📂Probability Theory

Definition of Random Fields

Definition ¹

Set-based Definition

A stochastic process is defined as a set of random variables. For a parameter set or an index set $T$, a stochastic process over $T$, denoted as $f$, is defined as a set of random variables over $t \in T$, denoted as $f(t)$. If $T$ is $n$-dimensional and $f$ is a $d$-dimensional vector-valued process, it is called a $\left( n , d \right)$ random field.

Functional Definition

Let $\mathbb{R}^{n}$ be a $n$-dimensional Euclidean space, and $\mathbb{X}$ be a set of $d$-dimensional random vectors. A function defined as follows $g$ is called a $\left( n , d \right)$ random field. $$ g : \mathbb{R}^{n} \to \mathbb{X} $$

Explanation

Essentially, the two definitions do not differ, only in how general or specific they are in detailed aspects. Conceptually, a random field is just a type of stochastic process; however, it should convey a sense of a field in its naming.

Gaussian Random Field ²

A Gaussian Random Field refers to a random variable corresponding to a coordinate $\mathbf{x} \in \mathbb{R}^{n}$ that follows a Gaussian distribution. For example, a random field on a 2-dimensional plane with a mean $0$ and a Gaussian kernel $k \left( x_{1} , x_{2} \right) = \exp \left( - \left| x_{1} - x_{2} \right|^{2} / 2 \sigma^{2} \right)$ as its variance can be expressed as follows. $$ U \sim N \left( 0 , k \left( x_{1} , x_{2} \right) \right) $$ Whatever the coordinates $\left( x_{1} , x_{2} \right) \in \mathbb{R}^{2}$ may be, $U$ follows a univariate normal distribution, and by the definition of a random field, $U$ is both a $(2, 1)$ random field and a Gaussian random field. From a graphical perspective, it can be visualized as a function $U$ where the random variable following the corresponding normal distribution of any given point is the function value itself.