## [ai-geostats] Pseudo variances and covariances?

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• Hello Readers, There s no need to stalk the lost variance because the correct formula is posted on ai-geostats.org/ documents. Also posted is an Excel
Message 1 of 1 , Feb 24, 2006
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There's no need  'to stalk the lost variance' because the correct formula is posted on ai-geostats.org/ documents. Also posted is an Excel template that shows how  degrees of freedom for sets of measured values with variable weights become positive irrationals. The correct formula gives the variances of area, count, density, distance, length, mass and volume weighted averages. However, there's an urgent need to study the validity of kriging variances and covariances.

Given that weighted averages are functionally dependent values of sets of measured values with variable weights, it follows that each weighted average has its own variance. Readers have a right to know why the variance of the distance-weighted average was replaced with the kriging variance of a subset of some infinite set of kriged estimates. David, on page 286 of his 1977 textbook, claims, '(W)riting all the necessary covariances for that set of equations is a good test to find out whether one really understands geostatistics'. His covariances are meaningless measures for spatial dependence because his set of sixteen functionally dependent (calculated!) boreholes has precisely zero degrees of freedom whereas the set of nine measured boreholes gives df(r)=n-1=9 for the randomized set and df(o)=2(n-1)=18 for the ordered set.

Stanford's Journel, in his letter of October 15, 1992, to the Editor-in-Chief of Mathematical Geology, postulates, '(T)he very reason for geostatistics or spatial dependence in general is the acceptance (a decision rather) that spatially distributed data should be considered a priori as dependent one to another, unless proven otherwise'. Readers have a right to know who decided that spatial dependence can be assumed a priori. In the same letter Journel speculates, 'Mr Merks' anger arises fro [sic] a misreading of geostatistical theory, or a reading too encumbered by classical Fischerian [sic] statistics'. Surely, Readers have a right to know how Journel proves otherwise.

Journel's 1992 prevarications were triggered by "Precision Estimates for Ore Reserves", a paper that explained how to verify spatial dependence by applying Fisher's F-test to the variance of gold grades of a set of rounds in a drift and the first variance term of the ordered set. Rejected by Mathematical Geology but reviewed and published by Erzmetall, it is posted on my website under 'Reviewed papers'. In March 2005, MG's current Editor-in-Chief was interested in a paper on testing for spatial and charting sampling variograms so I obliged. Early this year, he wanted me to put his degrees of freedom in my paper. So I withdrew the paper and suggested he and his reviewer study Fisher's F-test for spatial dependence as defined and approved by Technical Committee 69-Applications of Statistical Methods. Cerattepe gold, which gives the stats for one of the data sets on which my paper is based, is posted on ai-geostats.org/ documents. Shortly, I'll post the unadulterated version of this paper on my website.

Assuming spatial dependence between dense data in small sample spaces may well make scientific sense most of the time but it makes no sense between sparse data in large sample spaces. Assuming continued mineralization between widely spaced boreholes is a scientific fraud but verifying spatial dependence between ordered ore zones within lines of boreholes makes scientific sense. Bre-X's phantom gold resource was atypical but Fisher's F-test played a role in unraveling the salting scam. The F-test was also applied to a set of ordered blasthole grades at Hecla's Grouse Creek gold mine where it showed that spatial dependence dissipated into randomness between 10 m and 20 m.

Barrick-Placer's Deep South shrunk by a whopping 50%! Oversmoothed, I assume! To find out what smoothing is all about, click 'A Study on Kriging Small Blocks' under 'Articles and letters' on my website. Can the requirement of functional independence be violated a little but not a lot? Of course, Readers have a right to know!

Jan W Merks
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