• Communications for Statistical Applications and Methods
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• 제14권3호
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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.

• Wind and Structures
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• 제19권4호
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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.

• 한국수자원학회논문집
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• 제44권2호
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• pp.85-96
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• 2011

연최대수문량의 도시적 분석에 주로 이용되어 온 확률도시위치는 표본자료와 적정 확률분포형의 적합도를 표시하여 초과확률을 산정할 수 있도록 하며, 일부 적합도 검정에도 사용되기도 한다. 확률도시위치를 결정하는 도시위치공식은 오래 전부터 꾸준히 연구되어 왔는데, 특히 빈도해석에 널리 사용되는 GEV 분포에 대한 연구는 다른 분포형보다 더욱 활발히 이루어져 왔다. 본 연구에서는 GEV 분포에 적합한 도시위치공식을 추정하고자 GEV 분포의 순서통계량의 평균 개념을 이용하여 이론적 축소변량을 유도하였다. 또한 다양한 표본크기와 형상 매개변수와 연관이 있는 왜곡도 계수를 고려한 다양한 형태의 도시위치공식을 적용하고, 유전자 알고리즘을 적용하여 도시위치공식의 매개변수를 추정하였다. 유도된 도시위치공식의 정확성을 알아보기 위해 이론적 축소변량과 금회 유도된 도시위치공식을 포함한 다양한 도시위치공식에 의해 계산되는 축소변량 사이의 오차를 비교하였다. 그 결과, 본 연구에서 제안한 도시위치공식은 GEV 분포의 형상 매개변수가 -0.25~0.10의 범위를 가질 때 이론적 축소변량과 가장 작은 오차를 보이는 것으로 나타났다.

• 한국수자원학회논문집
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• 제42권5호
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• pp.365-374
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• 2009

확률도시위치는 주로 도시적 해석을 통한 연최대홍수량 또는 연최대강우량의 초과확률의 추정치 산정에 사용되며 빈도해석을 통해 선정된 적정 확률분포형과 표본자료의 적합도를 도시적으로 파악할 수 있도록 해주기 때문에 오래 전부터 수문 및 수자원 분야에 널리 이용되어 왔다. 본 연구에서는 Gumbel 분포에 적합한 도시위치공식을 추정하고자 Gumbel 분포의 순서통계량과 확률가중모멘트를 이용하여 다양한 표본크기에 대한 도시위치공식의 기본식을 유도하였고, 최적화 기법 중 하나인 유전자 알고리즘을 이용하여 도시위치공식의 매개변수를 추정하였다. 또한 Gumbel 분포에 적합한 도시위치공식을 검토하고자 Gumbel 분포의 이론적인 축소변량과 본 연구에서 추정한 도시위치공식과 기존의 도시위치공식에 의해 계산된 축소변량 간의 평균제곱근오차와 편의를 비교하였다. 그 결과, Gringorten이 제안한 도시위치공식을 적용한 경우의 축소변량간 평균제곱근오차와 순서별 편의가 가장 작은 것으로 분석되었다.

• 자원환경지질
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• 제22권2호
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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.

• 한국농공학회논문집
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• 제51권4호
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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.

• Wind and Structures
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• 제34권6호
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• pp.469-482
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• 2022

The generalized extreme value distribution (GEVD) is frequently used to fit the block maximum of environmental parameters such as the annual maximum wind speed. There are several methods for estimating the parameters of the GEV distribution, including the least-squares method (LSM). However, the application of the LSM with the expected order statistics has not been reported. This study fills this gap by proposing a fitting method based on the expected order statistics. The study also proposes a plotting position to approximate the expected order statistics; the proposed plotting position depends on the distribution shape parameter. The use of this approximation for distribution fitting is carried out. Simulation analysis results indicate that the developed fitting procedure based on the expected order statistics or its approximation for GEVD is effective for estimating the distribution parameters and quantiles. The values of the probability plotting correlation coefficient that may be used to test the distributional hypothesis are calculated and presented. The developed fitting method is applied to extreme thunderstorm and non-thunderstorm winds for several major cities in Canada. Also, the implication of using the GEVD and Gumbel distribution to model the extreme wind speed on the structural reliability is presented and elaborated.

• Communications for Statistical Applications and Methods
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• 제18권6호
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• pp.707-716
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• 2011

분위수와 분위(또는 타점위치)는 학계에서나 산업계에서 널리 사용되고 있다. 그런데 통계 소프트웨어에 구현되어 있는 표본 분위수 계산방법들과 표본 분위 계산방법들은 각각 적어도 일곱 가지가 있다. 분위수들이나 분위들을 정의하는 방법들 간의 사소해 보이는 차이가 그 값을 토대로 이루어지는 결정의 큰 차이를 가져올 수 있다. 이 논문에서는 경험적 누적확률을 사용한 기본 타점위치 방법과 Blom (1958)의 제안을 토대로 파생된 여섯 가지 타점위치 방법의 특징과 차이점을 논의하였다. 또 통계소프트웨어에 구현되어 있는 일곱 가지 표본분위수 계산방법들의 특징과 차이점들을 논의한 후 이들을 망라하는 하나의 일반식을 제시하였다. 이 논문에서는 이 일반식으로부터 얻어지는 통찰을 토대로 표본분위수에 대응되는 표본분위를 구하는 방법을 제안하였고, 이 제안을 각 표본분위수 방법에 적용하여 대응되는 표본분위 방법을 도출하였다. 이런 대응관계는 표본분위수와 표본분위에 대한 종합적 이해와 적용에 도움을 줄 수 있을 것이다.

• Wind and Structures
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• 제16권4호
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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%.

• 한국농공학회:학술대회논문집
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• 한국농공학회 1998년도 학술발표회 발표논문집
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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.