The Variogram: a spatial decomposition of variance

Suppose that we have measured the variables

z

N

locations, whose geographical positions are denoted

{\underline{x}}_{i}

. Consider all pairs of data locations, of which there are

N_{p} = \frac{N (N - 1)}{2}

Now each pair of data locations is placed into a "lag class", determined by a vector $\underline{h}$ (the "lag"), which means that the two points are separated by (roughly) $\underline{h}$ . Define $N_{c}$ classes, each described by a set $P_{\underline{h}}$ of pairs of indices, such that $(i, j) \in P_{\underline{h}} \overset{}{\Leftrightarrow} {\underline{x}}_{i} - {\underline{x}}_{j} \approx \underline{h} .$

The univariate variogram is defined as

γ (\underline{h}) = \frac{1}{2} E [(z (\underline{x} + \underline{h}) - z (\underline{x}))^{2}]

and the estimator of the variogram function for lag

\underline{h}

γ (\underline{h}) = \frac{1}{2 N_{\underline{h}}} \sum_{(i, j) \in P_{\underline{h}}}^{N_{\underline{h}}} (z_{i} - z_{j})^{2},

where

N_{\underline{h}}

is the number of distinct pairs of data values, placed in the set

P_{\underline{h}}

(pairs displaced by the vector

\underline{h}

γ (\underline{h})

is sometimes called the semi-variogram, because of the factor of 1/2 in its definition.

Now the sample variance, $s^{2}$ , can be written as a weighted sum

s^{2} = \sum_{c = 1}^{N_{c}} (\frac{N_{\underline{h}}}{N_{p}}) (\frac{1}{2 N_{\underline{h}}} \sum_{(i, j) \in P_{\underline{h}}}^{N_{\underline{h}}} (z_{i} - z_{j})^{2}),

s^{2} = \sum_{c = 1}^{N_{c}} γ (\underline{h}) (\frac{N_{\underline{h}}}{N_{p}})

It is this equation which leads me to declare that

The sample variogram is the spatial decomposition of the sample variance.