A Trivariate Extreme Value Distribution Applied to Flood Frequency Analysis

A trivariate extreme value distribution has been derived from the logistic model for the multivariate extreme value distribution. The construction of its corresponding probability distribution and density function is described. In order to obtain the parameters of such a trivariate distribution, a generalized maximum likelihood estimation procedure is described to allow for the cases of samples with different record lengths. Furthermore the reliability of the estimated parameters of the irivariate extreme value distribution is measured through the use of relative information ratios. A region in Northern Mexico with six gauging stations has been selected to apply the trivariate model. Results produced by the proposed model have been compared with those obtained by general extreme value (GEV) distribution functions.


Introduction
Flood frequency analysis has been carried out by using univariate distribution functions, the extreme value distributions being an important set of distributions used in this field of study.Generally, parameters of such distributions are estimated from a short record of flows.The variability of these estimates has prompted exploration of joint estimation models which use information from streamflow records of neighboring gauging stations.
in pioneering papers Finkelstein [1], Tiago de Oliveira [2], and Gumbel [3] gave the foundations for the multivariate approach to extreme value distributions.Following this work, several bivariate extreme value models began to appear in the literature.Rueda [4] explored the logistic and mixed models for bivariate extreme value distributions when both marginals are extreme value type I (EVl) distributions.He reported improvements in the estimation of parameters when the bivariate approach is used.Raynal [5] developed and applied three bivariate options from the logistic model of bivariate extreme value distribution for flood frequency analysis.He found that there exists an improvement in the parameter estimation phase, even in the case when both samples have the same record lengths.
Herein, the trivariate approach of multivariate extreme value distribution is presented with a view to its application to flood frequency analysis.
General characteristics, the procedure for estimation, and reliability of parameters of the trivariate extreme value distributions will be described in the following sections.An actual application of the proposed model to six gauging stations in Northern Mexico is presented in the paper.

Characteristics of the Trivariate Logistic Model
From the multivariate extension of the logistic model for bivariate extreme value distribution [3], the trivariate approach is: where m is the association parameter (msl) and F{s)=F(s, 6) is the marginal distribution function of s.Equation (1) must satisfy the following inequalities (Tiago de Oliveira [6,7]): Marginals in Eq. ( 1) can be either EVI distributions: or GEV distributions: The combinations have been named (Escalante [8] The particular form of Eq. ( 1), when the marginals are GEV distributions for the maxima, is (Escalante [8]); F{x, y, z, Ui, oi, ^1, u;, ai, ft, HI, OH, pi, m,) =

Estimation of Parameters
The method of maximum likelihood for estimating the parameters of trivariate extreme value distributions has been chosen due to its characteristics for consistency in large sample estimation and applicability in estimating the parameters of cumbersome density functions.
For the case of trivariate distribution functions, the sample arrangements couid allow having either an equal or different record length in any of the samples to be analysed.
In order to consider all possible combinations of data, it is required to have a sufficiently flexible formulation, therefore the following general form of the likelihood function will be used based on the generalization obtained by Anderson [9]: (8) where: ni, «2 =are respectively the univariate and bivariate record lengths before the common period ny, 04, rti -are respectively the bivariate and univariate record lengths after the common period lu, p =is the variable with univariate record before the common period, {p, g) =are the variables with bivariate record before the common period, (Xf y, z) = are the variables with trivariate record during the common period, {r, s) =are the variables with bivariate record after the common period, r = is the variable with univariate record after the common period, Ji =are indicator numbers such that: /. = 1 ifn, >Oand /, =0 ifn, =0.

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Given the complexity of the mathematical expressions in Eq. (10) and their partial derivatives with respect to the parameters, the constrained multivariable Rosenbrock method.Kuester and Mize [10], was applied to obtain the maximum likelihood estimators for the parameters by the direct maximization of Eq. ( 10).The required initial values of the parameters to start the optimization of Eq. ( 10) were provided by the univariate maximum likelihood estimators of the parameters for the case of the location, scale, and shape parameters.The initial values of the association parameters, bivariate and trivariate, were set equal to 2, following the procedure developed by Escalante [8].

Reliability of Estimated Parameters
The indicator selected to measure the reliability of estimated,parameters when using the trivariate distribution as compared with the univariate counterpart was the asymptotic relative information ratio.

Case Study
A region located in Northern Mexico, with a total of six gauging stations, was selected to apply the proposed methodology to the flood frequency analysis.Tables 2-4 show the results of the application of the trivariate extreme value distributions for the maxima to the data recorded in such gauging stations.
In order to compare the goodness of fit between the univariate and trivariate maximum likelihood estimates of the parameters in stations considered in the case study, the standard error of fit, as defined by Kite [11], was obtained and is displayed in Table 5.

Conclusions
The logistic model for trivariate general extreme value distribution for the maxima has been proposed.Asymptotic and data base results suggest that the proposed model is a suitable option to be considered when performing flood frequency analysis.

Table 1 .
Correlation cocfTiciBnu and relative sample sizes for the triplets of statiuns for the case study

Table 5 ,
Standard errors of fit for gauging stations of case study

Table 3 .
Univariate maximum likelihood estimate.*! of the parameters of the GEV distributions defined by the data of the gauging stations of the case study

Table 4 .
Trivariate maximum likelihood estimates of the parameters of the TEV222 distribution defined by the data of the gauging stations of the case study