**6. Spatial generalized extreme value model**

According to Coles (2001) [50], given *Y*1, … , *Yn* a sequence of independent random variables with a common distribution function *F* with *Mn* ¼ max f g *Y*1, … , *Yn* , if there a sequences of constants *an* > 0 and *bn* such that

$$P((M\_n - b\_n)/a\_n \le z) \to G(z),\tag{25}$$

when *n* ! ∞, for a non-degenerative distribution function *G*, then *G* is a member of the generalized extreme value (GEV) distribution family

$$G(\mathbf{y}; \eta, \tau, \xi) = \exp\left\{-\left[\mathbf{1} + \xi \left(\frac{\eta - \eta}{\tau}\right)\right]^{\frac{\omega}{\zeta}}\right\},\tag{26}$$

defined on f g *z* : 1 þ *ξ*ð Þ *z* � *η =τ* >0 , where �∞ <*η*< ∞, *τ* > 0 and �∞ < *ξ*< ∞.

Davison et al. in 2012 [51], describe spatial GVE as follows. For each *s* in <sup>2</sup> , suppose that *Y s*ð Þ is GEV distributed whose parameters *μ*ð Þ*s* , *σ*ð Þ*s* and *ξ*ð Þ*s* vary smoothly for *s* in <sup>2</sup> according to a stochastic process *W s*ð Þ. We assume that the processes for each GEV parameters are mutually independent Gaussian processes [52]. Then

$$\begin{aligned} \eta(\boldsymbol{s}) &= \boldsymbol{f}\_{\boldsymbol{\eta}}(\boldsymbol{s}; \boldsymbol{\beta}\_{\boldsymbol{\eta}}) + \mathcal{W}\_{\boldsymbol{\eta}}(\boldsymbol{s}; \boldsymbol{\sigma}\_{\boldsymbol{\eta}}, \boldsymbol{\phi}\_{\boldsymbol{\eta}}, \boldsymbol{\kappa}\_{\boldsymbol{\eta}}), \\ \tau(\boldsymbol{s}) &= \boldsymbol{f}\_{\boldsymbol{\tau}}(\boldsymbol{s}; \boldsymbol{\beta}\_{\boldsymbol{\tau}}) + \mathcal{W}\_{\boldsymbol{\tau}}(\boldsymbol{s}; \boldsymbol{\sigma}\_{\boldsymbol{\tau}}, \boldsymbol{\phi}\_{\boldsymbol{\tau}}, \boldsymbol{\kappa}\_{\boldsymbol{\tau}}), \\ \xi(\boldsymbol{s}) &= \boldsymbol{f}\_{\boldsymbol{\xi}}(\boldsymbol{s}; \boldsymbol{\beta}\_{\boldsymbol{\xi}}) + \mathcal{W}\_{\boldsymbol{\xi}}(\boldsymbol{s}; \boldsymbol{\sigma}\_{\boldsymbol{\xi}}, \boldsymbol{\phi}\_{\boldsymbol{\xi}}; \boldsymbol{\kappa}\_{\boldsymbol{\xi}}), \end{aligned} \tag{27}$$

where *f <sup>η</sup>*, *f <sup>τ</sup>* and *f xi* are deterministic functions depending on a regression parameters *βη*, *βτ* and *βξ* respectively. While *Wη*, *W<sup>τ</sup>* and *W<sup>ξ</sup>* are a zero mean stationary Gaussian process with correlation function *ρ h*, *ϕη*, *κη* � �, *<sup>ρ</sup> <sup>h</sup>*, *ϕτ* ð Þ , *κτ* and *ρ h*, *ϕξ*, *κξ* � � respectively, i.e.

$$\begin{split} \mathcal{W}\_{\eta} &\sim \mathcal{N}\left(0, \sigma\_{\eta}^{2}\rho\left(h, \phi\_{\eta}, \kappa\_{\eta}\right), \\ \mathcal{W}\_{\tau} &\sim \mathcal{N}\left(0, \sigma\_{\tau}^{2}\rho\left(h, \phi\_{\tau}, \kappa\_{\tau}\right), \\ \mathcal{W}\_{\xi} &\sim \mathcal{N}\left(0, \sigma\_{\xi}^{2}\rho\left(h, \phi\_{\xi}, \kappa\_{\xi}\right). \end{split} \right) . \tag{28}$$

Then conditional on the values of the tree Gaussian process at the sites ð Þ *s*1, … , *sk* , the maxima are assumed to follow GEV distributions

$$Y\_{s\_i} | \eta(s\_j), \tau(s\_j), \xi(s\_j) \sim \text{GEV}(\eta(s\_j), \tau(s\_j), \xi(s\_j)) \tag{29}$$

z independently for each location *s*1, … , *sk*, *j* ¼ 1, … , *k* and *i* ¼ 1, … , *n*.

Davison *et al.* in 2012 [51], proposed the construction of Bayesian hierarchical models for spatial extremes.

The *SpatialExtremes* package [53] allows modeling spatial extremes, through maxstable processes with the function *fitmaxstab*, which reports the values of the parameter estimates with their respective standard errors.

To implement hierarchical Bayesian models, the function *latent* is used, this reports the posterior median of the scale, shape and location parameters with their respective credible intervals.

Another package in the literature to model spatial extremes is *glmmfields* [34], with the function *glmmfields*, parameter estimation is performed under the Bayesian approach. The function *glmmfields* also allows modeling spatial extreme events incorporating temporally, that is, time, these models are known as spatio-temporal models.
