• Proceedings of the Korea Water Resources Association Conference
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• 2016.05a
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• pp.41-41
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• 2016

최근 다변량 확률모형을 이용한 빈도해석이 수문자료 등에 적용되면서 다양하게 연구되고 있으며 다변량 확률모형 중 copula 모형은 주변분포형에 대한 제약이 없어 여러 분야에 걸쳐 활발히 연구되고 있다. 강우자료는 기존 일변량 빈도해석을 수행하기 위하여 사용하던 block maxima 방법 대신 최소무강우시간(inter event time)을 통하여 강우사상을 추출하여 표본으로 사용한다. 또한 기후변화로 인한 강우량의 변화등에 대응하기 위하여 비정상성 Generalized Extreme Value(GEV)와 Gumbel 등의 확률분포형에 대한 연구도 많은 부분 이루어져 있다. 본 연구에서는, Archimedean copula 모형을 이용하여 이변량 확률모형을 구축하면서 여기에 사용되는 주변분포형에 정상성/비정상성 분포형을 적용하였다. 모형의 매개변수는 inference function for margin 방법을 이용하였으며 주변분포형으로는 정상성/비정상성 GEV, Gumbel 모형을 적용하였다. 결과로 정상성/비정상성 경향을 나타내는 지점을 구분하고 각 지점에 대한 정상성/비정상성 주변분포형을 적용한 이변량 확률분포형을 구하였다.

• Communications for Statistical Applications and Methods
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• v.30 no.5
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• pp.467-483
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• 2023

In this research, a conversion function and a distortion associated with the conversion function are defined and used to derive a unit power Gompertz distortion. A new family of copulas is built using the global distorted function. Four base copulas, namely Clayton, Gumbel, Frank, and Gaussian, are distorted into the family. Some properties including tail dependence coefficients and tail order are examined. Kendall's tau formula is derived for new copulas when the base copula is Clayton, Gumbel, or Frank. The maximum pseudo-likelihood estimation method is employed, and a simulation study was performed. The log-likelihood and AIC are reported to compare the performance of the fitted copulas. According to the applied data, the results indicate that new distorted copulas with additional parameters improve the fit.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.15 no.12
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• pp.4567-4583
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• 2021

This study proposes an analytical approximation algorithm based on extreme value theory (EVT) for the inverse of the power of the incomplete Gamma function. First, the Gumbel function is used to approximate the power of the incomplete Gamma function, and the corresponding inverse problem is transformed into the inversion of an exponential function. Then, using the tail equivalence theorem, the normalized coefficient of the general Weibull distribution function is employed to replace the normalized coefficient of the random variable following a Gamma distribution, and the approximate closed form solution is obtained. The effects of equation parameters on the algorithm performance are evaluated through simulation analysis under various conditions, and the performance of this algorithm is compared to those of the Newton iterative algorithm and other existing approximate analytical algorithms. The proposed algorithm exhibits good approximation performance under appropriate parameter settings. Finally, the performance of this method is evaluated by calculating the thresholds of space-time block coding and space-frequency block coding pattern recognition in multiple-input and multiple-output orthogonal frequency division multiplexing. The analytical approximation method can be applied to other related situations involving the maximum statistics of independent and identically distributed random variables following Gamma distributions.

• Proceedings of the Korea Water Resources Association Conference
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• 2016.05a
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• pp.201-201
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• 2016

The asymptotic extreme value distributions of maxima are a natural choice when designing against future extreme events like flood peaks or wave heights, given a stationary time series. The generalized extreme value distribution (GEV) is often utilised in this context because it is seen as a convenient single expression for extreme event analysis. However, the GEV has a drawback because the location of the distribution bound relative to the data is a discontinuous function of the GEV shape parameter. That is, for annual maxima approximated by the Gumbel distribution, the data is also consistent with a GEV distribution with an upper bound (no lower bound) or a GEV distribution with a lower bound (no upper bound). A more consistent single extreme value expression for design purposes is proposed as the Weibull distribution of smallest extremes, as applied to transformed annual maxima. The Weibull distribution limit holds here for sufficiently large sample sizes, irrespective of the extreme value domain of attraction applicable to the untransformed maxima. The Gumbel, Type 2, and Type 3 extreme value distributions thus become redundant, together with the GEV, because in reality there is only a single asymptotic extreme value distribution required for design purposes - the Weibull distribution of minima as applied to transformed maxima. An illustrative synthetic example is given showing transformed maxima from the normal distribution approaching the Weibull limit much faster than the untransformed sample maxima approach the normal distribution Gumbel limit. Some New Zealand examples are given with the Weibull distribution being applied to reciprocal transformations of annual flood maxima, where the untransformed maxima follow apparently different extreme value distributions.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.29 no.2B
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• pp.131-139
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• 2009

Recently frequent occurrences of heavy rainfall and increases of rainfall intensity resulted in severe flood damage in Korea. In order to mitigate the vulnerability of flood, it is necessary to estimate proper design rainfalls considering the increasing trend of extreme rainfalls for hydrologic planning and design. This study focused the estimation of design rainfalls in a design target year. Tests of trend indicated that there are 7 sites showing increasing trends among 56 sites which have hourly data more than 30 years in Korea. This study analyzed the relationship between mean of annual maximum rainfalls and parameters of the Gumbel distribution. Based on the relationship, this study estimated the probability density function and design rainfalls in a design target year, and then constructed the rainfall-frequency curve. The proposed method estimated the design rainfalls 6-20% higher than those from the stationary rainfall frequency analysis.

• Journal of Korea Water Resources Association
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• v.49 no.5
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• pp.411-422
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• 2016

Rainfall-runoff modeling in conjunction with rainfall frequency analysis has been widely used for estimating design floods in South Korea. However, uncertainties associated with underlying distribution and sampling error have not been properly addressed. This study applied a Bayesian method to quantify the uncertainties in the rainfall frequency analysis along with Gumbel distribution. For a purpose of comparison, a probability weighted moment (PWM) was employed to estimate confidence interval. The uncertainties associated with design rainfalls were quantitatively assessed using both Bayesian and PWM methods. The results showed that the uncertainty ranges with PWM are larger than those with Bayesian approach. In addition, the Bayesian approach was able to effectively represent asymmetric feature of underlying distribution; whereas the PWM resulted in symmetric confidence interval due to the normal approximation. The use of long period data provided better results leading to the reduction of uncertainty in both methods, and the Bayesian approach showed better performance in terms of the reduction of the uncertainty.

• Journal of Korea Water Resources Association
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• v.49 no.10
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• pp.823-833
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• 2016

This study developed a trivariate Copula function based drought frequency analysis model to better evaluate the recent 2014~2015 drought event. The bivariate frequency analysis has been routinely used for the drought variables of interest (e.g. drought duration and severity). However, the recent drought patterns showed that the intensity can be regarded as an important factor which is being characterized by short duration and severe intensity. Thus, we used the trivariate Copula function approach to incorporate the trivariate drought characteristics into the drought frequency analysis. It was found that the return periods based on the trivariate frequency analysis are, in general, higher than the existing bivariate frequency analysis. In addition, this study concludes that the increase in drought frequency claimed by the Gumbel copula function has been overestimated compared to the Student t Copula function. In other words, the selection of copula functions is rather sensitive to the estimation of trivariate drought return periods at a given duration, magnitude and intensity.

• International Journal of Reliability and Applications
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• v.11 no.1
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• pp.41-53
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• 2010

In the present paper we develop a mathematical model that facilitates the calculation of reliability of a complex repairable system having three units namely super priority, priority and ordinary. The system is analyzed with the application of Gumbel Hougaard copula when different types of repair possible at a particular state due to deliberate failure. Various reliability measures such as reliability, MTTF and profit function have been evaluated by using supplementary variable and Laplace transform techniques.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.30 no.3B
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• pp.257-267
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• 2010

A flood event can be characterized by three attributes such as peak discharge, total flood volume, and flood duration, which are correlated each other. However, the amount of peak discharge is only used to evaluate the flood events for the hydrological plan and design. The univariate analysis has a limitation in describing the complex probability behavior of flood events. Thus, the univariate analysis cannot derive satisfying results in flood frequency analysis. This study proposed bivariate flood frequency analysis methods for evaluating flood events considering correlations among attributes of flood events. Parametric distributions such as Gumbel mixed model and bivariate gamma distribution, and a non-parametric model using a bivariate kernel function were introduced in this study. A time series of annual flood events were extracted from observations of inflow to the Soyang River Dam and the Daechung Dam, respectively. The joint probability distributions and return periods were derived from the relationship between the amount of peak discharge and the total volume of flood runoff. Applicabilities of bivariate flood frequency analysis were examined by comparing the return period acquired from the proposed bivariate analyses and the conventional univariate analysis.

• The KIPS Transactions:PartD
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• v.8D no.4
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• pp.393-400
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• 2001

This paper presents a stochastic model for the software failure phenomenon based on a nonhomogeneous Poisson process (NHPP). The failure process is analyzed to develop a suitable mean value function for the NHPP ; expressions are given for several performance measure. Actual software failure data are compared with generalized model by Goel dependent on the constant reflecting the quality of testing. The performance measures and parametric inferences of the new models, Rayleigh and Gumbel distributions, are discussed. The results of the new models are applied to real software failure data and compared with Goel-Okumoto and Yamada, Ohba and Osaki models. Tools of parameter inference was used method of the maximun likelihood estimate and the bisection algorithm for the computing nonlinear root. In this paper, using the sum of the squared errors, model selection was employed. The numerical example by NTDS data was illustrated.