1 Introduction
Let us continue our understanding of the Kriging method. It is important to recall that while Kriging can be treated as a deterministic problem solvable through linear algebra, we are now shifting our focus to the more common (statistical) Bayesian perspective. The primary objective of Kriging is to utilize observed values \(y_1, \dots, y_N\) sampled at spatial locations \(\mathbf{s}_1, \dots, \mathbf{s}_N\) to predict an unknown value \(y_{\text{new}}\) at a specific location \(\mathbf{s}_{\text{new}}\) (where \(\mathbf{s}_{\text{new}} \notin \{\mathbf{s}_1, \dots, \mathbf{s}_N\}\)). In addition to the deterministic approach previously discussed, we can address this problem using stochastic and Bayesian frameworks. Under this view, we assume the data values \(y_i\) are observed realizations \(y(\mathbf{s}_i)\) of a random variable \(Y(\mathbf{s}_i)\) belonging to a spatial random field \(Y\). Since the collected data \(y(\mathbf{s}_1), \dots, y(\mathbf{s}_N)\) represent only a single realization of the random variables \(Y(\mathbf{s}_1), \dots, Y(\mathbf{s}_N)\), our predictions are formulated using these random variables as inputs. Consequently, the predictor \(\hat{Y}(\mathbf{s})\) is itself a random variable. Note: While our previous post used \(Z(\mathbf{s}_1), \dots, Z(\mathbf{s}_N)\) to denote data values, we will now use \(Y(\mathbf{s}_1), \dots, Y(\mathbf{s}_N)\). This change is strictly notational to help distinguish the stochastic approach from the deterministic one.
2 Random Fields
Let \((\Omega, \mathcal{F}, \mathbb{P})\) be a probability space. A function \(Y:\Omega \to \mathbb{R}\) is called a if it is \(\mathcal{F}/\mathcal{B}(\mathbb{R})\)–measurable, where \(\mathcal{B}(\mathbb{R})\) denotes the Borel \(\sigma\)–algebra on on \(\mathbb{R}\). Now, in the same space of probability and \(D\subseteq\mathbb{R}^d\) a domain in \(d\) dimensions a stochastic process, or more appropriately in our context a random field \(Y\) on \(D\) is a collection of random variables \(\{Y(\mathbf{s}) : \mathbf{s}\in D\}\), where for each fixed \(\mathbf{s}\in D\), the mapping \([Y(\mathbf{s}):\Omega\to\mathbb{R} ]\) is a random variable. Thus, a random field is just a a collection of random variables [@cressie1993spatial]. Now, the moments of a random field \(Y\) provide useful information, for example; the mean of a random field Y can be defined as:
\[\begin{equation} \mu(\mathbf{s}) = \mathbb{E}[Y(\mathbf{s})], \qquad \mathbf{s} \in D, \end{equation}\]
provided the expectation exists for all \(\mathbf{s} \in D\). The second central moment of the random field \(Y\) (variance) is given by:
\[\begin{equation} \operatorname{Var}(Y(\mathbf{s})) = \mathbb{E}\!\left[(Y(\mathbf{s}) - \mu(\mathbf{s}))^2\right], \qquad \mathbf{s} \in D, \end{equation}\]
where \(\mu(\mathbf{s}) = \mathbb{E}[Y(\mathbf{s})]\) is the mean function previously defined. And, in general, the covariance \(\mathbf{C}\) of the random field \(Y\) is defined as:
\[\begin{equation} \sigma^2 \mathbf{C}(\mathbf{s}, \mathbf{s'}) = \operatorname{Cov}(Y(\mathbf{s}), Y(\mathbf{s'})) = \mathbb{E}\!\left[(Y(\mathbf{s}) - \mu(\mathbf{s})) (Y(\mathbf{s'}) - \mu(\mathbf{s'}))\right], \quad \mathbf{s},\mathbf{s'}\in D. \end{equation}\]
where the scalar parameter \(\sigma^2\) is known as the process variance. In the statistics literature this parameter is often included in the definition of the covariance function. For example, for a Gaussian covariance, \(\mathbf{C}(\mathbf{s}, \mathbf{s'}) = \sigma^2\exp\left(-\frac{(x - x')^2}{2\ell^2}\right)\), however in the numerical analysis texts they don’t include this “amplification” factor. The reason is that in the deterministic/numerical analysis setting of scattered data interpolation, that factor is irrelevant since it will be absorbed in the expantions coefficients (fasshahuer2015kernel?). However, in stochastics methods the variance process \(\sigma^2\) plays an important role. Thus, considering this, the variance of the random field \(Y\) is the diagonal of the covariance
\[\begin{equation} \operatorname{Var}(Y) = \sigma^2 \mathbf{C}(\mathbf{s}, \mathbf{s'}) \end{equation}\]
There are different kinds of spatial fields (spatial stochastic processes) but we will be interested in the called Gaussian random fields (or also known as Gaussian processes).