Generalized Logistic Distribution Research Articles

Estimation of regional and at-site quantiles of extreme winds under flood index procedure

Extreme winds are becoming more common among environmental events with the most catastrophic societal consequences. A regional frequency analysis of Daily Annual Maximum Wind Speed (DAMWS) is necessary not only for a comprehensive understanding of wind hazards but also for infrastructure design and safety, wind energy potential, disaster risk reduction, insurance and risk assessment in a particular region of study. This study investigated regional frequency analysis of DAMWS of Baluchistan and Sindh provinces of Pakistan. L-moments regionalization techniques along with flood index procedure were applied to DAMWS records of 21 stations from 1990 to 2019 across the study area. We intended to find the regional frequency distribution for maximum winds and predict the returns for extreme winds events in the future. Only one station namely Lasbella was found to be discordant. With the help of cluster analysis, the remaining 20 stations were further divided into two homogeneous. Heterogeneity measures validate that both regions are homogenous with allotted stations. Regional quantiles for both regions are estimated through best-fit probability distribution among Generalized Normal (GNO), Generalized Logistic (GLO), Pearson Type 3 (P3), Generalized Pareto (GPA), and Generalized Extreme Value (GEV). Robustness of GLO distribution compared to GEV distribution is assessed through Monte Carlo simulations of relative bias and relative root mean square error. Findings clearly show that GLO distribution is the best for regional modeling. Furthermore, with the help of index flood procedure we determined at-site quantiles of all stations for various return periods. These estimated quantiles are of valuable information for various sectors, including infrastructure, energy, disaster management, and climate resilience, leading to improved planning, development, and risk reduction in the face of wind-related hazards in Sindh and Balochistan provinces of Pakistan.

The Partial L-Moment of the Four Kappa Distribution

Statistical analysis of extreme events such as flood events is often carried out to predict large return period events. The behaviour of extreme events not only involves heavy-tailed distributions but also skewed distributions, similar to the four-parameter Kappa distribution (K4D). In general, this covers many extreme distributions such as the generalized logistic distribution (GLD), the generalized extreme value distribution (GEV), the generalized Pareto distribution (GPD), and so on. To utilize these distributions, we have to estimate parameters accurately. There are many parameter estimation methods, for example, Method of Moments, Maximum Likelihood Estimator, L-Moments, or partial L-Moments. Nowadays, no researchers have applied the partial L-Moments method to estimate the parameters of K4D. Therefore, the objective of this paper is to derive the partial L-Moments (PL-Moments) for K4D, namely the PL-Moments of the K4D in order to estimate hydrological extremes from censored data. The findings of this paper are formulas of parameter estimation for K4D based on the PL-Moments approach. We have derived the Partial Probability-Weighted Moments (PPWMs) of the K4D (β'r) and derive the estimation of parameters when separated by shape parameters (k,h) conditions i.e., case k>-1 and h>0, case k>-1 and h=0 and case -1<k<-1/h and h<0. Finally, we expect that the parameter estimate for K4D from this formula will help to make accurate forecasts. Doi: 10.28991/ESJ-2023-07-04-06 Full Text: PDF

Determining optimal probability distributions for gridded precipitation data based on L-moments

Understanding the probability distributions of precipitation is crucial for predicting climatic events and constructing hydraulic facilities. To overcome the inadequacy of precipitation data, regional frequency analysis was commonly used by “trading space for time”. However, with the increasing availability of gridded precipitation datasets with high spatial and temporal resolutions, the probability distributions of precipitation for these datasets have been less explored. We used L-moments and goodness-of-fit criteria to identify the probability distributions of annual, seasonal, and monthly precipitation for a 0.5° × 0.5° dataset across the Loess Plateau (LP). We examined five 3-parameter distributions, namely General Extreme Value (GEV), Generalized Logistic (GLO), Generalized Pareto (GPA), Generalized Normal (GNO), and Pearson type III (PE3), and evaluated the accuracy of estimated rainfall using the leave-one-out method. We also presented pixel-wise fit-parameters and quantiles of precipitation as supplements. Our findings indicated that precipitation probability distributions vary by location and time scale, and the fitted probability distribution functions are reliable for estimating precipitation under various return periods. Specifically, for annual precipitation, GLO was prevalent in humid and semi-humid areas, GEV in semi-arid and arid areas, and PE3 in cold-arid areas. For seasonal precipitation, spring precipitation mainly conforms to GLO distribution, summer precipitation around the 400 mm isohyet prevalently follows GEV distribution, autumn precipitation primarily meets GPA and PE3 distributions, and winter precipitation in the northwest, south, and east of the LP mainly conforms to GPA, PE3 and GEV distributions, respectively. Regarding monthly precipitation, the common distribution functions are PE3 and GPA for the less-precipitation months, whereas the distribution functions of precipitation for more-precipitation months vary substantially across different regions of the LP. Our study contributes to a better understanding of precipitation probability distributions in the LP and provides insights for future studies on gridded precipitation datasets using robust statistical methods.

Regional Analysis of Consecutive Days Maximum Rainfall Using L-Moments: A Case Study for Brahmaputra Valley Region of Assam, India

In this study, Regional Rainfall Frequency analysis has been carried out based on L-moment in the Brahmaputra Valley. For this purpose, the consecutive 1 to 6 days maximum rainfall for 11 sites (data varying from 14 years to 23 years) have been considered for our study. The heterogeneity measures for 1 to 6 days of the study region are calculated. The five probability distributions namely, generalized extreme value (GEV), generalized logistic (GLO), generalized Pareto (GPA), generalized normal (GNO), and the Pearson Type III (PE3) have been considered. The Z-statistics criteria and L-moment ratio diagram for goodness of fit tests have been used for identifying the best fitting distributions for 1 to 6 days of the study area. The PE3 distribution is identified as the best fitting distribution for the consecutive 1-day and 6-days maximum rainfall while GLO distribution for the consecutive 2-days and 4-days, GNO distributions for the consecutive 3-days and 5-days maximum rainfall of the study region. The regional relationships for each season of the study regions are develope

State-of-the-Art Statistical Approaches for Estimating Flood Events.

Reliable quantile estimates of annual peak flow discharges (APFDs) are needed for the design and operation of major hydraulic infrastructures and for more general flood risk management and planning. In the present study, linear higher order-moments (LH-moments) and nonparametric kernel functions were applied to APFDs at 18 stream gauge stations in Punjab, Pakistan. The main purpose of this study was to evaluate the impacts of different quantile estimation methods towards water resources management and engineering applications by means of comparing the state-of-the-art approaches and their quantile estimates calculated from LH-moments and nonparametric kernel functions. The LH-moments (η = 0, 1, 2) were calculated for the three best-fitted distributions, namely, generalized logistic (GLO), generalized extreme value (GEV), and generalized Pareto (GPA), and the performances of these distributions for each level of LH-moments (η = 0, 1, 2) were compared in terms of Anderson–Darling, Kolmogorov–Smirnov, and Cramér–Von Mises tests and LH-moment ratio diagrams. The findings indicated that GPA and GEV distributions were best fitted for most stations, followed by GLO distribution. The quantile estimates derived from LH-moments (η = 0, 1, 2) had a lower relative absolute error, particularly for higher return periods. However, the Gaussian kernel function provided a close estimate among nonparametric kernel functions for small return periods when compared to LH-moments (η = 0, 1, 2), thus highlighting the importance of using LH-moments (η = 0, 1, 2) and nonparametric kernel functions in water resources management and engineering projects.

Linear Estimation in the Type II Generalized Logistic Distribution under Progressive Censoring

Generalized distributions have become increasingly popular in applications. They are highly flexible in data analysis, especially with skewed data, which are common in many applications. The Generalized Logistic Distribution (GLD) and its special cases have recently received a lot of interest in the literature. We derived estimators of the unknown parameters of type II Generalized Logistic Distribution (Type II GLD) based on progressively type II censored data. A variety of point estimation methods is employed. We considered the best linear unbiased estimator (BLUE) and the best (affine) linear equivariant estimator (BLEE). In addition, we considered Bayesian estimation. Simulation approaches were used to study the estimators and compare them with the maximum likelihood estimator (MLE) in a range of progressive censoring schemes. The mean squared error (MSE) and bias were employed as comparison criteria. An example based on real data is presented.

Frequency analysis of annual maximum rainfall series of fifteen meteorological observatories of Sindh, Pakistan

The study is an application of L-moments based regional frequency analysis using annual maximum rainfall series of fifteen sites of Sindh, Pakistan. Initial screening reveals that the observed data series at each site has no serial correlation and is randomly and identically distributed. The discordancy measure (Di) show that none of the site in a group of fifteen sites is discordant. The regional heterogeneity measure (H) shows that the region of fifteen sites is homogeneous. L-moment ratio diagram and |ZDIST| statistic are in favor of generalized logistic distribution (GLO) as best-fitted regional distribution. Parameters of GLO distribution and regional as well as at-site quantiles for various return periods have been estimated. Results show that the estimated quantiles of each site are greater than the average values of observed data series at corresponding sites. Random forest regression has been used for the estimation of quantiles at ungauged sites.

Generalized logistic distribution and its regression model

A new generalized asymmetric logistic distribution is defined. In some cases, existing three parameter distributions provide poor fit to heavy tailed data sets. The proposed new distribution consists of only three parameters and is shown to fit a much wider range of heavy left and right tailed data when compared with various existing distributions. The new generalized distribution has logistic, maximum and minimum Gumbel distributions as sub-models. Some properties of the new distribution including mode, skewness, kurtosis, hazard function, and moments are studied. We propose the method of maximum likelihood to estimate the parameters and assess the finite sample size performance of the method. A generalized logistic regression model, based on the new distribution, is presented. Logistic-log-logistic regression, Weibull-extreme value regression and log-Fréchet regression are special cases of the generalized logistic regression model. The model is applied to fit failure time of a new insulation technique and the survival of a heart transplant study.

Linear Combination and Reliability of Generalized Logistic Random Variables

Experimental random data, in general, present a skewed behaviour. Thus, asymmetrical generalized distributions are of interest. The generalized logistic distributions (GLDs) are good candidates to model skewed data because their probability density functions (p.d.f.) and characteristic functions are mathematically simple. In this paper, exact expressions in terms of the H-function are, for the first time, derived for the p.d.f. and for the cummulative distribution function of the linear combination of GLDs of type IV with different location, scale and shape parameters. Also, exact and approximate expressions are derived for $R=P(X<Y)$. Numerical examples illustrate the correctness of the expressions derived.

Estimation of wind speed using regional frequency analysis based on linear‐moments

The quantiles of annual maximum wind speed (AMWS) can be estimated for different meteorological stations of interest by using at‐site frequency analysis and extreme value theory. These estimates are of immense importance for the codification of wind speed. However, the historical data of wind speed at the number of meteorological stations are sometimes unavailable and often insufficient due to the shorter length, especially in developing countries like Pakistan. The scarcity of the data increases the uncertainty of the quantiles estimates regarding policy implications. To cope with the problem, an approach of Regional Frequency Analysis (RFA) is opted here. In this study, RFA of AMWS using linear‐moments (L‐moments) is carried out by considering wind speed data of nine meteorological stations of province Punjab, Pakistan. No station is found to be discordant. A single homogenous region is constituted from these nine stations using a subjective approach based on their geographical locations. Heterogeneity measures justify that these nine stations of Punjab form a single homogeneous region. Regional quantiles estimates are found through the most appropriate probability distribution among generalized normal (GNO), generalized logistic (GLO), Pearson Type 3 (P3), generalized Pareto (GPA), Weibull (WEI), log Pearson Type 3 (LP3) and generalized extreme value (GEV) distributions. Z‐statistic and L‐moment ratio diagram suggest that GLO and GNO distributions are better choices than others. Robustness of both distributions is evaluated through relative bias (RB) and relative root mean square error (RRMSE). Findings indicate that overall, GLO distribution is better than GNO. Further, we also find at‐site quantiles from dimensionless quantities (regional quantiles) using the sample mean and median as scaling factors. Quantiles' estimates calculated from this study can be used in codified structural designs for policy implications.

Stress Affection of Two Scale Truncated Generalized Logistic Parameters with Progressive Censoring

This paper deals with non-Bayesian estimation problem of constant-stress Accelerated Life Tests (ALTs) when the lifetime of the items follow truncated Generalized Logistic Distribution (GLD). Some considerations on inference based on the use of asymptotically normality of the ML estimators are presented considering the stress effects on the two scale parameters of the truncated GLD with a k-level constant-stress ALT under progressive type-I censored grouped data. The EM algorithm method is used to obtain the estimators of the unknown parameters. In addition, estimator of the two scale parameters, reliability function under usual conditions and Fisher information matrix of the estimators are given. Finally, we present a Simulation Study to illustrate the proposed procedure.

Regional analysis of maximum rainfall using L-moment and LH-moment: A comparative case study for the northeast India

Rainfall data of the northeast region of India has been considered for selecting best fit model for rainfall frequency analysis. The methods of L-moment has been employed for estimation of parameters five probability distributions, namely Generalized extreme value (GEV), Generalized Logistic(GLO), Pearson type 3 (PE3), 3 parameter Log normal (LN3) and Generalized Pareto (GPA) distributions. The methods of LH-moment of four orders (L1 L2, L3 & L4-moments) have also been used for estimating the parameters of three probability distributions namely Generalized extreme value (GEV), Generalized Logistic (GLO) and Generalized Pareto (GPA) distributions. PE3 distribution has been selected as the best fitting distribution using L-moment, GPA distribution using L1-moment and GLO distribution using L2, L3 & L4-moments. Relative root mean square error (RRMSE) and RBIAS are employed to compare between the results found from L-moment and LH-moment analysis. It is found that GPA distribution designated by L1-moment method is the most suitable and the best fitting distribution for rainfall frequency analysis of the northeast India. Also the L1-moment method is significantly more efficient than L-moment and other orders of LH-moment for rainfall frequency analysis of the northeast India.

Joint pattern of seasonal hydrological droughts and floods alternation in China’s Huai River Basin using the multivariate L-moments

Under the current condition of climate change, droughts and floods occur more frequently, and events in which flooding occurs after a prolonged drought or a drought occurs after an extreme flood may have a more severe impact on natural systems and human lives. This challenges the traditional approach wherein droughts and floods are considered separately, which may largely underestimate the risk of the disasters. In our study, the sudden alternation of droughts and flood events (ADFEs) between adjacent seasons is studied using the multivariate L-moments theory and the bivariate copula functions in the Huai River Basin (HRB) of China with monthly streamflow data at 32 hydrological stations from 1956 to 2012. The dry and wet conditions are characterized by the standardized streamflow index (SSI) at a 3-month time scale. The results show that: (1) The summer streamflow makes the largest contribution to the annual streamflow, followed by the autumn streamflow and spring streamflow. (2) The entire study area can be divided into five homogeneous sub-regions using the multivariate regional homogeneity test. The generalized logistic distribution (GLO) and log-normal distribution (LN3) are acceptable to be the optimal marginal distributions under most conditions, and the Frank copula is more appropriate for spring-summer and summer-autumn SSI series. Continuous flood events dominate at most sites both in spring-summer and summer-autumn (with an average frequency of 13.78% and 17.06%, respectively), while continuous drought events come second (with an average frequency of 11.27% and 13.79%, respectively). Moreover, seasonal ADFEs most probably occurred near the mainstream of HRB, and drought and flood events are more likely to occur in summer-autumn than in spring-summer.

REGIONAL FREQUENCY ANALYSIS OF STREAMFLOW BASED ON PARTIAL L-MOMENTS APPROACH

An attempt has been made to model the annual maximum streamflow in the East Coast of Peninsular Malaysia, utilizing the guidelines in regional flood frequency analysis. The L-moments and partial L-moments (PL-moments) at several censoring levels are employed to estimate the regional parameters of three extreme value distributions, namely, generalized extreme value (GEV), generalized logistic (GLO) and generalized Pareto (GPA) distributions. A total number of 18 streamflow stations located throughout the eastern region of Peninsular Malaysia are used as case study. The performances of the L-moments and PL-moments methods and their corresponding distribution functions are compared using Monte Carlo simulation. The results of relative root mean square error (RRMSE) and relative bias (RBIAS) show that the GLO distribution of PL-moments at censoring level 0.1 is appropriate to model the regional streamflow data in East Coast of Peninsular Malaysia compared to L-moments. The overall simulation results indicated that, in some situation, the PL-moments method improves the streamflow quantile prediction and provide useful tools for application in regional flood frequency analysis.

Annual Maximum Exchange Rate in Southeast Asia Based on Methods of L-moment and Maximum Likelihood

The main objective of this study is to determine the best fitting distribution to describe the annual series of maximum daily exchange rate from 1993 to 2013 for 4 countries in Southeast Asia based on L-moment and Maximum Likelihood (ML). Four three-parameter extreme-value distributions which are considered are Generalized Extreme Value (GEV), Generalized Logistic Distribution (GLD), Generalized Pareto Distribution (GPD) and Pearson III (P3). The estimation of parameters of these distributions is determined using the L-moment and maximum likelihood. The adequacy of the distributions based on parameter estimates computed using the two methods are evaluated using goodness-of-fit tests. When the goodness-of-fit results for these distributions are compared, it is found that, on the average, the performance of L-moment is better than the performance of maximum likelihood. Although the best fitting distribution may very according to the method of estimation and country considered, in most cases, data for the majority of the several countries are found to follow the generalized logistic distribution.

Spatial variability and cartography of maximum annual daily rainfall under different return periods in Northern Algeria

The estimation of precipitation quantileshas always been an area of great importance to meteorologists, hydrologists, planners and managers of hydrotechnical infrastructures. In many cases, it is necessary to estimate the values relating to extreme events for the sites where there is little or no measurement, as well as their return periods. A statistical approach is the most used in such cases. It aims to find the probability distribution that best fitsthe maximum daily rainfall values. In our study, 231 rainfall stations were used to regionalize and find the best distribution for modeling the maximum daily rainfall in Northern Algeria. The L-moments method was used to perform a regionalization based on discordance criteria and homogeneity test. It gave rise to twelve homogeneous regions in terms of L-Coefficient of variation(L-CV), L-Skewness (L-CS)and L-Kurtosis (L-CK). This same technique allowed us to select the regional probability distribution for each group using the Z statistic. The generalized extreme values distribution (GEV) was selected to model the maximum daily rainfall of 10 groups located in the north of the steppe region and the generalized logistic distribution (GLO) for groups representing the steppes of Central and Western Algeria. The study of uncertainty by the bias and RMSE showed that the regional approach is acceptable. We have also developed maximum daily rainfall maps for 2, 5, 10, 20, 50 and 100 years return periods. We relied on a network of 255 rainfall stations. The spatial variability of quantiles was evaluated by semi-variograms. All rainfallfrequencymodelshave a spatial dependence with an exponential model adjusted to the experimental semi-variograms. The parameters of the fitted semi-variogram for different return periods are similar,throughout, while the nugget is more important for high return periods. Maximum daily rainfall increases from South to North and from West to East, andis more significant in the coastal areas of eastern Algeria where it exceeds 170 mm for a return period of 100 years. However,it does not exceed 50 mm in the highlands of the west.

English

Estimation of flood magnitude is a crucial component in planning, designing, and managing of water resources projects. The main focus in hydrologic design is the estimation of high flow quantile. L-moments, popular among hydrologist in flood estimation is known to be oversensitive towards the lower part of the distribution and gives insufficient weight to large sample values. As an alternative, the method of partial L-moments (PL-moments) is proposed to give weightage to the upper part of distribution and large values in censored sample. In this paper, three widely used distributions are selected namely; generalized extreme value (GEV), generalized logistic (GLO) and generalized Pareto (GPA) distribution, for the analysis of censored flood samples. Monte Carlo simulations are conducted to illustrate the performance of PL-moments compared to simple L-moments in fitting each distribution to its samples. Finally, both simple L-moments and PL-moments are used to fit the GLO distribution to two data sets of annual maximum flow series of River Ketil in Kedah and River Gemencheh in Negeri Sembilan, Malaysia.

Modeling the distribution of extreme returns in the Chinese stock market

It is well known that extreme share returns on stock markets can have important implications for financial risk management. In this paper, we are concerned with the distribution of the extreme daily returns of the Shanghai Stock Exchange (SSE) Composite Index. Three well-known distributions in extreme value theory, i.e., Generalized Extreme Value (GEV), Generalized Logistic (GL) and Generalized Pareto distributions, are employed to model the SSE Composite index returns based on the data from 1991 to 2013. The parameters for each distribution are estimated by using the Power Weighted Method (PWM). Our results indicate that the GL distribution is a better fit for the minima series and that the GEV distribution is a better fit for the maxima series of the returns for the Chinese stock market. This is in contrast to the findings for other markets, such as the US and Singapore markets. Our results are robust regardless of the introduction of stock movement restriction and the global financial crisis. Further, the implications of our findings for risk management are discussed.

L-moments approach for flood frequency analysis of river Okhuwan in Benin-Owena River basin in Nigeria

The flood frequency analysis of River Okhuwan at Ugonoba site in Benin Owena River Basin in Nigeria was carried out using annual maximum stream flow series for 20 years (1989-2008). The objective of the study was to determine the best fit probability distribution model applicable to the site from amongst three selected 3-parameter distribution models namely: Generalized Extreme Value distribution (GEV), Generalized Logistic distribution (GLO) and generalized Pareto distribution (GPA) whose parameters were estimated by the method of L-moments. The best fit distribution was selected based on result of four goodness-of-fit tests applied to the candidate distributions. The results indicate that GPA is the best fit distribution model for analyzing the annual maximum flood series at the study site for return period less than or equal to 25 years but for return periods between 25years and 200 years, GLO or GEV is recommended in that order.Keywords: Flood frequency Analysis,L-moments, Probability distribution, Return period, Parameter estimation

Inferences for the Generalized Logistic Distribution Based on Record Statistics

Estimation for the parameters of the generalized logistic distribution (GLD) is obtained based on record statistics from a Bayesian and non-Bayesian approach. The Bayes estimators cannot be obtained in explicit forms. So the Markov chain Monte Carlo (MCMC) algorithms are used for computing the Bayes estimates. Point estimation and confidence intervals based on maximum likelihood and the parametric bootstrap methods are proposed for estimating the unknown parameters. A numerical example has been analyzed for illustrative purposes. Comparisons are made between Bayesian and maximum likelihood estimators via Monte Carlo simulation.