This paper discusses comparisons of the LH moments method with L moments method. LH moments, a generalization of L moments, based on linear combinations of higher-order statistics was introduced for charactering the upper part of distributions and larger events in data by Wang (1997). Analysis of observed data shows that using LH moments estimates of the upper part of distribution events are expected to be more reasonable than the L moments estimates. A comparison of the LH moment diagram and the L moment diagram of the data also shows that the GEV distribution describes the LH moment ratios better than the L moments

Shabri, Ani

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Universiti Teknologi Malaysia Institutional Repository

Matematika, 2002, Jilid 18, bil. 1, hlm. 33–43c©Jabatan Matematik, UTM.Comparisons of the LH Momentsand the L MomentsAni ShabriDepartment of MathematicsUniversiti Teknologi Malaysia,81310 UTM Skudai, Johor, MalaysiaAbstract This paper discusses comparisons of the LH moments method withL moments method. LH moments, a generalization of L moments, based onlinear combinations of higher-order statistics was introduced for charactering theupper part of distributions and larger events in data by Wang (1997). Analysisof observed data shows that using LH moments estimates of the upper partof distribution events are expected to be more reasonable than the L momentsestimates. A comparison of the LH moment diagram and the L moment diagramof the data also shows that the GEV distribution describes the LH moment ratiosbetter than the L moments.Keywords LH moments, GEV, PWM, Linear Combination, Higher-Order Sta-tisticsAbstrak Kertas ini membincangkan perbandingan antara kaedah LH momendengan kaedah L momen. LH momen adalah L-momen umum yang berasaskankepada gabungan linear bagi statistik tertib tinggi digunakan untuk dipadankandengan ciri-ciri hujung taburan dan data ekstrim. Analisis terhadap data meng-gunakan penganggar LH momen bagi hujung atas taburan didapati LH lebihbaik berbanding penganggar L momen. Perbandingan bagi gambarajah LHmomen dan gambarajah L momen ke atas data juga menunjukkan bahawa tabu-ran GEV dapat dipadankan dengan baik dengan nisbah LH momen berbandingdengan nisbah L momen.Katakunci LH Moments, GEV, PWM, Gabungan Linear, Statistik Peringkat-Tinggi1 IntroductionFloods are extreme events, which may kill people and destroy property. The probabilitythat such floods occur is an essential input for the design of hydraulic structures and alsofor estimating the risks involved. Statistical analysis of extreme events are often conducted34 Ani Shabrifor predicting large return period events. The relationship between flood magnitude andits return period is of great importance to avoid high damages. However, estimating floodsof large return periods is difficult because extreme events are by definition rare and therelevant data record is often short. Regional frequency using conventional moments [5],[8] and L-Moment analysis can solve this problem. However conventional moments havebeen criticized for being sensitive to the upper part of the distributions and thus sampleoutliers [12]. L moments were found as an alternative to product moments for characterizingdistributions and data [3]. Since 1990, many researchers have investigated application ofL moments to analyzing extremes. Examples of works L moments can be found in theworks by Ani [1], Al-Khudhairy [2], Hosking [3], Hosking and Wallis [4] and Vogel et al.[10,11]. LH moments are generalization of L moments have been introduced to characterizethe upper part of distributions and larger events in data [12, 13]. Wang [12, 13] foundthe method of LH moments provides an analytical means of fitting a distribution to thelarger sample events without explicit sample censoring. In this paper, L moments and LHmoments are used for characterizing the upper part of distributions and larger events indata. LH moments and L moments are compared by using real data. LH moments and Lmoments diagrams are constructed using annual maximum flow data from 31 streams inMalaysia to examine the appropriateness of the General Extreme Value (GEV) distributionin modeling the extreme events.2 General Extreme Value DistributionThe most widely accepted distribution for describing flood frequency data in U.K.[7], Aus-tralia and United Stated [9, 10] is the GEV. It was introduced by Jenkinson in 1955, andrecommended by the Natural Environmental Research Council [7]. The cumulative distri-bution function of the GEV can be writtenF (x) = exp{−[1−kα(x − ξ)]1/k}if k 6= 0 (1)= exp{− exp[− 1α(x− ξ)]}if k = 0 (2)or in inverse formx(F ) = ξ + αk[1− (− lnF )−k]for k 6= 0 (3)x(F ) = ξ − α ln− lnF for k = 0 (4)where ξ is a location parameter, α is a scale parameter and k is shape parameter. For k = 0,the distribution is unbounded both above and below, and it referred to as the extreme valueType I (EV1) or Gumbel distribution. For k < 0, the distribution has a finite lower boundat ξ+α/k and a thick upper tail, and it referred to as the Type II extreme value distribution.For k > 0, the distribution has a finite upper bound at ξ + α/k, and is referred to as theType III extreme value distribution.Normalized Probability Weighted Moments (PWM) of the GEV distribution are givenfor k 6= 0Comparisons of the LH Moments and the L Moments 35Br = ξ +αk[1− Γ(1 + k)(r + 1)−k](5)and for k = 0Br = ξ + α [+ ln(r + 1)] (6)where Γ() is Gamma function and  = 0.5772 is the Euler constant [12].3 Higher Probability Weighted Moments and LH mo-mentsThe LH moments are based on linear combination of higher probability weighted momentswas introduced by Wang 1997 [12]. For the first four LH moments are defined asλη1 = E[X(η+1) : (η+1)](7)λη2 =12E[X(η+2) : (η+2) −X(η+1) : (η+2)](8)λη3 =13E[X(η+3) : (η+3) − 2X(η+2) : (η+3) +X(η+1) : (η+3)](9)λη4 =14E[X(η+4) : (η+4) − 3X(η+3) : (η+4) + 3X(η+2) : (η+4)−X(η+2) : (η+4)] (10)whereE [Xj : m ] =m!(j − 1)!(m− j)!∫ 10x(F )F j−1dFWhen η = 0, LH moments become identical to L moments. As η increases LH momentsreflect more and more the characteristics of the upper part of distributions and larger eventin data. LH moments are called L1 moments, L2 moments, . . . for η = 1, 2, respectively.LH coefficient of variation, skewness, and kurtosis areτη2 =λη2λη1(11)τη3 =λη3λη2(12)τη4 =λη4λη2(13)36 Ani Shabrirespectively. Given a ranked sample x1 ≤ x2 ≤ . . . ≤ xn, the unbiased estimator of LHmoments are given byλˆη1 =1( nη+1)n∑i=1(i− 1η)xi (14)λˆη2 =12( nη+2)n∑i=1{(i− 1η + 1)−(i− 1η)(n− 11)}xi (15)λˆη3 =13( nη+3)n∑i=1{(i− 1η + 2)− 2(i− 1η + 1)(n− 11)+(i− 1η)(n− i2)}xi (16)λˆη4 =14( nη+4)n∑i=1{(i− 1η + 3)− 3(i− 1η + 2)(n− i1)+ 3(i− 1η + 1)(n− i2)−(i− 1η)(n− i3)}xi (17)where(mj)=m!j!(m− j)!(18)is equal to zero when j > m. The first four LH moments for GEV distribution are givenfor k 6= 0λη1 = ξ +αk[1− Γ(1 + k)(η + 1)−k](19)λη2 =(η + 2)αΓ(1 + k)2!k[−(η + 2)−k + (n+ 1)−k](20)λη3 =(η + 3)αΓ(1 + k)3!k[−(η + 4)(η + 3)−k + 2(η + 3)(η + 2)−k−(η + 2)(η + 1)−k](21)λη4 =(η + 4)αΓ(1 + k)4!k[3(η + 5)(η + 4)(η + 3)−k − 3(η + 4)(η + 3)(η + 2)−k+(η + 3)(η + 2)(η + 1)−k − (η + 6)(η + 5)(η + 4)−k](22)and for k = 0λη1 = ξ + α [+ ln(η + 1)] (23)λη2 =(η + 2)α2![ln(η + 2)− ln(η + 1)] (24)λη3 =(η + 3)α3![(η + 4) ln(η + 3)− 2(η + 3) ln(η + 2) + (η + 2) ln(η + 1)] (25)λη4 =(η + 4)α4![−3(η + 5)(η + 4) ln(η + 3) + 3(η + 4)(η + 3) ln (η + 2)−(η + 3)(η + 2) ln(η + 1) + (η + 6)(η + 5) ln(η + 4)] (26)The three parameters ξ, α and k in the GEV distribution can be estimated by matchingto the first three LH moments to their sample estimates for a selected η. By using theComparisons of the LH Moments and the L Moments 37approximationk = α0 + α1[τη3 ] + α2[τη3 ]2 + α3[τη3 ]3 (27)where η as given in the table 1.Table 1: Values of Coefficients Varywith η = 0 and 2η α0 α1 α2 α30 0.2849 −1.8213 0.8140 −0.28352 0.5914 −2.3351 0.7442 −0.1616k is obtained using (21) and (22) or (24) and (25). This gives a solution for α and ξ.4 Application To Annual Flood DataCharacteristics of stream flow records from 31 stations in Malaysia used this study aresummarized in Table 2. The data was obtained from the Department of Irrigation andDrainage Malaysia. The record length ranges are from 14 to 37 years, with average of 22years and median of 20 years. The data are used here to demonstrate the effect of L2moments (L2-mom), and L moments (L-Mom) in estimating the GEV distribution for floodfrequency analysis.Each of the data series is fitted to the GEV distribution by using L-Moment and L2-Moment and the GEV for k = 0 (EV1) distribution by using L-moment. As a first example,the annual maximum flows at site 9 was analysed. The data are plotted as filled circlesagainst the Gumbel reduced variable in the probability plot in Figure 1 (a).The nonexceedance probability is calculated using the Gringorten formula [13]Fi =i− 0.44n+ 0.12where i is the rank and n is the sample size. The Gumbel reduced variable is calculated asgi = − ln {− ln(Fi)}The fitted distribution is plotted in Figure 1(a) as the continuous line. There are threelines will give very different estimates of floods. The EV1 distribution has a straight line,while GEV distribution having a concave shape. The graph show that the fitting is in seriouserror, especially for larger flows. The GEV using L2-mom was found is the best followed byGEV using L-mom and EV1 using L-mom. Second and third examples are analyses usingdata from site number 16 and 29 respectively. The data are plotted against the Gumbelreduced variable in Figure 1 (b) and Figure 1 (c). The advantages of using L2 moments arealso evident here. Root mean square error (RMSE) was used to evaluate the precision ofthe methods of parameter estimate of GEV distribution using L-Mom (GEV-Lmom) and38 Ani ShabriTable 2: Characteristics of Streamflow Recordsof 31 Stations in MalaysiaStation Site Years ofL Moment Ratio LH MomentRatio η = 2Number Recordτ3 τ4 τ ′′3 τ ′′41732401 1 16 0.141 0.001 0.075 −0.0191737451 2 32 0.276 0.111 0.272 0.1061836402 3 18 0.286 0.156 0.292 0.0962130422 4 21 0.09 0.207 0.288 0.1522235401 5 18 0.358 0.140 0.281 0.0512237471 6 34 0.865 0.768 0.726 0.5642527411 7 22 0.189 0.156 0.318 0.2902928401 8 15 0.323 0.384 0.545 0.4503030401 9 16 −0.062 0.188 0.229 0.1353224433 10 21 0.243 0.189 0.334 0.1843329401 11 14 0.077 0.007 0.140 0.2463519426 12 29 0.160 0.157 0.286 0.1483629403 13 23 0.158 −0.086 −0.043 −0.1424019462 14 34 0.332 0.268 0.413 0.2674023412 15 26 0.096 0.012 0.119 0.1394121413 16 21 −0.04 0.016 0.024 0.0634131453 17 14 0.636 0.464 0.577 0.4234218416 18 15 −0.201 0.113 −0.023 0.0734219415 19 16 0.171 0.217 0.333 0.1634223450 20 19 0.427 0.147 0.316 0.1174232452 21 19 −0.018 0.095 0.181 0.2334732461 22 16 0.079 0.068 0.163 0.1554832441 23 25 0.769 0.589 0.623 0.3865129437 24 18 0.223 0.168 0.302 0.1265130432 25 32 0.440 0.237 0.404 0.2195229436 26 15 0.152 0.032 0.112 0.0105320443 27 23 0.365 0.285 0.440 0.2615428401 28 18 0.197 0.128 0.243 0.0775721442 29 37 0.368 0.168 0.332 0.1505724411 30 20 0.788 0.812 0.777 0.6926019411 31 31 0.062 0.183 0.240 0.086Comparisons of the LH Moments and the L Moments 39L2-Mom (GEV-L2Mom) and EV1 distribution using L-Mom (EV1-Lmom). RMSE can beexpressed asRMSE =√√√√ 1nn∑i=j(xi − xFixi)2where xi is ordered set observation values and xFi is computed observation values fora given value of Fi. For a complete data series (0 ≤ F ≤ 1), j = 1 while for j values wasdetermined based on sample size data and Gringorten formula.Table 3 gives the values of RMSE for a complete data series (0 ≤ F ≤ 1) and the upperpart of distributions (0.9 ≤ F ≤ 1) ).The 2 methods were ranked for all the stations according to the values of RMSE ona scale 1 to 3, with one being the best method and the result is given in Table 4 and 5.Clearly, the GEV using L-mom methods was the best of all, followed by the EV1 usingL-mom and GEV using L2-mom for a complete data series, (0 ≤ Fi ≤ 1). While for theupper part of distributions (0.9 ≤ Fi ≤ 1) the GEV using L2-mom is the best followed byGEV using L-mom and EV1 using L-mom.5 Comparison of L-Moment and LH moments Ratio Di-agramFigure 2 compares the kurtosis versus skewness curve of L moment diagram based onequations (12) and (13) for −0.65 ≤ k ≤ 0.65 and η = 0. Also plotted in the diagramis L skewness and L kurtosis for 31 stations. Most of the sample estimates of L kurtosis andL skewness are seen to fall far from the theoretical GEV curve. Figure 3 is the L2 momentdiagram of skewness and kurtosis for the same 31 stations. L2 moment diagram also foundbased on equations (12) and (13) for −0.65 ≤ k ≤ 0.65 but with η = 2. The L2 kurtosisestimates scatter much more evenly above and below and also close to the theoretical GEVcurve as compared to the L moments diagram of Figure 2. This shown that the GEVdistribution using L2 moment ratio better than L moment ratios.6 ConclusionLH moments are based on linear combination of higher order statistics can be used forcharacterizing the upper part of distributions and larger events in a sample. L momentsare a special case of LH moments. A comparison of the L2 moment diagram and the Lmoment diagram of annual maximum flow of 31 stations shows that the GEV distributionusing the L2 moment ratios better than the L moments ratios with respect to this data set.It is suggested that LH moments be used as a matter of routine for estimating floods oflarge return periods.7 AcknowledgmentsThe author would like to thank the Department of Irrigation and Drainage Malaysia forproviding the flood flow data of streams in Peninsular Malaysia. The computing facili-40 Ani ShabriTable 3: Values of RMSE for A Complete Data Series 0 ≤ F ≤ 1And The Upper Part of Distribution (0.9 ≤ F ≤ 1)Station0 ≤ F ≤ 1 0.9 ≤ F ≤ 1NoGEV EV1 GEV EV1L-MOM L2-MOM L-MOM L-MOM L2-MOM L-MON1 0.023 0.544 0.019 0.009 0.003 0.0122 0.024 0.146 0.061 0.014 0.006 0.0073 0.005 0.050 0.020 0.007 0.003 0.0044 0.016 0.077 0.034 0.001 0.002 0.0025 0.022 0.580 0.148 0.027 0.008 0.0126 0.076 0.341 26.600 0.391 0.323 0.8067 0.004 0.006 0.004 0.016 0.013 0.0178 0.005 0.020 0.008 0.016 0.016 0.0279 0.009 0.147 0.047 0.001 0.000 0.00510 0.005 0.004 0.021 0.009 0.009 0.00911 0.017 0.037 0.011 0.002 0.003 0.00212 0.013 0.015 0.012 0.000 0.001 0.00013 0.108 8.396 0.097 0.039 0.003 0.04214 0.005 0.006 0.007 0.010 0.009 0.02515 0.051 0.330 0.026 0.003 0.003 0.00816 0.000 0.000 0.000 0.000 0.000 0.00017 0.037 0.150 3.764 0.086 0.067 0.15118 0.003 0.066 0.013 0.008 0.002 0.00519 0.000 0.000 0.000 0.000 0.000 0.00020 0.280 7.647 2.854 0.028 0.001 0.01221 0.030 0.043 0.023 0.006 0.004 0.01022 0.034 0.077 0.022 0.006 0.006 0.00723 0.409 11.839 104.741 0.374 0.226 0.13724 0.078 0.116 0.150 0.005 0.004 0.00425 0.047 0.692 1.293 0.047 0.049 0.08126 0.022 1.404 0.016 0.007 0.002 0.00827 0.008 0.016 0.073 0.015 0.016 0.03228 0.050 5.090 0.304 0.009 0.006 0.00729 0.028 0.422 0.275 0.007 0.001 0.01630 0.378 2.283 12.843 0.628 0.256 4.54631 0.015 0.046 0.037 0.000 0.001 0.004Comparisons of the LH Moments and the L Moments 41Table 4: Ranking of the Methods of parameterEstimator for the 31 Stations by RMSEfor a full data series, (on a scale of 1 to 3with 1 being the best method)Number of Stations Receiving RankingMethod 1 2 3GEV - L MOM 21 8 2GEV - L2 MOM 2 12 17EV1 - L MOM 9 10 12Table 5: Ranking of the Methods of parameterEstimator for the 31 Stations by RMSEfor the upper part of distributions,(0.9 ≤ Fi ≤ 1) (on a scale of 1 to 3 with1 being the best method)Number of Stations Receiving RankingMethod 1 2 3GEV - L MOM 10 13 8GEV - L2 MOM 21 7 3EV1 - L MOM 0 12 1942 Ani ShabriComparisons of the LH Moments and the L Moments 43ties especially MATHCAD software provided by College University Tun Hussein Onn aregratefully acknowledged.References[1] S. Ani, Penggunaan Analisis Frekuensi Banjir, Matematika, 16 (2000), 47-59.[2] D. H. A. Al-Khudhairy, Regional Flood Frequency Analysis, European Commission,Joint Research Centre, Institute for System, Informatics and Safety, 1998.[3] J.U. Chowhury, J.R. Stedinger & L. Lu, Goodness-of Fit for Regional GeneralizedExtreme Value Flood Distributions, Water Resour. Res., 27(7)(1991), 1765-1776.[4] J.R.M. Hosking, Analysis and Estimation of Distributions Using Linear Combinationof Order Statistics, IBM Research Division, T.J. Watson Research Center YorktownHeights, New York 10598, 1989.[5] J.R.M. Hosking & J.R. Wallis, Some Statistics Useful In Regional Frequency Analysis,Water Resour. Res., 29(2)(1993), 271-281.[6] A. Sankarasubramaniam & K. Srinivasan, Investigation and Comparison of SamplingProperties of L moments and ConventionaL moments, Journal of Hydrology, 218(1999),13-34.[7] C.D. Sinclair and M.I. Ahmad, Location-Invariant Plotting Positions For PWM Esti-mation of the Parameters of the GEV Distribution, Journal of Hydrology, 99(1988),271-279.[8] J.R. Stedinger, R.M. Vogel & G.E. Foufoula, Frequency Analysis of Extreme Events,Handbook of Applied Hydrology, Mc-Graw Hill Book Co., New York, chapter 18, 1993.[9] R.M. Vogel & N.M. Fennessey, L-Moment Diagrams Should Replace Product Diagrams,Water Resour. Res. 29(6)(1993), 1745-1752.[10] R.M. Vogel, W.O. Thomas & T. A. McMahon, Floodflow frequency model Selection inSouthwestern U.S.A., J. Water Resour., Planning Manage., ASCE, 119(3)(1993).[11] R.M. Vogel, W.O. Thomas & T. A. McMahon, Floodflow frequency model Selection inAustralia., Journal of Hydrology, 146(1993), 421-449.[12] Q.J. Wang, Using Higher Probability Weighted Moments For Flood Frequency Analysis,Journal of Hydrology, 194(1997), 95-106.[13] Q.J. Wang, LH moments For Statistical Analysis of Extreme Events, Water Resour.Res. Vol.33, (1997), 2841-2848.

Comparisons of the LH Moments and the L Moments

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Comparisons of the LH Moments and the L Moments

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