📂Statistical Analysis

Spatial Processes

Definition ¹

Especially when it is $r > 1$, for a fixed subset $D \in \mathbb{R}^{r}$ of the Euclidean space, the following set of $p$-variate random vectors $Y(s) : \Omega \to \mathbb{R}^{p}$ is also referred to as a Spatial Process. $$ \left\{ Y(s) : s \in D \right\} $$ Especially when the spatial process is a finite set and represented as a vector like the following, it is also referred to as a Random Field. $$ \left( Y \left( s_{1} \right) , \cdots , Y \left( s_{n} \right) \right) $$

Description

Especially when dealing with spatial data including point-referenced data, $Y(s)$ is assumed to be continuously sampled with respect to $s$, but the actual realization $D = \left\{ s_{1} , \cdots , s_{n} \right\}$ would be a finite set.

In undergraduate courses on stochastic processes, one commonly studies only such stochastic processes concerning $r = p = 1$ and $[ 0 , \infty ) \subset \mathbb{R}$. $$ \left\{ Y_{t} : t \in [ 0 , \infty ) \right\} $$ If one has been introduced to stochastic processes only as a backdrop to time-series data, the definition of spatial processes might be somewhat perplexing. In reality, the general definition of a stochastic process as ‘a set of random elements’ is sufficient, so there’s no reason not to consider $\left\{ Y(s) \right\} _{s \in D}$ a stochastic process.

Rather than strictly calling spatial processes a generalization of temporal processes, it’s more accurate to say they were never distinctly separated to begin with. If this is hard to grasp, it might help to remember that the 1-dimensional axis of time $\mathbb{R}^{1}$ when dealing with time-series is indeed a genuine Euclidean space. Thinking it over, the term ‘process’ following the flow of time $t \in \mathbb{R}$ didn’t quite align with everyday language, so there’s no need to feel uneasy about the term ‘spatial process’.