Geostatistics: Test for spatial dependence

Sampling protocols are optimized and sampling variograms are constructed by applying analyis of variance (ANOVA). Fisher's F-test is the quintessence of ANOVA. A sampling protocol is optimized by partitioning the total variance of the measurement procedure (the sum of the variances of sample selection, preparation and analytical stages) into its components. Similary, a sampling variogram is constructed by plotting the variance terms of an ordered set of measured values against the variance of the randomly distributed set and the lower limits of its asymmetric 95% and 99% confidence ranges. Sampling variograms show where orderliness in sample spaces or sampling units dissipate into randomness.

Several ISO Technical Committees have accepted these ANOVA applications. Surely, those who participate in the activities of ISO Technical Commitees are not "too encumbered" with analysis of variance. Let's look at Fisher's F-test that so irked CIM's, JMG's and IMM's geostatistical reviewers. This numerical example can be found in "Precision Estimates for Ore Reserves" where Fisher's F-test is applied to the variances of test results for gold in a set of twelve (12) rounds from a drift. The calculated value of F=var(x)/var1(x)=33.18/10.07=3.29 between the variance of the randomly distributed set of rounds and the first variance term of the ordered set turned out to be statistically significant at 1% probability. Hence, the probability exceeds 99% that these test results for gold display a significant degree of spatial dependence.

The reason why geostatistical thinkers find Fisher's F-test so vexing is that it cannot be applied without taking degrees of freedom into account. The F-test is based on comparing the ratio between two variances with values tabulated in F-distributions at 5% and 1% probability as a function of the numbers of degrees of freedom for each of the variances. Since kriged estimates are functionally dependent variable, they are not awarded degrees of freedom and Fisher's F-test cannot be applied to kriging variances. It is not surprising then that geostatistical scholars are trying to make Fisher's F-test and degrees of freedom disappear by decision, dictate or doctrine.

Why does it make sense to either replace a set of independently measured values or to enhance it with functionally dependent values by kriging with the bizarre proviso to avoid oversmoothing? So violate the fundamental requirement of functional independence a little but don't let the incredible kriging machine run out of control.