Semivariance

In spatial statistics, the semivariance is described by


 * $$\gamma(h)=\sum_{i=1}^{n(h)}\frac{(z(x_i+h)-z(x_i))^2}{n(h)}$$

where z is a datum at a particular location, h is the distance between ordered data, and n(h) is the number of paired data at a distance of h. A plot of semivariances versus distances between ordered data in a graph is known as a semivariogram rather than a variogram, in which the sum of squared differences is divided by 2n(h).

The semivariance is calculated in the same manner as the variance but only those observations that fall below the mean are included in the calculation. It is sometimes described as a measure of downside risk in an investments context. For skewed distributions, the semivariance can provide additional information that a variance does not.

Controversy
In situ or temporally ordered sets give df(o) = 2(n &minus; 1) degrees of freedom for the first variance term. The semivariance is an invalid measure for variability, precision and risk because the sum of squared differences between x and x + h is divided by n, the number of data in the set, but it ought to be divided by df(o) = 2(n &minus; 1), the degrees of freedom for the first variance term (see Ref 2).

The statement that only measured values below the mean are included in the semivariance makes no statistical sense (see Ref 4). Clark, in her Practical Geostatistics, which can be downloaded from her website, proposed that the factor 2 be moved for mathematical convenience and berates those who refer to variograms rather than semi-variograms.