haa (994) proposed the estimatio of sale ad shape parameters usig the order statistis of exeedaes, {X -,,..., X, }, where X -, is the smallest data poit to exeed a give threshold. Based o a extesive mathematial aalysis, the shape parameter was derived as = + 2, where ( M ) () 2 2 = (2) 2 M = M ad () (3) i terms of momets of exesses obtaied from the log-trasformed data: r M [log( X ) log( X )] (4) ( r) = i= + i,, The sale parameter a be obtaied as () M a = u (5) ρ where ρ = if, ad ρ = /(-) if <. Geerally speaig, estimates the shape parameter of ubouded GPD, whereas 2 orrespods to the bouded ase (Reiss ad Thomas 997). The ofidee bouds for a estimate of were also derived based o the argumet of asymptoti ormality (de Haa 994). 3 METHOD OF L-MOMENTS 3. Order Statistis ad L-Momets Usig the desity futio of a r th order statisti, X r: ( Kedall ad Stuart 977), alog with a trasformatio u = F(x), its expetatio a be expressed i terms of the quatile futio, x(u), as r r E[X r: ] = r x( u) u ( u) du (6) r Expetatios of the maximum ad miimum of a sample of size a be easily obtaied from eq.(6) by settig r = ad r =, respetively. E[X : ] = x ( u) u du, ad E[X : ] = x( u)( u) du (7)

3 The probability-weighted momet (PWM) of a radom variable was formally defied by Greewood et al. (979) as i j i j M i, j, = E[ X u ( u) ] = x( u) u ( u) du (8) Certai liear ombiatios of PWMs, referred to as L-momets, are show to be aalogous to ordiary momets i a sese that they also provide measures of loatio, dispersio, sewess, urtosis, ad other aspets of the shape of probability distributios or data samples (Hosig 99). A r th order L- momet is mathematially defied as r * r = p r, β = λ (9) where p * r, represets the oeffiiets of shifted Legedre polyomials (Hosig 99). From a ordered radom sample of size, ubiased estimates b ad a of β ad α, respetively, a be obtaied as (Hosig ad Wallis 997) b a = = i= i= i X i i X i ad () Normalized forms of higher order L-momets, τ r = λ r /λ 2, (r = 3, 4,... ) are oveiet to wor with due to their bouded variatio, i.e., τ r <. Extesive simulatio-based studies ad ompariso with iformatio-theoreti measures, e.g., ross-etropy ad divergee, Padey et al. (999) have show that L- momets are very effetive i summarizig distributio properties ad extreme quatile estimatio. Figure 2 shows that the RMSE of λ ad λ 2 ireases from % to 2% with ireasig values of. The RMSE of τ 3 remais somewhat isesitive to, ad is the order of 2%. Usig 3 sample L-momets, the loatio (h), sale (a) ad shape () parameters of the GPD a be estimated as (3τ3 ) =, a = ( )(2 ) λ2 ad ( τ + ) 3 h= λ (2 ) λ () 2 I ase that the loatio parameter is ow, the first two L-momets a be used to estimate ad a as Bias ( λ h) λ = 2 ad λ 2% % -2% -4% 2 λ λ2 τ3 (Bouded GPD) a = ( )( h) (2) Shape Parameter () (Ubouded GPD) Figure : Normalized bias of sample estimates of L-momets (paret: GPD, sample size 5). 3.2 Parameter estimatio Hosig ad Wallis (997) illustrated that L- momets are effiiet i estimatig parameters of a wide rage of distributios from small samples. The mai advatage of L-momets is that, beig a liear ombiatio of data, they are less iflueed by outliers. I geeral, the bias of small sample estimates of higher-order L-momets is fairly small. Furthermore, the required omputatio is quite limited as ompared with other traditioal tehiques, suh as maximum lielihood ad least-squares. Figures ad 2 illustrate variatios of the bias ad root-mea-square error (RMSE), respetively, of sample estimates (size 5) of the first three GPD L-momets (λ, λ 2, τ 3 ) with respet to its shape parameter. The bias ad RMSE estimates are ormalized by exat values of respetive L-momets. As see from Figure, bias assoiated with λ ad λ 2 is almost zero ad that of τ 3 is fairly small ( 3%). RMSE 3% 2% % % (Bouded GPD) Shape Parameter () (Ubouded GPD) Figure 2: Normalized RMSE of sample estimates of L-momets (paret: GPD, sample size 5). λ λ2 τ3

4 4 COMPARISON OF SHAPE ESTIMATES 4. Three methods The paper presets a alterative approah to the estimatio of tail idex of extreme value distributio i the POT approah. The proposed estimatio method is based o sample L-momets istead of order statistis. The L-momets a be osidered as liear ombiatios of ertai expetatios of order statistis (Hosig 99). I otrast with ordiary statistial momets, the L-momets of higher order a be reliably estimated from small samples. It is expeted that L-momets ould provide more reliable estimates of the tail idex with a limited sesitivity to the threshold parameter. I this study, three methods are osidered for the estimatio of, amely, () de Haa (DHN) - eq.(3) (2) 2 L-momets (2-LM) - eq.(2) (3) 3 L-momets (3-LM) - eq.() To ompare the bias ad RMSE of various estimates of, a simulatio experimet was desiged i whih samples, eah of size of, were geerated from a GPD of presribed parameters. Cosiderig N p top order statistis ad orrespodig sample of exesses, the three methods were applied to estimate the shape parameter. The simulatio osisted of, yles. Numerial results for N p = 3 ad are summarized i Table for various values of the paret GPD. As see from Table, bias assoiated with all the three estimates is small (-6%). I the first ase of N p = 3, the bias of DHN estimate is higher tha those of L-momet estimates. I all the ases reported i Table, RMSE values of all the three estimates are almost idetial. A geeral observatio is that both bias ad RMSE are almost isesitive to the shape parameter of the paret GPD used i the simulatio. Cosiderig these results, it is atiipated that L-momet estimates of the tail idex () would at least be ompetitive to those obtaied from the de Haa method. I otrast with the de Haa method, the L-momet method is more isightful sie L-momets have ituitive meaig, suh as Gii's mea differee (λ 2 ) ad sewess. O the other had, de Haa's formulae based o logtrasformed data are of "bla-box" ature. Sie the distributio of L-momets ratios, λ 2 /λ ad τ 3, is asymptotially ormal (Hosig 99), the ofidee bouds o a also be ostruted i priiple. This topi is a area of future researh. Table : Bias ad RMSE the Pareto shape parameter estimated from POT samples (GPD paret, sample size ) 4.2 Simulatios To illustrate a trade-off assoiated with the seletio of threshold i the POT method, two searios were aalyzed via Mote Carlo simulatios. I the first ase, samples of size N= were geerated from the Gumbel distributio (µ=5 mph, σ=6.25), ad POT samples were ostruted from upper N p order statistis. Cosiderig the origial sample (N=) as a sample of aual maximum wid speeds, the exeedae rate was estimated as λ = N p /N. Wid speed quatiles for retur periods (R) of to 5 years were estimated from the POT sample by fittig the Pareto distributio usig de Haa ad L-momet methods. The simulatio osisted of, yles. The ormalized bias ad RMSE of quatile estimates are reported i Table 2 for various values of N p ad R. I all the ases, the bias is fairly small (< 3%) ad RMSE varies i a arrow rage from 2 - %, similar to results of Gross et al.(994). Table 2: Bias ad RMSE of Gumbel quatile estimates ( years of simulated data). Table 2 learly suggests that the GPD represetatio of POT data is fairly aurate.

5 I the seod ase, a more realisti seario was osidered by assumig that N= idepedet observatios of wid speed were olleted over m=25 years period suh that the exeedae rate beomes λ = N p /m (e.g., Dougherty ad Corotis 998). I the simulatio, samples were geerated from the Gumbel distributio (µ=3 mph, σ=6.5 mph) ad quatile bias ad RMSE were estimated for the same values of N p ad R as osidered before. As see from Table 3, the bias (2-4%) ad RMSE (3-6%) have ireased sigifiatly i ompariso to the previous ase. Table 3: Bias ad RMSE of Gumbel quatile estimates from 25 years of simulated data osistig of observatios. 4.3 Case study To ompare the performae of de Haa ad L- momet methods i a more realisti settig, atual wid speed data olleted at various statios the Uited States were aalyzed. The method of L- momet was added to a omputer program provided by Heert ad Simiu (996), whih geerates a uorrelated sample from the origial wid data based o eight-day (four-day) iterval ad applies the de Haa method for GPD parameters estimatio. Here, umerial results are preseted for oe statio, amely, Helea (MT). This statio is hose for o partiular reaso other tha the fat that it represets a typial log (35 years) term data i the available data-base of wid speeds. Estimates of the tail shape parameter,, obtaied from Helea, is plotted agaist the threshold veloity i Figure 3..4 Tail Parameter () LM 2-LM DHN CB CB The ireased error a be attributed to large values of the exeedae rate (λ -4), beause the previous ase orrespods to fairly small values of λ (λ<<). A ompariso of results preseted i Tables 2 ad 3 ofirms our ommets of Setio 2. about a tradeoff betwee the auray ad amout of data i the POT method. I ase of relatively short time series (2-5 years), the seletio of right threshold is importat beause ilusio of ueessary data ould result i the loss of auray of quatile estimates. I short, hyper-sesitivity of quatile estimates to the threshold value is a major pratial diffiulty assoiated with the POT method. Table 3 suggests de Haa method results i the least bias ad RMSE, followed by the 3 L-Momet method. However, the differees i bias ad RMSE assoiated with the three methods are limited (withi %), ad would hage differetly with variatio i the POT sample size. Similar simulatios a be performed for paret distributio other tha the Gumbel, though the qualitative olusio of the setio is ot expeted to hage Threshold (mph) Figure 3: Tail parameter estimates usig wid speed data from Helea, MT. The 95%-ofidee bouds o de Haa estimates obtaied as ±2s, s beig the stadard deviatio of, are also show i this figure. The formulae for s are already summarized by Heert ad Simiu (996). Note that ireasig values of the threshold veloity (u) o the X-axis implies dereasig size of the POT sample. It is iterestig to ote that 3-LM estimates of ted to be lose to the upper ofidee boud, whereas 2-LM estimates are fairly lose to de Haa estimates (DHN). The variatio of 3-LM estimates with the threshold exhibits more stable tred tha those obtaied from DHN ad 2-LM methods. It is somewhat expeted sie the L-sewess, whih determies i 3-LM method, teds to be less sesitive to variatios i the POT sample.

6 Quatile (mph) Figure 4: Quatile estimates usig wid speed data from Helea, MT (R= years) I Figure 4, quatiles estimates for year retur period are plotted agaist the threshold veloity. Note that 2-LM ad DHN estimates are i very lose agreemet. 3-LM quatile estimates follow a smoother tred, whih is helpful i idetifyig the represetative threshold ad quatile value. From Helea, a 3-LM quatile estimate orrespodig to horizotal-stable part of the urves is roughly 8 mph. 5 CONCLUSIONS Threshold (mph) 3-LM 2-LM DHN A aurate estimatio of parameters of the Pareto distributio (GPD) i the peas-over-threshold approah is of osiderable importae. The paper examies the method of L-Momets ad ompares its performae agaist a widely aeptable method of de Haa (994). I the de Haa method, the first two momets of peas (or exesses) of log-trasformed data are used for parameter estimatio, whereas the L-momet method utilizes liear ombiatios of expetatios of order statistis of peas i the origial data. Two variatios of the L-momet method are disussed, amely, 2-LM ad 3-LM, whih iorporate 2 ad 3 L-momets, respetively, i the parameter estimatio. Despite the proedural differees, it is iterestig to ote that the de Haa ad 2- LM estimates of the tail shape parameter () are i fairly lose agreemet i simulated data (Tables 2 ad 3) as well as i atual wid speed data olleted i the Uited States (Figures 3-5). I this respet, a equivalee betwee the 2-LM ad de Haa methods is a otable poit of the paper. Furthermore, the L- momet method is more isightful beause of ituitive meaig of L-momets, suh as Gii's dispersio oeffiiet (λ 2 ) ad sewess. I the 3-LM method, higher order estimates of the shape parameter are obtaied usig the L-sewess of POT data. The aalysis of atual wid speed data illustrates that 3-LM estimates of ted to follow a more stable tred i the proximity of the upper boud estimates obtaied from the de Haa method. Therefore, 3-LM method appears to be useful i idetifyig meaigful upper bouds of desig quatiles. 6 ACKNOWLEDGEMENT The ase study desribed i this paper is performed with the wid data provided by Dr. Simiu for whih he is greatfully aowledged. REFERENCES Carers, J. ad Maes, M.A. (998). Idetifyig tail bouds ad ed poits of radom variables. J. Strutural Safety, 2, -23. Daviso, A.C. ad Smith, R.L. (99). Models of exeedaes over high thresholds. J. Royal Statistial Soiety, Series B, 52, pp de Haa, L. (994). Extreme value Statistis. Extreme Value Theory ad Appliatios, J. Galambos, J. Leher ad E. Simiu (eds.), v., pp Dougherty, A.M. ad Corotis, R.B. (998). Extreme wid estimatio: Theoretial osideratios. Strutural Safety ad Reliability, N. Shirashi, M. Shiozua ad Y.K. We (eds.), v.2, pp Dougherty, A.M. Corotis, R.B ad Shwartz, L.M. (999). Extreme value theory ad mixed distributios - Appliatios. Appliatios of Statistis ad Probability, R.E. Melhers ad M.G. Stewart (eds.), v., pp Elligwood, B. (98). Developmet of a probability-based load riterio for Ameria Natioal Stadard A58. NBS Spe. Publ. 577, Natioal Bureau of Stadards, Washigto D.C. Greewood, J.A., Ladwehr, J.M. ad Matalas, N.C. (979). Probability weighted momets: Defiitio ad relatio to parameters of several distributios expressable i iverse form. Water Resoures Researh, 5(5), Gross, J., Heert, A., Leher, J. ad Simiu, E. (994). Novel extreme value estimatio proedures: Appliatio to extreme wid data. Extreme Value Theory ad Appliatios, J. Galambos, J. Leher ad E. Simiu (eds.), v., pp Heert, N.A. ad Simiu, E. (996). Extreme wid distributio tails: A peas over threshold approah. J. Strutural Egieerig, ASCE, 22(5),

7 Hosig, J.R.M. (99) L-momets aalysis ad estimatio of distributios usig liear ombiatio of order statistis. J. of the Royal Statistial Soiety, Series B, 52, Hosig, J.R.M. ad Wallis, J.R. (997). Regioal Frequey Aalysis: A Approah based o L- Momets. Cambridge Uiversity Press, Cambridge, UK. Kedall, M.G. ad Stuart, A. (977). Advaed Theory of Statistis. Griffi, Lodo, U.K. Padey, M.D., Va Gelder, P.H.A.J.M. ad Vrijlig, J.K. (999) Appliatio of iformatio theory to the assessmet of effetiveess of L-urtosis riterio for quatile estimatio. Aepted for publiatio i J. Hydrologi Egieerig, ASCE. Piad, J. (975). Statistial iferee usig order statistis. Aals of Statistis, 3, 9-3. Reiss, R.D. ad Thomas, M. (997). Statistial Aalysis of Extreme Values with Appliatios to Isurae, Fiae, Hydrology ad Other Fields. Birhauser Verlag, Basel, Switzerlad. Simiu, E. ad Heert, A. (996). Extreme Wid Distributio Tails: A Pea Over Threshold Approah. J. Strutural Egieerig, ASCE, 22(5),

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