• Journal of the Korean Data and Information Science Society
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• v.28 no.4
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• pp.733-742
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• 2017

There are many areas of applications where Gumbel distribution are employed such as environmental sciences, system reliability and hydrology. The goodness-of-fit test for Gumbel distribution is very important in environmental sciences, system reliability and hydrology data analysis. Therefore, we propose the two test statistics to test goodness-of-fit for the Gumbel distribution based on the generalized Lorenz curve. We compare the new test statistic with the Anderson - Darling test, Cramer - vonMises test, and modified Anderson - Darling test in terms of the power of the test through by Monte Carlo method. As a result, the new test statistics are more powerful than the other test statistics. Also, we propose new graphic method to goodness-of-fit test for the Gumbel distribution based on the generalized Lorenz curve.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.38 no.2
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• pp.249-259
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• 2018

More than half of annual rainfall occurs in summer season in Korea due to its climate condition and geographical location. A frequency analysis is mostly adopted for designing hydraulic structure under the such concentrated rainfall condition. Among the various distributions, univariate Gumbel distribution has been routinely used for rainfall frequency analysis in Korea. However, the distributional changes in extreme rainfall have been globally observed including Korea. More specifically, the univariate Gumbel distribution based rainfall frequency analysis is often fail to describe multimodal behaviors which are mainly influenced by distinct climate conditions during the wet season. In this context, we purposed a Gumbel mixture distribution based rainfall frequency analysis with a Bayesian framework, and further the results were compared to that of the univariate. It was found that the proposed model showed better performance in describing underlying distributions, leading to the lower Bayesian information criterion (BIC) values. The mixed Gumbel distribution was more robust for describing the upper tail of the distribution which playes a crucial role in estimating more reliable estimates of design rainfall uncertainty occurred by peak of upper tail than single Gumbel distribution. Therefore, it can be concluded that the mixed Gumbel distribution is more compatible for extreme frequency analysis rainfall data with two or more peaks on its distribution.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.13 no.4
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• pp.151-161
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• 1993

The log-Gumbel distribution in real space is defined by transforming the conventional log-Gumbel distribution in log space. For this model, the parameter estimation techniques are applied based on the methods of moments, maximum likelihood and probability weighted moments. The asymptotic variances of estimator of the quantiles for each estimation method are derived to find the confidence limits for a given return period. Finally, the log-Gumbel model is applied to actual flood data to estimate the parameters, quantiles and confidence limits.

• Wind and Structures
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• v.1 no.1
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• pp.25-42
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• 1998

Improvements to the Gumbel method of extreme value analysis of wind data made over the last two decades are reviewed and illustrated using sample data for Jersey. A new procedure for extending the Gumbel method to include M-th highest annual extremes is shown to be less effective than the standard method, but leads to a method for calibrating peak-over-threshold methods against the standard Gumbel approach. Peak-over-threshold methods that include at least the 3rd highest annual extremes, specifically the modified Jensen and Franck method and the "Method of independent storms" are shown to give the best estimates of extremes from observations.

• 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.

• Journal of Korea Water Resources Association
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• v.48 no.5
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• pp.331-343
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• 2015

Most nonstationary frequency models are defined as the probability models containing the time-dependent parameters. For frequency analysis of annual maximum rainfall data, the Gumbel distribution is generally recommended in Korea. For the nonstationary Gumbel models, the time-dependent location and scale parameters are defined as linear and exponential relationship, respectively. The exponentially time-varying scale parameter of nonstationary Gumbel model is generally used because the scale parameter should be positive. However, the exponential form of scale parameter occasionally provides overestimated quantiles. In this study, various forms of time-varying scale parameters such as exponential, linear, and logarithmic forms were proposed and compared. The parameters were estimated based on the method of maximum likelihood. To compare the accuracy of each scale parameter, Monte Carlo simulation was performed for various conditions. Additionally, nonstationary frequency analysis was conducted for the sites which have more than 30 years data with a trend in rainfall data. As a result, nonstationary Gumbel model with exponentially time-varying scale parameter generally has the smallest root mean square error comparing with another forms.

• Proceedings of the Korea Water Resources Association Conference
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• 2008.05a
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• pp.173-178
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• 2008

확률도시위치는 주로 도시적 해석을 통한 연최대홍수량 또는 연최대강우량의 초과확률의 추정치 산정에 사용되며 빈도해석을 통해 선정된 적정 확률분포형과 표본자료의 개략적인 적합도를 도시적으로 파악할 수 있도록 해주기 때문에 오래 전부터 널리 이용되어 왔다. 본 연구에서는 Gumbel 분포에 적합한 도시위치공식을 새롭게 추정하기 위해 Gumbel 분포의 order statistic과 확률가중모멘트를 이용하여 다양한 표본크기에 대한 도시위치공식의 기본식을 유도하였고, 최적화 기법 중 하나인 유전자 알고리즘을 이용하여 유도된 도시위치공식의 매개변수를 추정하였다. 또한 본 연구에서 추정된 도시위치공식과 기존에 널리 사용되고 있는 도시 치공식의 정확도를 비교하기 위해 reduced variate 간의 오차를 계산하여 비교 검토하였다. 그 결과, 금회 추정된 도시위치공식은 높은 순위에서는 기존의 도시위치공식에 비해 더 정확도가 높은 것으로 나타났고, 표본크기에 대한 순위를 모두 고려할 경우에는 기존의 도시위치공식에 비해 정확도가 높은 것으로 나타나 Gumbel 분포에 대해서 높은 정확도를 보이는 것으로 나타났다.

• Proceedings of the Korea Water Resources Association Conference
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• 2011.05a
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• pp.378-382
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• 2011

지난 2010년 9월 21일~22일 서울지역에 발생한 시간당 최고 250 mm의 집중호우는 짧은 지속 시간동안 강수총량이 컸기 때문에 재산 및 인명피해가 상당했다. 기존의 강우빈도해석은 각 사상의 강수총량 또는 지속시간별 연최대치강우량을 기준으로 산정하는 방식인데 이번 이상홍수의 경우에는 강수총량과 첨두강수량이 높음에도 불구하고 지속시간이 짧기 때문에 체감적 강도에 미치지 못하는 재현기간을 가질 것으로 생각된다. 그러므로 빈도해석 시 기존 일변량 빈도해석과 달리 이변량, 삼변량, 다변량의 빈도해석이 필요할 것으로 사료된다. 본 연구의 대상유역은 용담댐 유역으로 강수총량, 지속시간, 첨두값을 대상으로 상관관계를 비교하여 지속시간과 첨두강수량에 대한 Gumbel mixed 모형을 적용하여 그에 따른 재현기간을 산정하였다. 또한 단일 Gumbel 모형과의 비교를 통해 두 모형의 차이점을 밝힘으로써 Gumbel Mixed 모형에 대한 신뢰도를 높였다. 따라서 본 연구에서 제안한 Gumbel mixed 모형은 이상홍수 뿐 아니라 다양한 수문학적 설계와 관리에 유용할 것으로 기대된다.

• Journal of Korean Society of Steel Construction
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• v.22 no.6
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• pp.573-580
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• 2010

In this study, an assessment method that considers the effects of directional wind speeds on buildings or structures that are sensitive to wind is proposed. Also, the basic characteristics of directional wind speeds were assessed by means of local annual maximum wind speeds. From the method of assessment of the characteristics of directional wind speeds, their goodness-of-fit was verified by applying extreme value distribution to the data on annual maximum wind speeds from the Korea Meteorological Administration. To consider the characteristics of directional winds, an assessment method is suggested that divides the directional wind pattern of each directional wind speed into four groups. From the study results, all the data on directional wind speeds based on the Gumbel distribution were examined using data on annual maximum wind speeds from Seoul, Tongyung, and Incheon. Since the Gumbel model of all directional wind speeds has independent probability characteristics that govern the 4 directional wind pattern groups, the application ratio proposed was based on the assessment of these four groups. According to the goodness-of-fit of the data on the annual maximum wind speeds based on the Gumbel distribution, new application ratios were proposed that consider the directional wind speeds in Seoul, Tongyung, and Incheon.

• Proceedings of the Korea Water Resources Association Conference
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• 2015.05a
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• pp.147-147
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• 2015

전 세계적으로 이상기후로 인한 극한가뭄 및 이상홍수 등의 피해 발생이 확인되고 있으며 그 발생빈도 또한 급격히 증가하고 있다. 그러나 기존의 빈도해석은 시간의 변화에 따라 자료의 통계적 특성이 변하지 않는다는 정상성(stationarity)을 기본 가정으로 수행되기 때문에 극한 사상에 경향성이 있는 경우에 적용하기엔 한계가 있다. 비정상성 빈도해석을 위해 개발된 비정상성 확률 분포 모형들은 대부분 매개변수에 시간항을 포함하는 형태로 정의된다. 이중에서도 우리나라에 널리 사용되고 있는 Gumbel 모형에 대해 살펴보면, 비정상성 Gumbel 모형의 위치 및 규모 매개변수는 시간에 대해 선형(linear) 및 지수(exponential) 함수의 관계를 보이는 형태로 가정한다. 규모 매개변수의 지수함수의 형태는 음(-)의 값이 추정되는 것을 방지하기 위해 제안되어 널리 사용되고 있으나 이로 인해 확률수문량이 과다산정되는 문제가 발생하기도 한다. 본 연구에서는 이러한 문제를 해결하기 위해 비정상성 Gumbel 모형을 대상으로 규모 매개변수의 다양한 형태를 비교하고자 한다. 이를 위해 비정상성 Gumbel 모형 규모 매개변수를 지수함수, 선형, 로그, 로지스틱 형태로 가정하여 비교하였다. 각 모형의 매개변수의 추정은 최우도법을 적용하였으며, 규모 매개변수의 형태별 정확도 비교를 위해 모의실험을 수행하였다.