Hierarchical Bayesian Model for Assessing Extreme Daily Rainfall Through Combining Climatic and Topographic Information

Frequency analysis of the annual maximum rainfall time series is essential for designing infrastructures to provide protection against local floods and related events. However, the results of the frequency analysis obtained are ambiguous. In this study, we aimed to develop a spatial hierarchical Bayesian model framework through combining the climatic and topographic information. To confirm the applicability of the proposed method, the results of at-site frequency analysis and regional frequency analysis using the index flood method were compared in the Busan-Ulsan-Gyeongnam region. Furthermore, a hierarchical Bayesian model was developed, in which the parameters of the generalized logistic distribution comprised relatively simple covariate relationships upon considering the possibility of expansion into various probability distributions and more complex covariate structures. The uncertainty of this model was analyzed using the coefficient of variation of rainfall quantile ensemble. The results confirmed that the regional frequency analysis using the hierarchical Bayesian model combined with the climatic and topographic information could provide an accurate estimate of extreme daily rainfall with relatively good agreement with the estimate at a specific site, but is a more reliable approach.

A DISTRIBUTION LAW OF THE RESULTS OF MEASUREMENT OF COMPOSITION AND PROPERTIES OF PETROLEUM PRODUCTS

Идентификацию закона распределения результатов измерений состава и свойств нефтепродуктов проводили путем проверки сложной гипотезы с использованием пяти критериев согласия: χ2-Пирсона, Колмогорова, Смирнова, ω2 Крамера-Мизеса-Смирнова; Ω2 Андерсона-Дарлинга. В качестве возможных функций распределения вероятностей рассматривали 12 симметричных одномодальных законов распределения, а также 66 смесей этих же законов распределения. Целью идентификации являлось нахождение универсального закона распределения (смеси законов распределений) справедливого для всех рассматриваемых величин. Проверка сложной гипотезы о соответствии какому-либо симметричному одномодальному закону распределения показала, что не существует универсального закона распределения справедливого для всех методик измерений состава и свойств нефтепродуктов, наиболее часто не отвергалась гипотеза о соответствии данных обобщенному логистическому распределению, распределению Лапласа и двустороннему экспоненциальному распределению. Проверка сложной гипотезы о соответствии какой-либо смеси симметричных одномодальных законов распределения показала, что эмпирическая функция распределения результатов измерений состава и свойств нефтепродуктов может быть представлена в виде смеси двух нормальных распределений с разными значениями параметров положения и масштаба. При этом для одной и той же выборки значения достигаемого уровня значимости гипотезы о соответствии смеси законов распределений в несколько раз выше среднего значения достигаемого уровня значимости гипотезы о соответствии одному закону распределения. На основе проведенного исследования обоснована вероятностная модель процесса испытаний нефтепродуктов, в рамках которой результат испытаний нефтепродуктов рассматривается как случайная величина с функцией распределения в виде смеси нормальных законов распределения: «основного» с дисперсией, не превышающей установленные требования (при статистически управляемом состоянии процесса испытаний), и «засоряющего» с дисперсией значительно превышающей установленные требования (при статистически неуправляемом состоянии процесса испытаний). The identification of a distribution law of the results of measurements of the composition and properties of petroleum products was carried out by testing a complex hypothesis using five goodness-of-fit tests: χ2-Pearson, Kolmogorov, Smirnov, ω2Cramer-Mises-Smirnov; Ω2 Anderson-Darling. Twelve symmetric unimodal distribution laws and 66 mixtures of the same distribution laws were considered as possible probability distribution functions. The purpose of the identification was to find a universal distribution law (a mixture of distribution laws) that is valid for all considered quantities. Testing the complex hypothesis of compliance with any symmetric unimodal distribution law showed that there is no universal distribution law that is valid for all measurement techniques of the composition and properties of petroleum products; most often the hypothesis of the correspondence the data to the generalized logistic distribution, the Laplace distribution and the two-sided exponential distribution was not rejected. Testing a complex hypothesis about the correspondence of any mixture of symmetric unimodal distribution laws showed that the empirical distribution function of the results of measurements of the composition and properties of petroleum products can be represented as a mixture of two normal distributions with different values ​​of the position and scale parameters. At the same time, for the same sample, the values ​​of the achieved significance level of the hypothesis about the correspondence to the mixture of distribution laws is several times higher than the average value of the achieved significance level of the hypothesis about the correspondence to one distribution law. Based on this study, a chance model of the process of testing petroleum products was substantiated, within which the result of testing petroleum products is considered as a random variable with a distribution function in the form of a mixture of normal distribution laws: "basic" with a variance not exceeding the established requirements (with a statistically controlled state of the test process), and "fouling" with a variance significantly exceeding the established requirements (with a statistically uncontrolled state of the test process).

Comparison of stationary and non-stationary frequency models for assessing design discharges in variable climate: north-west England case study

<p>Successive occurrence of floods across north-west England over the course of the past few years has resulted in the need for the local authorities and decision makers to (re-) assess several flood management schemes. However, ongoing decision-making on how flood control measures are constructed, is frequently still made on the basis of the assumption that the flood characteristics of catchments have remained constant over time (i.e., stationarity). To verify the validity of this assumptions, non-parametric tests alongside change-permitting flood frequency frameworks based on Generalized Logistic distribution model (as the recommended model in the UK catchments) have been applied to a dataset of extreme peak river flow measurements across the region (39 catchments with up to 75 years of records). Allowing the location parameter of the model to change linearly with time, cumulative annual rainfall and cumulative annual temperature as covariates, one stationary as well as six non-stationary models have been introduced. The regional non-stationary frequency results indicate a notable improvement over the stationary predictions, estimating design flood quantiles (i.e., 100-year events) up to 75% larger than classic stationary estimates. Moreover, the vast majority of rivers demonstrate statistically significant changes (mainly driven by cumulative annual rainfall), specifically in the late 1990s. This indicates that non-stationary models should be taken into consideration, along with the traditional stationary ones to help understanding the changes in the peak river flow regimes across the north-west England.</p>

Regional Frequency Analysis of Rainfall Using L-Moment Method as A Design Rainfall Prediction

Frequency analysis is a method for predicting the probability of future hydrological events based on historical data. Frequency analysis of rain data and discharge data is generally carried out using the moment method, but the moment method has a large bias, variant, and slope so that it has the potential to produce inaccurate hydrological design magnitudes. The L-moment method is a linear combination of Probability Weighted Moment which processes data in a concise and linear manner. This research was conducted that L-moment method will obtain a regional probability distribution and design rainfall which can be used as a basis for calculating hydrological planning in anticipation of disasters. The location of the study in Mount Merapi area was chosen in order to more accurately predict the maximum rainfall that could cause cold lava in the area to reduce the risk of loss to the people living around Mount Merapi. The results showed that the entire rainfall stations homogeneous and no data was released. The L-moment regional ratio results τ2R = 0.203, τ3R = 0.166, dan τ4R = 0.169. The homogeneity and heterogeneity tests show that all rainfall stations are uniform or homogeneous. No data were released from the discordance test results. Growth factor value increases in each design rainfall return periods. The regional probability distribution that is suitable for the research area is Generalized Logistic distribution with design rainfall equation has been formulated. Test model showed the minimum RBias = 0.45%, maximum RBias = 41.583%, minimum RRSME = 0.45%, and maximum RRSME = 71.01%. The stability of L-moment method showed by model test minimum error = 1.64% and maximum error = 16.60%.

The generalization of the hyperbolic secant distribution and the logistic distribution in the single distribution with variable kurtosis

The generalization V of the logistic distribution is proposed and investigated. For a random variable having a generalized logistic distribution of type V, the characteristic function is found, the generating function of moments is formed, and the expression of dispersion is obtained. The dependence of the kurtosis coefficient of the generalized logistic distribution on the power parameter is found and investigated. The interval of values of the coefficient of kurtosis of the generalized logistic distribution is determined. It is found that the coefficient of kurtosis depends only on the power parameter.

Binomial Regression Models with a Flexible Generalized Logit Link Function

In binomial regression, a link function is used to join the linear predictor variables and the expectation of the response variable. This paper proposes a flexible link function from a new class of generalized logistic distribution, namely a flexible generalized logit (glogit) link. This approach considers both symmetric and asymmetric models, including the cases of lighter and heavier tails, as compared to standard logistic. The glogit is created from the inverse cumulative distribution function of the exponentiated-exponential logistic (EEL) distribution. Using a Bayesian framework, we conduct a simulation study to investigate the model performance compared to the most commonly used link functions, e.g., logit, probit, and complementary log–log. Furthermore, we compared the proposed model with several other asymmetric models using two previously published datasets. The results show that the proposed model outperforms the existing ones and provides flexibility fitting the experimental dataset. Another attractive aspect of the model are analytically tractable and can be easily implemented under a Bayesian approach.