Stationarity of Spatial Processes 📂Statistical Analysis

Stationarity of Spatial Processes

Definitions ¹

Consider a spatial process $\left\{ Y(s) \right\}_{s \in D}$ and direction vector $\mathbf{h} \in \mathbb{R}^{r}$, which is a set of random variables $Y(s) : \Omega \to \mathbb{R}^{1}$ in a fixed subset $D \subset \mathbb{R}^{r}$ of Euclidean space. Specifically, represent $n \in \mathbb{N}$ number of sites as $\left\{ s_{1} , \cdots , s_{n} \right\} \subset D$, and assume that $Y(s)$ has a variance for all $s \in D$.

$\left\{ Y(s) \right\}$ is said to have strong stationarity if the distribution of the following two random vectors is the same for all $\left\{ s_{1} , \cdots , s_{n} \right\}$ and all $\mathbf{h}$. $$ \left( Y \left( s_{1} \right) , \cdots , Y \left( s_{n} \right) \right) \\ \left( Y \left( s_{1} + \mathbf{h} \right) , \cdots , Y \left( s_{n} + \mathbf{h} \right) \right) $$
$\left\{ Y(s) \right\}$ is considered to have weak stationarity if for all $s \in D$, $\mu (s)$ is a constant function $\mu (s) := \mu$ and both $s , s + \mathbf{h}$ belong to $D$, and the covariance for all $\mathbf{h}$ can be expressed as a function of $\mathbf{h}$ only, independently from $s$, as some function $C$. $$ \operatorname{Cov} \left( Y (s) , Y \left( s + \mathbf{h} \right) \right) = C \left( \mathbf{h} \right) $$ Here, we call $C$ a covariance function or variogram, and especially when $\left\| \mathbf{h} \right\| \to \infty$, if $C \left( \mathbf{h} \right) \to 0$ then $\left\{ Y(s) \right\}$ is said to be ergodic.
If the mean of $\left[ Y \left( s + \mathbf{h} \right) - Y(s) \right]$ is $0$ and its variance depends only on $\mathbf{h}$, then $\left\{ Y(s) \right\}$ is said to have intrinsic stationarity. $$ \begin{align*} E \left[ Y \left( s + \mathbf{h} \right) - Y(s) \right] =& 0 \\ \operatorname{Var} \left[ Y \left( s + \mathbf{h} \right) - Y(s) \right] =& 2 \gamma ( \mathbf{h} ) \end{align*} $$

$2 \gamma \left( \mathbf{h} \right)$ is called a variogram.

Theorems

Strong stationary spatial processes are weak stationary, and weak stationary processes are intrinsic. $$ \text{Strong} \implies \text{Weak} \implies \text{Intrinsic} $$ Furthermore, if the random vector $\left( Y \left( s_{1} \right) , \cdots , Y \left( s_{n} \right) \right)$ follows a multivariate normal distribution for all $\left\{ s_{1} , \cdots , s_{n} \right\}$, then $\left\{ Y(s) \right\}$ is Gaussian. A necessary and sufficient condition for a weak stationary spatial process to be strong stationary is for the spatial process to be Gaussian. $$ \text{Strong} \overset{\text{gaussian}}{\impliedby} \text{Weak} $$

Explanation

Why Stationarity is Necessary

Just as stationarity in time series analysis became a common assumption for various models, stationarity in spatial processes refers to properties that logically must be satisfied before analyzing spatial data. If stationarity cannot be assumed, analysis becomes meaningless in many cases.

Strong stationarity is essentially stationarity itself without doubt. The issue is, even though this might be true stationarity in theory, it might be hard to find examples in reality, thus a retreat to the compromised condition of weak stationarity might be necessary.
Weak stationarity compromises that it’s alright if not all distributions at each site are known, at least the mean is constant and its covariance depends only on relative distance and direction $\mathbf{h}$.
The term intrinsic stationarity might be unfamiliar only to those who have studied statistics, but it’s called ‘intrinsic’ as its definition is similar to the viewpoint that the observed differences between two points depend solely on $\mathbf{h}$.

Definition of an intrinsic function: In differential geometry, a function that depends only on the coefficients of the first fundamental form $g_{ij}$, not on the unit normal $\mathbf{n}$, is called intrinsic.

Ergodicity

While the pronunciation of Ergodic is closer to [엘가딕] in Korean, let’s not dwell on it.

That a spatial process is ergodic, i.e., $$ \lim_{\left\| \mathbf{h} \right\| \to \infty} C \left( \mathbf{h} \right) = 0 $$ suggests that regardless of direction, as the distance between two sites increases, their correlation decreases. This is a fairly logical assumption. Not all data may be ergodic, but intuitively, it’s common for relationships to weaken as they become more distant, unless $C \left( \mathbf{h} \right)$ has periodicity or is an exceptionally unique case. It seems plausible to expect at least up to $C \left( \mathbf{h} \right) \searrow \varepsilon$ in a limit sense.

Ergodicity in stochastic processes, generally considered dependent on time $t$, approaches the concept in a manner similar to how, after a long time ($t \to \infty$), a specific state returns to its initial state. Similarly, in spatial processes, it deals with how their correlation decreases beyond a distant range ($\left\| \mathbf{h} \right\| \to \infty$) instead of over time. Although many fields explain ergodicity in connection with time and initial states, it’s not an entirely forced naming.