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...et al., 2012; Shaby, 2012). In contrast, the HKEVP model (1) and (2) does not suffer this deficiency. It enables Bayesian formulation and attendant Markov chain Monte Carlo (MCMC) sampling (see, e.g. =-=Robert and Casella, 2004-=-) by relying on a random effects representation and considering a large but finite number of Gaussian densities. 2.1 Rounding out the model Returning now from the generic case to the task of modeling ...

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...µ+ σ ξ [ϑ(s)ξ − 1] σ∗(s) = ασϑ(s)ξ ξ∗(s) = αξ Al |α iid ∼ PS(α) for l = 1, . . . , L (2) where ϑ(s) is defined as in (1) and GEV(µ, σ, ξ) denotes the generalized extreme value distribution (see, e.g. =-=Coles, 2001-=-, page 47) with location parameter µ, scale parameter α, and shape parameter ξ. 7 Figure 1: Realization of isotropic hierarchical Gaussian extreme value processes with three different values of α. The...

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...ical model is the quantity ϑ(s) = ( L∑ l=1 Alϕw,l(s |v,Σ) 1/α )α , (1) where Al ∼ PS(α) independently for l = 1, . . . , L, and PS(α) denotes a positive stable distribution with parameter α ∈ (0, 1) (=-=Samorodnitsky and Taqqu, 1994-=-). The construction (1) is similar to that of Stephenson (2009) and Fougères et al. (2009), and induces spatial dependence through the (scaled) kernel functions ϕw,l, l = 1, . . . , L. The hierarchic...

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...ing linear trend with temperature for all of the six most important worldwide food crops. This study regressed global yields on maximum and minimum temperatures extracted from a gridded data product (=-=Mitchell and Jones, 2005-=-). They find that although yields have increased, the increase has been slowed by rising temperature. We note that the interpolation scheme that produced the gridded product used in this study (Mitche...

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...du/landcover/ This 1-km resolution global landcover classification map was produced by passing retrievals from the MODIS instrument aboard the Terra satellite through a classification tree algorithm (=-=Friedl et al., 2002-=-). Obviously land use changes throughout the years. For our purposes, focusing attention on locations that are currently used for agriculture seems most natural, since our primary interest is in risk ...

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...evidence of extremal dependence, revealed no obvious form for the time evolution of γ, but suggested a slight increase in time. Figure 4 shows empirical F-madogram estimates of extremal coefficients (=-=Smith, 1990-=-) for 30 years of data around 4 reference years in the study period. The curves are clearly below 2 at moderate spatial lags indicating the presence of extremal dependence. In addition, it appears tha...

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... are currently used for agriculture seems most natural, since our primary interest is in risk to agriculture as it is currently practiced. Elevation data is from the Shuttle Radar Topography Mission (=-=Farr et al., 2007-=-) dataset SRTM30v2.1, downloaded from USGS at http://dds.cr.usgs.gov/srtm/. This is a 30 arc-second resolution data product derived from instruments flown aboard the space shuttle Endeavour in 1994. T...

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...tributions are not known for more than a trivial number of locations. This precludes maximum likelihood and Bayesian estimation, although methods based on composite likelihoods have had some success (=-=Padoan et al., 2010-=-; Cooley et al., 2012; Shaby, 2012). In contrast, the HKEVP model (1) and (2) does not suffer this deficiency. It enables Bayesian formulation and attendant Markov chain Monte Carlo (MCMC) sampling (s...

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...ce against keeping σ and ξ constant in time (see Figures S-2 and S-3 in the online supplement), but the plots in Figure 3 hinted at a possible linear time trend for µ. Similarly, F-madogram analysis (=-=Cooley et al., 2006-=-), which showed clear evidence of extremal dependence, revealed no obvious form for the time evolution of γ, but suggested a slight increase in time. Figure 4 shows empirical F-madogram estimates of e...

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...space shuttle Endeavour in 1994. The prediction grid was generated by transforming the vertices of a square grid into geographical longitude/latitude coordinates using the inverted Hammer projection (=-=Snyder, 1987-=-). Since this is an equal area projection, each grid point in geographical coordinates represents the same land area. The resultant coordinates were then checked against the MODIS classification map a...

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...as daily temperature, precipitation, and cloud cover, as well as several indices of extremes, such as yearly high temperatures, number of frost days, and drought index, on a station-by-station basis (=-=Tank et al., 2009-=-). This dataset has undergone rigorous quality control to flag, for example, inhomogeneities due to station relocation or changing urban heating effects. Annual maximum temperatures for the years 1900...

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...rocess models, spatial prediction is accomplished through the classical geostatistical technique Kriging. Under previous max-stable process models, spatial prediction is possible but complicated (see =-=Wang and Stoev, 2010-=-, e.g.). The HKEVP, in contrast, permits straightforward spatial prediction at unobserved sites. Prediction at an unobserved location sp and time tp proceeds as follows. At each MCMC iteration i 1. Co...

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...own for more than a trivial number of locations. This precludes maximum likelihood and Bayesian estimation, although methods based on composite likelihoods have had some success (Padoan et al., 2010; =-=Cooley et al., 2012-=-; Shaby, 2012). In contrast, the HKEVP model (1) and (2) does not suffer this deficiency. It enables Bayesian formulation and attendant Markov chain Monte Carlo (MCMC) sampling (see, e.g. Robert and C...

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... First, the GEVP has been criticized as appearing implausibly smooth and has been diagnosed as fitting several datasets less satis8 factorily than more recently-proposed spatial max-stable processes (=-=Smith and Stephenson, 2009-=-; Davison et al., 2012). The HKEVP described above adds flexibility by including the parameter α ∈ (0, 1) which allows the resultant process to vary from very smooth process realizations to very rough...

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...ic hierarchical Gaussian extreme value processes with three different values of α. The HKEVP model defined by (1) and (2) is max-stable both marginally and conditionally on the spatial random effect (=-=Reich and Shaby, 2012-=-). Marginally over A1, . . . , AL, Y (s0) ∼ GEV(µ, σ, ξ) at any location s0. The parameter α controls the smoothness of the process in the following way. As α→ 1, the latent variables A1, . . . , AL, ...

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... Westra and Sisson (2011). While it is possible to model temperature fields with flexible models that can capture broad ranges of tail behaviors, spatial quantile regression for example (Reich, 2012; =-=Tokdar and Kadane, 2011-=-), an extreme value approach is preferred for this application. For very rare events, the scarcity of available data is likely insufficient to accurately fit the far tail of a non-parametric or semi-p...

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.... (2011), and Westra and Sisson (2011). While it is possible to model temperature fields with flexible models that can capture broad ranges of tail behaviors, spatial quantile regression for example (=-=Reich, 2012-=-; Tokdar and Kadane, 2011), an extreme value approach is preferred for this application. For very rare events, the scarcity of available data is likely insufficient to accurately fit the far tail of a...

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...e GEVP has been criticized as appearing implausibly smooth and has been diagnosed as fitting several datasets less satis8 factorily than more recently-proposed spatial max-stable processes (Smith and =-=Stephenson, 2009-=-; Davison et al., 2012). The HKEVP described above adds flexibility by including the parameter α ∈ (0, 1) which allows the resultant process to vary from very smooth process realizations to very rough...

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...ze both the tail of the marginal temperature distributions at locations of interest and the spatial dependence of rare events. To this end, we look to the family of spatial max-stable process models (=-=Davison et al., 2012-=-, for a review). These models provide the tail behavior required by classical extreme value theory and explicitly represent spatial dependence among the extreme events. Previous applications of spatia...

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...hical Gaussian extreme value processes with three different values of α. The HKEVP model defined by (1) and (2) is max-stable both marginally and conditionally on the spatial random effect (Reich and =-=Shaby, 2012-=-). Marginally over A1, . . . , AL, Y (s0) ∼ GEV(µ, σ, ξ) at any location s0. The parameter α controls the smoothness of the process in the following way. As α→ 1, the latent variables A1, . . . , AL, ...