• Journal of the Korean Operations Research and Management Science Society
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• v.28 no.4
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• pp.145-153
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• 2003

In this Paper, we examine the limiting distributional behaviour of extreme values of mixed Erlang random variables. We show that, in the finite mixture of Erlang distributions, the component distribution with an asymptotically dominant tail has a critical effect on the asymptotic extreme behavior of the mixture distribution and it converges to the Gumbel extreme-value distribution. Normalizing constants are also established. We apply this result to characterize the asymptotic distribution of maxima of sojourn times in M/M/s queuing system. We also show that Erlang mixtures with continuous mixing may converge to the Gumbel or Type II extreme-value distribution depending on their mixing distributions, considering two special cases of uniform mixing and exponential mixing.

• Journal of Korea Water Resources Association
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• v.52 no.11
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• pp.895-904
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• 2019

It has been well recognized that extreme rainfall process often features a nonstationary behavior, which may not be effectively modeled within a stationary frequency modeling framework. Moreover, extreme rainfall events are often described by a two (or more)-component mixture distribution which can be attributed to the distinct rainfall patterns associated with summer monsoons and tropical cyclones. In this perspective, this study explores a Mixture Distribution based Nonstationary Frequency (MDNF) model in a changing rainfall patterns within a Bayesian framework. Subsequently, the MDNF model can effectively account for the time-varying moments (e.g. location parameter) of the Gumbel distribution in a two (or more)-component mixture distribution. The performance of the MDNF model was evaluated by various statistical measures, compared with frequency model based on both stationary and nonstationary mixture distributions. A comparison of the results highlighted that the MDNF model substantially improved the overall performance, confirming the assumption that the extreme rainfall patterns might have a distinct nonstationarity.

• Journal of the Korean Society of Marine Environment & Safety
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• v.21 no.3
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• pp.253-258
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• 2015

The purpose of this study is to find suitable probability distribution function of complex distribution data like multimodal. Normal distribution is broadly used to assume probability distribution function. However, complex distribution data like multimodal are very hard to be estimated by using normal distribution function only, and there might be errors when other distribution functions including normal distribution function are used. In this study, we experimented to find fit probability distribution function in multimodal area, by using AIS(Automatic Identification System) observation data gathered in Mokpo port for a year of 2013. By using chi-squared statistic, gaussian mixture model(GMM) is the fittest model rather than other distribution functions, such as extreme value, generalized extreme value, logistic, and normal distribution. GMM was found to the fit model regard to multimodal data of maritime traffic flow distribution. Probability density function for collision probability and traffic flow distribution will be calculated much precisely in the future.

• The Korean Journal of Applied Statistics
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• v.28 no.5
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• pp.847-858
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• 2015

Deep-sea fishery in the Antarctic Ocean has been actively progressed by the developed countries including Korea. In order to prevent the environmental destruction of the Antarctic Ocean, related countries have established the Commission for the Conservation of Antarctic Marine Living Resources (CCAMLR) and have monitored any illegal unreported or unregulated fishing. Fishing of tooth fish, an expensive fish, in the Antarctic Ocean has increased recently and high catches per unit effort (CPUE) of fishing boats, which is suspicious for an illegal activity, have been frequently reported. The data of CPUEs in a fishing area of the Antarctic Ocean often show an extreme Distribution or a mixture of two extreme distributions. This paper proposes an algorithm to detect an outlier of CPUEs by using the mixture of two extreme distributions. The parameters of the mixture distribution are estimated by the EM algorithm. Log likelihood value and posterior probabilities are used to detect an outlier. Experiments show that the proposed algorithm to detect outlier of the data can be adopted instead of simple criteria such as a CPUE is greater than 1.

• Structural Engineering and Mechanics
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• v.74 no.1
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• pp.55-67
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• 2020

The common practice to predict the characteristic structural load effects (LEs) in long reference periods is to employ the extreme value theory (EVT) for building limit distributions. However, most applications ignore that LEs are driven by multiple loading events and thus do not have the identical distribution, a prerequisite for EVT. In this study, we propose the composite extreme value modeling approach using clustering to (a) cluster initial blended samples into finite identical distributed subsamples using the finite mixture model, expectation-maximization algorithm, and the Akaike information criterion; (b) combine limit distributions of subsamples into a composite prediction equation using the generalized Pareto distribution based on a joint threshold. The proposed approach was validated both through numerical examples with known solutions and engineering applications of bridge traffic LEs on a long-span bridge. The results indicate that a joint threshold largely benefits the composite extreme value modeling, many appropriate tail approaching models can be used, and the equation form is simply the sum of the weighted models. In numerical examples, the proposed approach using clustering generated accurate extrema prediction of any reference period compared with the known solutions, whereas the common practice of employing EVT without clustering on the mixture data showed large deviations. Real-world bridge traffic LEs are driven by multi-events and present multipeak distributions, and the proposed approach is more capable of capturing the tendency of tailed LEs than the conventional approach. The proposed approach is expected to have wide applications to general problems such as samples that are driven by multiple events and that do not have the identical distribution.

• Journal of the Korean Society of Safety
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• v.32 no.5
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• pp.157-167
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• 2017

Millions of People die every year from diseases caused by exposure to outdoor air pollution. Especially, one of the most severe types of air pollution is fine particulate matter (PM10, PM2.5). South Korea also has been suffered from severe PM. This paper analyzes regional risks induced by PM10 and PM2.5 that have affected domestic area of Korea during 2014~2016.3Q. We investigated daily maxima of PM10 and PM2.5 data observed on 284 stations in South Korea, and found extremely high outlier. We employed extreme value distributions to fit the PM10 and PM2.5 data, but a single distribution did not fit the data well. For theses reasons, we implemented extreme mixture models such as the generalized Pareto distribution(GPD) with the normal, the gamma, the Weibull and the log-normal, respectively. Next, we divided the whole area into 16 regions and analyzed characteristics of PM risks by developing the FN-curves. Finally, we estimated 1-month, 1-quater, half year, 1-year and 3-years period return levels, respectively. The severity rankings of PM10 and PM2.5 concentration turned out to be different from region to region. The capital area revealed the worst PM risk in all seasons. The reason for high PM risk even in the yellow dust free season (Jun. ~ Sep.) can be inferred from the concentration of factories in this area. Gwangju showed the highest return level of PM2.5, even if the return level of PM10 was relatively low. This phenomenon implies that we should investigate chemical mechanisms for making PM2.5 in the vicinity of Gwangju area. On the other hand, Gyeongbuk and Ulsan exposed relatively high PM10 risk and low PM2.5 risk. This indicates that the management policy of PM risk in the west side should be different from that in the east side. The results of this research may provide insights for managing regional risks induced by PM10 and PM2.5 in South Korea.

• Communications for Statistical Applications and Methods
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• v.30 no.3
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• pp.259-272
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• 2023

In this paper, we suggest a new method for the prediction of sharp changes in particulate matter (PM₁₀) using quantile mapping. To predict the current PM₁₀ density in Seoul, we consider PM₁₀ and precipitation in Baengnyeong and Ganghwa monitoring stations observed a few hours before. For the PM₁₀ distribution estimation, we use the extreme value mixture model, which is a combination of conventional probability distributions and the generalized Pareto distribution. Furthermore, we also consider a quantile generalized additive model (QGAM) for the relationship modeling between precipitation and PM₁₀. To prove the validity of our proposed model, we conducted a simulation study and showed that the proposed method gives lower mean absolute differences. Real data analysis shows that the proposed method could give a more accurate prediction when there are sharp changes in PM₁₀ in Seoul.

• 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.

• The Korean Journal of Applied Statistics
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• v.25 no.5
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• pp.757-773
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• 2012

Traditional value at risk(S-VaR) has a difficulity in predicting the future risk of financial asset prices since S-VaR is a backward looking measure based on the historical data of the underlying asset prices. In order to resolve the deficiency of S-VaR, an economic value at risk(E-VaR) using the risk neutral probability distributions is suggested since E-VaR is a forward looking measure based on the option price data. In this study E-VaR is estimated by assuming the generalized gamma distribution(GGD) as risk neutral density function which is implied in the option. The estimated E-VaR with GGD was compared with E-VaR estimates under the Black-Scholes model, two-lognormal mixture distribution, generalized extreme value distribution and S-VaR estimates under the normal distribution and GARCH(1, 1) model, respectively. The option market data of the KOSPI 200 index are used in order to compare the performances of the above VaR estimates. The results of the empirical analysis show that GGD seems to have a tendency to estimate VaR conservatively; however, GGD is superior to other models in the overall sense.

• The Journal of Asian Finance, Economics and Business
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• v.8 no.1
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• pp.665-673
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• 2021

Default has become an extreme concern in the current world due to the financial crisis. The previous prediction of companies' bankruptcy exhibits evidence of decision assistance for financial and regulatory bodies. Notwithstanding numerous advanced approaches, this area of study is not outmoded and requires additional research. The purpose of this research is to find the best classifier to detect a company's default risk and bankruptcy. This study used secondary data from the Pakistan Stock Exchange (PSX) and it is time-series data to examine the impact on the determinants. This research examined several different classifiers as per their competence to properly categorize default and non-default Pakistani companies listed on the PSX. Additionally, PSX has remained consistent for some years in terms of growth and has provided benefits to its stockholders. This paper utilizes machine learning techniques to predict financial distress in companies listed on the PSX. Our results indicate that most multi-stage mixture of classifiers provided noteworthy developments over the individual classifiers. This means that firms will have to work on the financial variables such as liquidity and profitability to not fall into the category of liquidation. Moreover, Adaptive Boosting (Adaboost) provides a significant boost in the performance of each classifier.