• Wind and Structures
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• v.19 no.4
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• pp.371-387
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• 2014

Probability plotting positions are popular and used as the basis for distribution fitting and for inspecting the quality of the fit because of its simplicity. The plotting positions that lead to excellent approximation to the mean of the order statistics should be used if the objective of the fitting is to estimate quantiles. Since the mean depends on the sample size and is not amenable for simple to use closed form solution, many plotting positions have been presented in the literature, including a new plotting position that is derived based on the weighted least-squares method. In this study, the accuracy of using the new plotting position to fit the Gumbel distribution for estimating quantiles is assessed. Also, plotting positions derived by fitting the mean of the order statistics for all ranks is proposed, and an approximation to the covariance of the order statistics for the Gumbel (and Weibull) variate is given. Relative bias and root-mean-square-error of the estimated quantiles by using the proposed plotting position are shown. The use of the proposed plotting position to estimate the quantiles of annual maximum wind speed is illustrated.

• Communications for Statistical Applications and Methods
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• v.14 no.3
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• pp.561-576
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• 2007

Five plotting positions are applied to the computation of probability weighted moments (PWM) on the parameters of the generalized logistic distribution. Over a range of parameter values with some finite sample sizes, the effects of five plotting positions are investigated via Monte Carlo simulation studies. Our simulation results indicate that the Landwehr plotting position frequently tends to document smaller biases than others in the location and scale parameter estimations. On the other hand, the Weibull plotting position often tends to cause larger biases than others. The plotting position (i - 0.35)/n seems to report smaller root mean square errors (RMSE) than other plotting positions in the negative shape parameter estimation under small samples. In comparison to the maximum likelihood (ML) method under the small sample, the PWM do not seem to be better than the ML estimators in the location and scale parameter estimations documenting larger RMSE. However, the PWM outperform the ML estimators in the shape parameter estimation when its magnitude is near zero. Sensitivity of right tail quantile estimation regarding five plotting positions is also examined, but superiority or inferiority of any plotting position is not observed.

• Magazine of the Korean Society of Agricultural Engineers
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• v.40 no.2
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• pp.59-68
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• 1998

This study was conducted to derive optimal design floods by generalized gamma distribution model of the annual maximum series at eight watersheds along Geum, Yeongsan and Seomjin river systems. Design floods obtained by different methods for evaluation of parameters and for plotting positions in the generalized gamma distribution model were compared by the relative mean errors and graphical fit along with 95% confidence limits plotted on gamma probability paper. The results were analyzed and summarized as follows. 1. Basic statistics and parameters were calculated by the generalized gamma distribution model using different methods for parameters. 2. Design floods according to the return periods were obtained by different methods for evaluation of parameters and for plotting positions in the generalized gamma distribution model. 3. It was found that design floods derived by sundry averages method for parameters and Cunnane method for plotting position in the generalized gamma distribution are much closer to those of the observed data in comparison with those obtained by the other methods for parameters and for plotting positions from the viewpoint of relative mean errors. 4. Reliability of design floods derived by sundry averages method in the generalized gamma distribution was acknowledged within 95% confidence interval.

• Proceedings of the Korean Society of Agricultural Engineers Conference
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• 1998.10a
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• pp.318-324
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• 1998

This study was conducted to derive optimal design floods by Generalized Extreme-value(GEV) distribution for the annual maximum series at ten watersheds along Han, Nagdong, Geum, Yeongsan and Seomjin river systems. Adequacy for the analysis of flood data used in this study was established by the tests of Independence, Homogeneity, detection of Outliers. L-coefficient of variation, L-skewness and L-kurtosis were calculated by L-moment ratio respectively. Parameters were estimated by the Methods of Moments and L-Moments. Design floods obtained by Methods of Moments and L-Moments using different methods for plotting positions in GEV distribution were compared by the relative mean and relative absolute error. It was found that design floods derived by the method of L-moments using weibull plotting position formula in GEV distribution are much closer to those of the observed data in comparison with those obtained by method of moments using different formulas for plotting positions in view of relative mean and relative absolute error.

• Proceedings of the Korean Society of Agricultural Engineers Conference
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• 1998.10a
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• pp.331-337
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• 1998

This study was carried out to derive optimal design floods by Weibull-3 distribution with the annual maximum series at seven watersheds along Man, Nagdong, Geum, Yeongsan and Seomjin river systems. Adequacy for the analysis of flood data used in this study was acknowledged by the tests of Independence, Homogeneity, detection of Outliers. Parameters were estimated by the Methods of Moments and L-Moments. Design floods obtained by Methods of Moments and L-Moments using different methods for plotting positions in Weibull-3 distribution were compared by the rotative mean error and relative absolute error. It has shown that design floods derived by the method of L-moments using Weibull plotting position formula in Weibull-3 distribution are much closer to those of the observed data in comparison with those obtained by method of moments using different formulas for plotting positions in view of relative mean and relative absolute error.

• Magazine of the Korean Society of Agricultural Engineers
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• v.42 no.3
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• pp.45-55
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• 2000

This study was carried out to derive optimal design low flows bythe Weibull-3 and Wakeby distributions for the partial consecutive duration series at seven watersheds along Han. nagdong, Geum Yeongsan and Seomjin river systems. L-coefficient of variation L-skewness and L-kurtosis were calculated by the L-moment ratio respectively. Parameters were estimated by the method of L-Moments with consecutive duration. Design low flows obtained by method of L-Moments using with consecutive duration, Design low flows obtained by method of L-Moments using different methods for plotting positions formulas in the Weibull-3 and Wakeby distributions were compared by the Root Mean Square Errors(RMSE). It has shown that design low flows derived by the method of L-moments using Weivull plotting position formula in Wakeby distribution were much closer to those of the observed data in comparison with those obtained by the methods of L-moments with the different formulas for plotting positions in Weibull-3 distribution from the viewpoint of Root Mean Square Errors.

• Economic and Environmental Geology
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• v.22 no.2
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• pp.167-171
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• 1989

Palaeomagnetic studies accompany calculation and plotting of pole position. This paper explains three graphical methods for the determination of pole position and plotting problem. It also derives the numerical formula for pole calculation and explains the method how the pole plotted on the rear hemisphere can be transformed to the frontal hemisphere, which is not clarified elsewhere.

• Journal of Korea Water Resources Association
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• v.42 no.5
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• pp.365-374
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• 2009

Probability plotting positions are used for the graphical display of annual maximum rainfall or flood series and the estimation of exceedance probability of those values. In addition, plotting positions allow a visual examination of the fitness of probability distribution provided by frequency analysis for a given data. Therefore, the graphical approach using plotting position has been applied to many fields of hydrology and water resources planning. In this study, the plotting position formula for the Gumbel distribution is derived by using the order statistics and the probability weight moment of the Gumbel distribution for various sample sizes. And then, the parameters of plotting position formula for the Gumbel distribution are estimated by using genetic algorithm. The appropriate plotting position formulas for the Gumbel distribution are examined by the comparison of root mean square errors and biases between theoretical reduced Gumbel variates and those calculated from derived and existing plotting position formulas. As the results, Gringorten's plotting position formula has the smaller root mean square errors and biases than any other formulas.

• Communications for Statistical Applications and Methods
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• v.18 no.6
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• pp.707-716
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• 2011

Quantiles and quantile ranks(or plotting positions) are widely used in academia and industry. Sample quantile methods and sample quantile methods implemented in some major statistical software are at least seven, respectively. Small looking differences between the methods can make big differences in outcomes that result from decisions based on them. We discussed the characteristics and differences of the basic plotting position using the empirical cumulative probability and the six plotting positions derived from the suggestion of Blom (1958). After discussing the characteristics and differences of seven quantile methods used in the some major statistical software, we suggested a general expression covering all seven quantile methods. Using the insight obtained from the general expression, we proposed four propositions that make it possible to find the plotting position method that correspond to each of the seven quantile methods. These correspondences may help us to understand and apply quantile methodology.

• Journal of The Korean Society of Agricultural Engineers
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• v.51 no.3
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• pp.53-62
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• 2009

This study was carried out to select optimal probability distribution based on design accumulated monthly mean inflow from the viewpoint of drought by Gamma (GAM), Generalized extreme value (GEV), Generalized logistic (GLO), Generalized normal (GNO), Generalized pareto (GPA), Gumbel (GUM), Normal (NOR), Pearson type 3 (PT3), Wakeby (WAK) and Kappa (KAP) distributions for the observed accumulative monthly mean inflow of Chungjudam. L-moment ratio was calculated using observed accumulative monthly mean inflow. Parameters of 10 probability distributions were estimated by the method of L-moments with the observed accumulated monthly mean inflow. Design accumulated monthly mean inflows obtained by the method of L-moments using different methods for plotting positions formulas in the 10 probability distributions were compared by relative mean error (RME) and relative absolute error (RAE) respectively. It has shown that the design accumulative monthly mean inflow derived by the method of L-moments using Weibull plotting position formula in WAK and KAP distributions were much closer to those of the observed accumulative monthly mean inflow in comparison with those obtained by the method of L-moment with the different formulas for plotting positions in other distributions from the viewpoint of RME and RAE.