A nonparametric kernel methods is proposed and evaluated performance for estimating annual maximum stream flow quantiles. The bandwidth of the estimator is estimated by using an optimal technique and a cross-validation technique. Results obtained from a limited amount of real data from Malaysia show that quantiles estimated by nonparametric method using these techniques have small root mean square error and root mean absolute error. Based on correlation coefficient test shown that the nonparametric model approach is accurate, uniform and flexible alternatives to parametric models for flood frequency analysis

Shabri, Ani

English

Universiti Teknologi Malaysia Institutional Repository

Matematika, 2002, Jilid 18, bil. 2, hlm. 99–107c©Jabatan Matematik, UTM.Nonparametric Kernel Estimation of AnnualMaximum Stream Flow QuantilesAni ShabriDepartment of Mathematics, Faculty of ScienceUniversiti Teknologi Malaysia81310 UTM Skudai, Johor, MalaysiaAbstract A nonparametric kernel methods is proposed and evaluated perfor-mance for estimating annual maximum stream flow quantiles. The bandwidth ofthe estimator is estimated by using an optimal technique and a cross-validationtechnique. Results obtained from a limited amount of real data from Malaysiashow that quantiles estimated by nonparametric method using these techniqueshave small root mean square error and root mean absolute error. Based on corre-lation coefficient test shown that the nonparametric model approach is accurate,uniform and flexible alternatives to parametric models for flood frequency anal-ysis.Keywords Bandwidth, cross-validation, kernel, correlation coefficient.Abstrak Abstrak Kaedah Nonparametric Kernel dicadangkan dan dinilaikanperlaksanaannya dalam menganggar kuantil aliran tahunan maksimum. Pen-ganggar bagi bandwidth dianggarkan menggunakan teknik optimal dan ’cross-validation’. Hasil keputusan menggunakan data sebenar yang terhad dari Malay-sia, menunjukkan bahawa penganggar kuantil berdasarkan model nonparametricmenggunakan kedua-dua teknik ini menghasilkan nilai punca min ralat kuasadua dan punca min ralat mutlak yang kecil. Berdasarkan ujian pekali korelasimenunjukkan bahawa pendekatan model nonparametric adalah tepat, seragamdan ianya boleh dijadikan sebagai kaedah alternatif bagi model parametric dalamanalisis frekuensi banjir.Katakunci Bandwidth, ’cross-validation’, kernel, pekali korelasi.1 IntroductionIn flood frequency analysis, one of the more important issues is the choice of the bestprobability distribution. The true distribution is always unknown in practice and oftenarbitrary choice of or preference for a given distribution increase the estimation uncertainty.Some countries have tried to find standard distributions for flood frequency analysis inorder to avoid the arbitrariness in the selection of the distribution types. Many distributions100 Ani Shabrihave been used to estimate flood flow frequencies from observed annual flood series. Thegeneral extreme value (GEV) distribution is recommended as a base model in the UnitedKingdom. After appraising many distributions, the U.S. Water resources Council (WRC),issued a series of Bulletins recommending the Log-Pearson Type III (LP3) a base methodfor use by all U.S. Federal Agencies [1]. While the most commonly used in Canada beingthe three parameter log-normal (LN3), the Generalized Extreme Value (GEV) and thelog-Pearson III (LP3) [2].Recently, nonparametric density estimation methods have gained popularity in a varietyof field of science, including hydrology. This model has the advantage. The shapes ofnonparametric density functions are directly determined by the data [3]. It is not requiringassumptions about the distribution of the population of interest [1] and the estimation ofparameters (i.e. mean, variance and skew) are not needed. The parametric distributions arelimited to certain shapes, however nonparametric densities can adapt to the often-irregularempirical distribution of random variables found in nature [3].In the present study, the nonparametric kernel method was compared with the GEV,LNIII, GPA and EV1 distributions. The objective of the present study is to introduce andevaluate the performance of annual maximum series using nonparametric techniques andparametric distributions based on real data.2 Nonparametric Frequency AnalysisThe probability density function f(x) estimated by a nonparametric method is given by [2]f(x) = 1nhn∑j=1K(x− xjh)(1)where x1 to xn are the observations, K() is a kernel function, itself a probability densityfunction, and h is a bandwidth or smoothing factor to be estimated from the data.The following condition are impose on the kernel [2]:∫K(z) dz = 1∫zK(z) dz = 0 (2)∫z2K(z) dz = C 6= 0where C is the kernel variance.The kernel distribution function is the integration of the density Eq. (1) from −∞ to x[2]:F (x) = 1nn∑j=1Kl(x− xjh)(3)where Kl(u) =∫ u−∞K(w) dw.The kernel distribution function may serve to estimate percentiles corresponding to agiven probability of exceedance. The flood quantile xT with a return period of T years, ofNonparametric Kernel Estimation of Annual Maximum Stream Flow Quantiles 101the kernel distribution function is [2]xT = F−1(1− 1T)(4)where F−1() represents the inverse of F (). The value of xT can be determined by solvingEq. (3) numerically.In nonparametric frequency analysis, the choice of kernel function is not critical sincevarious kernels lead to comparable estimates [2]. Gaussian, Gumbel, and Epanechnikovkernels were tested in flood frequency analysis and it was found that the choice of thekernel does not important, and the shape of kernel does not affect extrapolation accuracy[2]. However the calculation and to choice of the bandwidth, h in Eq. 1 is critical. Table 1shows some of the commonly kernels functions.Table 1: Some Kernel Function Wheret = h−1(x− xi)Kernel K(t)Epanechnikov K(t) = 34 (1− t2), |t| < 1Rectangular K(t) = 12 , |t| < 1Biweight K(t) = 1516 (1− t2)2, |t| < 1Gaussian K(t) = 1√2pi e− 12 t2Cauchy K(t) = 1pi(1+t2)EV1 K(t) = e−t−e−tIn this study only the Gaussian kernel is considered.3 Estimation Of Smoothing Factor h3.1 Optimal Technique (OPT)Several bandwidth estimation techniques are based on minimization of an estimate of themean square error (MSE) or of the IMSE function. IMSE is obtained by integrating theMSE function over the domain of x.The criterion of optimality is based on minimizing the IMSE given by [4]IMSE =∫ ∞−∞[fˆ(x) − f(x)]2dx (5)where fˆ(x) is an estimate of the known density function f(x). An optimal value of h canbe obtained by minimizing the IMSE for a given density f(x) and sample. The value ofh has to be derived empirically from the observed data. The optimal value of h, can be102 Ani Shabridetermined numerically by differentiating the objective function Eq. (5) with respect to hand then equating it to zero, is given by [4]h =n∑i=2i−1∑j=1(xi − xj)√5n(n− 10/3)(6)Based on numerical studies have indicated that the computation of h is also optimal forsmall samples where n = 25 was the smallest sample considered [1].3.2 Cross-Validation Technique (CV)Other method of computing h is to minimize the integrated mean of a cross-validationtechnique [2]:IMSE =∫ ∞−∞E[fˆ(x) − f(x)]2dxIn cross-validation, estimates of f(x) are constructed each time using every data pointexcept one. Let fˆ−i(x) be the nonparametric kernel estimate ignoring a single data point,xi; that isfˆ−i(x) =1(n− 1)hn−1∑j 6=iK [(x − xj)/h] (7)The smoothing factor h can be determined numerically from cross-validation procedureby solving Eq. (8) [2]n∑i=1,i=jn∑j=1exp(dij4){(1− 4√2nn− 1exp(dij4)(xi − xjh)− 1)(xi − xjh+ 1)− 1}= 0(8)where dij = −((xi − xj)/h)2. The cross-validation procedure leads to consistent and as-ymptotically optimal nonparametric density estimates.4 Data Used For Numerical AnalysisThe annual maximum stream flows for the 10 stations in Malaysia was selected for numericaldemonstration. The statistical characteristics of these data are given in Table 1. These datawere selected on the basis of length completeness and independence of record. The lengthsof record are between 15 to 32 lengths and mean of annual maximum stream flows are from6 to 560 m3/s. The data was obtained from the Department of Irrigation and DrainageMalaysia.Nonparametric Kernel Estimation of Annual Maximum Stream Flow Quantiles 103Table 2: Characteristic of Annual Maximum Stream Flow DataNo Stations River/States Records Mean Skewness KurtosisLength m3s−1(years)1 1737451 Rantau Panjang, Johor 32 259.2 1.066 3.1282 1836402 Sayong, Johor 18 120.5 1.042 2.9163 2527411 Muar, Johor 22 171.6 1.308 4.9514 2928401 Keratong, Johor 15 561.8 2.053 6.5325 3224433 Triang, Pahang 21 42.0 1.166 3.7636 3519426 Bentong, Pahang 29 186.2 0.760 3.0207 3629403 Lepar, Pahang 23 197.1 0.409 1.4278 4232452 Kemaman, Terengganu 19 354.3 0.071 2.1879 4732461 Paka, Terengganu 16 6.2 0.368 2.13510 5229436 Nerus, Terengganu 15 435.2 0.450 1.787To assess how well the proposed nonparametric method fit to the observed annual maxi-mum flow data, the XT −T relationships using Eq. (4) are shown in Figure 1. The observedflood discharge corresponds to a return period which was computed by using the Gringortonplotting position formula [5]T = n+ 0.12i− 0.44(9)in which i is the rank assigned to each data point in the sample with a value of one thehighest flow, two for the second highest and so on.5 Results and Discussion5.1 Root Mean Square Error And Root Mean Absolute ErrorTwo criteria were used for comparing the two methods of bandwidth h of nonparametricmethod. The first criteria is defined as the root mean square error (RMSE) between observedand computed flood discharge for entire sample. The RMSE is calculated by the equationsRMSE =√√√√ 1nn∑i=1(xT − xˆTxT)2(10)104 Ani Shabriwhere xT and xˆT are observed and computed flood values, respectively for a given value ofT . The other criteria is the root mean absolute error and can be calculated by the equationsRMAE =√√√√ 1nn∑i=1∣∣∣∣xT − xˆTxT∣∣∣∣ (11)5.2 Goodness-Of-Fit TestThe goodness-of-fit test using correlation coefficient (CC) test is used to test the correlation,r between the observations xT and computed flood value, xˆT . The CC test is measureof linearity of a probability plot. If the sample to be tested is actually drawn from thehypothesized distribution is expected to be nearly linear and the correlation coefficient willbe near to one. If x¯ denotes the average value of the observations and w¯ denotes the averagevalue of computed flood, then correlation coefficient sample can be defined as [6]r =∑(xT − x¯)(xˆT − w¯)√∑(xT − x¯)2∑(xˆT − w¯)2(12)A comparison of the bandwidths for the Gaussian Kernel using OPT and CV for eachstations is shown in Table 3.Table 3: Comparison of Bandwidth for a Normal Kernel Baseon Values RMSE, RMAE and rStation No r(Correlaation) RMSE RMAEOPT CV OPT CV OPT CV1 0.996 0.996 0.012 0.013 0.065 0.0672 0.994 0.999 0.011 0.004 0.054 0.0383 0.995 0.998 0.034 0.007 0.099 0.0524 0.996 1.000 0.025 0.0004 0.084 0.0135 0.994 1.000 0.877 0.0007 0.435 0.0156 0.998 0.996 0.005 0.014 0.045 0.0667 0.995 0.999 0.122 0.017 0.146 0.0698 0.997 0.996 0.008 0.014 0.057 0.0709 0.997 0.997 0.0017 0.00118 0.029 0.03110 0.998 0.997 0.0014 0.0019 0.029 0.033Table 3 shows that the nonparametric method using OPT and CV techniques provide agood fit to data because the all stations have produce correlation coefficient, r nearly one.For all 10 stations, Figure 1 shows that the nonparametric kernel estimator using thesetechniques can fit the real data points closely and provided a sufficiently good approximationto all data.Nonparametric Kernel Estimation of Annual Maximum Stream Flow Quantiles 105Figure 1: Comparison of Observed and Computed Frequency Curves for the Annual Maxi-mum Stream Flows in Malaysia.106 Ani ShabriFigure 2: (Continue Figure 1)Figure 3: Plot of RMSE and RMAE for OPT and CV Techniques of Nonparametric Methodfor 10 Stations.Nonparametric Kernel Estimation of Annual Maximum Stream Flow Quantiles 107The value of RMSE and RMAE for each stations using OPT and CV technique werepresented in Figure 2. It can be seen that the value of RMSE and RMAE fall in nearly zeroat all stations except at stations 5 and stations 7 for the OPT technique.6 ConclusionsThe correlation coefficient test statistic as a goodness of fit test is introduced for testing thenonparametric to fit the annual maximum flood flow data. The CC test statistic was foundto be useful tool for discriminating among competing bandwidth estimation techniques.Overall these nonparametric method using the OPT and CV techniques consistently pro-duced linear probability plots with r nearly one as measured by the CC test statistics. Thenonparametric models with these bandwidth estimation techniques are readily applicableto annual maximum flow frequency analysis.Based on RMSE and RMSE, clearly, the CV technique is more accurate than the OPtechnique. However those differences between two methods were not too great and thereforethese bandwidth techniques could be considered comparable for practical purpose.It can be seen from the results presented here, that the nonparametric procedure isuseful as it automatically gives stable and accurate estimates of density. It offers a practicaland uniform approach to flood frequency analysis.AcknowledgmentsThe author would like to thank the Department of Irrigation an Drainage in Malaysia forproviding the flood flow data of streams in the Peninsular of Malaysia.References[1] K. Adamowski. A Monte Carlo Comparison of Parametric And Nonparametric Esti-mation of Flood Frequencies. Journal of Hydrology, 108(1989), 295-308.[2] K. Adamowksi. Regional Analysis of Annual Maximum and Partial Duration FloodData by Nonparametric and L-Moment Methods, Journal of Hydrology, 229(2000),219-231.[3] D. Faucher, P. F. Rasmussen & B. Bobee. A Distribution Function Based BandwithSelection Method for Kernel Quantile Estiamtion. Journal of Hydrology, 250(2001),1-11.[4] S.L. Guo, R.K. Kachroo & R.J. Mngodo. Nonparametric Kernel Estimation of LowQuantiles, Journal of Hydrology, 185 (1996), 335-348.[5] D. Jain & V.P. Singh. Estimating Parameters of EV1 Distribution For Flood Fre-quency Analysi, Water Res. Resources, 23(1987), 59-70.[6] R.M. Vogel, The Probability Plot Correlation Coefficient Test for the Normal, Lognor-mal, and Gumbel Distribution Hpotheses, Water Resour. Res., 22(4) (1986), 587-590.

Nonparametric Kernel estimation of annual maximum stream flow quantiles.

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Nonparametric Kernel estimation of annual maximum stream flow quantiles.

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