Higher moments method for generalized Pareto distribution in flood frequency analysis

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1 IOP Confeence Seies: Eath and Envionmental Science PAPER OPEN ACCESS Highe moments method fo genealized Paeto distibution in flood fequency analysis To cite this aticle: C R Zhou et al IOP Conf. Se.: Eath Envion. Sci. 8 View the aticle online fo updates and enhancements. Related content - Identification of Climate Change with Genealized Exteme Value (GEV Distibution Appoach Anita Rahayu - Copula-based modeling of degeecoelated netwoks Mathias Raschke, Makus Schläpfe and Konstantinos Tantopoulos - An Empiical Fitting Method fo Type Ia Supenova Light Cuves. II. Estimating the Fist-light Time and Rise Time WeiKang Zheng, Patick L. Kelly and Alexei V. Filippenko This content was downloaded fom IP addess on 4//8 at 8:5

2 d Intenational Confeence on Wate Resouce and Envionment (WRE IOP Conf. Seies: Eath and Envionmental Science ( doi :.88/55-5/8// Highe moments method fo genealized Paeto distibution in flood fequency analysis C R Zhou, Y F Chen,, Q Huang and S H Gu College of Hydology and Wate Resouces Hohai Univesity, No. Xikang Road, Gulou Distict in Nanjing, Jiangsu Povince, China Shanghai Hydological Geneal Station, No. 4 East Longhua Road, Distict Xuhui in Shanghai, China @hhu.edu.cn Abstact. The genealized Paeto distibution (GPD has poven to be the ideal distibution in fitting with the peak ove theshold seies in flood fequency analysis. Seveal moments -based estimatos ae applied to estimating the paametes of GPD. Highe linea moments (LH moments and highe pobability weighted moments (HPWM ae the linea combinations of Pobability Weighted Moments (PWM. In this study, the elationship between them will be exploed. A seies of statistical expeiments and a case study ae used to compae thei pefomances. The esults show that if the same PWM ae used in LH moments and HPWM methods, the paamete estimated by these two methods is unbiased. Paticulaly, when the same PWM ae used, the PWM method (o the HPWM method when the ode equals shows identical esults in paamete estimation with the linea Moments (L-Moments method. Additionally, this phenomenon is significant when that the same ode PWM ae used in HPWM and LH moments method.. Intoduction Peak Ove Theshold (POT seies is an altenative to the annual maximum seies when studying chaacteistics of exteme events. Seveal distibutions can be used to descibe exteme events of dischage obsevations (annual floods. Fo POT seies, the genealized Paeto distibution (GPD has been poven to be a pefect candidate fo modeling POT samples [, ]. The GPD can descibe the patial duation seies (o POT seies which ae obtained by letting a theshold select the highest values fom an odinay hydological time seies []. Howeve, the POT appoach emains unde-employed ecently, mainly due to the complexities associated with its usage [4], such as the choice of thesholds, the selection of citeia fo etaining floods peaks, and so on. Paamete estimation is a cucial pat of flood fequency analysis (FFA. Many types of estimatos have been developed to impove the efficiency and obustness of the fitting techniques. Moment methods, including the method of moments (MOM, the pobability weighted moments (PWM method, and the linea moments (L-Moments method, use sample moments to estimate population moments. The law of lage numbes is the basic discipline of these methods. The PWM method showed advantages fo small datasets due to its smalle uncetainty than odinay moments []. This chaacteistic suggested that the PWM method was a moe suitable method fo elatively shot length of flood seies. The L-Moments method was fist put fowad by Hosking [5] and it has been widely Content fom this wok may be used unde the tems of the Ceative Commons Attibution. licence. Any futhe distibution of this wok must maintain attibution to the autho(s and the title of the wok, jounal citation and DOI. Published unde licence by Ltd

3 d Intenational Confeence on Wate Resouce and Envionment (WRE IOP Conf. Seies: Eath and Envionmental Science ( doi :.88/55-5/8// applied to estimate vaious distibutions. L-Moments, which ae linea combinations of the PWM, tend to shae simila chaacteistics with PWM. Additionally, Wang [6] developed LH moments which ae genealizations of L-Moments. Highe L-Moments (LH moments ely on the linea combinations of highe-ode statistics with the capability of a moe detailed analysis of annual flood peak data []. Pevious eseaches have concluded that highe moments gave moe consideation to significant flood peaks and extacted infomation existing in the POT seies to the geatest possible extent. Seveal studies have attempted to deduce calculation pocedues fo odinay moments, such as the emegence of, LH moments o highe PWM fo common distibutions (genealized Exteme Value, genealized Paeto, Peason Type-III, Gumbel and Weibull distibutions [8], to name a few. This pape is stuctued as follows: in the fist section, a bief intoduction to GPD and estimatos based on PWM will be pesented; in section, statistical expeiments wee given to investigate the pefomances of all methods studied in this pape. In section, these methods wee applied to a case study of the POT seies of the Yangtze Rive, China.. Backgounds The Paeto distibution (PD was fist put fowad by Vilfedo Paeto (an Italian civil enginee, economist, and sociologist. Pickands [9] extended it to the genealized Paeto distibution (GPD, and evealed its geat impotance to exteme value theoy. In this study, GPD with thee paametes,,,, which efe to the location, scale and shape of GPD espectively, was analyzed. The shape of GPD diffes with the value of shape paamete. The definition of GPD is given by: x (, G (,, x exp(, Consideing pactical application, adequate flood quantile estimates ae cucial. The quantile function is given by equation (., ( G x( G ( ln( G, The pobability weighted moments (PWM method is a widely used estimato in paamete estimation fo hydological distibutions, including GPD. Howeve, one should note that the PWM method is only applicable when.5. Consideing X is the andom vaiable and its distibution is F (x, the PWM of X is given by: M p,, s [ x( F ] F ( F df ( p Whee x(f is the quantile function; p,, s ae eal numbes; and efes to the ode of PWM. To obtain excellent statistics chaacteistics of GPD, p, s wee used. The PWM of X shown as equation (4. s ( M,, x( F F df; (,,,..., n (4 Let { x, x, x,... xn} be andom samples selected fom X. Sot the seies ascending, then it will become a new seies x (, x (,..., x ( n (note x( x(... x( n. Theefoe, PWM can be calculated by

4 d Intenational Confeence on Wate Resouce and Envionment (WRE IOP Conf. Seies: Eath and Envionmental Science ( doi :.88/55-5/8// equation (5. The b of samples ae estimatos to the of population. b n ( i ( i...( i x n (5 ( i ( n ( n...( n i If, the quantile function of X is x( G [ ( G ]. Thus, is given by: x( G G dg [ ( ( G ] G dg G G dg ( (6 ( Accoding to the GPD-based method of moments (MOM, the equations below (see equations ( to (9 put a constaint condition on the use of moment methods. Consequently, they ae theoetically applicable only when, and must not exceed.5 fo the eason that when. 5, the vaiance will tend to be infinite. Howeve, eseaches have peviously suggested that moments estimatos fo <.5. E (X ( Va ( X (8 ( ( If, then G ( G dg B (,, whee B (, be expessed as p q B ( p, q x ( x dx. Highe moments methods ( Skew ( X (9 is the Beta-function which can. Equation (6 can be changed into equation (., ( (.. Highe pobability weighted moments method Highe PWM (HPWM give moe weights to lage flows [] because the powe function F inceases with the values of samples. The HPWM method has been developed to be an estimato fo many hydological distibutions [, ]. These applications have futhe poven the availability and obustness of HPWM method. In this pape, the HPWM method was developed fo GPD, and the estimato to each paamete was listed. Accoding to the desciptions above, if,, the PWM fo GPD can be calculated as equation (. ( (, ( If, one can calculate as follows accoding to the HPWM method based on GPD.

5 d Intenational Confeence on Wate Resouce and Envionment (WRE IOP Conf. Seies: Eath and Envionmental Science ( doi :.88/55-5/8// ( (, ( ( ( (, (, ( ( ( ( (, (, (, (, ( (4 [( ( ] (, (, (5 ( (, (6 The key, clealy, is to establish a method to estimate. To calculate the estimato ˆ of, let the left side of equation (4 be Z ; then, its estimato can be expessed by ˆ ( b ( b Z ( b ( b. Then, the elationship between Z and can be investigated. Consideing that is a andom numbe within ~, so equation ( is given below. (, (, Z (, (, ( If and ae given, Z in equation ( can be easily calculated. Relationship between Z and wee dawn, and then these cuves wee fitted by polynomial. The esults appaently showed linea elationships between Z and (see figue. The linea fitting coefficients and linea fitting fomulas wee shown in table. Figue. Relationship diagam of Z ~. Ode Table. Linea fitting coefficients and linea fitting fomulas. Fitting coefficients and coelation coefficients R Linea fitting fomulas Z Z Z 5. 4

6 d Intenational Confeence on Wate Resouce and Envionment (WRE IOP Conf. Seies: Eath and Envionmental Science ( doi :.88/55-5/8// Z Z Z In conclusion, when,, HPWM can be used to estimate the paametes of GPD accoding to the following steps: Step : calculate PWM of the samples ( b, b, b, and apply them to compute Ẑ as the estimato of Z ; Step : substitute Ẑ into the linea fitting fomulas in table to calculate ˆ, such as we can calculate the estimate of ˆ when the ode of the HPWM equals : ˆ.Zˆ.. Step : substituteˆ into equation (5 to computeˆ ; Step 4: useˆ, ˆ to calculate ˆ by equation (6... L-Moments Hosking intoduced L-Moments as an altenative to Moments Method fo chaacteizing distibutions and data []. They ae linea combinations of PWM and have shown excellent pefomances as estimatos [-5]. PWM ae essential components of L-Moments, and PWM ( with,, can be calculated by equation (8. -,, (8, Given the elationship between L-Moments and PWM, L-Moments can be expessed by equation (9. (9 6 6 Refeing to the intoduction of PWM and HPWM, the L-moments of samples can be estimated by its PWM as ˆ b; ˆ b ˆ b; 6b 6b b. Theefoe, accoding to equations (8 and (9, L-M of the GPD samples with,, can easily lead to the expession of,, (see equation (., [,, ] ( ( ( 5

7 d Intenational Confeence on Wate Resouce and Envionment (WRE IOP Conf. Seies: Eath and Envionmental Science ( doi :.88/55-5/8// ( [6, 6,, ] ( ( ( Thus, equations ( to ( can be used to calculate the estimation of each paamete. ˆ ˆ ˆ ; ˆ ˆ ˆ ( ˆ ( ˆ( ˆ ˆ ( ˆ ˆ ( ˆ ˆ (.. LH Moments Wang [] was the fist to suggest LH moments as the genealization of L-Moments. Pevious eseaches have applied it into fitting with hydological distibutions [6-8], which have evealed the limitation in the usage of the poposed LH moments. Many studies intoduced the LH moments, but thee ae still ae studies about its coesponding calculation methods. Simila to L-Moments, the fist thee odes of LH moments can be defined by equations (4 and (5. ( [(! ( ] (4 [( 4(! ( ( ( ( ] ( 4( ( ( ( ( ( ( ( Accoding to the GPD-based PWM, in equation (5 above can be easily obtained. So equations above can be tansfomed into equations (6 and (. ( (, [(, (, ] [( 4(, ( (, ( (, ] 6 (5 (6 ( ( 4(, ( (, ( (, * (, (, ( Refeing to equation (, is the only unknown paamete coesponding to a given ode. 6

8 d Intenational Confeence on Wate Resouce and Envionment (WRE IOP Conf. Seies: Eath and Envionmental Science ( doi :.88/55-5/8// Because the calculation expession of is not that eady to get, it leads one to seek the numeical methods to it, and find the appoximate elationship between and. Let the value of change within the ange of [-,, and one can calculate each coesponding to diffeent. Then daw the points ( i, i coesponding to each given ode. Fit these points with the polynomial functions. Finally, cubic polynomial pefoms well in fitting with the points. So the appoximate elationship between and can be expessed as by equation (8. ( ( (8 Figue shows the esults of polynomial fitting and the elationship between and ; Table gives out the coefficients of polynomial fitting and coelation coefficients. Ode Figue. Appoximate elationship between τ and ζ. Table. The coefficients of polynomial fitting and coelation coefficients. The coefficients of polynomial fitting Coelation coefficients R The value of can be estimated by the PWM of samples (see equation (9. Thus the estimation of can be obtained by the polynomial elationship between them (equation (8. ˆ ( 4( b ( ( ( b ( b ( b ( ( b When the estimation of has been calculated, the othe two unknown paametes estimated by equations ( to (. ˆ ˆ ( [(, ˆ (, ˆ, (9 would be (

9 d Intenational Confeence on Wate Resouce and Envionment (WRE IOP Conf. Seies: Eath and Envionmental Science ( doi :.88/55-5/8// ˆ ˆ ˆ (, ˆ ˆ ˆ ( 4. Statistical expeiments and esults Statistical expeiments wee used to evaluate and compae the methods of PWM, HPWM, L-Moments and LH moments. Cunnane [9] demonstated that the unbiased plotting position, which was the mean of the ode statistics in samples fom the educed vaiety population, had the smallest mean squae eo among all such estimates. In ode to educe sampling eos as much as possible, a andom-numbe geneato was used in Matlab, to geneate andom numbes to follow the GPD. Two andom datasets, each with a size of * (paametes:.,.,.,.,, ae geneated. The data was ascendinglysoted. Accoding to theoy of ode statistics, the mean of each ode statistics was calculated as an unbiased plotting position. The paametes of the samples wee estimated by the method given o deduced above. All esults ae given in table. Fom table, it was clea that if the fist thee PWM wee used, the esults of diffeent methods wee identical to each othe. This was clea to see fom the esults of PWM, HPWM (, L-Moments and LH moments (. Moeove, when the same PWMs wee used, LH moments methods and HPWM method (when showed almost the same pefomances. Table. Estimatos and thei pefomances. Distibution Function: GPD; Paametes:.,, Paametes PWM HPWM Estimatos Paametes L-M LH moments Estimatos Distibution Function: GPD; Paametes:.,, Paametes PWM HPWM Estimatos Paametes L-M LH moments Estimatos Case study and esults Daily unoff seies of Cuntan and Hankou gauged stations (Yangtze Rive in China was selected fo futhe study to investigate the applicability of GPD and HPWM to the POTS obsevations and test the pefomance of HPWM and LH moments method in paamete estimation. Cuntan gauged station was 8

10 d Intenational Confeence on Wate Resouce and Envionment (WRE IOP Conf. Seies: Eath and Envionmental Science ( doi :.88/55-5/8// set up in the yea of 89. It monitos the fluxes and sand fom the uppe steam of Yangtze Rive, and this makes it be an impotant station to contol the inflow of the Thee Geoge Poject []. Hankou gauged station is set up in the yea of 865, but the daily unoff time seies fom 95 to 5 is included in this eseach due to they can be taken as stationay(pass the statistical test. Figue. The location of Cuntan Station (Download fom the pape of Yang, et al.. As a esult of anthopogenic pocesses as well as the constuction and opeation of hydoelectolical pojects, the hydologic popeties of Cuntan station have damatically changed. Thus, a consistency test must be studied befoe applying it to the FFA seies. Theefoe, the unoff data fom 99 to 4 was selected in this study. Accoding to the esults of statistical test (significance level=.5, the seies is stationay. Pevious studies [] showed moe easonable esults will be obtained when the numbe of annual aveage peaks to be to. So the theshold value was detemined to be m / s in this study. Thee ae 95 peaks occued in the seies, so appoximately.6 peaks pe yea. The same wok was done to Hankou gauged station, the theshold is settled as 4 m /s, and peaks can be obtained in total, o.46 peaks each yea (within to All methods mentioned in this pape ae used to fit with the peak unoff points. The esults wee listed in table 4, and all theoetical cuves and empiical points ae dawn in figue 4. Table 4 shows that all of the methods pefomed well fo paamete estimation. Paticulaly, the HPWM method and LH moments method seem to povide a boad ange of selections because they can give out diffeent esults that coespond to diffeent odes. Thus, one can select a bette set of paametes to descibe GPD in fitting with empiical points. The distibutions used to fit with peak unoff points wee dawn whee the best fit was achieved. In ode to descibe the goodness of fit, sum elative squae vaiation ( is used in this study: n xip x p ( % x ( i p Whee is the sum elative squae deviation and fequency p. xip is the estimato of x p coesponding to the 9

11 d Intenational Confeence on Wate Resouce and Envionment (WRE IOP Conf. Seies: Eath and Envionmental Science ( doi :.88/55-5/8// Figue 4. Relative diagams of empiical points and theoetical cuves, fo which paametes estimate by all methods mentioned in this pape, take Cuntan gauged station as an example. Table 4. Results of paamete estimation by seveal methods. PWM Cuntan gauged station: HPWM Paametes Estimatos L-M Cuntan gauged station: LH moments Paametes Estimatos PWM Hnakou gauged station: HPWM Paametes Estimatos

12 d Intenational Confeence on Wate Resouce and Envionment (WRE IOP Conf. Seies: Eath and Envionmental Science ( doi :.88/55-5/8// L-M Hankou gauged station: LH moments Paametes Estimatos Table 5 lists the means of between theoetical cuves (fo which the paametes ae estimated by diffeent methods and empiical points, especially lage empiical points (fo which the fequency is not ove 5% ( P ( X 5%. Accoding to the infomation shown in table 5 and figue 4, the pefomance of method does not significantly impove as highe PWM ae used. It is clea that when the ode anges fom to 4, deceases; howeve, it inceases afte the ode is moe than 4. The elative shows the pecentage of of P( X 5% takes in the sum. Clealy to see that with the ode inceases, the elative deceases damatically. So the theoetical cuves which paametes ae estimated by highe moments methods can fit with the lage flood points bette. Table 5. Sum elative squae vaiations ( between flood peaks and theoetical values. PWM Cuntan gauged station: HPWM Methods 4 5 ( P 5% (Summay Relative 5.59% 5.59%.6% 58.58% 4.46% 8.8%.% L-M Cuntan gauged station: LH moments Methods 4 5 ( P 5% (Summay Relative 5.59% 5.59%.% 5.45% 4.4% 8.6%.69% PWM Hankou gauged station: HPWM Methods 4 5 ( P 5% (Summay Relative 85.% 85.% 8.4% 8.68% 84.68% 8.9% 8.6% L-M Hankou gauged station: LH moments Methods 4 5 ( P 5% (Summay Relative 85.% 85.% 9.% 8.8% 84.44% 86.9% 8.54% Note: Relative = ( P 5% / (summay.

13 d Intenational Confeence on Wate Resouce and Envionment (WRE IOP Conf. Seies: Eath and Envionmental Science ( doi :.88/55-5/8// 6. Conclusions Seveal estimatos, like PWM and L-Moments, wee fist intoduced in this pape. In addition, the GPD-based appoximate calculation methods of HPWM and LH moments wee mainly deduced. A statistical expeiment and a case study wee applied to investigate and the was the feasible measuement of these methods. Following conclusions can be dawn fom the esults of this study. Both the statistical expeiment and case study poved that the GPD-based HPWM and LH moments deduced in this pape wee feasible and applicable in paamete estimation. When the same ode PWM wee used in HPWM and LH moments methods, the esults wee identical. Fo example, the HPWM method (when showed simila pefomance to the methods of PWM, L-Moments and the LH moments (when ; the LH moments method (when pefomed as well as the HPWM method (when when the same PWM wee used in calculation. Howeve, fom the standpoint of which was used to measue the goodness-of-fit of each method, the ode of the PWM affected the pefomances of the highe moments methods. Fo example, when the ode inceased fom to 4, the pefomances of the HPWM method and LH moments method became bette. But when exceeded 4, the pefomances of eithe the HPWM method o LH moments method became noticeably wose. The methods with highe moments tended to show moe flexibility in fitting flood seies on the up tail fo example, the theoetical cuves fit well with the empiical points which fequency wee less than 5%. Compaing all of the methods outlined in this pape, it became clea that the HPWM method showed the same pefomance as the LH moments method and they used much moe infomation in the flood seies than the PWM method and L-Moments method. Howeve, unlike LH moments method, the HPWM method was much easie to calculate. So HPWM method is a moe pefeable one. Refeences [] Wang J and Song S Application of genealized Paeto distibution in flood seies fequency analysis J. Nothwest A. & F. Univ. (Nat. Sci. Ed (in Chinese [] Bemudez P D Z and Kotz S Paamete estimation of the genealized Paeto distibution pat - II J. Stat. Plan. Infe [] Diedda R and Puliga M 9 Pefomances of some paamete estimatos of the genealized Paeto distibution ove ound-off samples Phys. Chem. Eath [4] Lang M, Ouada T B M J and Bobée B 999 Towads opeational guidelines fo ove- theshold modeling J. Hydol. 5 - [5] Hosking J R M 99 L-moments: Analysis and estimation of distibutions using linea combinations of ode statistics J. Roy. Stat. Soc. B [6] Wang Q J 99 LH moments fo statistical analysis of exteme events Wate Resou. Res [] Meshgi A and Khalili D 9 Compehensive evaluation of egional flood fequency analysis by L- and LH moments I. A e-visit to egional homogeneity Stoch. Envion. Res. Risk Assess. 9-5 [8] Busababodhin P, Seo Y A, Pak J, et al 6 LH-moment estimation of Wakeby distibution with hydological applications Stoch. Envion. Res. Risk Assess. 5-6 [9] Pickands J 95 Statistical infeence using exteme ode statistics Ann Stat 9- [] Wang Q J 99 Using highe pobability weighted moments fo flood fequency analysis J. Hydol. 94: 95-6 [] Song S, Li Y and Kang Y Calculation highe pobability weighted moments fo genealized Exteme value in hydology Adv. Mate. Res (in Chinese [] Li Y and Song S Application of highe pobability weighted moments of P-III distibution to flood fequency analysis J. Hydoele. Eng. -8 (in Chinese

14 d Intenational Confeence on Wate Resouce and Envionment (WRE IOP Conf. Seies: Eath and Envionmental Science ( doi :.88/55-5/8// [] Pilon P J, Adamowski K and Alila Y 99 Regional analysis of annual maxima pecipitation using L-moments Atmos. Res. 8-9 [4] Hosking J R M and Wallis J R 99 Some statistics useful in egional fequency analysis Wate Resou. Res. 9-8 [5] Chen Y, Wang Q, Sha Z, et al Application of L-moments based egional flood fequency analysis method to middle and lowe Yangtze Rive J. Hohai Univ. (Nat. Sci. Ed. - (in Chinese [6] Bhuyan A, Boah M and Kuma R Regional flood fequency analysis of Noth-bank of the ive Bahmaputa by using LH-Moments Wate Resou. Manag [] Hewa G A, Wang Q J, McMahon T A, et al Genealized exteme value distibution fitted by LH moments fo low-flow fequency analysis Wate Resou. Res. 4 - [8] Mushed M S, Pak B J, Jeong B Y, et al 9 LH-Moments of some distibutions useful in hydology Communications of the Koean Statistical Society [9] Cannane C 98 Unbiased plotting positions - a eview J. Hydol. 5- [] Yang S L, Zhang J and Xu X J Influence of the Thee Goged Dam on downsteam delivey of sediment and its envionmental implications, Yangtze Rive Geogaphs. Res. Lett. 4 Doi:.9/GL94 [] Fang B, Guo S L, Chai X L, et al 5 Flood fequency analysis based on FPOT method Wate Powe 9- (in Chinese