📂Statistical Analysis

Definition of Variogram

Definition ¹

In a fixed subset $D \subset \mathbb{R}^{r}$ of Euclidean space , consider a space process $\left\{ Y(s) \right\}_{s \in D}$ which is a set of random variables $Y(s) : \Omega \to \mathbb{R}^{1}$ and a direction vector $\mathbf{h} \in \mathbb{R}^{r}$. Specifically, represent $n \in \mathbb{N}$ sites as $\left\{ s_{1} , \cdots , s_{n} \right\} \subset D$, and assume that $Y(s)$ has variance existing for all $s \in D$. The following defined $2 \gamma ( \mathbf{h} )$ is called a Variogram. $$2 \gamma ( \mathbf{h} ) := E \left[ Y \left( s + \mathbf{h} \right) - Y(s) \right]^{2}$$ Especially, half of the variogram $\gamma ( \mathbf{h} )$ is called a Semivariogram.

Explanation

Definition of Regular Spatial Process:

2. If in all $s \in D$, $\mu (s)$ is a constant function $\mu (s) := \mu$ and both belong to $D$ for all $\mathbf{h}$ such that the covariance is expressed as a function of $\mathbf{h}$ only $C : \mathbb{R}^{r} \to \mathbb{R}$, regardless of $s$, through some function $C$ then $\left\{ Y(s) \right\}$ is said to have Weak Stationarity. $$\operatorname{Cov} \left( Y (s) , Y \left( s + \mathbf{h} \right) \right) = C \left( \mathbf{h} \right)$$ Here, $C$ is called the Covariance Function or Covariogram.
3. If the mean of $\left[ Y \left( s + \mathbf{h} \right) - Y(s) \right]$ is $0$ and the variance depends only on $\mathbf{h}$ then $\left\{ Y(s) \right\}$ is said to have Intrinsic Stationarity. $$\begin{align*} E \left[ Y \left( s + \mathbf{h} \right) - Y(s) \right] =& 0 \\ \Var \left[ Y \left( s + \mathbf{h} \right) - Y(s) \right] =& 2 \gamma ( \mathbf{h} ) \end{align*}$$

Intrinsic Stationarity

Just seeing from the definition, although the variogram $2 \gamma ( \mathbf{h} ) = E \left[ Y \left( s + \mathbf{h} \right) - Y(s) \right]^{2}$ is a function that depends also on $s$, it’s usually assumed that the given space process is intrinsically stationary. Conversely, because the definition of intrinsic stationarity itself doesn’t depend on $s$, these two cannot be thought of separately.

Weak Stationarity

It is natural to call $C \left( \mathbf{h} \right)$ the covariance function in the definition of weak stationarity, and the reason why it is specifically called a Covariogram despite it can be defined alone without $\gamma$ is due to the following relationship.

Theorem

For a weakly stationary spatial process $\left\{ Y (s) \right\}_{s \in D}$, the semivariogram $\gamma \left( \mathbf{h} \right)$ and the covariogram $C \left( \mathbf{h} \right)$ satisfy the following. $$\Var Y = \gamma \left( \mathbf{h} \right) + C \left( \mathbf{h} \right)$$

Proof

Following the weak stationarity of the space process $\left\{ Y \right\}$, by substituting the zero vector into the direction vector $\mathbf{h} \in \mathbb{R}^{r}$ in $\operatorname{Cov} \left( Y (s) , Y \left( s + \mathbf{h} \right) \right) = C \left( \mathbf{h} \right)$, we obtain the following. $$C \left( \mathbf{0} \right) = \operatorname{Cov} \left( Y (s) , Y (s) \right) = \Var Y (s)$$

Relationship of Stationarity: A strongly stationary spatial process is a weakly stationary spatial process, and a weakly stationary spatial process is intrinsic. $$\text{Strong} \implies \text{Weak} \implies \text{Intrinsic}$$

Meanwhile, since a weakly stationary spatial process is intrinsically stationary, for all $\mathbf{h} \in \mathbb{R}^{r}$ $$\Var \left[ Y \left( s + \mathbf{h} \right) - Y(s) \right] = 2 \gamma ( \mathbf{h} )$$ holds. If we unravel this backwards, $$\begin{align*} & 2 \gamma \left( \mathbf{h} \right) \\ =& \Var \left[ Y \left( s + \mathbf{h} \right) - Y (s) \right] \\ =& \Var \left[ Y \left( s + \mathbf{h} \right) \right] + \Var \left[ Y (s) \right] - 2 \operatorname{Cov} \left[ Y \left( s + \mathbf{h} \right) , Y (s) \right] \\ =& \operatorname{Cov} \left[ Y \left( s + \mathbf{h} \right) , Y \left( s + \mathbf{h} \right) \right] + \operatorname{Cov} \left[ Y (s) , Y (s) \right] - 2 \operatorname{Cov} \left[ Y \left( s + \mathbf{h} \right) , Y (s) \right] \\ =& C ( \mathbf{0} ) + C ( \mathbf{0} ) - 2 C ( \mathbf{h} ) \\ =& 2 \left[ C ( \mathbf{0} ) - C ( \mathbf{h} ) \right] \\ =& 2 \left[ \Var Y - C ( \mathbf{h} ) \right] \end{align*}$$ thus, we obtain the following equality. $$\gamma \left( \mathbf{h} \right) = \Var Y - C \left( \mathbf{h} \right)$$

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See Also

• Isotropic Variogram: When the variogram does not depend on direction and only on distance, we say the variogram is isotropic.
• Semivariogram Models: When the semivariogram is isotropic, plotting the scatter diagram with x-axis as $d := \left\| \mathbf{h} \right\|$ and y-axis as $\gamma (h)$ and fitting it to a specific model can give a sense of how variance changes with distance. It’s from this graphical examination that $2 \gamma$ and $C$ are called Variograms.
• Empirical Variogram $\gamma^{\ast}$: In actual data, there may not be many observations that exactly match $\mathbf{h}$. Before analysis, it’s advisable to look into whether the data meets certain assumptions through $\gamma^{\ast}$.