Estimation of the generalized logistic distribution of extreme events using partial L-moments

Abstract

Statistical analysis of extreme events is often carried out to predict large return period events. In this paper, the use of partial L-moments (PL-moments) for estimating hydrological extremes from censored data is compared to that of simple L-moments. Expressions of parameter estimation are derived to fit the generalized logistic (GLO) distribution based on the PL-moments approach. Monte Carlo analysis is used to examine the sampling properties of PL-moments in fitting the GLO distribution to both GLO and non-GLO samples. Finally, both PL-moments and L-moments are used to fit the GLO distribution to 37 annual maximum rainfall series of raingauge station Kampung Lui (3118102) in Selangor, Malaysia, and it is found that analysis of censored rainfall samples of PL-moments would improve the estimation of large return period events.

Editor D. Koutsoyiannis; Associate editor K. Hamed

Citation Zakaria, Z.A., Shabri, A. and Ahmad, U.N., 2012. Estimation of the generalized logistic distribution of extreme events using partial L-moments. Hydrological Sciences Journal, 57 (3), 424–432.     

Beta-�� distribution and its application to hydrologic events

The beta-κ distribution is a distinct case of the generalized beta distribution of the second kind. In previous studies, beta-p and beta-κ distributions have played important roles in representing extreme events, and thus, the present paper uses the beta-κ distribution. Further, this paper uses the method of moments and the method of L-moments to estimate the parameters from the beta-κ distribution, and to demonstrate the performance of the proposed model, the paper presents a simulation study using three estimation methods (including the maximum likelihood estimation method) and beta-κ and non beta-κ samples. In addition, this paper evaluates the performance of the beta-κ distribution by employing two widely used extreme value distributions (i.e., the GEV and Gumbel distributions) and two sets of actual data on extreme events.     

Trimmed L-moments (1,0) for the generalized Pareto distribution

Abstract

Statistical analysis of extremes is often used for predicting the higher return-period events. In this paper, the trimmed L-moments with one smallest value trimmed—TL-moments (1,0)—are introduced as an alternative way to estimate floods for high return periods. The TL-moments (1,0) have an ability to reduce the undesirable influence that a small value in the statistical sample might have on a large return period. The main objective of this study is to derive the TL-moments (1,0) for the generalized Pareto (GPA) distribution. The performance of the TL-moments (1,0) was compared with L-moments through Monte Carlo simulation based on the streamflow data of northern Peninsular Malaysia. The result shows that, for some cases, the use of TL-moments (1,0) is a better option as compared to L-moments in modelling those series.

Citation Ahmad, U.N., Shabri, A. & Zakaria, Z.A. (2011) Trimmed L-moments (1,0) for the generalized Pareto distribution. Hydrol.Sci. J. 56(6), 1053–1060.     

Trivariate generalized extreme value distribution in flood frequency analysis

Abstract

The multivariate extension of the logistic model with generalized extreme value (GEV) marginals is applied to provide a regional at-site flood estimate. The maximum likelihood estimators of the parameters were obtained numerically by using a multivariable constrained optimization algorithm. The asymptotic results were checked by distribution sampling techniques in order to establish whether or not those results can be utilized for small samples. A region in northern Mexico with 21 gauging stations was selected to apply the model. Results were compared with those obtained by the most popular univariate distributions, the bivariate approach of the logistic model and three regional methods: station-year, index flood and L-moments. These show that there is a reduction in the standard error of fit when estimating the parameters of the marginal distribution with the trivariate distribution instead of its univariate and bivariate counterpart, and differences between at-site and regional at-site design events can be significant as return period increases.     

Partial L-moments for the analysis of censored flood samples / Utilisation des L-moments partiels pour l’analyse d’échantillons tronqués de crues

Abstract

Abstract A parameter estimation method is proposed for fitting the generalized extreme value (GEV) distribution to censored flood samples. Partial L-moments (PL-moments), which are variants of L-moments and analogous to ?partial probability weighted moments?, are defined for the analysis of such flood samples. Expressions are derived to calculate PL-moments directly from uncensored annual floods, and to fit the parameters of the GEV distribution using PL-moments. Results of Monte Carlo simulation study show that sampling properties of PL-moments, with censoring flood samples of up to 30% are similar to those of simple L-moments, and also that both PL-moment and LH-moments (higher-order L-moments) have similar sampling properties. Finally, simple L-moments, LH-moments, and PL-moments are used to fit the GEV distribution to 75 annual maximum flow series of Nepalese and Irish catchments, and it is found that, in some situations, both LH- and PL-moments can produce a better fit to the larger flow values than simple L-moments.     

LH-moment estimation of a four parameter kappa distribution with hydrologic applications

The study of distribution tails is a fundamental research in statistical frequency analysis relevant to many research fields, such as insurance, hydrological events, earthquake, etc. Here, we describe and investigate the effect and feasibility of the high-order L-moment (LH-moment) method for estimating heavy-tail conditions by fitting a four parameter kappa distribution. Details of parameter estimation using LH-moments for the four parameter kappa distribution (K4D) are described and formulated. Monte-Carlo simulation is performed to illustrate the performance of the LH-moment method in terms of heavy-tail quantiles over all quantiles using K4D and non K4D samples, respectively. The result suggests that the method is either useful (when the method of L-moment estimation fails to give a feasible solution) or as effective as the L-moment approach in handling data following K4D. Applications to the annual maximum flood and sea level data are presented.     

Alternative PWM-estimators of the Gumbel distribution

Probability weighted moments (PWM) are widely used in hydrology for estimating parameters of statistical distributions, including the Gumbel distribution. The classical PWM-approach considers the moments β_i=E[XFⁱ] with i=0,1 for estimation of the Gumbel scale and location parameters. However, there is no reason why these probability weights (F⁰ and F¹) should provide the most efficient PWM-estimators of Gumbel parameters and quantiles. We explore an extended class of PWMs that does not impose arbitrary restrictions on the values of i. Estimation based on the extended class of PWMs is called the generalized method of probability weighted moments (GPWM) to distinguish it from the classical procedure. In fact, our investigation demonstrates that it may be advantage to use weight functions that are not of the form Fⁱ. We propose an alternative PWM-estimator of the Gumbel distribution that maintains the computational simplicity of the classical PWM method, but provides slightly more accurate quantile estimates in terms of mean square error of estimation. A simple empirical formula for the standard error of the proposed quantile estimator is presented.     

A Bayesian analysis of the Gumbel distribution: an application to extreme rainfall data

With the objective of modelling annual rainfall maximum intensities in different geographical zones of Chile, we have created a Bayesian inference method for the generalized extreme value type I distribution (Gumbel distribution). We considered an uninformative prior distribution for the location parameter, μ, and three different prior distributions for the scale parameter, σ. Under these conditions we obtained the posterior distribution of (μ, σ) and associated summary statistics such as modes, expected values, quantiles and credibility intervals. In order to predict and estimate return periods, we obtained the posterior distribution of future observations, its expected value, quantiles and credibility intervals. To obtain several of these posterior summary measures it was necessary to utilize both numerical and Laplace approximations. Furthermore we estimate return period curves and intensity–duration–frequency curves.     

Statistics of extremes and estimation of extreme rainfall: I. Theoretical investigation / Statistiques de valeurs extrêmes et estimation de précipitations extrêmes: I. Recherche théorique

Abstract

Abstract The Gumbel distribution has been the prevailing model for quantifying risk associated with extreme rainfall. Several arguments including theoretical reasoning and empirical evidence are supposed to support the appropriateness of the Gumbel distribution. These arguments are examined thoroughly in this work and are put into question. Specifically, theoretical analyses show that the Gumbel distribution is quite unlikely to apply to hydrological extremes and its application may misjudge the risk, as it underestimates seriously the largest extreme rainfall amounts. Besides, it is shown that hydrological records of typical length (some decades) may display a distorted picture of the actual distribution, suggesting that the Gumbel distribution is an appropriate model for rainfall extremes while it is not. In addition, it is shown that the extreme value distribution of type II (EV2) is a more consistent alternative. Based on the theoretical analysis, in the second part of this study an extensive empirical investigation is performed using a collection of 169 of the longest available rainfall records worldwide, each having 100–154 years of data. This verifies the inappropriateness of the Gumbel distribution and the appropriateness of EV2 distribution for rainfall extremes.     

Robust flood statistics: comparison of peak over threshold approaches based on monthly maxima and TL-moments

ABSTRACT

Flood quantile estimation based on partial duration series (peak over threshold, POT) represents a noteworthy alternative to the classical annual maximum approach since it enlarges the available information spectrum. Here the POT approach is discussed with reference to its benefits in increasing the robustness of flood quantile estimations. The classical POT approach is based on a Poisson distribution for the annual number of exceedences, although this can be questionable in some cases. Therefore, the Poisson distribution is compared with two other distributions (binomial and Gumbel-Schelling). The results show that only rarely is there a difference from the Poisson distribution. In the second part we investigate the robustness of flood quantiles derived from different approaches in the sense of their temporal stability against the occurrence of extreme events. Besides the classical approach using annual maxima series (AMS) with the generalized extreme value distribution and different parameter estimation methods, two different applications of POT are tested. Both are based on monthly maxima above a threshold, but one also uses trimmed L-moments (TL-moments). It is shown how quantile estimations based on this “robust” POT approach (rPOT) become more robust than AMS-based methods, even in the case of occasional extraordinary extreme events.

Editor M.C. Acreman Associate editor A. Viglione     

Comprehensive evaluation of regional flood frequency analysis by L- and LH-moments. I. A re-visit to regional homogeneity

As part I of a sequence of two papers, previously developed L-moments by Hosking (J R Stat Soc Ser B Methodol 52(2):105–124, 1990), and the LH-moments by Wang (Water Resour Res 33(12):2841–2848, 1997) are re-visited. New relationships are developed for regional homogeneity analysis by the LH-moments, and further establishment of regional homogeneity is investigated. Previous works of Hosking (J R Stat Soc Ser B Methodol 52(2):105–124, 1990) and Wang (Water Resour Res 33(12):2841–2848, 1997) on L-moments and LH-moments for the generalized extreme value (GEV) distribution are extended to the generalized Pareto (GPA) and the generalized logistic (GLO) distributions. The Karkhe watershed, located in western Iran is used as a case study area. Regional homogeneity was investigated by first assuming the entire study area as one regional cluster. Then the entire study area was designated “homogeneous” by the L-moments (L); and was designated “heterogeneous” by all four levels of the LH-moments (L₁ to L₄). The k-means method was used to investigate the case of two regional clusters. All levels of the L- and LH-moments designated the upper watershed (region A), “homogeneous”, and the lower watershed (region B) “possibly-homogeneous”. The L₃ level of the GPA and the L₄ level of the GLO were selected for regions A and B, respectively. Wang (Water Resour Res 33(12):2841–2848, 1997) identified a reversing trend in improved performance of the GEV distribution at the LH-moments level of L₃ (during the goodness-of-fit test). Similar results were also obtained in this research for the GEV distribution. However, for the case of the GPA distribution the reversing trend started at L₄ for region A; and at L₂ for region B. As for the case of the GLO, an improved performance was observed for all levels (moving from L to L₄); for both regions.     

Modelling bivariate extreme precipitation distribution for data‐scarce regions using Gumbel–Hougaard copula with maximum entropy estimation

A new method of parameter estimation in data scarce regions is valuable for bivariate hydrological extreme frequency analysis. This paper proposes a new method of parameter estimation (maximum entropy estimation, MEE) for both Gumbel and Gumbel–Hougaard copula in situations when insufficient data are available. MEE requires only the lower and upper bounds of two hydrological variables. To test our new method, two experiments to model the joint distribution of the maximum daily precipitation at two pairs of stations on the tributaries of Heihe and Jinghe River, respectively, were performed and compared with the method of moments, correlation index estimation, and maximum likelihood estimation, which require a large amount of data. Both experiments show that for the Ye Niugou and Qilian stations, the performance of MEE is nearly identical to those of the conventional methods. For the Xifeng and Huanxian stations, MEE can capture information indicating that the maximum daily precipitation at the Xifeng and Huanxian stations has an upper tail dependence, whereas the results generated by correlation index estimation and maximum likelihood estimation are unreasonable. Moreover, MEE is proved to be generally reliable and robust by many simulations under three different situations. The Gumbel–Hougaard copula with MEE can also be applied to the bivariate frequency analysis of other extreme events in data‐scarce regions.     

Comprehensive evaluation of regional flood frequency analysis by L- and LH-moments. II. Development of LH-moments parameters for the generalized Pareto and generalized logistic distributions

As part II of a sequence of two papers, previously developed L-moments by Hosking (1990), and the LH-moments by Wang (1997) are further investigated. The LH-moments (L to L₄) are used to develop the regional parameters of the generalized extreme value distribution, generalized Pareto (GPA) distribution and the generalized logistic (GLO) distributions. These respective probability distribution functions (PDFs) are evaluated in terms of their performances. Flood peaks by the corresponding PDFs are compared with those generated by Monte Carlo simulation of randomized data, considering the respective LH-moments. The influence of the LH-moments on estimated PDFs are studied by evaluating the relative bias (RBIAS) in quantile estimation due to variability of the k parameter. Karkhe watershed located in western Iran was used as a case study area. Part I of this study identified the study area as regions A and B. The minimum calculated relative root mean square error (RRMSE) and RBIAS between simulated flood peaks and flood peaks by the corresponding PDFs were used in PDF selection, considering the respective LH-moments. The boxplots of the RRMSE tests identified the L₃ level of the GPA distribution as the suitable PDF for sample sizes 20 and 80; for region A. Similar results were found for the RBIAS test. As for region B, the boxplots of the RRMSE tests indicated similar results for the three PDFs. However, the boxplots of the RBIAS tests identified the L₄ level of the GLO most suitable for sample sizes 20 and 80. Relative efficiencies of the LH-moments were investigated, measured as RRMSE ratios of L-moments over the respective LH-moments. For the most parts the findings of this part of the study were similar to those of part I.     

Graphical estimation of extreme value prediction functions

Graphical application of the Type 1 (Gumbel) extreme value distribution is very simple since the distribution inverse gives a linear x-y plot. In contrast, the Type 2 and Type 3 extreme value distributions have nonlinear functions with respect to the same axes. A simple three-point graphical estimation procedure is described for these two distributions. This approach allows the nonlinear flood magnitude prediction functions to be located in any desirable position relative to the plotted annual maxima, subject to the constraint of having an extreme value form. The computation is very simple and requires only the location of a unique zero of a one-parameter function within a defined interval.     

The null distribution of sample serial correlation coefficient

 The null distribution of the lag-k sample serial correlation coefficient (r _k , k=1,2,3) was investigated by Monte Carlo simulation. For a time series with normal, exponential, Pearson 3, EV1 (Gumbel), or generalized Pareto (GP) distribution type, the null distribution of its r _k can be approximated by the normal distribution with mean −1/(n−k) and variance 1/(n−1). But for a time series with the lognormal, EV2 or EV3 (Weibull) distribution type, the null distribution of r _k is skewed distributed. In such cases, a simulation technique is suggested to construct percentile confidence intervals at a given significance level.     

Analysis of largest earthquakes in Turkey and its vicinity by application of the Gumbel III distribution

A study of the spatial distribution of seismicity parameters is undertaken along Turkey and its vicinity, using the Gumbel’s third asymptotic distribution of extreme values (GIII). The data set used spans of 111 years (1900–2010). The seismicity of the whole region is subdivided into equal area mesh of 1° lat. × 1° long. Various seismicity parameters examined, resulted from the application of the GIII method. The results show a quite good correlation between the seismicity parameters and the tectonic regime of the studied area. For instance high values concentrated around North Anatolian Fault. The x ²-test is applied throughout the whole process and in every stage of GIII, in order to check the accuracy of the obtained results. The spatial distribution of upper-bound (ω) formed a W-shape pattern, which shows the difference in the mechanical structure of the materials in the examined area.     

Depth distribution of interrill overland flow and the formation of rills

A Gumbel distribution for maxima is proposed as a model for the depths of interrill overland flow. The model is tested against three sets of field measurements of interrill overland flow depths obtained on shrubland and grassland hillslopes at Walnut Gulch Experimental Watershed, southern Arizona. The model is found to be a satisfactory fit to 81 of the 90 measured distributions. The shape δ and location λ parameters of all fitted distributions are strongly correlated with discharge. However, whereas a common relationship exists between discharge and δ for all depth distributions, the relationships with λ vary systematically downslope. Using the Gumbel distribution as a model for the distribution of overland flow depths, a probabilistic model for the initiation of rills is developed, drawing upon the previous work of Nearing. As an illustration of this approach, we apply this model to the shrubland and grassland hillslopes at Walnut Gulch. It is concluded that the presence of rills on the shrubland, but not on the grassland, is due to the greater runoff coefficient for the shrubland and/or the greater propensity of the shrubland for soil disturbance compared with the grassland. Finally, a generalized conceptual model for rill initiation is proposed. This model takes account of the depth distribution of overland flow, the probability of flow shear stress in excess of local soil shear strength, the spatial variability in soil shear strength and the diffusive effect of soil detachment by raindrops. Copyright © 2005 John Wiley & Sons, Ltd.