**2. Gaussian processes**

A stochastic process *W t*ð Þ, *t*∈ *T* is a collection of random variables. That is, for each *t*∈*T*, *W t*ð Þ is a random variable [24]; if the stochastic process is indexed by a coordinate space *s*∈ *A* ⊂ *<sup>d</sup>*, then the stochastic process is called a random field [25]. A realization of the random field, *W s*ð Þ, *s*∈ *A*, is given by ð Þ *W s*ð Þ¼ <sup>1</sup> *y s*ð Þ<sup>1</sup> , … ,*W s*ð Þ¼ *<sup>n</sup> y s*ð Þ*<sup>n</sup>* . Generally from the sample *y s*ð Þ<sup>1</sup> , … , *y s*ð Þ*<sup>n</sup>* one tries to know the characteristics of the process *W* in *si*, *i* ¼ 1, … , *n*; and with this information to make inference of the process *W s*ð Þ on all *<sup>A</sup>* <sup>⊂</sup> *<sup>d</sup>* , a convex set where *s* varies continuously. To the geo-referenced data *y s*ð Þ<sup>1</sup> , … , *y s*ð Þ*<sup>n</sup>* is often referred to as geocoded, geostatistical data or point-referenced data. The study of this type of data is known as geostatistics, which is a part of spatial statistics that studies phenomena with continuous variation in space, a convex region denoted *A* [26].

A process *W* is second order stationary if it has finite variance, constant mean and its covariance function depends only on distance. Having second-order stationarity in a stochastic process implies having intrinsic stationarity, i.e., second-order stationarity is stronger than intrinsic stationarity. On the other hand, weak stationarity and second-order stationarity are equivalent in the space [27]. The following defines what is known as a Gaussian process (field).

**Definition 1.** A stochastic process *W s*ð Þ : *<sup>s</sup>*<sup>∈</sup> *<sup>A</sup>* <sup>⊂</sup> <sup>2</sup> , where *<sup>s</sup>* varies continuously on a fixed subset *A* content in <sup>2</sup> , is a Gaussian process if for any collection of locations *s*1, … , *sn* with *si* ∈ *A*, the joint distribution of ð Þ *W s*ð Þ<sup>1</sup> , … ,*W s*ð Þ*<sup>n</sup>* is multivariate Gaussian [1].

What is known as a stationary Gaussian process is defined below.

**Definition 2.** A Gaussian process *W s*ð Þ : *<sup>s</sup>*<sup>∈</sup> *<sup>A</sup>* <sup>⊂</sup> <sup>2</sup> , is stationary if <sup>∀</sup>*s*<sup>∈</sup> *<sup>A</sup>*:

$$E(\mathcal{W}(\mathfrak{s})) = \mathbf{0},\tag{1}$$

$$\operatorname{Var}(\mathcal{W}(\mathfrak{s})) = \sigma^2,\tag{2}$$

and its correlation function depends only on the distance, i.e.

$$\operatorname{Corr}(\mathcal{W}(\mathfrak{s}), \mathcal{W}(\mathfrak{s'})) = \rho(h), \tag{3}$$

where *h* ¼ *s* � *s* <sup>0</sup> k k is the Euclidean distance that exists between *s* and *s* 0 .

That is, the mean and variance of *W s*ð Þ are constant and its correlation function only depends on the distance, so that

$$\mathcal{W} \sim \mathcal{N}\left(\mathbf{0}, \sigma^2 \rho(h)\right) \tag{4}$$

Given **W** ¼ ð Þ *W*1, … ,*Wn* , where *Wi* ¼ *W s*ð Þ*<sup>i</sup>* , the distribution of **W** is normal multivariate ð Þ NM , i.e.


**Table 1.**

*Models for the spatial correlation structure of a spatial process.*

$$\mathbf{W} \sim \mathcal{N} \mathcal{M}(\mathbf{0}, \sigma^2 \mathbf{R}),\tag{5}$$

where the ð Þ *i*, *j* element of **R** is given by ð Þ **R** *ij* ¼ *Corr W s*ð Þ*<sup>i</sup>* ,*W sj* � � � � <sup>¼</sup> *<sup>ρ</sup> hij* � �, *hij* <sup>¼</sup> *si* � *sj* � � � � is the Euclidean distance between *si* and *sj*. Note that the covariance of the Gaussian process is given by *Cov*ð Þ¼ **<sup>W</sup>** *<sup>σ</sup>*<sup>2</sup>**R**.

In this way, the correlation structure of a stationary Gaussian process can be studied through the *ρ*ð Þ *h* function. Several parametric expressions for this function are shown in the **Table 1**. In these correlation functions, *ϕ*>0 is a range parameter controlling the spatial decay over distance; *h* ¼ *s* � *s* <sup>0</sup> k k is the Euclidean distance between *s* and *s* <sup>0</sup> and *h*≥0; Γð Þ� denotes the gamma function. *κ* >0, in theory of spatial extremes *Jκ*ð Þ� and *Kκ*ð Þ� are the Bessel and modified Bessel function of the third kind of order *κ* [28], while in the spatial survival analysis and generalized linear models *Kκ*ð Þ� is the modified Bessel function of the second kind of order *κ* [29]; *κ* is a shape parameter that determines the analytic smoothness of the underlying process *W* [1]. In the powered exponential correlation function 0 <*κ* ≤2 and in the Bessel correlation function *κ* ≥0.
