Fahim Ashkar

The use of geometric and gamma-related distributions for frequency analysis of water deficit

AbstractThis paper presents an approach to perform statistical frequency analysis of water deficit duration and severity using respectively the geometric and exponential distributions. Monthly mean water discharges are compared to a given threshold and classified in two mutually exclusive ways. This leads to a two state random variable such that: a success represents the absence of a water deficit event (mean monthly discharge exceeds threshold), and a failure, a water deficit event (mean monthly discharge is below threshold). If we suppose that this random variable gives rise to a Markov process of order 1, then the duration of a water deficit event X (consecutive months in deficit) will have a geometric distribution. In turn, the summation of discharges in deficit will give the severity of a water deficit event which can be represented by a one-parameter exponential distribution. The threshold or base level is taken as a percentile of the observed mean discharges of a given month. This base level, which varies from month to month, can be viewed as the limit of an acceptable deficit (or energetic failure) associated to a given empirical probability of being in deficit. The second step of the approach is to estimate the value of the parameter for each distribution using the maximum likelihood method. Expressions for the estimator of a given percentile,

ON SOME METHODS OF FITTING THE GENERALIZED PARETO DISTRIBUTION

\hat x_q

A Multiple Criteria Decision Modelling approach to selection of estimation techniques for fitting extreme floods

, as well as its variance are deduced. Finally, the presented models are applied to observed data.

Study of hydrological phenomena by extreme value theory

Abstract The estimation of flood quantiles from available streamflow records has been a topic of extensive research in this century. However, the large number of distributions and estimation methods proposed in the scientific literature has led to a state of confusion, and a gap prevails between theory and practice. This concerns both at-site and regional flood frequency estimation. To facilitate the work of “hydrologists, designers of hydraulic structures, irrigation engineers and planners of water resources”, the World Meteorological Organization recently published a report which surveys and compares current methodologies, and recommends a number of statistical distributions and estimation procedures. This report is an important step towards the clarification of this difficult topic, but we think that it does not effectively satisfy the needs of practitioners as intended, because it contains some statements which are not statistically justified and which require further discussion. In the present paper we review commonly used procedures for flood frequency estimation, point out some of the reasons for the present state of confusion concerning the advantages and disadvantages of the various methods, and propose the broad lines of a possible comparison strategy. We recommend that the results of such comparisons be discussed in an international forum of experts, with the purpose of attaining a more coherent and broadly accepted strategy for estimating floods.

Choice between competitive pairs of frequency models for use in hydrology: a review and some new results

The two-parameter generalized Pareto distribution (GPD) has been recommended for the frequency analysis of environmental extreme events. In the present paper, we concentrate on one form of the GPD (which we will call GPDB) which can be useful in the frequency analysis of two types of hydrological variables: (1) when the shape parameter α is positive, the distribution (denoted by GPDB-2) can be used to study phenomena such as flood flows, which are bounded from below but have a long right tail; (2) when α is negative, the resulting distribution (GPDB-3) could be used to study variables such as low flows which are bounded from both sides but with a left tail. Six versions of the generalized method of moments (GMM) for fitting GPDB are investigated. The flexibility of the GMM provides the user with the possibility of choosing a version of the method (i.e. the moment pair that is used in fitting the distribution) which assigns larger weight to the larger elements of the sample, or another version which gives more weight to the smaller elements, depending on the problem at hand. A general formula for the asymptotic variance of the T-year event XT obtained by combining any two moments of GPDB is presented and applied. It is shown that the adequate choice of the order of these two moments to fit the distribution can lead, in some cases, to a considerable reduction in the variance of the estimator of XT, in comparison with estimation by the traditional method of moments (which uses moments of order one and two). The performance indices that are used to compare the different versions of the GMM are based on root mean square error, bias, and variance—both asymptotic and observed (based on simulation)—of GPDB quantiles and parameters. It is shown that moments of order (0),−1) and (2,−1) lead to the best results when the shape parameter a is positive (GPDB − 2), and the traditional method of moments can be considered as most efficient for negative values of α (GPDB −3). The GMM with moments of order 0 and one is shown to be rather consistent and moderately satisfactory for both GPDB-2 and GPDB-3.

Risk Analysis of Hydrologic Data: Review and New Developments Concerning the Halphen Distributions

We refocus attention on moment ratio diagrams and their uses in hydrology with four major objectives: (1) to summarize the information available in the literature about possible uses of the traditional moment ratio diagram introduced by Karl Pearson, which uses the coefficient of skewness and of kurtosis to compare the shapes of various distributions commonly used in hydrology; (2) to complete this traditional MRD by integrating into it the regions occupied by the log-Pearson Type III and generalized gamma distributions which are more and more used in hydrology; (3) to present another MRD which uses ratios of moments of orders −1 (harmonic mean), quasi zero (geometric mean) and 1 (arithmetic mean); (4) to stress the need to consider the different MRDs (along with the more recently introduced L-moment ratio diagrams) as complementary tools for choosing between distributions fitted to hydrologic data. Finally, using Monte Carlo simulation we compare the two types of diagrams as tools to identify and discriminate between different distributions.

Parameter and quantile estimation of the 2-parameter kappa distribution by maximum likelihood

AbstractThe problem of fitting a probability distribution, here log-Pearson Type III distribution, to extreme floods is considered from the point of view of two numerical and three non-numerical criteria. The six techniques of fitting considered include classical techniques (maximum likelihood, moments of logarithms of flows) and new methods such as mixed moments and the generalized method of moments developed by two of the co-authors. The latter method consists of fitting the distribution using moments of different order, in particular the SAM method (Sundry Averages Method) uses the moments of order 0 (geometric mean), 1 (arithmetic mean), −1 (harmonic mean) and leads to a smaller variance of the parameters. The criteria used to select the method of parameter estimation are:- the two statistical criteria of mean square error and bias;- the two computational criteria of program availability and ease of use;- the user-related criterion of acceptability. These criteria are transformed into value functions or fuzzy set membership functions and then three Multiple Criteria Decision Modelling (MCDM) techniques, namely, composite programming, ELECTRE, and MCQA, are applied to rank the estimation techniques.

Cramér-von Mises and Anderson-Darling Goodness-of-Fit Tests for the Two-Parameter Kappa Distribution

We describe and give hydrological applications of a probabilistic model based on extreme value theory which can be used to study the values of a hydrologic process that exceed a certain threshold level QB.This model is useful in estimating extreme events XTof return period T based on N years of available hydrologic record. We also present easy-to-use tables which give confidence intervals for XT.The hydrologic applications reported are a flood frequency analysis, a methodology for estimating flood damage, an estimation of precipitation probabilities, and a prediction of extreme tide levels.