Linear combination of Gumbel random variables

The recent paper by Loaiciga and Leipnik (Stoch Environ Res Risk Environ 13:251–259, 1999) derived the probability distribution of the sum of two independent Gumbel random variables. The results given are of little practical use because they are given in terms of characteristic functions. In this note, we consider the more general problem of deriving the linear combination of two independent Gumbel random variables. Explicit expressions are given for the probability density function and the cumulative distribution function of the linear combination. Various particular cases are also considered.     

Stochastic simulation of bivariate gamma distribution: a frequency-factor based approach

A frequency-factor based approach for stochastic simulation of bivariate gamma distribution is proposed. The approach involves generation of bivariate normal samples with a correlation coefficient consistent with the correlation coefficient of the corresponding bivariate gamma samples. Then the bivariate normal samples are transformed to bivariate gamma samples using the well-known general equation of hydrological frequency analysis. We demonstrate that the proposed bivariate gamma simulation approach is capable of generating random sample pairs which not only have the desired marginal densities of component random variables but also their correlation coefficient. Scatter plots of simulated bivariate sample pairs also exhibit appropriate linear patterns (dependence structure) that are commonly observed in environmental and hydrological applications. Caution should also be exercised when specifying combinations of coefficients of skewness and the correlation coefficient for bivariate gamma simulation.     

A bivariate gamma distribution for use in multivariate flood frequency analysis

A gamma distribution is one of the most frequently selected distribution types for hydrological frequency analysis. The bivariate gamma distribution with gamma marginals may be useful for analysing multivariate hydrological events. This study investigates the applicability of a bivariate gamma model with five parameters for describing the joint probability behavior of multivariate flood events. The parameters are proposed to be estimated from the marginal distributions by the method of moments. The joint distribution, the conditional distribution, and the associated return periods are derived from marginals. The usefulness of the model is demonstrated by representing the joint probabilistic behaviour between correlated flood peak and flood volume and between correlated flood volume and flood duration in the Madawask River basin in the province of Quebec, Canada. Copyright © 2001 John Wiley & Sons, Ltd.     

Correlated gamma variables in the analysis of microbial densities in water

The probability density function (p.d.f.) of the ratio of two correlated gamma variables is derived and used to fit aquatic microbial-density data. The ratio p.d.f. is tackled by first taking the Fourier transform of a generalized Kibble–Gaver, unsymmetrical, characteristic function (c.f.) to obtain the corresponding bivariate p.d.f. of two correlated gamma variables with different shape and scale parameters. The ratio p.d.f. follows by weighted integration of the bivariate p.d.f. The derivation of the gamma bivariate and ratio p.d.f.s relies on the use of weighted Laguerre–Charlier polynomials that lead to p.d.f.s amenable to computation. The bivariate gamma p.d.f. and the ratio p.d.f. of correlated gamma variables are useful statistical tools in the analysis of skewed water-resources data. Computational examples illustrate the calculation of bivariate p.d.f.s for positive and negative correlation and the fitting of the ratio p.d.f. to correlated bacterial densities in stream water.     

Frequency analysis of droughts using the Plackett copula and parameter estimation by genetic algorithm

This study aims to model the joint probability distribution of drought duration, severity and inter-arrival time using a trivariate Plackett copula. The drought duration and inter-arrival time each follow the Weibull distribution and the drought severity follows the gamma distribution. Parameters of these univariate distributions are estimated using the method of moments (MOM), maximum likelihood method (MLM), probability weighted moments (PWM), and a genetic algorithm (GA); whereas parameters of the bivariate and trivariate Plackett copulas are estimated using the log-pseudolikelihood function method (LPLF) and GA. Streamflow data from three gaging stations, Zhuangtou, Taian and Tianyang, located in the Wei River basin, China, are employed to test the trivariate Plackett copula. The results show that the Plackett copula is capable of yielding bivariate and trivariate probability distributions of correlated drought variables.     

Derivation and assessment of a bivariate IDF relationship using paired rainfall intensity–duration data

A procedure is presented for developing a rainfall intensity–duration–frequency (IDF) relationship that is consistent with bivariate normal distribution modeling. The Box–Cox transformation was used to derive the relation and two methods of determining the parameters of this transformation were evaluated. To assess the uncertainty of the parameters, a confidence interval was constructed and verified with the non-parametric bootstrap method. Additionally, the effect of sample size on the bivariate normality assumption was examined. Case studies, based on data from significant gauge stations in Korea, were performed. The result shows that the use of the bivariate normal model as an IDF relationship is particularly recommended when the available data size is small.     

Modelling bivariate rainfall distribution and generating bivariate correlated rainfall data in neighbouring meteorological subdivisions using copula

The rainfall patterns of neighbouring meteorological subdivisions of India are similar because of similar climatological and geographical characteristics. Analysing the rainfall pattern separately for these meteorological subdivisions may not always capture the correlation and tail dependence. Furthermore, generating the multivariate rainfall data separately may not preserve the correlation. In this study, copula method is used to derive the bivariate distribution of monsoon rainfall in neighbouring meteorological subdivisions. Different Archimedean copulas are used for this purpose and the best copula is selected based on nonparametric test and tail dependence coefficient. The fitted copula is then applied to derive the bivariate distribution, joint return period and conditional distribution. Bivariate rainfall data is generated with the fitted copula and it is observed with the increase of sample size, the generated data is able to capture the correlation as well as tail dependence. The methodology is demonstrated with the case study of two neighbouring meteorological subdivisions of North‐East India: Assam and Meghalaya meteorological subdivision and Nagaland, Manipur, Mizoram and Tripura meteorological subdivision. Copyright © 2010 John Wiley & Sons, Ltd.     

Application of bivariate frequency analysis to the derivation of rainfall–frequency curves

Bivariate distributions have been recently employed in hydrologic frequency analysis to analyze the joint probabilistic characteristics of multivariate storm events. This study aims to derive practical solutions of application for the bivariate distribution to estimate design rainfalls corresponding to the desired return periods. Using the Gumbel mixed model, this study constructed rainfall–frequency curves at sample stations in Korea which provide joint relationships between amount, duration, and frequency of storm events. Based on comparisons and analyses of the rainfall–frequency curves derived from univariate and bivariate storm frequency analyses, this study found that conditional frequency analysis provides more appropriate estimates of design rainfalls as it more accurately represents the natural relationship between storm properties than the conventional univariate storm frequency analysis.     

Derivation of a curve number and kinematic-wave based flood frequency distribution

Abstract

The physically-based flood frequency models use readily available rainfall data and catchment characteristics to derive the flood frequency distribution. In the present study, a new physically-based flood frequency distribution has been developed. This model uses bivariate exponential distribution for rainfall intensity and duration, and the Soil Conservation Service-Curve Number (SCS-CN) method for deriving the probability density function (pdf) of effective rainfall. The effective rainfall-runoff model is based on kinematic-wave theory. The results of application of this derived model to three Indian basins indicate that the model is a useful alternative for estimating flood flow quantiles at ungauged sites.     

A new bivariate Gamma distribution generated from functional scale parameter with application to drought data

Univariate and bivariate Gamma distributions are among the most widely used distributions in hydrological statistical modeling and applications. This article presents the construction of a new bivariate Gamma distribution which is generated from the functional scale parameter. The utilization of the proposed bivariate Gamma distribution for drought modeling is described by deriving the exact distribution of the inter-arrival time and the proportion of drought along with their moments, assuming that both the lengths of drought duration (X) and non-drought duration (Y) follow this bivariate Gamma distribution. The model parameters of this distribution are estimated by maximum likelihood method and an objective Bayesian analysis using Jeffreys prior and Markov Chain Monte Carlo method. These methods are applied to a real drought dataset from the State of Colorado, USA.     

Uncertainty and variability in bivariate modeling of hydrological droughts

There are two kinds of uncertainty factors in modeling the bivariate distribution of hydrological droughts: the alteration of predefined critical ratios for pooling droughts and excluding minor droughts and the temporal variability of univariate and/or bivariate characteristics of droughts due to the impact of human activities. Daily flow data covering a period of 56 hydrological years from two gauging stations from a humid region in South China are used. The influences of alterations of threshold values of flow and critical ratios of pooling droughts and excluding minor droughts on drought properties are analyzed. Six conventional univariate models and three Archimedean copulas are employed to fit the marginal and joint distributions of drought properties, the Kolmogorov–Smirnov and Anderson–Darling methods are used for testing the goodness-of-fit of the univariate model, and the Cramer-von Mises method based on Rosenblatt’s transform is applied for the test of the bivariate model. The change point analysis of the copula parameter of bivariate distribution of droughts is first made. Results demonstrate that both the statistical characteristics of each drought property and their bivariate joint distributions are sensitive to the critical ratio of excluding minor droughts. A model can be selected to fit the marginal distribution for drought deficit volume or maximum deficit, but it is not determined for drought duration with the lower ratios of the pooling and excluding droughts. The statistical uncertainty of drought duration makes the modeling of bivariate joint distribution of drought duration and deficit volume or of drought duration and maximum deficit undermined. Change points significantly occurred in the period from the late 1970s to the middle 1980s for a single drought property and the copula parameter of their joint distribution due to the impact of human activities. The difference between two subseries separated by the change point is remarkable in the magnitudes of drought properties and the joint return periods. A copula function can be selected to optimally fit the bivariate distribution, provided that the critical ratios of pooling and excluding droughts are great enough such as the optimal value of 0.4 in the case study. It is valuable that the modeling and designing of the bivariate joint correlation and distribution of drought properties can be performed on the subseries separated by the change point of the copula parameter.     

Application of copulas for derivation of drought severity–duration–frequency curves

This study presents copula‐based multivariate probabilistic approach to model severity–duration–frequency (S‐D‐F) relationship of drought events in western Rajasthan, India. Drought occurrences are analysed using standardized precipitation index computed on monthly mean areal precipitation, aggregated at a time scale of 6 months. After testing with a series of probability density functions, the drought variable severity is found to be better represented with log‐normal distribution, whereas duration is well fitted with exponential distribution. Four different classes of bivariate copulas – Archimedean, extreme value, Plackett, and elliptical families are evaluated for modelling joint distribution of drought characteristics. It is observed that the extreme value copula – Gumbel–Hougaard copula – performed better as compared with other classes of copulas, based on results of various statistical tests and upper tail dependence coefficient. The joint distribution obtained from best performing copula is then employed to determine conditional return period and to derive drought severity‐duration‐frequency (S‐D‐F) curves for the study region. The results of the study suggests that the copula method can be used effectively to derive the drought S‐D‐F curves, which can be helpful in planning and adopting suitable drought mitigation strategies in drought‐prone areas. Copyright © 2011 John Wiley & Sons, Ltd.     

Assessment of hydrological droughts for the Yellow River,China, using copulas

Droughts are one of the normal and recurrent climatic phenomena on Earth. However, recurring prolonged droughts have caused far‐reaching and diverse impacts because of water deficits. This study aims to investigate the hydrological droughts of the Yellow River in northern China. Since drought duration and drought severity exhibit significant correlation, a bivariate distribution is used to model the drought duration and severity jointly. However, drought duration and drought severity are often modelled by different distributions; the commonly used bivariate distributions cannot be applied. In this study, a copula is employed to construct the bivariate drought distribution. The copula is a function that links the univariate marginal distributions to form the bivariate distribution. The bivariate return periods are also established to explore the drought characteristics of the historically noticeable droughts. The results show that the return period of the drought that occurred in late 1920s to early 1930s is 105 years. The significant 1997 dry‐up phenomenon that occurred in the downstream Yellow River (resulting from the 1997–1998 drought) only has a return period of 4·4 years and is probably induced by two successive droughts and deteriorated by other factors, such as human activities. Copyright © 2007 John Wiley & Sons, Ltd.     

Determination of probability distributions for Strahler stream lengths based on Poisson process and DEM

Abstract

One of the basic tasks in geomorphologic analysis is to know the probability distributions of the stream lengths of different orders. In practical applications, this information is useful for basin rainfall-runoff modelling. The objective of this study is to determine the length distributions of the Strahler streams. A Poisson process was used to derive the theoretical distributions. The result showed that the length distribution of the first-order stream is an exponential distribution and the second-order or higher order stream length is a gamma distribution. In order to verify the theoretical distributions, a digital elevation model (DEM) was adopted to calculate the stream lengths of four basins in Taiwan. Kolmogorov-Smirnov and chi-square tests were used to test the goodness-of-fit of the data. Results showed that the length distributions of the first- and second-order streams analysed by using DEM correspond with those from the derived distribution method.     

Sums, products, and ratios for downton’s bivariate exponential distribution

Motivated by environmental applications, we derive the exact distributions of R = X+Y, P = X Y and W = X/(X+Y) and the corresponding moment properties when X and Y follow Downton’s bivariate exponential distribution. The expressions turn out to involve several special functions. For practical purposes, we also provide extensive tabulations of the percentage points associated with the distributions.     

Return period of bivariate distributed extreme hydrological events

 Extreme hydrological events are inevitable and stochastic in nature. Characterized by multiple properties, the multivariate distribution is a better approach to represent this complex phenomenon than the univariate frequency analysis. However, it requires considerably more data and more sophisticated mathematical analysis. Therefore, a bivariate distribution is the most common method for modeling these extreme events. The return periods for a bivariate distribution can be defined using either separate single random variables or two joint random variables. In the latter case, the return periods can be defined using one random variable equaling or exceeding a certain magnitude and/or another random variable equaling or exceeding another magnitude or the conditional return periods of one random variable given another random variable equaling or exceeding a certain magnitude. In this study, the bivariate extreme value distribution with the Gumbel marginal distributions is used to model extreme flood events characterized by flood volume and flood peak. The proposed methodology is applied to the recorded daily streamflow from Ichu of the Pachang River located in Southern Taiwan. The results show a good agreement between the theoretical models and observed flood data. The author wishes to thank the two anonymous reviewers for their constructive comments that improving the quality of this work.     

Regional bivariate modeling of droughts using L-comoments and copulas

The regional bivariate modeling of drought characteristics using the copulas provides valuable information for water resources management and drought risk assessment. The regional frequency analysis (RFA) can specify the similar sites within a region using L-comoments approach. One of the important steps in the RFA is estimating regional parameters of the copula function. In the present study, an optimization-based method along with the adjusted charged system search are introduced and applied to estimate the regional parameters of the copula models. The capability of the proposed methodology is illustrated by copula functions on drought events. Three commonly used copulas containing Clayton, Frank and Gumbel are employed to derive the joint distribution of drought severity and duration. The result of the new method are compared to the method of moments and after applying several goodness-of-fit tests, the results indicate that the new method provides higher accuracy than the classic one. Furthermore, the results of the upper tail dependence coefficient indicate that the Gumbel copula is the best-fitted copula among the other ones for modeling drought characteristics.     

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.     

Joint probability distribution of annual maximum storm peaks and amounts as represented by daily rainfalls

Abstract

A procedure is presented for using the bivariate normal distribution to describe the joint distribution of storm peaks (maximum rainfall intensities) and amounts which are mutually correlated. The Box-Cox transformation method is used to normalize original marginal distributions of storm peaks and amounts regardless of the original forms of these distributions. The transformation parameter is estimated using the maximum likelihood method. The joint cumulative distribution function, the conditional cumulative distribution function, and the associated return periods can be readily obtained based on the bivariate normal distribution. The method is tested and validated using two rainfall data sets from two meteorological stations that are located in different climatic regions of Japan. The theoretical distributions show a good fit to observed ones.     

Bivariate drought frequency curves and confidence intervals: a case study using monthly rainfall generation

Although water resources management practices recently use bivariate distribution functions to assess drought severity and its frequency, the lack of systematic measurements is the major hindrance in achieving quantitative results. This study aims to suggest a statistical scheme for the bivariate drought frequency analysis to provide comprehensive and consistent drought severities using observed rainfalls and their uncertainty using synthesized rainfalls. First, this study developed a multi-variate regression model to generate synthetic monthly rainfalls using climate variables as causative variables. The causative variables were generated to preserve their correlations using copula functions. This study then focused on constructing bivariate drought frequency curves using bivariate kernel functions and estimating their confidence intervals from 1,000 likely replica sets of drought frequency curves. The confidence intervals achieved in this study may be useful for making a comprehensive drought management plan through providing feasible ranges of drought severity.