Generalized Logistic Research Articles

A regional frequency analysis of precipitation extremes in Mainland China with fuzzy c‐means and L‐moments approaches

ABSTRACTOwing to their essential influence on human society and natural environment exerted by inducing disasters, such as floods and droughts, further studies on precipitation extremes in China are needed. This study presents the regional frequency and spatial‐temporal patterns of precipitation extremes in China based on a high‐resolution (0.5° × 0.5°) daily precipitation dataset from 1961 to 2013. With fuzzy c‐means, L‐moments methods and other scientific statistical tests, a regional frequency analysis (RFA) is conducted, aiming to further understand the regional and spatial distribution of precipitation extremes across China. The results show that: (1) the whole Mainland China can be divided into 50 homogeneous regions on the basis of the characteristics of mean annual precipitation and location indices. (2) For most of the regions, Generalized Extreme Value (GEV), Generalized Normal (GNO) and Pearson type III (PE3) distributions of precipitation extremes fit well, according to the results of goodness‐of‐fit (GOF) test. (3) For RX1DAY, GEV has the best‐fit distribution in the east, northeast and southwest of China, whereas GNO distribution mostly fits the northern and parts of southwest and southeast; in addition, regions which fit PE3 and Generalized Logistic (GLO) distribute dispersedly across the country. (4) For RX5DAY, GEV mainly fits in the middle, southwestern and southern; GNO and PE3 apply best to the northeastern and northern, respectively. (5) Return periods of 20, 50 and 100 years for their best‐fit distributions decrease gradually from southeastern China to northwestern China. Compared with the results of GEV distribution fitted to each grid, RFA may provide more accurate estimates of rainfall quantiles. Definitely, the study results will not only benefit further understanding of the unique and complex features of extreme precipitation in the whole Mainland China but also contribute to the nation‐scale flood prevention, control and management in the backdrop of the changing climate.

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.

Regional Frequency Analysis of Extremes Precipitation Using L-Moments and Partial L-Moments

Extremes precipitation may cause a series of social, environmental, and ecological problems. Estimation of frequency of extreme precipitations and its magnitude is vital for making decisions about hydraulic structures such as dams, spillways, and dikes. In this study, we focus on regional frequency analysis of extreme precipitation based on monthly precipitation records (1999–2012) at 17 stations of Northern areas and Khyber Pakhtunkhwa, Pakistan. We develop regional frequency methods based on L-moment and partial L-moments (L- and PL-moments). The L- and PL-moments are derived for generalized extreme value (GEV), generalized logistic (GLO), generalized normal (GNO), and generalized Pareto (GPA) distributions. The Z-statistics and L- and PL-moments ratio diagrams of GNO, GEV, and GPA distributions were identified to represent the statistical properties of extreme precipitation in Northern areas and Khyber Pakhtunkhwa, Pakistan. We also perform a Monte Carlo simulation study to examine the sampling properties of L- and PL-moments. The results show that PL-moments perform better than L-moments for estimating large return period events.

Regional intensity–duration–frequency analysis in the Eastern Black Sea Basin, Turkey, by using L-moments and regression analysis

The analysis of rainfall frequency is an important step in hydrology and water resources engineering. However, a lack of measuring stations, short duration of statistical periods, and unreliable outliers are among the most important problems when designing hydrology projects. In this study, regional rainfall analysis based on L-moments was used to overcome these problems in the Eastern Black Sea Basin (EBSB) of Turkey. The L-moments technique was applied at all stages of the regional analysis, including determining homogeneous regions, in addition to fitting and estimating parameters from appropriate distribution functions in each homogeneous region. We studied annual maximum rainfall height values of various durations (5 min to 24 h) from seven rain gauge stations located in the EBSB in Turkey, which have gauging periods of 39 to 70 years. Homogeneity of the region was evaluated by using L-moments. The goodness-of-fit criterion for each distribution was defined as the ZDIST statistics, depending on various distributions, including generalized logistic (GLO), generalized extreme value (GEV), generalized normal (GNO), Pearson type 3 (PE3), and generalized Pareto (GPA). GLO and GEV determined the best distributions for short (5 to 30 min) and long (1 to 24 h) period data, respectively. Based on the distribution functions, the governing equations were extracted for calculation of intensities of 2, 5, 25, 50, 100, 250, and 500 years return periods (T). Subsequently, the T values for different rainfall intensities were estimated using data quantifying maximum amount of rainfall at different times. Using these T values, duration, altitude, latitude, and longitude values were used as independent variables in a regression model of the data. The determination coefficient (R2) value indicated that the model yields suitable results for the regional relationship of intensity–duration–frequency (IDF), which is necessary for the design of hydraulic structures in small and medium sized catchments.

Study on quantile estimates of extreme precipitation and their spatiotemporal consistency adjustment over the Huaihe River basin

The quantile estimates and spatiotemporal consistency of extreme precipitation are studied by regional linear frequency analysis for Huaihe River basin in China. Firstly, the study area can be categorized into six homogeneous regions by using cluster analysis, heterogeneity measure, and discordancy measure. In the next step, we determine the optimum distribution for each homogeneous region by using two criteria of Monte Carlo simulations and the root-mean-square error (RMSE) of the sample L-moments. A diagram of L-moments ratio is used to further judge and validate the optimum distribution. The generalized extreme value (GEV), generalized normal (GNO), and generalized logistic (GLO) for 24-h duration are determined to be the more appropriate distribution based on the two criteria, L-moments ratio plot, and the tail thickness of curve in adjacent regions. A summary assessment can provide the more reasonable distribution, which avoids arbitrary results from single test. An important practical element of this study that was missing from previous works is the quantile spatiotemporal consistency analysis, which helps identify non-monotonicity among quantiles at different durations and reduces the gradient of estimates in the adjacent regions. Abnormality and spatial discontinuation can be removed by distributing the surplus of the ratio and twice different interpolation. A complete set of spatiotemporal consistent quantile estimates for various duration (24 h, 3 days, 5 days, and 7 days) and return periods (from 2 to 1000 years) can be obtained by using the abovementioned method in the study area, which are in the agreement with the observed precipitation extremes. It will provide important basis for hydrometeorological research, which is of significant scientific and practical merit.

A robust data scaling algorithm to improve classification accuracies in biomedical data.

BackgroundMachine learning models have been adapted in biomedical research and practice for knowledge discovery and decision support. While mainstream biomedical informatics research focuses on developing more accurate models, the importance of data preprocessing draws less attention. We propose the Generalized Logistic (GL) algorithm that scales data uniformly to an appropriate interval by learning a generalized logistic function to fit the empirical cumulative distribution function of the data. The GL algorithm is simple yet effective; it is intrinsically robust to outliers, so it is particularly suitable for diagnostic/classification models in clinical/medical applications where the number of samples is usually small; it scales the data in a nonlinear fashion, which leads to potential improvement in accuracy.ResultsTo evaluate the effectiveness of the proposed algorithm, we conducted experiments on 16 binary classification tasks with different variable types and cover a wide range of applications. The resultant performance in terms of area under the receiver operation characteristic curve (AUROC) and percentage of correct classification showed that models learned using data scaled by the GL algorithm outperform the ones using data scaled by the Min-max and the Z-score algorithm, which are the most commonly used data scaling algorithms.ConclusionThe proposed GL algorithm is simple and effective. It is robust to outliers, so no additional denoising or outlier detection step is needed in data preprocessing. Empirical results also show models learned from data scaled by the GL algorithm have higher accuracy compared to the commonly used data scaling algorithms.

Modeling of Annual Maximum Storm Intensity with Bayesian Markov Chain Monte Carlo (MCMC) and L-moment

This study presents the best fitting distribution to describe the siries MSI based on hourly rainfall form 1970 to 2008 for three rain gauge stations in Peninsular Malaysia namely Bertam, Dungun and Pekan. Two three-parameter extreme value distributions which are considered are Generalized Extreme Value (GEV) and Generalized Logistic (GL). The parameters of these distributions are determined using the Bayesian MCMC with non-informative prior distribution and L-moments (LMOM) method. The Goodness-Of-Fit (GOF) between empirical data and theoretical distributions are then evaluated for each stations. The result show that the majority of the stations are found that the L-moment method can give the best modelling for MSI, specified for GEV distribution. Based on the model that has been identified, we can reasonably predict the risks associated the MSI for various return periods.

A comparative study to determine the optimal copula model for the wind speed and precipitation of typhoons

ABSTRACTTyphoon is usually one of main reasons to damage the buildings, infrastructures, water operations, and human life in summer in East Asia, which includes China, Japan, and Korea. Typhoons can be considered multivariate extremes that consists of climate variables that affect wind and precipitation. Copula models can be used to investigate the dependence between random variables. In this study, we used copula models to investigate the optimal marginal distribution for the relations between the wind speed and the precipitation of typhoons at the Jeju weather station in South Korea. We used the 10‐min average wind speed (W10) and the total precipitation (PT) from 65 typhoon events that had affected the Jeju station. Two case studies were considered using the typhoon data. Case one included all typhoon events, and case two included 39 typhoon events with the following thresholds:W10 ≥ 10 m s−1andPT ≥ 50 mm. The marginal distributions for the copula models included the generalized extreme value (GEV), generalized logistic (GLO), generalized Pareto (GPA), and Weibull (WBU) distributions as well as three copula models: Clayton, Frank, and Gumbel copulas. Each step was checked by the robust diagnostic, the probability plot correlation coefficient (PPCC) test, and the goodness‐of‐fit test (Snstatistic). The Frank copula model withGLO(W10) – GPA(PT) had the best performance in both cases. The results of this study present the key steps needed to identify an optimum copula for a bivariate distribution in applications related to atmospheric sciences and provides an example of how this applies to wind velocity and precipitation from typhoons.

Application of artificial neural network, multiple-regression and index-flood techniques in regional flood frequency estimation

Flood frequency estimation is fundamental in both engineering science and engineering hydrology science. Comparing the efficiencies of artificial neural network (ANN), multiple-regression (MR) and index-flood (IF) techniques based on L-moments in Qazvin Province of Iran was the main objective of this study. Using the main variables affecting flood magnitude, the study area was divided into two regions based on the clustering approach. The homogeneity of these regions was confirmed using the homogeneity test of L-moments approach. Using the L-moment ratios and the Z-statistic criteria, generalised logistic (GLO) and generalised Pareto (GPA) distributions were identified respectively for the first and second homogen regions as the most robust distributions among five candidate distributions. Relative root mean square error (RRMSE) measure was applied for evaluating the performance of three methods in comparison with the curve fitting method. In general, ANN method gives more reliable estimations.

Statistical Modeling of Extreme Rainfall Processes (SMExRain): A Decision Support Tool for Extreme Rainfall Frequency Analyses

Many probability distributions have been developed to model the extreme rainfall processes. However, there is no general agreement as to which distribution should be used. In the present study, a decision support tool for statistical modeling of extreme rainfall processes (SMExRain) was therefore developed to identify the most suitable distribution that could provide accurate extreme rainfall estimates. More specifically, the proposed tool can be used to assess the descriptive and predictive abilities of ten commonly-used probability models, Beta-K, Beta-P, Generalized Extreme Value (GEV), Generalized Logistic (GLO), Generalized Normal (GNO), Generalized Pareto (GPA), Gumbel, Log-Pearson Type III (LP3), Pearson Type III (P3), and Wakeby, for their accuracy and robustness in the estimation of annual maximum precipitations. The proposed decision support tool was tested using 5-minute, 1-hour, and 24-hour annual maximum precipitation data from a network of 21 raingages located in the Ontario region in Canada. Results based on various numerical and graphical goodness-of-fit criteria have indicated that the GEV, GNO, and P3 models were the best models for describing the distribution of annual maximum precipitations in this region. The GEV distribution, however, was preferred to the GNO and P3 because it requires a simpler parameter estimation method and it was based on a more solid theoretical basis for representing the distribution of extreme random variables. Therefore, the GEV could be recommended as the most suitable model for describing the distribution of annual maximum precipitations in Ontario region based on the proposed decision support tool.

Mixture Distribution Fitting and Modeling for Regional Frequency Analysis of Maximum Wind Speed Over South Korea

본 연구에서는, L-moments와 Harmony Search 파라미터 추정법을 사용하여 지역빈도해석을 통한 한국 68개소의 설계풍속을 도출하는 연구를 실시하였다. 기존에 일반적으로 풍속 산정에 적용된 Generalized Logistic (GLOG), General Extreme Value (GEV), Generalized Pareto (GPAR), Weibull의 분포형에 적용하여 설계 풍속값을 산정하였을 뿐만 아니라 Gamma와 Gumbel의 분포를 조합시킨 혼합분포형을 이용하여서도 설계값을 산정하였다. 산정결과, 한국에서는 Weibull분포가 가장 적절한 분포모형임을 알 수 있었다. 기존에 적용되지 않았던 새로운 혼합형 분포형도 특정지역에서 충분히 사용가능한 분포형이 될 수 있음을 알 수 있었다. In this study, we tried to determine and investigate meteorologically homogeneous regions and to derive regional wind frequency estimates for 68 gauged sites in South Korea using an index wind method with L-moments and harmony search parameter estimation. Comparative regional wind frequency estimates were made for the tested region using various distributions, namely the Generalized Logistic (GLOG), General Extreme Value (GEV), Generalized Pareto (GPAR), Weibull. Furthermore, we applied three types of two-component mixture distributions, built by combination of Gumbel and Gamma distributions with five-parameters such as Gumbel-Gumbel (GumGum), Gamma-Gamma (GamGam), and Gamma-Gumbel (GamGum) distribution models. Their parameters are estimated by harmony search method (HS). The results of the current study showed that the Weibull distribution is identified as the bestfit distribution in Korea. In other words, Weibull as a suitable distribution with three parameters for other Korean stations.

Evaluation of the Index-Flood Approach Related Regional Frequency Analysis Procedures

AbstractIndex-flood related regional frequency analysis (RFA) procedures are in use by hydrologists to estimate design quantiles of hydrological extreme events at data sparse/ungauged locations in river basins. There is a dearth of attempts to establish which among those procedures is better for RFA in the L-moment framework. This paper evaluates the performance of the conventional index flood (CIF), the logarithmic index flood (LIF), and two variants of the population index flood (PIF) procedures in estimating flood quantiles for ungauged locations by Monte Carlo simulation experiments and a case study on watersheds in Indiana in the U.S. To evaluate the PIF procedure, L-moment formulations are developed for implementing the procedure in situations where the regional frequency distribution (RFD) is the generalized logistic (GLO), generalized Pareto (GPA), generalized normal (GNO) or Pearson type III (PE3), as those formulations are unavailable. Results indicate that one of the variants of the PIF procedu...

Comparison of Annual Maximum and Partial Duration Series for Derivation of Rainfall Intensity-Duration-Frequency Relationships in Peninsular Malaysia

AbstractThe use of partial duration series (PDS) with generalized Pareto (GPA) distribution in hydrological frequency analysis has been popular during the last few decades because PDS is capable of including more extreme events than annual maximum series (AMS). In Malaysia, the most updated rainfall intensity-duration-frequency (IDF) relationships that serve as the official hydrological design guideline are based on the fitting of AMS to the generalized extreme value (GEV) distribution. The purpose of this paper is to identify the statistical model that best represent Peninsular Malaysia for deriving rainfall IDF relationships. Besides GEV and GPA distributions, other distributions such as generalized logistic (GLO), Gumbel (EV1), lognormal (LN3), and log-Pearson Type III (LP3) are also included in this study. Rainfall data with record length varying between 10 and 41 years from 60 rainfall stations in Peninsular Malaysia are computed into AMS and PDS, and fitted to the selected distributions. The L-momen...

Comparison of two methods of regional flood frequency analysis by using L-moments

A regionalized relationship to estimate flood magnitudes for ungauged and poorly gauged catchments can be established using regional flood frequency analysis (RFFA). Comparison of index-flood and multiple-regression analyses based on L-moments was the main objective of this study. Factor analysis was applied to determine main variables influencing flood magnitude. Results showed that the main variables are perimeter, equivalent diameter, time of concentration, length of main waterway and area. The study area was divided into two regions based on the Ward’s method of clustering approach and site characteristics. The homogeneity test based on L-moments showed that two regions were acceptably homogeneous. Five distributions were fitted to the annual peak flood data of two homogeneous regions. Using the L-moment ratios and the Z-statistic criteria, generalized logistic (GLO) and generalized Pareto (GPA) distributions were identified for the first and second homogeneous regions respectively as the most robust distributions among five potential distributions. The relative root mean square error (RRMSE) measure was applied for evaluating the performance of the index-flood and the multiple-regression methods in comparison with the curve fitting (plotting position) method. In general, the index-flood method gives more reliable estimations for various flood magnitudes of different recurrence intervals.

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.

The Copula Function-Based Probability Characteristics Analysis on Seasonal Drought & Flood Combination Events on the North China Plain

Drought & flood events, especially the drought & flood combination events (DFCEs) on the North China Plain (NCP), known as an important grain production region in China, constitute a serious threat to China’s food security. Studies on DFCEs in this region are of great significance for the rational allocation of water resources and the formulation of integrated response strategy for droughts and floods. In this study, L-moments theory and bivariate copula method were used to evaluate the probability characteristics of seasonal DFCEs (continuous drought, continuous flood, and alternation between drought and flood) on the NCP, based on the daily precipitation data (1960–2012) at 19 meteorological stations. Results indicate the following: (1) On the NCP, the precipitation in summer accounts for 56.45%–72.02% of mean annual precipitation, and the precipitation in autumn and spring come second. The winter precipitation is the smallest (less than 4%); (2) The best-fit distribution for precipitation anomaly percentages in spring, summer and autumn are Generalized Normal (GNO), Generalized Logistic (GLO) and Pearson III (P-III) in sub-region I, respectively. While in sub-region II, they are respectively the P-III, P-III and Generalized Extreme-Value (GEV); (3) Compared with the Gumbel copula and Clayton copula, Frank copula is more suitable for spring-summer and summer-autumn precipitation anomaly percentage sequences on the NCP; (4) On the time scale, continuous drought respectively dominate in spring-summer DFCEs and in summer-autumn DFCEs on the NCP. Summer-autumn DFCEs prevail in sub-region I with the average probability value 0.34, while spring-summer DFCEs dominate in sub-region II, of which average probability value is 0.42; (5) On the spatial scale, most areas where the probability of continuous drought in spring-summer and spring drought & summer flood is relatively high are located in the northwest, northeast, and coastal parts of sub-region II; all the events with high probability of continuous drought in summer-autumn and summer flood & autumn drought occurred at the central part in the northwest of sub-region II.

The spatial return level of aggregated hourly extreme rainfall in Peninsular Malaysia

This paper is intended to ascertain the spatial pattern of extreme rainfall distribution in Peninsular Malaysia at several short time intervals, i.e., on hourly basis. Motivation of this research is due to historical records of extreme rainfall in Peninsular Malaysia, whereby many hydrological disasters at this region occur within a short time period. The hourly periods considered are 1, 2, 3, 6, 12, and 24 h. Many previous hydrological studies dealt with daily rainfall data; thus, this study enables comparison to be made on the estimated performances between daily and hourly rainfall data analyses so as to identify the impact of extreme rainfall at a shorter time scale. Return levels based on the time aggregate considered are also computed. Parameter estimation using L-moment method for four probability distributions, namely, the generalized extreme value (GEV), generalized logistic (GLO), generalized Pareto (GPA), and Pearson type III (PE3) distributions were conducted. Aided with the L-moment diagram test and mean square error (MSE) test, GLO was found to be the most appropriate distribution to represent the extreme rainfall data. At most time intervals (10, 50, and 100 years), the spatial patterns revealed that the rainfall distribution across the peninsula differ for 1- and 24-h extreme rainfalls. The outcomes of this study would provide additional information regarding patterns of extreme rainfall in Malaysia which may not be detected when considering only a higher time scale such as daily; thus, appropriate measures for shorter time scales of extreme rainfall can be planned. The implementation of such measures would be beneficial to the authorities to reduce the impact of any disastrous natural event.

Option Implied Tail Index and Volatility Based on Heavy-tailed Distributions: Evidence from KOSPI 200 Index Options Market

This paper compares the option implied tail indexes and volatilities from two option pricing formulas based on heavy-tailed distributions: generalized extreme value (GEV) distribution and generalized logistic (GLO) distribution. Option pricing models based on heavy-tailed distributions with three parameters overcome some well-known drawbacks of the Black–Scholes model when the realized underlying asset returns are not normally distributed. Both GEV-based and GLO-based option pricing formulas extract the implied volatilities successfully, indicating that they are compatible with the Black–Scholes formulas. However, GEV-based pricing model shows more unexpected patterns when extracting the implied tail indexes for put options than GLO-based pricing model including the credit crisis in 2008, implying that GEV-based pricing model is less capable of measuring the market sentiment during the extreme crisis events.

Modelling of Hydrological Drought Events in the Upper Tana Basin of Kenya

Drought is a recurring hazard which affects many parts of Kenya. In most countries in Sub-Saharan Africa, agriculture which is predominantly rain-fed is the main stay of the economies is highly prone to the impacts of drought which, whenever it occurs, leads to serious socioeconomic challenges at various levels. The study of drought duration, magnitude and severity have relevance in many areas such as waste load allocations, issuance of pollution discharge permits, location of treatment plants and sanitary landfills, determination of allowable water transfers and withdrawals both within, between and outside the affected areas and determination of minimum downstream release requirements for hydropower water supply, cooling plants and other facilities. Knowledge of the frequency distribution of the drought events is useful as it contributes to the assessment of drought risks which have implications on the long term ecological, economic and social well being of the biological and human communities that make use of water from the various streams in a basin. In this study five frequency distributions were fitted to drought duration and severity as determined from discharge data from representative river gauge stations in the upper Tana Basin of Kenya. The frequency distributions fitted to the two drought events were the Generalized Normal (GN) or 3-parameter Lognormal, Generalized Extreme Value (GEV) or the Extreme Value Type III, Generalized Pareto (GPA), Pearson Type III (P3) and Generalized Logistic (GL). The distributions of best fit for the drought events were identified using the Z value obtained from the average L-moment statistics of a particular candidate distribution and the average L-moment statistics. The Z value for each homogenous region was determined from sample estimates of Lcv , Lcs and Lck that were determined from probability weighted moment estimators and the weighted means of Lcv, Lcs and Lck using records from the river gauging stations representing each hydrologically homogenous region. A frequency distribution of best fit was selected if ׀Z Dis ׀ ≤ 1.64 and the one with the lowest ׀Z Dis ׀ value selected as the distribution