Suitability of Sample Size for Identifying Distribution Function in Regional Frequency Analysis

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1 京都大学防災研究所年報第 5 号 B 平成 9 年月 Auals of Disas. Prev. Res. Ist., Kyoto Uiv., No. 5 B, 7 Suitability of Sample Size for Idetifyig Distributio Fuctio i Regioal Frequecy Aalysis Biaya Kumar MISHRA*, Yasuto TACHIKAWA* ad Kaoru TAKARA *Graduate school of Urba ad Eviromet Egieerig, Kyoto Uiversity Syopsis Estimatio of evirometal extremes usig regioal frequecy aalysis eeds fittig of a appropriate frequecy distributio fuctio represetig a homogeeous regio. Best fit frequecy distributio depeds o the computed momet coefficiets usig either covetioal method of momet or recetly developed method of L-momet. These momet coefficiets deped o the legth of observed data record. This paper examies the deviatio of momet coefficiets with the legth of record i terms of root mea square error (RMSE). It shows that percetage deviatio is about 5% for sample size of ad decreases to ear or below % for sample size of 5. Keywords: Distributio fuctio, L-momets, Sample size, Momet coefficiets. Itroductio Reliable estimatio of the hydro-metrological extremes for the give retur period is of particular importace i plaig ad desig of hydraulic structures. Frequecy aalysis relates the magitude of extreme evets to their frequecy of occurrece through the use of probability distributios. It eeds a large umber of historical observed data at the place of iterest. I practice, either there is o data or is of very short legth. I such cases, regioal frequecy aalysis ca be a effective tool. Estimatio of frequecy distributio fuctio represetig a homogeeous regio is oe of the importat major steps i regioal frequecy aalysis. Best fittig frequecy distributio fuctio represetig the regioal data ca be determied from the plot of skewess versus correspodig kurtosis usig either covetioal method of momets or recetly developed L-momet method agaist various frequecy distributio fuctio represetig lies. A diagram based o C s ad C k, such as that i Figure, ca be used to idetify appropriate distributios i covetioal methods. A diagram based o LC s ad LC k, such as that i Figure, ca be used similar to covetioal momet ratio diagrams to idetify appropriate distributios i the method of L-momet. The locatio of sample estimate with respect to the distributio gives a idicatio of the suitability of a distributio of the data. A suitable paret distributio is that which averages the scattered ad aroud which the poits spread cosistetly. However, if the sample size is small, the bias i the values of higher momets may be large eough to give misleadig results (Rao & Hamed, ). Misleadig of best fittig distributio fuctio takes place sice momet ratios (coefficiet of variace, skewess ad kurtosis) varies largely with the legth of data or sample size. Therefore, it is very much importat to have the suitable legth of data (sample size) for idetifyig best-fit distributio fuctio i regioal frequecy aalysis

2 Fig. C s -C k momet ratio diagram (Rao & Hamed, ). Data screeig Aual maximum daily raifall values of Hirakata, Gojyou, Kameoka ad Katada rai-gauge statios lyig i the Yodo River basi regio, Japa have bee used to carry out the job. All these data were made available from the data bak of Iovative Disaster Prevetio Techology ad Policy Research Lab, Disaster Prevetio Research Istitute, Kyoto Uiversity, Japa. At every cosidered rai-gauge statio, sub-samples were formed from the whole sample cosiderig cotiuous dataset of differet sample size varyig from to 8 at a icremet of. Test for idepedece ad homogeeity (assumptio that the whole set data come from the same distributio) of the data was performed as discussed by Ma-Whitey (97) ad checked at 5% sigificace level. Test for outlier (a observatio that deviates largely from the bulk of the data) was performed as discussed by Grubbs (969) ad checked at 5% sigificace level. Fig. LC s -LC k momet ratio diagram (Rao & Hamed, 99) I the preset paper, it is iteded to show the fluctuatio patter of momet coefficiets (skewess ad kurtosis) i terms of root mea square error (RMSE) with the chage i sample size usig covetioal method of momets ad recetly developed method of L-momet. This fluctuatio patter will provide guidace for determiig sample size i regioal frequecy aalysis.. Methodology Data screeig of the collected data to check for idepedece, homogeeity, outliers etc., calculatio of momet coefficiet for data set of differet sample size ad fially calculatio of root mea square error (RMSE) are the major steps ivolved to carry out the preset study. These steps have bee discussed i the followig sectios.. Momet coefficiets The computatio relatioships for mostly used momet ratios amely coefficiet of variace (C v or LC v ), coefficiet of skewess (C s or LC s ) ad coefficiet of kurtosis (C k or LC k ) have bee discussed i followig subsectios... Covetioal momets Momet coefficiets, particularly coefficiet of skewess ad coefficiet of kurtosis, are used to characterize regioal probability distributios. If x i is observed raifall data at statio i ad is the o. of raifall records, these momet coefficiets are calculated as: Sample mea, m = x = x i i= Variace, m = ( x i x) i= Stadard deviatio, σ = (variace) / Third momet, m = ( x i x) ( )( ) i= Fourth momet, m = ( x i x) ( )( )( ) i= Momet coefficiets i.e. momet ratios are calculated as give below: Coefficiet of variatio, C v = σ/ mea = m / / x - 7 -

3 Coefficiet of skewess, C s =m / σ Coefficiet of Kurtosis, C k = m / σ.. L-momets L-momets (Hoskig ad Wallis, 997) based method is a simpler ad effective alterative as compared to covetioal method to select the best-fit distributio fuctio for a regio. For the ordered sample x j= x j= ---- x j= at ay statio, the first four L-momets are calculated as: l = b, l = b b l = 6b 6b + b l = b b + b b Where, bo = x j j= b = ( j ) x j ( ) j= b = ( j )( j ) x j ( )( ) j= b = ( j )( j )( j ) x j ( )( )( ) j= L-momet ratios are expressed as follows: L-coefficiet of variatio, LC v =l /l L-coefficiet of skewess, LC s =l /l L-coefficiet of kurtosis, LC k =l /l. Error calculatio Bias ad root mea square error (RMSE) are commo measures of the performace of a estimator (Hoskig & Wallis, 997). Bias is expressed as mea of the differece betwee parameter obtaied from available sample ad the parameter represetig the populatio data set. But RMSE is expressed as mea of ((θ s -θ p ) ) / where θ s ad θ p represets sample parameter ad populatio parameter respectively. Both the bias ad RMSE of parameter have the same uits of measuremet as the parameter. It is coveiet to express bias ad RMSE as ratios with respect to the parameter itself ad hece, we obtai dimesioless measures, the relative bias ad relative RMSE. Cotributios of egative ad positive biases may cacel out to give a misleadigly small value of the bias. Hece, the preset study has bee dealt i terms of RMSE oly as the average relative RMSE measures the overall deviatio of sample quatiles from populatio quatiles.. Results ad discussio As discussed i previous sectio, data were first aalyzed to check for idepedece, homogeeity ad outliers. Data sets of all the cosidered statios were foud idepedet ad homogeeous at 5% sigificace level. For example, value of z, a idicator for idepedece ad homogeeity test by Ma-Whitey (97) method, was foud as.9. This value is lesser tha.96 which is the critical value for sample size greater tha at 5% sigificace level. Hece, the populatio data set was assumed as idepedet ad homogeeous. The presece of outliers i the data causes difficulties whe fittig a distributio to the data. Presece of eve a sigle outlier, may be o upper or lower side, affects the momet coefficiet much. For example, deviatio i momet coefficiet values i presece ad absece of outlier at Hirakata statio has bee compared ad show i table. The fall i LC s ad LC k was foud as % ad %; whereas i C s ad C k was foud as 6% ad 69% after deletig the sigle outlier i upper side. The table shows that covetioal momet coefficiets are much deviatig with the presece of outlier as compared to L-momet coefficiets. I other words, covetioal momet coefficiets are more sesitive to the presece of outliers as compared to method of L-momets. Figures -8 shows variatio patter of momet coefficiets (skewess ad kurtosis) at Hirakata as a example. I geeral, similar variatio patter were obtaied at all cosidered rai-gauge statios. Cosiderig all the results, error i both covetioal ad L-momet coefficiet of skewess ad kurtosis is very large (poits are deviated largely from oe aother) for sample size of. Error i momet coefficiets is reduced largely to a sample size of 5 or more ad hece, deviatios i betwee poits are smaller as show i Figures,, 5 ad 6. As show i Figures 7 & 8, poits plotted for skewess versus kurtosis for dataset of smaller sample size deviated very much from each other as well as from the poit of whole sample size ad got cocetrated as the sample size icreased to 5. Deviatio i momets coefficiets for differet sub-samples size from that of the whole sample (9 years) has bee show umerically i table as a example at Hirakata i terms of RMSE

4 Table Compariso of momet coefficiet due to presece of outliers at Hirakata, Japa Momet Coeff. Without outlier With outlier Error % error LC s LC k C s C k L-skewess vs Sample size Covetioal skewess vs Sample size.5.8 L-skewess (LCs) Skewess (Cs) Fig. L-momet skewess versus sample size at Hirakata, Japa Fig. Covetioal momet skewess versus sample size at Hirakata, Japa L-kurtosis vs Sample size Covetioal kurtosis vs Sample size L-kurtosis (LCk) Kurtosis (Ck) Fig. 5 L-momet kurtosis versus sample size at Hirakata, Japa Fig. 6 Covetioal momet kurtosis versus sample size at Hirakata, Japa Sample size Table Momet coefficiet for differet sample size dataset at Hirakata, Japa L-momet Covetioal momet L-momet Covetioal momet RMSE i RMSE i RMSE i RMSE i % error i %ge error % error i % error i LC s LC k C s C k LC s i LC k C s C k

5 L-skewess vs L-kurtosis Skewess vs Kurtosis L-kurtosis (LCk) L-skewess (LCs) =9 = = = =5 =6 =7 =8 Kurtosis (Ck) Skewess (Cs) =9 = = = =5 =6 =7 =8 Fig. 7 LC s -LC k for differet sample size at Hirakata, Japa While calculatig the value of root mea square error (RMSE) for each data set, populatio coefficiets were assumed as that resulted from the whole data set which has legth of more tha 9. Coclusios Fidigs of the paper are as follows:. Covetioal momet coefficiets are more sesitive to outlier observatio as compared to L-momet coefficiet.. Deviatio patter of momet coefficiets (skewess ad kurtosis) with respect sample size is similar for both methods showig there is ot much comparative advatages of L-momets method over covetioal i terms of sample size.. Idetified frequecy distributio fuctio with the help of data set of sample size or less may ot be best-fit frequecy distributio represetig a regio. Fig. 8 C s -C k for differet sample size at Hirakata, Japa years. It shows that average % deviatio is about 5% for sample size ad decreases ear or below % as the sample icreases to 5. Refereces Grubbs, F.E. (969): Procedures for detectig outlyig observatios i samples, Techometrics, Vol. (), pp. -. Hoskig, J.R.M. ad Wallis, J.R. (997): Regioal Frequecy Aalysis, Cambridge Uiversity Press, Uited Kigdom. Ma, H.B. ad Whitey, D.R. (97): O a test of whether oe of two radom variables is stochastically larger tha the other, The auals of mathematical statistics, Vol. 8, pp Rao, A. R. ad Hamed, K.H. (99): Frequecy aalysis of upper Cauvery Flood data by L-momets, Water resources maagemet 8:8-. Rao, A. R. ad Hamed, K.H. (): Flood Frequecy Aalysis, CRC Press LLC, Florida

6 地域頻度解析における分布関数同定のためのサンプルサイズの適切性 Biaya Kumar MISHRA * 立川康人 * 宝馨 * 京都大学大学院工学研究科都市環境工学専攻要旨地域頻度解析において極値水文量を推定するためには, 同質と考えられる地域を代表する適切な頻度分布関数を決定する必要がある頻度分布の母数推定には, 積率法やL 積率法が用いられ, 推定された母数の値は同定に用いる時系列データの期間に依存する本研究では, 観測時系列データの長さによって現れる母数推定値の違いを RMSEを用いて分析するその結果,サンプルを用いた場合には5% 近い母数推定値の違いが現れ,5サンプルとするとそれが約 % に減少することがわかったキーワード : 分布関数,L 積率法, サンプルサイズ, モーメント係数 - 7 -

These characteristics are expressed in terms of statistical properties which are estimated from the sample data.

These characteristics are expressed in terms of statistical properties which are estimated from the sample data. 0. Key Statistical Measures of Data Four pricipal features which characterize a set of observatios o a radom variable are: (i) the cetral tedecy or the value aroud which all other values are buched, (ii)