PUBLICATIONS. Water Resources Research. Assessing the performance of the independence method in modeling spatial extreme rainfall

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1 PUBLICATIONS Water Resources Research RESEARCH ARTICLE Key Points: The utility of the independence method is explored in modeling spatial extremes Four spatial models are compared in estimating marginal distributions The independence method is more robust in estimating marginal parameters Correspondence to: F. Zheng, Citation: Zheng, F., E. Thibaud, M. Leonard, and S. Westra (2015), Assessing the performance of the independence method in modeling spatial extreme rainfall, Water Resour. Res., 51, , doi:. Received 6 JAN 2015 Accepted 2 SEP 2015 Accepted article online 7 SEP 2015 Published online 26 SEP 2015 Assessing the performance of the independence method in modeling spatial extreme rainfall Feifei Zheng 1, Emeric Thibaud 2, Michael Leonard 3, and Seth Westra 3 1 College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, Zhejiang, China, 2 Department of Statistics, Colorado State University, Fort Collins, Colorado, USA, 3 School of Civil, Environmental and Mining Engineering, University of Adelaide, Adelaide, South Australia, Australia Abstract Spatial statistical methods are often employed to improve precision when estimating marginal distributions of extreme rainfall. Methods such as max-stable and copula models parameterize the spatial dependence and provide a continuous spatial representation. Alternatively, the independence method can be used to estimate marginal parameters without the need for parameterizing the spatial dependence, and this method has been under-utilized in hydrologic applications. This paper investigates the effectiveness of the independence method for marginal parameter estimation of spatially dependent extremes. Its performance is compared with three spatial dependence models (max-stable Brown-Resnick, max-stable Schlather, and Gaussian copula) by means of a simulation study. The independence method is statistically robust in estimating parameters and their associated confidence intervals for spatial extremes with various underlying dependence structures. The spatial dependence models perform comparably with the independence method when the spatial dependence structure is correctly specified; otherwise they exhibit considerably worse performance. We conclude that the independence method is more appealing for modeling the marginal distributions of spatial extremes (e.g., regional estimation of trends in rainfall extremes) due to its greater robustness and simplicity. The four statistical methods are illustrated using a spatial data set comprising 69 subdaily rainfall series from the Greater Sydney region, Australia. VC American Geophysical Union. All Rights Reserved. 1. Introduction Extreme rainfall-derived floods are widely recognized as one of the most costly and dangerous classes of natural hazard worldwide due to their catastrophic impact on the environment and society [Hallegatte et al., 2013]. For example, floods in 2011 resulted in approximately $70 billion losses globally and more than 6000 fatalities [Centre for Research on the Epidemiology of Disasters, 2014]. To effectively design systems and strategies to mitigate or avoid flooding, it is crucial to accurately quantify the occurrence probabilities and return levels of extreme rainfall events [Westra et al., 2014]. Point-wise methods have often been used to statistically model extreme rainfall, typically using data from a single location; for example, the log Pearson type III distribution or generalized extreme value (GEV) distribution are commonly applied to extreme rainfall data from a single gauging station [Kuczera, 1999; Coles, 2001]. Despite their simplicity, the parameter estimates are often imprecise due to short records of extreme events [e.g., Westra and Sisson, 2011]; this is especially problematic when predicting events beyond the range of the observed data (e.g., the 100 year return period). It is possible to reduce parametric uncertainty by combining data from multiple sites across a spatial domain, using a model that either assumes identical parameter values across the domain, or relates the parameters to spatially varying covariates (R. L. Smith, Regional estimation from spatially dependent data, University of Chapel Hill, unpublished data, 1990a). Although the parameterization of the spatial process can be more approximate than the point-wise methods (e.g., point-wise GEV) because of the additional need to specify a spatial model, it allows more data to be used for model fitting and hence the precision of the estimated parameters is usually improved. However, due to the inherent spatial structure of rainfall events, the observed series from multiple sites are statistically dependent, especially when the spatial domain is small. Ignoring such dependence results in likelihood misspecification, and the resultant confidence intervals associated with estimated parameters typically would be misleading [Padoan et al., 2010]. ZHENG ET AL. SPATIAL RAINFALL MODELING 7744

2 There are a number of statistical methods for modeling spatial extremes, including copulas and max-stable models. The Gaussian copula is popularly used because it provides a full likelihood specification under high-dimensional multivariate settings, although it represents extremes as asymptotically independent [Sang and Gelfand, 2010]. Unlike Gaussian copula methods, max-stable models are able to represent asymptotic dependence [Padoan et al., 2010], making them an appealing alternative for many geophysical problems. However, for this model, it is not possible to specify the full likelihood, necessitating the use of composite likelihoods for parameter estimation [Varin et al., 2011]. Max-stable and copula-based models have been frequently adopted as a means of modeling spatial extreme rainfall over the past few years. For instance, Sang and Gelfand [2010] proposed a hierarchical spatial method to model the daily annual maximum rainfall in a region from South Africa, with spatial dependence represented by a Gaussian copula. Shang et al. [2011] and Westra and Sisson [2011] fitted max-stable process models to maximum daily rainfall time series in California (United States) and east Australia, respectively. Subsequently, Davison et al. [2012] applied max-stable models to rainfall gauges in Switzerland, and Thibaud et al. [2013] developed threshold max-stable models for daily rainfall series from the Val Ferret watershed in the Swiss Alps. However, the use of max-stable and copula models is not without difficulty. Issues include (i) computational challenges and unstable convergence problems caused by the high inference complexity, especially dealing with large and complex spatial data sets [Chandler and Bate, 2007; Thibaud et al., 2013]; and (ii) the difficulty in specifying parametric models to adequately represent the underlying dependence structures for observed spatial extremes [Davison et al., 2012]. In many cases, the primary interest is in the marginal distribution of the spatial extremes rather than the spatial dependence, such as for trend analyses of rainfall extremes [Westra and Sisson, 2011; Seidou et al., 2006]. In this case, it is natural to appeal to simpler approaches for inference in terms of both the computational efficiency and dependence parameterizations. The independence method [Eicker, 1963; White, 1982] is such an approach, where independence is assumed among the spatial extremes, so that the marginal structure can be inferred from the independence likelihood without specifying a dependence model. In the independence method, the standard asymptotic formulae for parameter variances and covariances are adjusted to take into account the spatial dependence [Eicker, 1963], with details given in section 2.4. Although the statistical theory related to the independence method is well-developed, the independence method has received far less attention in the hydrological literature, despite its potential to improve marginal inference [Chandler and Bate, 2007] relative to point-wise approaches. Huber [1967] proved the consistency and asymptotic normality of the independence method, and White [1982] examined the consequences of model misspecification when using this inference approach. In terms of model application in hydrology community, the limited examples include Northrop and Jonathan [2011] who used the independence method to model spatially dependent nonstationary extremes of wave heights induced by a hurricane, Blanchet and Lehning [2010] who implemented this approach to model snow depth return levels in Switzerland, and Van de Vyver [2012] who employed this method to model rainfall extremes in Belgium. In addition to modeling spatial extremes, the independence method has been employed to deal with temporally clustered extremes [Fawcett and Walshaw, 2006, 2007]. This paper investigates the effectiveness of the independence method for estimating marginal parameters of spatial extremes, and compares its performance with the Brown-Resnick, Schlather and Gaussian copula models by means of simulation studies (section 3). These four statistical techniques are illustrated using a spatial data set comprising 69 subdaily rainfall series from the Greater Sydney region, Australia (section 4). We next provide an overview of univariate extreme value theory, max-stable models, the Gaussian copula, and the independence method. 2. Spatial Extreme Modeling Approaches 2.1. Univariate Extreme Value Theory Univariate extreme value theory describes the statistical behavior of the random variable M n 5max fx 1 ; ::::::X n g for a sequence of independent and identically distributed (IID) random variables X 1 ; ::::::; X n. If the limiting ZHENG ET AL. SPATIAL RAINFALL MODELING 7745

3 distribution (n!1) of the appropriately linearly rescaled maxima M n (denoted as F) exists and is nondegenerate, then F must be max-stable [Fisher and Tippett, 1928], i.e., F n ða n y1b n Þ5FðyÞ; (1) for all n > 1 and for some constants a n > 0 and b n 2 R. The only distribution F that satisfies equation (1) is the generalized extreme-value, GEV (l; r; nþ, distribution [Jenkinson, 1955] 8 h exp 2 11n y2l i 21=n >< ; n 6¼ 0 r FðyÞ5 n h exp 2exp 2 y2l io ; (2) >: ; n50 r defined on the set {y: 11nðy2lÞ=rÞ > 0}, where l 2 R is the location parameter, r > 0 is the scale parameter, and n is the shape parameter. The value of n determines the type of the tail behavior, with n < 0; n50, and n > 0 corresponding to the Weibull, Gumbel, and Frechet distributions, respectively [Coles, 2001]. The GEV distribution is temporally stationary, although it can be easily modified to incorporate nonstationarity. For example, variations through time can be modeled as a linear trend in the location parameter by lðtþ5l 0 1bt; t50; ::::; ðn21þ where b corresponds to the rate of change over time [Westra et al., 2013]. The (non)stationary GEV distribution has been widely employed to model rainfall extremes, and in this paper its parameters are estimated using the maximum likelihood method [Coles, 2001; Zheng et al., 2013]. It is noted that the standard properties of the maximum likelihood estimator are only valid for n > 21/2 [Coles, 2001] Max-Stable Process Models Max-stable processes [de Haan, 1984] extend univariate max-stable distributions that satisfy equation (1) to the spatial context for its main application. By analogy with univariate extreme value theory (equation (1)), a max-stable process ZðÞ describes the limiting process of maxima from IID random fields Y i ðsþ 2R. That is, for suitable sequences of functions a n ðsþ > 0 and b n ðsþ 2R, max n i51 ZðsÞ5 lim Y iðsþ2b n ðsþ ; s 2 R d ; (3) n!11 a n ðsþ provided that the limits exist. The physical interpretation of extending the univariate max-stable model to the spatial context is that the marginal distribution at each point s follows a univariate GEV distribution, while the marginal distributions for multiple points are jointly modeled through continuous spatial functions of the GEV parameters: lðsþ, rðsþ, and nðsþ. The theory of max-stable processes is based on standard Frechet margins, but applies more generally as any GEV-distributed variable can be transformed to a standard Frechet distribution. The Smith model (R. L. Smith, Max-stable processes and spatial extremes, unpublished manuscript, 1990b) represents the first attempt to characterize spatial extremes, but realizations from this model are too smooth to be useful for many environmental variables such as rainfall [Davison et al., 2012; Thibaud et al., 2013]. More complex max-stable models were proposed by Schlather [2002] and Kabluchko et al. [2009]. These two models are constructed from Gaussian processes parameterized by correlation functions and variograms [see Davison et al., 2012 for details]. In this paper, we focus on the Schlather model [Schlather, 2002] with the powered exponential correlation function qðhþ5exp ½2ðh=kÞ t Š; k > 0; 0 < t 2 (4) and the Brown-Resnick [Kabluchko et al., 2009] model with variogram a 2 ðhþ52ðh=kþ t ; k > 0; 0 < t 2: (5) where k and t control the range and the smoothness of the processes, respectively. The full likelihood of max-stable models is often analytically unknown or computationally prohibitive to calculate; in its place the model parameters are typically estimated using a pairwise likelihood method [Padoan et al., 2010]. Given a spatial data set with n independent replicates from a max-stable process ðy1 k; :::; yk D Þ k51;:::;n observed at D sites, the pairwise log likelihood is ZHENG ET AL. SPATIAL RAINFALL MODELING 7746

4 ð/þ5 Xn X log fðyi k ; yk j ; /Þ; i; j 2 D; (6) k51 i<j where /5½h; aš, with h and a5½k; tš representing marginal and dependence parameters, respectively, and f is a probability density function of the bivariate max-stable process, representing the likelihood contribution of two different observations from the same replicate (e.g., the rainfall observations from two different sites at the same year). Under the same regularity conditions as for the asymptotic normality of the standard maximum likelihood, the maximum pairwise likelihood estimator ^h has a limiting multivariate normal distribution with ^h Nðh; H 21 VH 21 Þ; (7) when n!1 (the covariance H 21 VH 21 is often referred to as the sandwich matrix). H is the negative expected value of the second derivative of the log-likelihood, 2E½r 2 ðhþš, which can be estimated by the Hessian of the log-likelihood at its maximum, and V is the variance of the first derivative of the loglikelihood, obtained by the empirical variance of the score contribution from each observation [Varin et al., 2011]. Model selection can be guided by minimizing the Takeuchi information criterion TIC 522 ð^hþ12trðvh 21 Þ, an equivalent of the Akaike information criterion for misspecified models [Varin and Vidoni, 2005] Gaussian Copula The Gaussian copula model is equivalent to fitting a Gaussian process to data for which the marginal distributions have been transformed to standard Gaussian distributions. Here we consider GEV marginal distributions, while the dependence parameters are those of the Cauchy correlation function [Davison et al., 2012] qðhþ5½11ðh=kþ 2 Š 2t ; k > 0; t > 0: (8) The full likelihood method was used to estimate parameters for the Gaussian copula model Independence Method Marginal parameters h are estimated first by maximizing the independence log likelihood ðhþ5 Xn k51 X log ~ f ðyi k ; hþ (9) i where ~ f is a probability density function of the marginal distribution. The independence likelihood ignores the spatial dependence among the sites, although in practice the extremes within the hydrological processes are spatially dependent. This results in model (likelihood) misspecification and underestimation of the confidence intervals associated with the estimated parameters. To solve this problem, the robust covariance matrix H 21 VH 21 given in equation (7) is used to adjust the standard error estimates and account for the spatial dependence [Huber, 1967]. When the data are spatially independent, H 21 5V, and equation (7) becomes ^h Nðh; H 21 Þ, which is the covariance matrix used in the standard maximum likelihood method. Analogous to the max-stable models, the TIC can be used for model selection. The pairwise and independence likelihoods are both examples of composite likelihoods, and they both use the sandwich estimator (equation (7)) to estimate the standard errors [Varin et al., 2011]. However, the pairwise likelihood of a max-stable process characterizes the spatial dependence through a parametric model, whereas the independence likelihood implicitly allows for the spatial relationship by adjusting the standard error estimates. Therefore, when the goal of the analysis is inference about the marginal parameters (i.e., the dependence structure is of no interest), the independence method is a compelling alternative to parametric spatial dependence models due to its simplicity and computational efficiency. 3. Simulation Studies 3.1. Outline of the Simulation Experiment A synthetic experiment was designed to compare the performance of five statistical models when dealing with spatial extremes with different spatial dependence structures. Synthetic data sets were generated ZHENG ET AL. SPATIAL RAINFALL MODELING 7747

5 Table 1. Parameter Values of the Models Used for Generating Synthetic Data Sets Generating Model Max-stable Brown-Resnick Max-stable Schlather Gaussian copula Dependence Structure Parameter Values of the Dependence Model Brown-Resnick k50:5; t50:2 (weak dependence) Powered exponential k50:5; t50:5 (moderate dependence) Cauchy k51; t51 (strong dependence) Data Length (n) and Spatial Domain n530, 40, sites with latitude (lat) and longitude (lon) randomly generated within a spatial domain of R 2 5[0, 1] 2 Parametric Models of GEV Margins lðs; tþ5a 0 1a 1 3lat1 a 2 3lon1bt; t50; ::::; ðn21þ rðsþ5b 0 1b 1 3lat1b 2 3lon nðsþ5n using three spatial dependence models: the max-stable Brown-Resnick and Schlather models, and the Gaussian copula. The following five models were then fitted to each synthetic data set: the independence method (M1), the Brown-Resnick model (M2), the Schlather model (M3), the Gaussian copula (M4), and the standard maximum likelihood approach ignoring spatial dependence (M5). The independence and standard maximum likelihood methods are identical in parameter estimates except for the adjustment of the standard errors for the independence method. Parameters for all models were estimated using the R package SpatialExtremes [Ribatet, 2013], and their performance was compared in terms of estimates of marginal parameters and their associated confidence intervals. Table 1 outlines the parameter values used for the three generating models. For each generating model, three different parameter sets were used in the dependence structure, representing different spatial dependence strength [Ribatet, 2013]. We used 50 sites randomly located within a spatial domain of R 2 5[0,1] 2 for all simulations, representing latitude (lat) and longitude (lon) for each site. Three replicates with 30, 40, and 50 respective data points ( years ) were considered for each site (analogous to time series of annual maximum rainfall). The simulated data at each site followed a GEV distribution with spatial covariates for the location and scale parameters (l and r). Furthermore, a trend parameter was considered for l. The shape parameter n was considered to be constant over time and space when modeling spatial rainfall extremes [Westra and Sisson, 2011]. A total of eight parameters were estimated, i.e., h 5 [a 0, a 1, a 2, b 1, b 2, b 3, n, b] as shown in Table 1. For each parameter set, 1000 spatial data sets were simulated to enable statistically rigorous analysis. To provide realistic values of h for generating synthetic data sets, the case study rainfall data set in section 4 was fitted using the independence method, with spatial covariates given in Table 1. The resultant estimates were a 0 555, a 1 587, a , b 1 525, b 2 553, b , n50:2, and b We noted that different choices of parameters in the generating model to those outlined above would be expected to yield similar relative performances of the statistical methods. In addition to n50:2, n520:2 and 0 were also considered to assess the effect of different types of GEV distributions. Given that eight parameters are estimated, it is not straightforward to compare the model performance according to the parameter estimates. To explore the aggregate effect of the eight parameter estimates, we also compared model performance in terms of the corresponding 100 year return levels. Such a comparison is meaningful as predicting magnitudes of rare events is typically the aim of the model application in practice Results of Parameter Estimates The prediction accuracy for rare events is sensitive to estimates of the shape parameter n [Coles, 2001]. Thus, Figure 1 presents results of this parameter (^n-n) from each method when moderate dependence strength (k50:5; t50:5) was used in the synthetic data sets. This figure shows that the independence method (M1) consistently produces approximately unbiased estimates for the synthetic data from the three different generating models. The three explicit models (M2, M3, and M4) yielded comparable performance with the independence method when the model dependence structure was correctly specified (i.e., the fitting model was the same as the generating model), otherwise they exhibited significantly worse performance. For example, when the generating model was the Gaussian copula (M4), both max-stable models (M2 and M3) produced biases in n. To compare the estimation efficiency, the root mean square error (RMSE) was calculated, accounting for both the bias and variance of the estimates from each method [Zheng et al., 2014] ZHENG ET AL. SPATIAL RAINFALL MODELING 7748

6 Figure 1. Errors of the shape parameter estimates. M1: the independence method, M2: Brown-Resnick model, M3: Schlather model, M4: the Gaussian copula. n50:2, k50:5; t50:5, and data length n5 50 were used in the generating models. The orange dotted line represents ^n-n50. RMSE5fvar s ð^hþ1½e s ð^hþ2hš 2 g 1=2 ; (10) where var s ð^hþ is the sample variance of ^h, E s ð^hþ is the sample mean of the estimates and ½E s ð^hþ2hš 2 is the square of the bias. The RMSE value was calculated for each parameter, with results of ^n showing in Figure 2. In terms of the RMSE of ^n, overall the independence method performed similarly to the three explicit models despite not having specified a dependence model. Although the independence method had less biased estimates than the spatial dependence models when the dependence structure was misspecified (Figure 1), it produced a slightly larger variance. The same observation was also made for the other seven parameters (not shown). Interestingly, the Gaussian copula model yielded low RMSE values when it matched the generating model (Figure 2, right). This is because the full likelihood was used in the Gaussian copula model, resulting in a reduced estimation variance compared to the composite likelihoods used to fit the independence and the two max-stable models. For each statistical method, the estimates of the eight parameters were used to predict the 100 year return levels at the centre (0.5, 0.5) of the spatial domain, with results shown in Figure 3. Since the generating models incorporated a temporal trend, the 100 year prediction was based on the 10th year of the period (n 5 10 years) in order to account for the trend parameter estimates. Similar to Figure 1, the estimates produced by the independence method were consistently unbiased, while explicit models performed the best when the dependence structure was correctly specified. For instance, the median values of 100 year return level estimates from the two max-stable models were approximately 465, which is 27.4% above the true value (365.3) when the Gaussian copula was used to generate the synthetic data sets (Figure 3, right). Figure 2. RMSE values of ^n (the true n50:2). M1, M2, M3, M4, k, t, and n are shown in the caption of Figure 1. ZHENG ET AL. SPATIAL RAINFALL MODELING 7749

7 Figure 3. Estimates of 100 year return levels, with red dotted line representing the true value of M1, M2, M3, M4, n, k, t, and n are shown in the caption of Figure 1. The RMSE values of the estimated 100 year return levels are shown in Figure 4. As before, the independence method performed similarly to the more complex explicit models, and better when the models were misspecified. This suggests that, relative to the explicit approaches, the independence method is more reliable and robust for estimating the 100 year return levels. The bias and standard deviation of the 100 year return level estimates from each method are outlined in Table 2, showing the superior performance of the independence method than the two max-stable models and the Gaussian copula Comparison of Estimated Confidence Intervals We further compared the estimates of confidence intervals from different methods as they are often of great interest in statistical modeling, and the direct comparison of the standard error estimates is another alternative for assessing model performance [Fawcett and Walshaw, 2007]. Confidence intervals of the parameters were computed based on a normal approximation to the distribution of the sandwich estimator (equation (7)); the 95% confidence intervals of a parameter are 61:96SE where SE is the standard error calculated from equation (7). This method is used because of its simplicity, although other alternatives are also available that can account for asymmetrical uncertainty estimates. These are the profile likelihood method [Coles, 2001] and adjusted log-likelihood approaches [Chandler and Bate, 2007]. We calculated the actual coverage probability of the 95% confidence interval for each true parameter value; namely, the proportion of the time that the true parameter lies within the estimated 95% interval over the 1000 simulations (R. L. Smith, Regional estimation from spatially dependent data, University of Chapel Hill, unpublished data, 1990a). If the actual coverage probability is less than or greater than the nominal coverage probability (95%), this implies that the interval from the estimated standard error is too small or too Figure 4. RMSE values of the 100 year return level estimates (true value is 365.3). M1, M2, M3, M4, n, k, t, and n are shown in the caption of Figure 1. ZHENG ET AL. SPATIAL RAINFALL MODELING 7750

8 Table 2. Bias and Standard Deviation (SD) of the 100 Year Return Level Estimates From Each Fitting Model a Generating Models The Brown-Resnick model Parameter Values of Dependence Model and the Data Length b Independence Method Four Fitting Models Brown-Resnick Model Schlather Model Gaussian Copula Bias SD Bias SD Bias SD Bias SD P P P P P P P The Schlather model P P P P P P P The Gaussian copula P P P P P P P a The true values for n520:2; 0; 0:2 are 192.8, 253.3, and respectively. b P1 : n520:2; k50:5; t50:5; n550, P2 : n50; k50:5; t50:5; n550, P3 : n50:2; k50:5; t50:5; n550, P4 : n50:2; k50:5; t50:5; n540, P5 : n50:2; k50:5; t50:5; n530, P6 : n50:2; k50:5; t50:2; n550, P7 : n50:2; k51; t51; n550. large, respectively [Wang, 2008]. Therefore, a smaller discrepancy between the actual and nominal coverage probability indicates a more reliable estimate in confidence interval. Figure 5 shows the means of the actual coverage probabilities of the eight estimated parameters obtained from the independence method (M1), the three spatial dependence models (M2, M3, and M4), and the standard maximum likelihood method (M5) that ignores spatial dependence. Not surprisingly, M5 has substantially lower coverage probability for all cases considered, emphasising the importance of accounting for spatial dependence in correctly determining the precision of the estimated marginal parameters. The coverage probabilities estimated by the independence method are consistently close to the nominal value regardless of the underlying dependence structure among spatial extremes. This agrees with other studies [Kauermann and Carroll 2001] in that the independence method generally slightly overestimates the Figure 5. Means of the coverage probabilities (95% confidence intervals) of the eight estimated parameters. M1, M2, M3, M4, n, k, t, and n are shown in the caption of Figure 1. The reddotted lines represent the nominal coverage probability of 95%. ZHENG ET AL. SPATIAL RAINFALL MODELING 7751

9 Figure 6. Means of estimated 100 year return levels. The red-dotted lines represent the true values. The red and green solid lines (M1, M3) overlap with the red-dotted line in the middle plot. variances (Figures 2 and 4, Table 2), but leads to coverage of the 95% confidence intervals near the nominal level (Figure 5). The three spatial dependence models exhibited similar performance with the independence method when the dependence structure was correctly specified; however, when the dependence model was misspecified, they all produced poorer performance than the independence method Impacts of Different Shape Parameters, Data Length, and Spatial Dependence Results presented in sections 3.2 and 3.3 were based on synthetic data sets with n50:2, k50:5; t50:5 (moderate dependence strength), and data length n5 50. We also considered the performance of the five methods in modeling spatial extremes with n520:2 and 0, different spatial dependence strength and varying data length (n 5 30 and 40) in the simulation study as outlined in Table 1. For different values of n and data length, the relative performance (bias, variance, and coverage probability) among the four methods was similar to that based on n50:2 and n 5 50, although the variance became larger for greater values of n as shown in Table 2. The influence of the spatial dependence on the model performance is illustrated in Figure 6. This figure shows that dependence strength does not influence the performance of the independence method in terms of return level estimates. This is also the case for the spatial dependence models when the dependence model is correctly specified, but there is some influence when the dependence model is misspecified. The Gaussian copula produced larger biases (underestimation) for the 100 year return level estimates when spatial dependence was stronger in the synthetic data generated from the two max-stable models (see also Table 2). In contrast, when the synthetic data was simulated from the Gaussian copula, the two max-stable models exhibited better performance for moderate and strong dependence, although they consistently overestimated the return levels Observations From the Simulation Study Based on the results of the simulation study, the following observations can be made: 1. The standard maximum likelihood method significantly underestimated the standard errors, resulting in overconfidence in parameter estimates. This highlights the need to take spatial dependence into account when estimating the marginal parameters for spatially dependent data. 2. The independence method showed good performance in estimating marginal parameters, return levels and confidence intervals due to its reliable performance in modeling spatial extremes with various underlying dependence structures. Furthermore, unlike the spatial dependence methods, the independence method did not need a parametric model for the dependence structure to be specified, leading to greater simplicity and robustness in practice. 3. The spatial dependence models (max-stable and Gaussian models) produced comparable performance to the independence method in terms of parameter and confidence interval estimates when the spatial dependence structure was correctly specified. For the misspecified dependence structure, the independence method outperformed the spatial dependence models in all cases. This implies that the ZHENG ET AL. SPATIAL RAINFALL MODELING 7752

10 Figure 7. Spatial rainfall data sets from the Greater Sydney region (black solid dots in the left plot) and the data availability for each year of record for each rainfall site (right). performance of the spatial dependence models depends on whether the underlying dependence structure is adequately specified, posing a challenge in practical applications as it is difficult to correctly specify a dependence structure for a realistic spatial data set. We conclude that the independence method is more appealing than the max-stable models for modeling the marginal distributions of spatially dependent extremes due to its relative ease of implementation, greater reliability, and greater robustness. 4. Case Study The independence method and the three explicit models were illustrated using a set of 69 rainfall gauges covering a 15,000 km 2 area in the vicinity of Sydney, Australia (Figure 7, left). Most stations have records of approximately 30 years over the period from 1981 to 2010 (Figure 7, right). The rainfall data are at a resolution of 5 min, and were aggregated to 24 h annual maximum series for analysis. A preliminary analysis was conducted on the case study data set using a nonstationary GEV model and the autocorrelogram. We found that the shape parameter ^n for each site was greater than 20.5 and there was no significant temporal dependence within the time series. Following Westra and Sisson [2011], the location parameter (l) was allowed to vary temporally and spatially while the scale parameter (r) was allowed to vary only spatially. The shape (n) and trend (b) parameters were assumed to be constant over the study domain. A nonstationary GEV model, i.e., lðtþ5l 0 1bt; t50; ::::; ðn21þ, was initially fit separately to each gauge, with the spatial distributions of ^l 0 and ^r given in Figure 8 (top). Based on a visual inspection, the spatial distributions of ^l 0 and ^r appear to be similar, so the same spatial covariates were used for both parameters. The latitude, longitude, and elevation values of the 69 rainfall sites were used as spatial covariates, and were each rescaled to the range [0,1] to enable a reliable convergence. Ten spatial models were considered, using different combinations of the latitude, longitude, and elevation. While other studies have allowed for a wider variety of covariates [Chandler, 2005], the covariates were selected based on previous studies of this region [Westra and Sisson, 2011]. The performance of the models was evaluated using the TIC (Table 3) and the outcome was that Model 5 yielded the best overall performance. This model was therefore used for remaining analyses. A comparison of the independence method with Model 5 to the point-wise nonstationary GEV model (Figure 8, bottom) in terms of parameter estimates shows reasonable consistency and supports the selection of this model. This is also confirmed by the reasonable fit verified by the quantile-quantile plot (QQ plot) for each separate site (not shown). In addition to modeling the rainfall extremes from gauged locations, model M5 also provides statistics at ungauged locations for the study region. ZHENG ET AL. SPATIAL RAINFALL MODELING 7753

11 Figure 8. The values of ^l 0 and ^r from the point-wise nonstationary GEV model (top) versus those from the independence method (bottom) for the spatial data set. Circles with larger sizes correspond to larger values in the top plot. The 95% confidence intervals of the trend parameter ^b and the shape parameter ^n from the independence method are presented in Figure 9 (dashed lines). The estimates from the point-wise method are also shown in grey with 95% confidence intervals. It is clear that there is considerably more uncertainty around estimates from the point-wise approach compared to the remaining models (the results of the three spatial dependence models are not shown), indicating the benefit of spatial modeling. The relationship between estimates and their longitude is shown in Figure 9, and similar observations were made when using the latitude or elevation. The parameter estimates and their associated standard errors for b and n from the four spatial approaches are presented in Table 4. The four models all produced positive estimates for the shape parameter n, suggesting that the 24 h annual extremes in the Greater Sydney region have a heavy-tailed distribution. In terms of the trend in annual extreme 24 h rainfall, the independence method produced a negative estimate, but this was not statistically significant, which is consistent with the observations in Zheng et al. [2015]. To investigate the impact of the trend parameter b on the estimates of other marginal parameters (l; r; n), the four models were applied to the case study data set again without the inclusion of b. Results showed that the variation of marginal parameter estimates was negligible. Even so, we still included ^b as it can assist in evaluating the fitting quality of the independence method applied to the case study data set as displayed in Figure 9 (left). The simulation study (section 3) demonstrated that the spatial dependence models (max-table and Gaussian copula models) only can produce comparable performance when the dependence model is ZHENG ET AL. SPATIAL RAINFALL MODELING 7754

12 Table 3. TIC Values of Different Spatial Covariates Used in the Independence Method for the Location and Scale Parameters Index Covariates TIC 1 lðs; tþ5a 0 1a 1 ðlonþ1a 2 ðlatþ1bt, rðsþ5b 0 1b 1 ðlonþ1b 2 ðlatþ 2 lðs; tþ5a 0 1a 1 ðlonþ1a 2 ðlatþ1a 3 ðelevationþ1bt, rðsþ5b 0 1b 1 ðlonþ1b 2 ðlatþ1b 3 ðelevationþ 3 lðs; tþ5a 0 1a 1 ðlonþ1a 2 ðlon 2 Þ1a 3 ðlatþ1a 4 ðelevationþ1bt, rðsþ5b 0 1b 1 ðlonþ1b 2 ðlon 2 Þ b 3 ðlatþ1b 4 ðelevationþ 4 lðs; tþ5a 0 1a 1 ðlonþ1a 2 ðlon 2 Þ1a 3 ðlatþ1a 4 ðlat 2 Þ1a 5 ðelevationþ1bt;rðsþ5b 0 1b 1 ðlonþ1b 2 ðlon 2 Þ1b 3 ðlatþ b 4 ðlat 2 Þ1b 5 ðelevationþ 5 lðs; tþ5a 0 1a 1 ðlonþ1a 2 ðlon 2 Þ1a 3 ðlatþ1a 4 ðlat 2 Þ1a 5 ðelevationþ1a 6 ðelevation 2 Þ1bt;rðsÞ5b 0 1b 1 ðlonþ b 2 ðlon 2 Þ1b 3 ðlatþ1b 4 ðlat 2 Þ1b 5 ðelevationþ1b 6 ðelevation 2 Þ 6 lðs; tþ5a 0 1a 1 ðlonþ1a 2 ðlon 2 Þ1a 3 ðlon 3 Þ1a 4 ðlatþ1a 5 ðlat 2 Þ1a 6 ðelevationþ1a 7 ðelevation 2 Þ bt;rðsÞ5b 0 1b 1 ðlonþ1b 2 ðlon 2 Þ1b 3 ðlon 3 Þ1b 4 ðlatþ1b 5 ðlat 2 Þ1b 6 ðelevationþ1b 7 ðelevation 2 Þ 7 lðs; tþ5a 0 1a 1 ðlonþ1a 2 ðlon 2 Þ1a 3 ðlatþ1a 4 ðlat 2 Þ1a 5 ðlat 3 Þ1a 6 ðelevationþ1a 7 ðelevation 2 Þ1bt;rðsÞ5b b 1 ðlonþ1b 2 ðlon 2 Þ1b 3 ðlatþ1b 4 ðlat 2 Þ1b 5 ðlat 3 Þ1b 6 ðelevationþ1b 7 ðelevation 2 Þ 8 lðs; tþ5a 0 1a 1 ðlonþ1a 2 ðlon 0:5 Þ1a 3 ðlatþ1a 4 ðlat 0:5 Þ1a 5 ðelevationþ1a 6 ðelevation 0:5 Þ1bt;rðsÞ5b 0 1b 1 ðlonþ b 2 ðlon 0:5 Þ1b 3 ðlatþ1b 4 ðlat 0:5 Þ1b 5 ðelevationþ1b 6 ðelevation 0:5 Þ 9 lðs; tþ5a 0 1a 1 ðlonþ1a 2 ðlon 0:5 Þ1a 3 ðlatþ1a 4 ðlat 2 Þ1a 5 ðelevationþ1a 6 ðelevation 2 Þ1bt;rðsÞ5b 0 1b 1 ðlonþ b 2 ðlon 0:5 Þ1b 3 ðlatþ1b 4 ðlat 2 Þ1b 5 ðelevationþ1b 6 ðelevation 2 Þ 10 lðs; tþ5a 0 1a 1 ðlonþ1a 2 ðlon 2 Þ1a 3 ðlatþ1a 4 ðlat 0:5 Þ1a 5 ðelevationþ1a 6 ðelevation 0:5 Þ1bt;rðsÞ5b 0 1b 1 ðlonþ 1b 2 ðlon 2 Þ1b 3 ðlatþ1b 4 ðlat 0:5 Þ1b 5 ðelevationþ1b 6 ðelevation 0:5 Þ correctly specified in terms of estimating marginal parameters and standard errors. This suggests that whether the spatial dependence models can adequately characterize the spatial dependence structure would heavily affect their final performance. We therefore investigated how well the spatial dependence models can represent the overall spatial dependence of the case study. Figure 10 shows the empirical pairwise extremal coefficients (1 < g 2) for the 24 h annual maxima data sets (grey dots), with g51 and 2, respectively, representing complete dependence and independence between two sites [Schlather and Tawn, 2003]. Significant variation in the pairwise dependence is clear: locations of similar distance show large differences in their dependence strength. For example, for the locations with spatial distances of approximately 50 km, the value of g ranges from 1.2 (strong dependence) to 2.0 (complete independence). This indicates that the dependence structure in spatial rainfall extremes is noisy and complex. The g values from the fitted max-stable Brown-Resnick (dotted line) and Schlather (solid line) models are shown in Figure 10. g is not a useful measure for asymptotically independent processes, and hence the result of the Gaussian Copula is not shown. The Schlather model has a worse fit than the Brown-Resnick model for larger distances due to its upper limit on the extremal coefficient, i.e., g 1:707 [Huser and Figure 9. The estimates for the spatial data set from the Greater Sydney region. The grey dots and lines are, respectively, nonstationary GEV estimates and 95% confidence intervals. The red solid lines are estimates from the independence method (dotted lines are 95% confidence intervals). ZHENG ET AL. SPATIAL RAINFALL MODELING 7755

13 Table 4. Parameter Estimates and Standard Errors (in Parentheses) for the Four Methods Parameter The Independence Method The Brown2Resnick Model The Schlather Model The Gaussian Copula ^b(mm/yr) (0.18) 0.03 (0.16) (0.17) 0.03 (0.16) ^n 0.13 (0.05) 0.15 (0.04) 0.16 (0.04) 0.14 (0.05) Davison, 2014]. This differs to the observations made in Thibaud et al. [2013] that the Schlather model showed a satisfactory performance in modeling spatial rainfall extremes, which may be attributable to differences in domain size (20.4 km 2 in Thibaud et al. [2013] compared to 15,000 km 2 in this study). The scatter plot in Figure 10 suggests significant challenges in specifying a parametric model that adequately represents the structure in the real data. Furthermore, the dependence structure could be spatially nonstationary (R. Huser, and M. G. Genton, Non-stationary dependence structure for spatial extremes, arxiv: v1 [stat.me], submitted manuscript, 2014), adding further difficulty in its parametric specification. The independence method is an attractive alternative for studies that focus exclusively on marginal distributions. However, spatial dependence models have to be adopted when the spatial dependence structure itself is the primary interest. 5. Summary and Conclusions Estimating rainfall quantiles with low exceedance probabilities with short and sparse records of extreme rainfall events is an ongoing challenge. Many analyses of extreme rainfall focus on estimating distributional parameters at individual locations, but this can lead to large uncertainties due to short record lengths. Spatial extreme value models are therefore becoming an increasingly popular tool to increase estimation precision, by pooling data from multiple gauges and thus borrow strength from neighboring gauges. This has motivated the development of numerous statistical techniques for modeling spatially referenced rainfall extremes over the past few years. Within the hydrological literature on spatial rainfall extremes, there are a large class of problems concerned more with marginal parameter values rather than the spatial dependence itself, such as trend detection or regional parameter estimation. Parametric models such as max-stable models and copulas have been often used to characterize the spatial dependence for estimating marginal parameters, while relatively few studies have been undertaken to implement the independence method that models the spatial dependence by adjusting standard error estimates. Figure 10. The empirical pairwise extremal dependence coefficients (grey dots) for the case study data set. The solid and dotted lines, respectively, represent the extremal dependence coefficients calculated from the max-stable Schlather and Brown-Resnick models. To demonstrate the effectiveness of the independence method, we compared it to two max-stable models (the Brown-Resnick and Schlather models) and the Gaussian copula by means of a simulation study. Our study demonstrated that the independence method was robust in accounting for the spatial dependence, regardless of the underlying dependence structure and the dependence strength. It consistently produced low RMSE values of the parameter estimates and return level estimates, as well as reliable estimates of the confidence intervals. When the dependence structure was correctly specified, the max-stable ZHENG ET AL. SPATIAL RAINFALL MODELING 7756

14 models and the Gaussian copula behaved similarly to the independence method, but when it was misspecified, they showed significantly poorer performance. Furthermore, the independence method is computationally efficient, whereas the pairwise likelihood method used in the max-stable models and the full likelihood used in the Gaussian copula can be computationally demanding especially when a large number of sites are considered [Padoan et al., 2010; Huser and Davison, 2014]. The independence method is therefore an appealing alternative to max-stable models and the Gaussian copula when estimating marginal parameters of spatial extremes, as with studies of regional trends in extreme rainfall [Zheng et al., 2015]. Applications of two max-stable models, the Gaussian copula and the independence method were applied to a real data set with 69 subdaily rainfall sites from the Greater Sydney region, Australia. The results showed that the independence method significantly improved the precision of the marginal parameter estimates for the rainfall extremes compared to the point-wise method. This ultimately leads to improved confidence in identifying trend parameters and predicting quantiles for rarer rainfall extremes. Acknowledgments This project was made possible by funding from the Federal Government through Geoscience Australia as part of the revision of Australian Rainfall and Runoff by Engineers Australia. This paper and the associated project are the result of a significant amount of in kind hours provided by Engineers Australia members. Dr Westra s time was funded in part by Australian Research Council Discovery Project DP We thank Professor Chandler for sharing his insight regarding the independence method. For details of the rainfall data set used in this paper, please contact feifeizheng@zju.edu.cn. References Bates, B. C., R. E. Chandler, S. P. Charles, and E. P. 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