Semivariance: Difference between revisions
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#REDIRECT [[Variogram#Semivariogram]] |
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{{POV}} |
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In [[spatial statistics]], the '''semivariance''' is described by |
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:<math>\gamma(h)=\sum_{i=1}^{n(h)}\frac{(z(x_i+h)-z(x_i))^2}{n(h)}</math> |
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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). |
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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 distribution]]s, the semivariance can provide additional information that a variance does not. |
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==Controversy== |
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In mathematical statistics, sets of '''n''' measured values give ''df=n-1'' degrees of freedom whereas ''in situ'' or temporally ordered sets give ''df(o)=2(n-1)'' degrees 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). |
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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. |
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Added comment 17/12/2006: in mathematical statistics, the degrees of freedom are n-1 when the true population mean is unknown. If the mean is known or can be realistically hypothesised, no degrees of freedom are lost and the variance may be divided by n. When defining the semi-variogram, Matheron showed theoretically that the true mean difference would be zero in the absence of significant drift in the values. If this assumption is valid, then no degrees of freedom are lost. This assumption can be verified by standard statistical techniques. |
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==See also== |
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* [[variance]] |
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* [[sampling variogram]] |
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* [[geostatistics]] |
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* [[kriging]] |
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== References == |
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* Clark, I, 1979, Practical Geostatistics, Applied Science Publishers |
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* David, M, 1978, Geostatistical Ore Reserve Estimation, Elsevier Publishing |
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* Hald, A, 1952, Statistical Theory with Engineering Applications, John Wiley & Sons, New York |
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* Journel, A G and Huijbregts, Ch J, 1978 Mining Geostatistics, Academic Press |
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* [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. |
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* [http://www.geostatscam.com/Adobe/Abuse_stats.pdf Merks, J W], Abuse of Statistics, CIM Bulletin, Jan 1993. |
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* [http://www.geostatscam.com/Adobe/Stats_exploration.pdf Merks, J W], Applied Statistics in Mineral Exploration, Mining Engineering, Feb 1997 |
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* [http://www.geostatscam.com/Adobe/Borehole_stats.pdf Merks, J W], Borehole Statistics with Spreadsheet Software, SME Transactions 2000, Vol 308 |
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==External links== |
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* 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 |
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* [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.] |
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[[Category:Geostatistics]] |
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[[Category:Statistical deviation and dispersion]] |
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[[de:Semivarianz]] |
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Latest revision as of 04:25, 16 March 2021
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