Semivariance: Wikis

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Encyclopedia

In spatial statistics, the empirical semivariance is described by

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

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. The semivariance is half the variance of the increments z(xi + h) − z(xi), but the whole variance of z-values at given separation distance h (Bachmaier and Backes, 2008).

A plot of semivariances versus distances between ordered data in a graph is known as a semivariogram rather than a variogram. Many authors call $2\hat\gamma(h)$ a variogram, others use the terms variogram and semivariogram synonymously. However, Bachmaier and Backes (2008), who discussed this confusion, have shown that $\hat\gamma(h)$ should be called a variogram, terms like semivariogram or semivariance should be avoided. This also shall become the guideline of the agricultural journal Precision Agriculture.

Controversy

In situ or temporally ordered sets give df(o) = 2(n − 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 − 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.

References

• Bachmaier, M and Backes, M, 2008, "Variogram or Semivariogram — Explaining the Variances in a Variogram". Article DOI: 10.1007/s11119-008-9056-2, Precision Agriculture, Springer-Verlag, Berlin, Heidelberg, New York.
• Clark, I, 1979, Practical Geostatistics, Applied Science Publishers
• David, M, 1978, Geostatistical Ore Reserve Estimation, Elsevier Publishing
• Hald, A, 1952, Statistical Theory with Engineering Applications, John Wiley & Sons, New York
• Journel, A G and Huijbregts, Ch J, 1978 Mining Geostatistics, Academic Press
• Merks, J W and Merks E A T, "Precision Estimates for Ore Reserves", Erzmetall, Vol 44, Nov 1991.
• Merks, J W, "Abuse of Statistics", CIM Bulletin, Jan 1993.
• Merks, J W, Applied Statistics in Mineral Exploration, Mining Engineering, Feb 1997
• Merks, J W, "Borehole Statistics with Spreadsheet Software", SME Transactions 2000, Vol 308