3.1.1. Generalized Extreme Value Model
[17] Extreme value theory provides a basis for the modeling of data maxima (or minima). On the basis of an underlying asymptotic argument, the theory allows for the extrapolation of the observed to the unobserved [Coles, 2001]. When the data maxima of random variables are taken over sufficiently long blocks of time, it is appropriate to fit the family of GEV distributions. The cumulative distribution function for the GEV is defined as
where in our application z denotes a value of the monthly streamflow maxima and θ = [μ, σ, ξ] and [1 + ξ()] ≥ 0. The location parameter μ indicates where the distribution is centered, the scale parameter σ > 0 indicates the spread of the distribution, and the shape parameter ξ indicates the behavior of the distribution's upper tail [Coles, 2001]. On the basis of the shape parameter, the GEV can assume three possible types, known as the Gumbel, Fréchet, and Weibull. The Gumbel is an unbounded lighttailed distribution (ξ = 0), whereby the tail decreases relatively rapidly (i.e., exponential decay). The Fréchet is a heavytailed distribution (ξ > 0), whereby the tail's rate of decrease is relatively slower (i.e., polynomial decay). The Weibull is a bounded distribution (ξ < 0), whereby the tail has a finite value at z = μ − [Coles, 2001]. Although several methods can be used for the estimation of the GEV model parameters, here the Maximum Likelihood Estimation (MLE) was utilized for its ability to easily incorporate covariate information [Katz et al., 2002]. Here the unknown parameters θ were estimated by maximizing the loglikelihood (llh) equation, which is defined as
where g(z;θ) is the derivative of G(z;θ) with respect to z and N is the sample size. Equation (2) can be expanded as
where ≥ 0. For the purpose of optimization, it is convenient to minimize nllh(θ) = − llh(θ) instead of directly maximizing llh(θ).
[18] It is generally useful to analyze the probabilistic quantiles of the associated GEV. Quantiles, z_{(1−p)}, are obtained by inverting equation (1):
with 0 < p < 1. That is, there is a p × 100% chance of exceeding z_{(1−p)} in the time block chosen. When the annual maxima are being examined, z_{(1−p)} corresponds to the return period (1/p), where the level z_{(1 − p)} is expected to be exceeded on average once every 1/p years [Coles, 2001]. For time scales other than annual, the interpretation is analogous but must be adjusted appropriately. Confidence intervals can be calculated for the quantiles utilizing techniques such as the delta method or the profile likelihood method (see sections 2.6.4 and 2.6.5, respectively, in the study by Coles [2001]).
[19] For the case study, we fitted the GEV to the historic winter monthly streamflow maxima. Typically, extreme values are analyzed on an annual time scale, thus resulting in one maximum (or minimum) value per year, but for data sets of limited length, this discards much of the available data. As such, here we examined the winter monthly maxima (the time period of importance in this application); thus, there were four values per year for 37 years, resulting in a sample size of 148. This relatively large sample size helped to justify the use of the MLE, as this estimation technique can be problematic for small sample sizes [Hosking and Wallis, 1988; Hosking, 1990].
[20] We refer to the above model as the unconditional (Uncond) model, and in section 3.1.2, we describe the modification of the GEV to include covariates (referred to as “conditional”).
3.1.2. Nonstationarity in GEV Model
[21] Traditionally, the GEV distribution assumes that observations are IID, but this assumption can be relaxed, with the introduction of covariates to account for nonstationarity. For instance, the parameters of the GEV distribution could be dependent on a given predictor, x, or more generally, for a suite (i.e., vector) of predictors, X. Theoretically, this could apply to any of the three aforementioned model parameters. Here, because of its intuitive appeal, we examined this dependence only for the location parameter:
where the β values are the intercept and predictor coefficients, which were fitted so as to maximize the likelihood function (equation (2)). We note that now θ = [β, σ, ξ] and that for each time block, μ and the resulting GEV will change with the covariate(s).
[22] The strong linear relationship between precipitation and streamflow for the winter months [Towler et al., 2010] is indicative of a rainfallrunoff mechanism for the streamflow, which provides 90%–95% of Bull Run River's water (PWB, 2007). Hence, R was examined as a covariate, as well as T, which influences evaporation and soil moisture. Furthermore, these covariates were readily available and downscaled from the AOGCMs. This resulted in the testing of four conditional (Cond) model combinations (see Table 1): (1) T as a covariate (Cond_{T} model), (2) R as a covariate (Cond_{R} model), (3) the product of R and T as a covariate (Cond_{RT} model), and (4) both R and T as covariates (Cond_{R}_{+ T}). The significance of the covariates was evaluated using a likelihood ratio test comparing the nested models that is adapted from section 2.6.6 in the study by Coles [2001] and summarized below.
Table 1. Generalized Extreme Value Coefficients and Goodness of Fit Statistics for Models of Winter Monthly Maximum StreamflowVariable  GEV Model 

Uncond β_{0}  Cond_{T}β_{0} + β_{1}T  Cond_{R}β_{0} + β_{1}R  Cond_{RT}β_{0} + β_{1}(RT)  Cond_{R + T}β_{0} + β_{1}R + β_{2}T 


β_{0} (SE)  1924 (120)  1930 (1000)  1739 (410)  611.4 (150)  1911 (880) 
β_{1} (SE)    −0.8914 (27)  61.08 (32)  3.716 (0.36)  141.2 (14) 
β_{2} (SE)          −36.45 (24) 
σ (SE)  1245 (84)  1220 (81)  1246 (160)  923.7 (69)  968.5 (74) 
ξ (SE)  −0.02246 (0.065)  −0.01286 (0.065)  −0.06180 (0.084)  0.07009 (0.082)  0.01619 (0.075) 
llh  −1289  −1289  −1274  −1250  −1250 
K  1  2  2  2  3 
AIC  2580  2582  2552  2504  2506 
M_{0}^{a}    Uncond  Uncond  Uncond  Cond_{R} 
D    0  30  78  48 
Sig^{b}    No (0.635)  Yes (0.000)  Yes (0.000)  Yes (0.000) 
ρ^{c}      0.5516  0.5989  0.5918 
[23] Consider a model, M_{0}, which is a sub (i.e., nested) model of M_{1}, and llh_{0}(M_{0}) and llh_{1}(M_{1}) are the maximized values of the loglikelihood for the models, respectively. The deviance statistic D can be calculated as:
If D > c_{α}, where c_{α} is the (1 − α) quantile of the χ_{k}^{2} distribution, then M_{0} can be rejected in favor of M_{1}. Here, α is the level of significance, the χ_{k}^{2} distribution is a largesample approximation, and k are the degrees of freedom associated with the test. Models were tested at the α = 0.05 significance level against the appropriate submodels. For Cond_{T}, Cond_{R}, and Cond_{RT}, the appropriate submodel was the Uncond model. For Cond_{R}_{+ T}, the submodel that was tested against was Cond_{R}. For each test, the degrees of freedom k = 1, with c_{α} = 3.84.
[24] The “best model” was selected from the conditional models based on minimizing the Akaike Information Criterion (AIC) [Akaike, 1974], which can be calculated as:
where llh was estimated from equation (3) with K being the number of parameters estimated.
[25] The conditional and unconditional GEV models were fitted using the extRemes package (http://cran.rproject.org/web/packages/extRemes/extRemes.pdf) in the statistical package “R” (http://www.rproject.org/). The best fit model was used in conjunction with future climate projections of R and T to estimate the corresponding streamflow quantiles. Furthermore, leaveoneout crossvalidation of the quantile estimates was calculated. This is carried out in the following manner: (1) the first response (z_{1}) and predictors (R_{1}, T_{1}) are removed from the observed data set, (2) the GEV model is fitted to the remaining (N − 1) responses and predictors, (3) the predictors (R_{1}, T_{1}) are used to estimate the first quantile response from the model developed in (2), and (4) the procedure is repeated for each of the remaining paired responses and predictors. Correlation between the observed maxima and crossvalidated conditional flood quantiles were calculated, with four quantiles considered: z_{99}, z_{90}, z_{50}, and z_{10}.