# To have or not to have variances

Not a word from CRIRSCO’s Chairman. I just want to know whether or not functions do have variances at Rio Tinto’s operations. Surely, Weatherstone wouldn’t toss a coin to make up his mind, would he? My functions do have variances. I work with central values such as arithmetic means and all sorts of weighted averages. It would be off the wall if the variance were stripped off any of those functions. But that’s exactly what had come to pass in Agterberg’s work. I’ve tried to find out what fate befell the variance of the distance-weighted average. I did find out who lost what and when.  And it was not pretty in the early 1990s! When Matheron’s seminal work was posted on the web it became bizarre. The geostatistocrats turned silent and resolved to protect their turf and evade the question. They do know what’s true and what’s false. And I know scientific truth will prevail in the end.

Agterberg talked about his distance-weighted average point grade for the first time during a geostatistics colloquium on campus at The University of Kansas in June 1970. He did so in his paper on Autocorrelation functions in geology. The caption under Figure 1 states; “Geologic prediction problem: values are known for five irregularly spaced Points P1 –P5. Value at P0 is unknown and to be predicted from five unknown values.”

Agterberg’s 1970 Figure 1 and 1974 Figure 64

Agterberg’s 1970 sample space became Figure 64 in Chapter 10. Stationary Random Variables and Kriging of his 1974 Geomathematics. Now his caption states, “Typical kriging problem, values are known at five points. Problem is to estimate value at point P0 from the known values at P1 –P5”. Agterberg seemed to imply his 1970 geologic prediction problem and his 1974 typical kriging problem do differ in some way. Yet, he applied the same function to derive his predicted value as well as his estimated value. His symbols suggest a matrix notation in both his paper and textbook.

The following function sums the products of weighting factors and measured values to obtain Agterberg’s distance-weighted average point grade.

Agterberg’s distance-weighted average

Agterberg’s distance-weighted average point grade is a function of his set of measured values. That’s why the central value of this set of measured values does have a variance in classical statistics. Agterberg did work with the Central Limit Theorem in a few chapters of his 1974 Geomathematics. Why then is this theorem nowhere to be found in Chapter 10 Stationary Random Variables and Kriging? All the more so because this theorem can be brought back to the work of Abraham de Moivre (1667-1754).

David mentioned the “famous” Central Limit Theorem in his 1977 Geostatistical Ore Reserve Estimation. He didn’t deem it quite famous enough to either work with it or to list it in his Index. Neither did he grasp why the central limit theorem is the quintessence of sampling theory and practice. Agterberg may have fumbled the variance of the distance-weighted average point grade because he fell in with the self-made masters of junk statistics. What a pity he didn’t talk with Dr Jan Visman before completing his 1974 opus.

The next function gives the variance of Agterberg’s distance-weighted average point grade. As such it defines the Central Limit Theorem as it applies to Agterberg’s central value. I should point out that this central value is in fact the zero-dimensional point grade for Agterberg’s selected position P0.

Agterberg’s long-lost variance

Agterberg worked with symbols rather than measured values. Otherwise, Fisher’s F-test could have been applied to test for spatial dependence in the sample space defined by his set. This test verifies whether var(x), the variance of a set, and var1(x), the first variance term of the ordered set, are statistically identical or differ significantly. The above function shows the first variance term of the ordered set. In Section 12.2 Conditional Simulation of his 1977 work, David brought up some infinite set of simulated values. What he talked about was Agterberg’s infinite set of zero-dimensional, distance-weighted average point grades. I do miss some ISO Standard on Mineral Reserve and Resource Estimation where a word means what it says, and where text, context and symbols make for an unambiguous read.

But I digress as we tend to do in our family. Do CRIRSCO’s Chairman and his Crirsconians know that our sun will have bloated to a red giant and scorched Van Gogh’s Sunflowers to a crisp long before Agterberg’s infinite set of zero-dimensional point grades is tallied? And I don’t want to get going on the immeasurable odds of selecting the least biased subset of some infinite set. Weatherstone should contact the International Association of Mathematical Geosciences and request IAMG’s President to bring back together his distance-weighted average and its long-lost variance. That’s all. At least for now!