Method Of L-moments Research Articles

Parameter Estimation of the Inverted Kumaraswamy Distribution by Using L-Moments Method: An Application on Precipitation Data

Modeling precipitation data plays a critical role in water resource and flood management. Statistical distributions are frequently used in describing hydrological variables. Different distributions and estimation methods have been presented in previous studies for modeling precipitation data. In this study, the inverted Kumaraswamy distribution is considered for its advantageous properties, and the L-moments and maximum likelihood methods are employed in estimating the parameters of the inverted Kumaraswamy distribution. In the application part, the annual maximum monthly precipitations recorded in the Rize, Türkiye are modeled with the inverted Kumaraswamy distribution. To the best of the author’s knowledge, the L-moment method is considered for the first time to estimate the parameters of the inverted Kumaraswamy distribution. In addition, the efficiencies of the estimation methods are compared with a Monte-Carlo simulation study. For evaluating the performances of the estimation methods, the goodness of fit criteria including root mean square error, Kolmogorov Smirnov test, and coefficient of determination (R^2) are used in the application part of the study. The results show that for the data considered, the L-moments method yields more accurate results than the maximum likelihood method in estimating the parameters when the sample size is small. Accordingly, the corresponding distribution with L-moments estimations provides a better fit to precipitation data obtained from the Rize station.

Possibilities of Using Regional Index-Flood Method and Annual Maximum and Partial Duration Series: A Case Study of Susurluk River Basin, Turkey

Floods are defined as disasters that occur as a result of a natural formation and cause negative effects on society life and economy. Floods that cause the greatest loss of life and property after earthquakes among the natural disasters experienced in our country, unplanned urbanization caused by the increasing population, uncontrolled settlements in river beds, and changing climate have increased its impact and frequency of occurrence over time. Therefore, it is important to estimate the magnitude of floods and the frequency of their occurrence in the most accurate way. This research aims to perform at-site and regional frequency analysis with L-moment methods in the Susurluk Basin. For regional frequency analysis, annual maximum and partial duration flood series were obtained using the data from 22 flow observation stations selected from the basin. In the first stage, the stations in the region were considered as a whole and the stations were found discordant with each other. Hierarchical cluster analysis was performed with the help of various physiological-hydrological characteristics of the stations located in the basin with geographic information systems, and the basin was divided into 2 regions, but homogeneity was still not achieved. Therefore, the basin is divided into 3 regions according to the dendrogram. There was no discordancy in all three regions and homogeneity was achieved. After achieving homogeneity, the most suitable frequency distribution was determined for each region. In addition, L-moments and L-moment ratios, probability distributions, and quantile discharges of the annual maximum and partial duration flood series were estimated for at-site frequency analysis. As a result, in regional flood frequency analysis, generalized logistic distribution provided the best fit in the annual maximum series, while Generalized Pareto and Pearson type 3 provided the best-fit distributions for the first and second regions in the partial duration series, respectively, in addition, generalized normal provided the best fit in the third region. It was concluded that it would be useful to use partial duration flood series as well as annual maximum series in the flood frequency analysis of the Susurluk basin.

Revisiting the Use of the Gumbel Distribution: A Comprehensive Statistical Analysis Regarding Modeling Extremes and Rare Events

The manuscript presents the applicability of the Gumbel distribution in the frequency analysis of extreme events in hydrology. The advantages and disadvantages of using the distribution are highlighted, as well as recommendations regarding its proper use. A literature review was also carried out regarding the methods for estimating the parameters of the Gumbel distribution in hydrology. Thus, for the verification of the methods, case studies are presented regarding the determination of the maximum annual flows and precipitations using nine methods for estimating the distribution parameters. The influence of the variability of the observed data lengths on the estimation of the statistical indicators, the estimation of the parameters, and the quantiles corresponding to the field of small exceedance probabilities (p < 1%) is also highlighted. In each case, the results are analyzed compared to those obtained with the Generalized Extreme Value distribution, the four-parameter Burr distribution, and the five-parameter Wakeby distribution estimated using the L-moments method. The results of the case studies highlight and reaffirm the statistical, mathematical, and hydrological recommendations regarding the avoidance of applying the Gumbel distribution in flood frequency analysis and its use with reservations in the case of maximum precipitation analysis, especially when the statistical indicators of the analyzed data are not close to the characteristic ones and unique to the distribution.

On the uncertainty of long-period return values of extreme daily precipitation

Methods for calculating return values of extreme precipitation and their uncertainty are compared using daily precipitation rates over the Western U.S. and Southwestern Canada from a large ensemble of climate model simulations. The roles of return-value estimation procedures and sample size in uncertainty are evaluated for various return periods. We compare two different generalized extreme value (GEV) parameter estimation techniques, namely L-moments and maximum likelihood (MLE), as well as empirical techniques. Even for very large datasets, confidence intervals calculated using GEV techniques are narrower than those calculated using empirical methods. Furthermore, the more efficient L-moments parameter estimation techniques result in narrower confidence intervals than MLE parameter estimation techniques at small sample sizes, but similar best estimates. It should be noted that we do not claim that either parameter fitting technique is better calibrated than the other to estimate long period return values. While a non-stationary MLE methodology is readily available to estimate GEV parameters, it is not for the L-moments method. Comparison of uncertainty quantification methods are found to yield significantly different estimates for small sample sizes but converge to similar results as sample size increases. Finally, practical recommendations about the length and size of climate model ensemble simulations and the choice of statistical methods to robustly estimate long period return values of extreme daily precipitation statistics and quantify their uncertainty.

Assessing Flood Risk: LH-Moments Method and Univariate Probability Distributions in Flood Frequency Analysis

This study examines all of the equations necessary to derive the parameters for seven probability distributions of three parameters typically used in flood frequency research, namely the Pearson III (PE3), the generalized extreme value (GEV), the Weibull (W3), the log-normal (LN3), the generalized Pareto Type II (PG), the Rayleigh (RY) and the log-logistic (LL3) distributions, using the higher-order linear moments method (LH-moments). The analysis represents the expansion of previous research whose results were presented in previous materials, and is part of hydrological research aimed at developing a standard for calculating maximum flows based on L-moments and LH-moments. The given methods for calculating the parameters of the examined distributions are used to calculate the maximum flows on Romania’s Prigor River. For both methods, the criterion for selecting the most suitable distribution is represented by the diagram of the L-skewness–L-kurtosis and LH-skewness–LH-kurtosis. The results for Prigor River show that the PG distribution is the best model for the L-moments method, the theoretical values of the statistical indicators being 0.399 and 0.221. The RY distribution is the best model for the LH-moments technique, with values of 0.398 and 0.192 for the two statistical indicators.

Extreme Events Analysis Using LH-Moments Method and Quantile Function Family

A direct way to estimate the likelihood and magnitude of extreme events is frequency analysis. This analysis is based on historical data and assumptions of stationarity, and is carried out with the help of probability distributions and different methods of estimating their parameters. Thus, this article presents all the relations necessary to estimate the parameters with the LH-moments method for the family of distributions defined only by the quantile function, namely, the Wakeby distribution of 4 and 5 parameters, the Lambda distribution of 4 and 5 parameters, and the Davis distribution. The LH-moments method is a method commonly used in flood frequency analysis, and it uses the annual series of maximum flows. The frequency characteristics of the two analyzed methods, which are both involved in expressing the distributions used in the first two linear moments, as well as in determining the confidence interval, are presented. The performances of the analyzed distributions and the two presented methods are verified in the following maximum flows, with the Bahna river used as a case study. The results are presented in comparison with the L-moments method. Following the results obtained, the Wakeby and Lambda distributions have the best performances, and the LH-skewness and LH-kurtosis statistical indicators best model the indicators’ values of the sample (0.5769, 0.3781, 0.548 and 0.3451). Similar to the L-moments method, this represents the main selection criterion of the best fit distribution.

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

Impact of Climate Variability on Rainfall Characteristics in the Semi-Arid Shashe Catchment (Botswana) from 1981–2050

Futuristic rainfall projections are used in scale and various climate impact assessments. However, the influence of climate variability on spatial distribution patterns and characteristics of rainfall at the local level, especially in semi-arid catchments that are highly variable and are not well explored. In this study, we explore the influence of climate variability on the spatial distribution and rainfall characteristics at a local scale in the semi-arid Shashe catchment, Northeastern Botswana. The LARS-WG, Long Ashton Research Station Weather Generator downscaling method, three representative scenarios (RCP 2.6, RCP 4.5, and RCP 4.5), three trend detection methods (Mann-Kendall, Sen’s slope, and innovative trend analysis) and L-moment method were used to assess climate change impacts on rainfall. Two data sets were used; one with 40 years of observed data from 1981–2020 and the other with 70 years from 1981–2050 (40 years of observed and 30 years of projected data from 2021–2050). Generally, the study found trend inconsistencies for all the trend detection methods. In most cases, Sen’s Slope has a high estimate of observed and RCP 2.6, while ITA overestimates rainfall totals under RCP 4.5 and RCP 8.5. The trend is increasing for annual total rainfall in most gauging stations while decreasing for annual maximum rainfall. The catchment is homogeneous, and Generalized Logistic distribution is the dataset’s best-fit distribution. Spatial coverage of a 100-year rainfall between 151–180 mm will be 81% based on observed data and 87% based on projected data under RCP 2.6 scenario when it happens. A 200-year rainfall between 196–240 mm under RCP 4.5 and 8.5 has high spatial areal coverage, at least 90% of the total catchment. The outcomes of this study will provide insightful information for water resource management and flood risk assessment under climate change. There is a need, however, to assess the transferability of this approach to other catchments in the country and assess the performance of other advanced modelling systems, such as machine learning, in this region.

Analysis of the upper tail of the short-term extreme tension distribution of mooring line by the peaks-over-threshold method

The consistency of the upper tail of the extreme response distribution plays an important role in the accuracy of the extreme response prediction. Although the peaks-over-threshold method has a high data utilization of the response time series, the consistency of the upper tail of the extreme response distribution was not thoroughly investigated, especially for the inherent randomness of the short-term sea state. This study aims to estimate the upper tail of the short-term extreme tension distribution of the mooring line considering the short-term variability by the peaks-over-threshold method. The consistency of the extreme tension distribution tail is quantified by a relative error compared with the global maximum method, and the theoretical optimal threshold is defined by minimizing the largest relative error among the tail quantiles for different thresholds. Results from a case study for the mooring line of the fish cage indicate that the effect of the random seed is significant for consistency in the prediction of the distribution tail of the short-term extreme tension. And the generalized Pareto distribution model combined with the L-moments method is recommended for the theoretical optimal threshold. These findings provide some useful suggestions for the preliminary design of the mooring line for offshore structures.

Applications of Burr III-Weibull quantile function in reliability analysis

This paper introduces a new family of distributions defined in terms of quantile function. The quantile function introduced here is the sum of quantile functions of life time distributions called Burr III and Weibull. Different distributional characteristics and reliability properties are included in the study. Method of Least Square and Method of L-moments are applied to estimate the parameters of the model. Two real life data sets are used to illustrate the performance of the model.

Flood Frequency Analysis Using the Gamma Family Probability Distributions

This article presents six probability distributions from the gamma family with three parameters for the flood frequency analysis in hydrology. The choice of the gamma family of statistical distributions was driven by its frequent use in hydrology. In the Faculty of Hydrotechnics, the improvement of the estimation of maximum flows, including the methodological bases for the realization of a regionalization study with the linear moments method with the corrected parameters, was researched and is an element of novelty. The linear moments method performs better than the method of ordinary moments because it avoids the choice of skewness depending on the origin of the flows, and is the method practiced in Romania. The L-moments method conforms to the current trend for estimating the parameters of statistical distributions. Observed data from hydrometric stations are of relatively short length, so the statistical parameters that characterize them are of a sample that requires correction. The correction of the statistical parameters is proposed using the method of least squares based on the inverse functions of the statistical distributions expressed with the frequency factor for L-moments. All the necessary elements for their use are presented, such as quantile functions, the exact and approximate relations for estimating parameters, and frequency factors. A flood frequency analysis case study was carried out for the Ialomita river to verify the proposed methodology. The performance of this distributions is evaluated using Kling–Gupta and Nash–Sutcliff coefficients.

ASYMPTOTIC NORMALITY OF TRIMMED L-MOMENTS ESTIMATOR FOR ARCHIMEDEAN COPULAS

In order to present a new estimation approach for multi-parameter distributions without a mean or for heavy tailed distributions, in which the L-moments method proposed by Gumbel, (1960), is invalid due to the absence of theoretical L-moments, Trimmed L-moments were first introduced by Elamir and Seheult (2003). In this paper, a new estimation method based on multi-parameter copulas' Trimmed L-moments is proposed with a simulation study. The consistency and the asymptotic normality of the new estimator also established.

Variations of extreme precipitation events with sub-daily data: a case study in the Ganjiang River basin

Abstract. Climate warming increases the intensity of extreme precipitation. Studying extreme precipitation patterns and changes is vital to reducing risk. This paper investigates thresholds, changes and timescales for extreme precipitation using sub-daily records from meteorological stations in the Ganjiang River basin. We use the gamma distribution and select the L-moment method to estimate the parameters α and β. Results show that (1) continuous precipitation events of 36 h contributed the most precipitation to the total but with lower frequency, which would be key events for flood monitoring; (2) the intensity and the occasional probability of extreme precipitation will increase in spring in the future in stations like Yifeng, Zhangshu and Ningdu, which will in turn increase the risk of storm floods; and (3) spatial distribution of extreme precipitation risk shows that the risk increases as elevation increases in the northern lowland and the Jitai Basin in the midstream region, while the risk in the southern mountainous region decreases as elevation increases. These findings will facilitate emergency preparedness, including risk management and disaster assistance, in the study areas.

Reliability Aspects of Quantile-Based Residual Coefficient of Variation

In reliability studies, the data is often truncated and hence the residual random variable plays a vital role in the modelling and analysis of lifetime data. Like various reliability measures such as hazard rate, mean residual life function, variance residual life function, the residual coefficient of variation is also an important tool considered by many. In this paper, we study a quantile version of coefficient of variation of residual life, an alternative to the traditional distribution function measure. We study various properties of it and obtain characterizations based on some well-known probability models. We introduce a class of distributions with linear quantile-based residual coefficient of variation and study their basic reliability properties. We also obtain some stochastic comparison and aging properties using quantile-based residual coefficient of variation. The L-moment method of estimation for the class of distributions with linear quantile-based residual coefficient of variation has also been illustrated using two real data sets.

Regional precipitation frequency analysis for 24-h duration using GPM and L-moments approach in South China

There are little researchers that combined satellite-based products and L-moments methods in frequency analysis. The objective of this study was to explore and evaluate the satellite precipitation data applied in univariate L-moments method. The Shaoguan City of South China was selected as the study area. Annual maximum precipitation (AMP) respectively from observational rainfall and Integrated Multi-satellitE Retrievals for Global Precipitation Mission (IMERG) gridded rainfall was used for the calculation of estimates using regional L-moments analysis (RLMA) method. Then, the quantiles at grids were corrected by mean bias correction (MBC) method with the estimates at sites. The results showed that the number of homogeneous regions as well as the spatial distribution of parameters ({C}_{v}, {C}_{s}, and {C}_{k}) for two data sets are different. When the return periods are larger than 40 years, the estimates at sites are also larger than those at grids and the difference will increase as return period increases. After being modified, the quantiles from IMERG are more reasonable and combine the spatial distribution characteristic of estimates from two cases, which indicates that the estimation calculated by satellite products with L-moments approach is applicable and more reasonable. This study can also provide reference for solving the issues regarding the density and coverage of rain gauges that affect the accuracy of frequency analysis, especially in remote areas of China or most parts of the word.

Correction to: Regionalization of rainfall intensity–duration–frequency (IDF) curves with L-moments method using neural gas networks

Correction to: Regionalization of rainfall intensity–duration–frequency (IDF) curves with L-moments method using neural gas networks

An Assessment of Uncertainties in Flood Frequency Estimation Using Bootstrapping and Monte Carlo Simulation

Reducing uncertainty in design flood estimates is an essential part of flood risk planning and management. This study presents results from flood frequency estimates and associated uncertainties for five commonly used probability distribution functions, extreme value type 1 (EV1), generalized extreme value (GEV), generalized pareto distribution (GPD), log normal (LN) and log Pearson type 3 (LP3). The study was conducted using Monte Carlo simulation (MCS) and bootstrapping (BS) methods for the 10 river catchments in eastern Australia. The parameters were estimated by applying the method of moments (for LP3, LN, and EV1) and L-moments (for GEV and GPD). Three-parameter distributions (e.g., LP3, GEV, and GPD) demonstrate a consistent estimation of confidence interval (CI), whereas two-parameter distributions show biased estimation. The results of this study also highlight the difficulty in flood frequency analysis, e.g., different probability distributions perform quite differently even in a smaller geographical area.

Evaluation of probability distributions for estimation of peak flood discharge using FFA approach

The estimation of Peak Flood Discharge (PFD) for a specific return period is of paramount importance in hydrological studies for the planning, design, and management of civil and hydraulic structures. This research investigates the evaluation of probability distributions through statistical tests, viz., the Chi-Square, Kolmogorov-Smirnov, and D-index. These tests aid in estimating the PFD using the Flood Frequency Analysis (FFA) approach for the Tapi River at Prakasha Barrage. In this analysis, various distributions from the extreme value family such as Extreme Value Type-1, Extreme Value Type-2, Generalized Extreme Value (GEV), and Pareto, were employed in the FFA. The distribution parameters are determined by standard techniques, such as the method of moments, Maximum Likelihood Method (MLM), and the Method of L-Moments (LMO). A qualitative assessment, along with the D-index value, was employed to select the most suitable distribution for the analyzed data. This study demonstrates that among the four distributions adopted in FFA, the GEV (LMO) distribution exhibits the best fit. Furthermore, this research suggests the use of PFDs derived from the GEV (LMO) distribution for design purposes.

Application of L-moment method for regional frequency analysis of meteorological drought across the Loess Plateau, China.

Water shortages have always been the primary bottleneck for the healthy and sustainable development of the ecological environment on the Loess Plateau (LP). Proper water resource management requires knowledge of the spatiotemporal characteristics of precipitation frequency. This paper employed the gridded precipitation dataset obtained from the China Meteorological Data Service Centre to present a spatially explicit characterization of precipitation frequencies in tandem with their return periods on the LP based on the L-moment method. The 60% and 80% of the mean annual precipitation from 1981 to 2010 were synonymous with severe and moderate droughts, respectively. Droughts occurred more frequently in the northwest than in the southeast of the LP. Moreover, the frequencies of moderate drought showed a slight difference throughout the area, while those of severe droughts demonstrated considerable differences between the northwestern arid zone and the southeastern semi-humid zone. The maps associated with various return periods of precipitation deficits can be used to produce drought risk maps together with drought vulnerability maps. These findings could also provide useful information for drought management, water resource management and the development of food security policies.

Regionalization of rainfall intensity–duration–frequency (IDF) curves with L-moments method using neural gas networks

Floods are one of the most frequent and destructive natural events which lead to lots of human and financial losses with damage to the houses, farms, roads, and other buildings. Intensity–duration–frequency (IDF) curves are the main and practical tools that have been used for flood control studies, including the design of the water structures. In many cases, there is no measuring device at the desired place, or their information is not helpful if there is any available. In this case, it is not possible to extract these curves through conventional methods. Regionalizing the IDF curves is a method that has solved the issues mentioned in the common methods. In this research, the regionalized IDF curves are extracted in Khuzestan province, Iran using 21 rain gauge stations through L-moments and neural gas networks. Clustering is one of the most effective steps and a prerequisite for regional frequency analysis (RFA) that divides the region and existing stations into hydrologically homogenous regions. In this study, clustering is done using two new models named neural gas (NG) and growing neural gas (GNG) network. Comparing the regional IDF curves with at-site curves, it was found that neural gas network models had a more accurate performance and higher efficiency, so they had the lowest estimate error amount among other models. Also, due to the acceptable difference between regional and at-site curves, the efficiency of L-moments in RFA was evaluated as appropriate.