Semivariance

In spatial statistics, the empirical semivariance is described by

:$hatgamma(h)=frac\{1\}\{2\}cdotfrac\{1\}\{n(h)\}sum\_\{i=1\}^\{n(h)\}(z(x\_i+h)-z(x\_i))^2$

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 $z(x\_i+h)-z(x\_i)$, 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 $2hatgamma(h)$ a variogram, others use the terms variogram and semivariogram synonymously. However, Bachmaier and Backes (2008), who discussed this confusion, have shown that $hatgamma(h)$ 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.

ee also

* Variance
* Sampling variogram
* Geostatistics
* Kriging

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
* [http://www.geostatscam.com/Adobe/Precision_reserves.pdf Merks, J W and Merks E A T] , Precision Estimates for Ore Reserves, Erzmetall, Vol 44, Nov 1991.
* [http://www.geostatscam.com/Adobe/Abuse_stats.pdf Merks, J W] , Abuse of Statistics, CIM Bulletin, Jan 1993.
* [http://www.geostatscam.com/Adobe/Stats_exploration.pdf Merks, J W] , Applied Statistics in Mineral Exploration, Mining Engineering, Feb 1997
* [http://www.geostatscam.com/Adobe/Borehole_stats.pdf Merks, J W] , Borehole Statistics with Spreadsheet Software, SME Transactions 2000, Vol 308

External links

* Shine, J.A., Wakefield, G.I.: A comparison of supervised imagery classification using analyst-chosen and geostatistically-chosen training sets, 1999, http://www.geovista.psu.edu/sites/geocomp99/Gc99/044/gc_044.htm
* [http://www.geostatscam.com This website explains why the true variance of a single distance-weighted average should not be replaced with the false kriging variance of a set of kriged estimates.]
* [http://www.actuaries.org/AFIR/colloquia/Orlando/Ferguson_Rom.pdf Post-Modern Portfolio Theory Comes of Age by Brian M. Rom and Kathleen W. Ferguson]