Semivariogram

The semivariogram captures the second-moment structure of the data.

\[\gamma(s,h) = \frac{1}{2} [var(Z(s) - Z(s+h))] \] \[= \frac{1}{2} [var(Z(s)) + var(Z(s+h)) - 2cov(Z(s),Z(s+h))].\]

Semivariogram

If second-order stationarity holds: \(\gamma(h) = \frac{1}{2} (2\sigma^2 -2 C(h)) = C(0) - C(h)\).

If positive spatial correlation is present:

• \(\gamma(0) = C(0) - C(0) = 0\). However due to measurement error and microscale variation there may be a nugget effect - this is often observed in sample variograms.
• The variogram attains \(\gamma(h^*) = \sigma^2\) when the lag exceeds the distance \(h^*\) for which the data points are correlated.
• The range is \(h^*\) (where \(C(h^*) = 0\) or at \(95\% \sigma^2\))

Semivariogram

Empirical semivariogram:

\[\hat{\gamma}(h) = \frac{1}{2 |N(h)|} \sum_{N(h)} \left(Z(s_i) - Z(s_j) \right)^2 \]

Empirical covariance estimation:

\[\hat{C}(h) = \frac{1}{|N(h)|} \sum_{N(h)} (Z(s_i)-\bar{Z})(Z(s_j)-\bar{Z})\]

Semivariogram

knitr::include_graphics('variogram.jpg')

Semivariogram and Covariance Functions

Semivariogram: \(\gamma(h)\) vs. \(h\)

Variogram: \(2\gamma(h)\) (but naming is not consistant)

Covariance function: \(C(h)\) vs. \(h\)

Semivariogram and Covariance Functions

Under stationarity \(\gamma(h) = \gamma(s_i-s_j) = C(0) - C(s_i-s_j)\). Recall that \(C(0) = var(Z(s))\).

ESTIMATION (Matheron 1962, 1963): \[\gamma(s_i-s_j) = \frac{1}{2|N(s_i-s_j)|} \sum_{N(s_i-s_j)} ({Z(s_i) - Z(s_j)})^2\]

\[C(s_i-s_j) = \frac{1}{2|N(s_i-s_j)|} \sum_{N(s_i-s_j)} (Z(s_i) - \bar{Z})(Z(s_j)-\bar{Z})\] \[\textrm{ where } \bar{Z} = \frac{1}{n} \sum_{i=1}^n Z(s_i). \]