log-Gumbel Distribution Research Articles

Analysis of Probability Distributions for Modelling Extreme Rainfall Events and Detecting Climate Change: Insights from Mathematical and Statistical Methods

Exploring the realm of extreme weather events is indispensable for various engineering disciplines and plays a pivotal role in understanding climate change phenomena. In this study, we examine the ability of 10 probability distribution functions—including exponential, normal, two- and three-parameter log-normal, gamma, Gumbel, log-Gumbel, Pearson type III, log-Pearson type III, and SQRT-ET max distributions—to assess annual maximum 24 h rainfall series obtained over a long period (1972–2022) from three nearby meteorological stations. Goodness-of-fit analyses including Kolmogorov–Smirnov and chi-square tests reveal the inadequacy of exponential and normal distributions in capturing the complexity of the data sets. Subsequent frequency analysis and multi-criteria assessment enable us to discern optimal functions for various scenarios, including hydraulic engineering and sediment yield estimation. Notably, the log-Gumbel and three-parameter log-normal distributions exhibit superior performance for high return periods, while the Gumbel and three-parameter log-normal distributions excel for lower return periods. However, caution is advised regarding the overuse of log-Gumbel, due to its high sensitivity. Moreover, as our study considers the application of mathematical and statistical methods for the detection of extreme events, it also provides insights into climate change indicators, highlighting trends in the probability distribution of annual maximum 24 h rainfall. As a novelty in the field of functional analysis, the log-Gumbel distribution with a finite sample size is utilised for the assessment of extreme events, for which no previous work seems to have been conducted. These findings offer critical perspectives on extreme rainfall modelling and the impacts of climate change, enabling informed decision making for sustainable development and resilience.

Empirical and analytical methods to characterize the efficiency of floods to move sediment in a small semi-arid basin

Abstract. Over a long multi-year period, flood events can be classified according to their effectiveness in moving sediments. Efficiency depends both on the magnitude and frequency with which events occur. The effective (or dominant) discharge is the water discharge which corresponds to the maximum sediment supply. If its calculation is well documented in temperate or humid climates and large basins, it is much more difficult in small and semi-arid basins in which short floods with high sediment supplies occur. Using the example of 31 years of measurements in the Wadi Sebdou (north-west Algeria), this paper compares the two main statistical approaches to calculate the effective discharge (the empirical method based on histograms of sediment supply by discharge classes and an analytical calculation based on a hydrological probability distribution and on a sediment rating curve) to a very simple proxy: the half-load discharge, i.e. the flow rate corresponding to 50 % of the cumulative sediment yield. Three types of discharge subdivisions were tested. In the empirical approach, two subdivisions provided effective discharge close to the half-load discharge. Analytical solutions based on log-normal and log-Gumbel probability distributions were assessed but they highly underestimated the effective discharge, whatever the subdivision used to adjust the flow frequency distribution. Furthermore, annual series of maximum discharge and half-load discharge enabled the return period of hydrological years with discharge higher than the effective discharge (around 2 years) to be inferred and showed that more than half of the yearly sediment supply is carried by flows higher than the effective discharge only every 7 hydrological years. This study was the first to adapt the analytical approach in a semi-arid basin and to show the potentiality and limits of each method in a such climate.

Best-fit probability distribution model for predicting annual maximum rainfall of Pusa (Bihar)

In present study an attempt has been made to evaluate the suitable probability distribution models for predicting 1, 2, 3, 4, 5, 6 and 7-days annual maximum rainfall amounts based on 39 years (1964 to 2002) daily rainfall data. Three probability distribution models namely: Log Normal distribution, Log Pearson Type-III distribution and Gumbel distribution models were considered to evaluate their goodness of fit. The Weibull’s method was used for computation of observed rainfall values at1, 5, 20, 30, 50, 95 and 99 percent probability levels. The Log Pearson type –III distribution was found suitable for 1 and 2 days maximum annual rainfall, while Gumbel distribution was found to be the best for predicting 3, 4, 5, 6 and 7- days annual maximum rainfall amounts. The relationships between annual maximum rainfall and return periods were also developed. The non – linear relationships (i.e. logarithmic) were found to be most suitable for all the cases.

Probabilistic Analysis of Peak Daily Rainfall for Prediction purposes in Selected Areas of Northern Nigeria

In this study, probability analysis was performed on peak daily rainfall data in order to predict rainfall interval values and to determine the best fit functions in some parts of Nigeria. The selected towns are Kaduna, Kano, Yola, Jos, Damaturu and Maiduguri. The obtained peak daily rainfall values were subjected to Gumbel, Log-Gumbel, Normal, Log-Normal, Pearson and Log-Pearson probability distributions. Mathematical equation for probability distribution functions were established for each town and used to predict peak rainfall. The predicted values were subjected to goodness of fit tests such as Chi-square, Correlation Coefficient, Coefficient of Determination and Errors of Estimates to determine how best the fits are. The model that satisfies the tests adequately was selected as the best fit model. The study revealed that the peak rainfall at Kaduna, Jos, Kano, Yola and Damaturu are best fitted by log-Gumbel, while log- Pearson distribution is suitable for predicting peak rainfall in Maiduguri. The result also shows that the occurrences of peak daily rainfall depth of 100 mm and above are rare in the selected areas. Keywords: Peak rainfall, Probability distribution models, Return period, Probability interval and goodness of fit tests

FLOOD FREQUENCY ANALYSIS OF RIVER KOSI, UTTARAKHAND, INDIA USING STATISTICAL APPROACH

In the present study, flood frequency analysis has been applied for river Kosi in Uttarakhand. The river Kosi is an important tributary of Ganga river system, which arising from Koshimool near Kausani, Almora district flows on the western side of the study area and to meet at Ramganga River. The annual flood series analysis has been carried out to estimate the flood quantiles at different return period at Kosi barrage site of river Kosi. The statistical approach provided a significant advantage of estimation of flood at any sites in the homogenous region with very less or no data. In the at –site analysis of annual flood series the Normal, Log normal, Pearson type III, Log Pearson type III, Gumbel and Log Gumbel distribution were applied using method of moments . From the analysis of different goodness of fit tests, it has been found that the Log Gumbel distribution with method of moment as parameters estimation found to be the best-fit distribution for Kosi River and other sites in the region. It is recommended that the regional parameters for Kosi Basin may be used only for primary estimation of flood and should be reviewed when more regional data available.

On accuracy of upper quantiles estimation

Abstract. Flood frequency analysis (FFA) entails the estimation of the upper tail of a probability density function (PDF) of annual peak flows obtained from either the annual maximum series or partial duration series. In hydrological practice, the properties of various methods of upper quantiles estimation are identified with the case of known population distribution function. In reality, the assumed hypothetical model differs from the true one and one cannot assess the magnitude of error caused by model misspecification in respect to any estimated statistics. The opinion about the accuracy of the methods of upper quantiles estimation formed from the case of known population distribution function is upheld. The above-mentioned issue is the subject of the paper. The accuracy of large quantile assessments obtained from the four estimation methods is compared to two-parameter log-normal and log-Gumbel distributions and their three-parameter counterparts, i.e., three-parameter log-normal and GEV distributions. The cases of true and false hypothetical models are considered. The accuracy of flood quantile estimates depends on the sample size, the distribution type (both true and hypothetical), and strongly depends on the estimation method. In particular, the maximum likelihood method loses its advantageous properties in case of model misspecification.

Statistical analysis of indoor radon data for the Walloon region (Belgium)

Two databases of indoor radon measurements in the Walloon region in Belgium can be used to establish a radon risk map of this region, taking into account the geological information. The data were collected, respectively, by long-term track-etch measurements and by short-term charcoal measurements. Here we present a preliminary study showing that the two databases can be merged. We begin our study by testing two types of distributions: the log-normal distribution and the log-Gumbel distribution, with the purpose of selecting the best to fit our databases. The paper then describes a methodology that allows us to combine them. The composite database will be used to establish a radon risk map for the Walloon region based on geology.

The PWM large quantile estimates of heavy tailed distributions from samples deprived of their largest element / Estimation des grands quantiles de distributions à queue à décroissance lente par la méthode des moments pondérés par les probabilités à partir d'échantillons amputés de leur plus grande valeur

Extremes of streamflow are usually modelled using heavy tailed distributions. While scrutinising annual flow maxima or the peaks over threshold, the largest elements in a sample are often suspected to be low quality data, outliers or values corresponding to much longer return periods than the observation period. In the case of floods, since the interest is focused mainly on the estimation of the right-hand tail of a distribution function, sensitivity of large quantiles to extreme elements of a series becomes the problem of special concern. This study investigated the sensitivity problem using the log-Gumbel distribution by generating samples of different sizes and different values of the coefficient of L-variation by means of Monte Carlo experiments. Parameters of the log-Gumbel distribution were estimated by the probability weighted moments (PWM) method, both for complete samples and the samples deprived of their largest element. In the latter case Hosking's concept of the “A” type PWM with Type II censoring was employed. The largest value was censored above the random threshold T corresponding to the non-exceedence probability F T. The effect of the F T value on the performance of the quantile estimates was then examined. Experimental results show that omission of the largest sample element need not result in a decrease in the accuracy of large quantile estimates obtained from the log-Gumbel model by the PWM method.

On the informative value of the largest sample element of log-Gumbel distribution

Extremes of stream flow and precipitation are commonly modeled by heavytailed distributions. While scrutinizing annual flow maxima or the peaks over threshold, the largest sample elements are quite often suspected to be low quality data, outliers or values corresponding to much longer return periods than the observation period. Since the interest is primarily in the estimation of the right tail (in the case of floods or heavy rainfalls), sensitivity of upper quantiles to largest elements of a series constitutes a problem of special concern. This study investigated the sensitivity problem using the log-Gumbel distribution by generating samples of different sizes (n) and different values of the coefficient of variation by Monte Carlo experiments. Parameters of the log-Gumbel distribution were estimated by the probability weighted moments (PWMs) method, method of moments (MOMs) and maximum likelihood method (MLM), both for complete samples and the samples deprived of their largest elements. In the latter case, the distribution censored by the non-exceedance probability threshold, F T , was considered. Using F T instead of the censored threshold T creates possibility of controlling estimator property. The effect of the F T value on the performance of the quantile estimates was then examined. It is shown that right censoring of data need not reduce an accuracy of large quantile estimates if the method of PWMs or MOMs is employed. Moreover allowing bias of estimates one can get the gain in variance and in mean square error of large quantiles even if ML method is used.

On robustness of large quantile estimates of log-Gumbel and log-logistic distributions to largest element of the observation series: Monte Carlo results vs. first order approximation.

Robustness of large quantile estimates to the largest element in a sample of methods of moments (MOM) and L-moments (LMM) was evaluated and compared. Quantiles were estimated by log-logistic and log-Gumbel distributions. Both are lower bounded and two-parameter distributions, with the coefficient of variation (CV) serving as the shape parameter. In addition, the results of these two methods were compared with those of the maximum likelihood method (MLM). Since identification and elimination of the outliers in a single sample require the knowledge of the sample’s parent distribution which is unknown, one estimates it by using the parameter estimation method which is relatively robust to the largest element in the sample. In practice this means that the method should be robust to extreme elements (including outliers) in a sample.

A note on the applicability of log-Gumbel and log-logistic probability distributions in hydrological analyses: II. Assumed pdf

Applicability of log-Gumbel (LG) and log-logistic (LL) probability distributions in hydrological studies is critically examined under real conditions, where the assumed distribution differs from the true one. The set of alternative distributions consists of five two-parameter distributions with zero lower bound, including LG and LL as well as lognormal (LN), linear diffusion analogy (LD) and gamma (Ga) distributions. The log-Gumbel distribution is considered as both a false and a true distribution. The model error of upper quantiles and of the first two moments is analytically derived for three estimation methods: the method of moments (MOM), the linear moments method (LMM) and the maximum likelihood method (MLM). These estimation methods are used as methods of approximation of one distribution by another distribution. As recommended in the first of this two-part series of papers, MLM turns out to be the worst method, if the assumed LG or LL distribution is not the true one. It produces a huge bias of upper quantiles, which is at least one order higher than that of the other two methods. However, the reverse case, i.e. acceptance of LN, LD or Ga as a hypothetical distribution, while the LG or LL distribution is the true one, gives the MLM bias of reasonable magnitude in upper quantiles. Therefore, one should avoid choosing the LG and LL distributions in flood frequency analysis, especially if MLM is to be applied.

A note on the applicability of log-Gumbel and log-logistic probability distributions in hydrological analyses: I. Known pdf

Two probability density functions (pdf), popular in hydrological analyses, namely the log-Gumbel (LG) and log-logistic (LL), are discussed with respect to (a) their applicability to hydrological data and (b) the drawbacks resulting from their mathematical properties. This paper—the first in a two-part series—examines a classical problem in which the considered pdf is assumed to be the true distribution. The most significant drawback is the existence of the statistical moments of LG and LL for a very limited range of parameters. For these parameters, a very rapid increase of the skewness coefficient, as a function of the coefficient of variation, is observed (especially for the log-Gumbel distribution), which is seldom observed in the hydrological data. These probability distributions can be applied with confidence only to extreme situations. For other cases, there is an important disagreement between empirical data and theoretical distributions in their tails, which is very important for the characterization of the distribution asymmetry. The limited range of shape parameters in both distributions makes the analyses (such as the method of moments), that make use of the interpretation of moments, inconvenient. It is also shown that the often-used L-moments are not sufficient for the characterization of the location, scale and shape parameters of pdfs, particularly in the case where attention is paid to the tail part of probability distributions. The maximum likelihood method guarantees an asymptotic convergence of the estimators beyond the domain of the existence of the first two moments (or L-moments), but it is not sensitive enough to the upper tails shape.

Estimation of quantiles and confidence intervals for the log-Gumbel distribution

The log-Gumbel distribution is one of the extreme value distributions which has been widely used in flood frequency analysis. This distribution has been examined in this paper regarding quantile estimation and confidence intervals of quantiles. Specific estimation algorithms based on the methods of moments (MOM), probability weighted moments (PWM) and maximum likelihood (ML) are presented. The applicability of the estimation procedures and comparison among the methods have been illustrated based on an application example considering the flood data of the St. Mary's River.

On testing that the distribution of extremes is of type I when type II is the alternative

Distributions of largest extremes are normally used to estimate tail probabilities. When dealing with unlimited extremes, the question arises which type of distributions of extremes is adequate. Sometimes the Gumbel distribution (type I), which can be regarded as a restriction of the log-Gumbel distribution (type II), is used without sufficient support for this choice; estimated extreme tail probabilities can then be more influenced by the choice of the type than by the data. This paper gives a quick test with high power where type I is the null hypothesis and type II is the alternative; it is a result based on a study to be published elsewhere dealing with the reduction of power as a consequence of using numerically simplified tests.