The whole contents of [Spatial Interpolation Theory] series are from the text book ‘Michael L. Stein, Interpolation of Spatial Data, New York: Springer, 1999.’
2.1 The basic Assumption of Random Fields
The random fields is a generalization version of stochastic process. Remark that ordinary stochastic process is a sequence of random variables such as ${Z(t): t\in \mathbb{R}}$. Usually, the “t” means time point or processing index. The random fields are defined as set of random variables ${Z(x) : x \in \mathbb{R}^d }$. Usually, we want to predict $Z(x_0)$ using $Z(x_1),Z(x_2),…,Z(x_n)$.
To make the inference about random fields, we define special statistical assets.
Mean Function : \(m(\textbf{x}) = E(Z(\textbf{x}))\)
Mean Vector : \(\textbf{m} = \{m(\textbf{x}_i)\}_{i=1}^n\)
Covariance Function: \(k(\textbf{x},\textbf{y}) = Cov(Z(\textbf{x}),Z(\textbf{y}))\)
**Covariance Matrix : ** $\textbf{K} = {k(\textbf{x}i,\textbf{x}_j)}{i,j=1}^n$
We can build Hilbert Space using the kernel matrix and mean vector.
\[\def\bf#1{\textbf{#1}}\]To make the inference about real world, people usually add assumption to make it tractable.
One of the assumption is Stationarity. The strong stationarity is defined as follow.
Strong Stationarity
\(Z(\bf x)\) is said to be strongly stationary if for any finite \(n \in \mathbb{R}, \bf{x}, \bf{x}_1,\bf{x}_2,...,\bf{x}_n \in \mathbb{R}^d,t_1,t_2,...,t_n \in \mathbb{R}\),
\[Pr(Z(\bf{x}_1 + \bf{x}) \leq t_1,Z(\bf{x}_2 + \bf{x}) \leq t_2, ..., Z(\bf{x}_n + \bf{x}) \leq t_n) = Pr(Z(\bf{x}_1 )\leq t_1,Z(\bf{x}_2 ) \leq t_2, ..., Z(\bf{x}_n ) \leq t_n)\]
Notice that, \(Pr(Z(\bf{x}) \leq t) = F(Z(\bf{x}))\) is Cumulative Distribution Function (CDF) and CDF is uniquely defined for each random variable. Therefore, for \(n=1\), \(Pr(Z(\bf{x}_1 + \bf{x}) \leq t_1 ) = Pr(Z(\bf{x}_1))\) is \(F(Z(\bf {x}_1 + \bf{x})) = F(Z(\bf{x}_1))\) which means transformation invariance property.