• Title, Summary, Keyword: plotting position

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On the Effects of Plotting Positions to the Probability Weighted Moments Method for the Generalized Logistic Distribution

  • Kim, Myung-Suk
    • 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.

Plotting positions and approximating first two moments of order statistics for Gumbel distribution: estimating quantiles of wind speed

  • Hong, H.P.;Li, S.H.
    • 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.

Derivation of Plotting Position Formulas Considering the Coefficients of Skewness for the GEV Distribution (왜곡도 계수를 고려한 GEV 분포의 도시위치공식 유도)

  • Kim, Soo-Young;Heo, Jun-Haeng;Choi, Min-Young
    • Journal of Korea Water Resources Association
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    • v.44 no.2
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    • pp.85-96
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    • 2011
  • Probability plotting position is generally used for the graphical analysis of the annual maximum quantile and the estimation of exceedance probability to display the fitness between sample and an appropriate probability distribution. In addition, it is used to apply a specific goodness of fit test. Plotting position formula to define the probability plotting position has been studied in many researches. Especially, the GEV distribution which is an important probability distribution to analyze the frequency of hydrologic data was popular. In this study, the theoretical reduced variates are derived using the mean value of order statistics to derived an appropriate plotting position formula for the GEV distribution. In addition, various forms of plotting position formula considering various sample sizes and coefficients of skewness related with shape parameters are applied. The parameters of plotting position formulas are estimated using the genetic algorithm. The accuracy of derived plotting position formula is estimated by the errors between the theoretical reduced variates and those by various plotting position formulas including the derived ones in this study. As a result, the errors by derived plotting position formula is the smallest at the range of shape parameter with -0.25~0.10.

Comparison of Plotting Position Formulas for Gumbel Distribution (Gumbel 분포에 대한 도시위치공식의 비교)

  • Kim, Soo-Young;Heo, Jun-Haeng;Shin, Hong-Joon;Kho, Youn-Woo
    • 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.

Methods in Palaeomagnetism (II): Calculation and Plotting of Palaeomagnetic Pole Positions (고자기학(古磁氣學)의 방법(方法)(II) : 고자기학적(古磁氣學的) 측정자료로부터 자북(磁北)의 위치를 도시하는 방법)

  • Kim, In-Soo
    • 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.

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Estimation of Design Floods Using 3 and 4 Parameter Kappa Distributions (3변수 및 4변수 Kappa 분포에 의한 설계홍수량 추정)

  • Maeng, Seung-Jin;Kim, Byeoung-Jun;Kim, Hyung-San
    • Journal of The Korean Society of Agricultural Engineers
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    • v.51 no.4
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    • pp.49-55
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    • 2009
  • This paper is to induce design floods through L-moment with 3 and 4 parameter Kappa distributions including test of independence by Wald-Wolfowitz, homogeneity by Mann-Whitney and outlier by Grubbs-Beck on annual maximum flood flows at 9 water level gaging stations in Han, Nakdong and Geum Rivers of South Korea. After analyzing appropriateness of the data of annual maximum flood flows by Kolmogorov-Smirnov test, 3 and 4 Kappa distributions were applied and the appropriateness was judged. The parameters of 3 and 4 Kappa distributions were estimated by L-moment method and the design floods by water level gaging station was calculated. Through the comparative analysis using the relative root mean square errors (RRMSE) and relative absolute errors (RAE) calculated by 3 and 4 parameter Kappa distributions with 4 plotting position formulas, the result showed that the design floods by 4 parameter Kappa distribution with Weibull and Cunnane plotting position formulas are closer to the observed data than those obtained by 3 parameter Kappa distribution with 4 plotting position formulas and 4 parameter Kappa distribution with Hazen and Gringorten plotting position formulas.

Relationship between the Sample Quantiles and Sample Quantile Ranks (표본분위수와 표본분위의 관계)

  • Ahn, Sung-Jin
    • 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.

The Gringorten estimator revisited

  • Cook, Nicholas John;Harris, Raymond Ian
    • Wind and Structures
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    • v.16 no.4
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    • pp.355-372
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    • 2013
  • The Gringorten estimator has been extensively used in extreme value analysis of wind speed records to obtain unbiased estimates of design wind speeds. This paper reviews the derivation of the Gringorten estimator for the mean plotting position of extremes drawn from parents of the exponential type and demonstrates how it eliminates most of the bias caused by the classical Weibull estimator. It is shown that the coefficients in the Gringorten estimator are the asymptotic values for infinite sample sizes, whereas the estimator is most often used for small sample sizes. The principles used by Gringorten are used to derive a new Consistent Linear Unbiased Estimator (CLUE) for the mean plotting positions for the Fisher Tippett Type 1, Exponential and Weibull distributions and for the associated standard deviations. Analytical and Bootstrap methods are used to calibrate the bias error in each of the estimators and to show that the CLUE are accurate to better than 1%.

Comparative Analysis of Flood Frequncy by Moment and L-moment in Weibull-3 distribution (Weibull-3 분포모형의 모멘트법 및 L-모멘트법에 의한 홍수빈도비교분석)

  • 이순혁;맹승진;송기헌;류경식;지호근
    • Proceedings of the Korean Society of Agricultural Engineers Conference
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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.

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Derivation of Optimal Design Flood by L-Moments (L-모멘트법에 의한 적정 설계홍수량의 유도)

  • 이순혁;박명근;맹승진;정연수;김동주;류경식
    • Proceedings of the Korean Society of Agricultural Engineers Conference
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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.

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