Fighting factoids with facts

Niall Weatherstone of Rio Tinto and Larry Smith of Vale Inco have been asked to study a geostatistical factoid and a statistical fact. I asked them to do so by email on July 8, 2008. Next time they chat I want them to discuss whether or not geostatistics is an invalid variant of classical statistics. I’ve asked Weatherstone to transmit my question to all members of his team. CRIRSCO’s Chairman has yet to confirm whether he did or not. I just want to bring to the attention of his Crirsconians my ironclad case against the junk science of geostatistics.

Not all Crirsconians assume, krige, and smooth quite as much as do Parker and Rendu. The problem is nobody grasps how to derive unbiased confidence intervals and ranges for contents and grades of reserves and resources. Otherwise, Weatherstone would have blown his horn when he talked to Smith. A few geostatistical authors referred per chance to statistical facts. Nobody has responded to my questions about geostatistical factoids. The great debate between Shurtz and Parker got nowhere because the question of why kriging variances “drop off” was never raised. So I’ll take my turn at explaining the rise and fall of kriging variances.

In the 1990s I didn’t geostat speak quite as well as did those who assume, krige and smooth. I did assume Matheron knew what he was writing about but he wasn’t. Bre-X proved it makes no sense to infer gold mineralization between salted boreholes. The Bre-X fraud taught me more about assuming, kriging, and smoothing than I wanted to know. And I wasn't taught to blather with confidence about confidence without limits. It reminds me of another story I’ll have to blog about some other day. It’s easy to take off on a tangent because I have so many factoids and facts to pick and choose from.

Functions have variances is a statistical fact I’ve quoted to Weatherstone and Smith. Not all functions have variances I cited as a geostatistical factoid. Factoid and fact are mutually exclusive but not equiprobable. One-to-one correspondence between functions and variances is a condition sine qua non in classical statistics. Therefore, factoid and fact have as much in common as do a stuffed dodo and a soaring eagle. My opinion on the role of classical statistics in reserve and resource estimation is necessarily biased.

The very function that should never have been stripped off its variance is the distance-weighted average. For this central value is in fact a zero-dimensional point grade. All the same, its variance was stripped off twice on Agterberg’s watch. David did refer to “the famous central limit theorem.” What he didn’t mention is the central limit theorem defines not only the variance of the arithmetic mean of a set of measured values with equal weights but also the variance of the weighted average of a set of measured values with variable weights. It doesn’t matter that a weighted average is called an honorific kriged estimate. What does matter is that the kriged estimate had been stripped off its variance.

Two or more test results for samples taken at positions with different coordinates in a finite sample space give an infinite set of distance-weighted average point grades. The catch is that not a single distance-weighted average point grade in an infinite set has its own variance. So, Matheron’s disciples had no choice but to contrive the surreal kriging variance of some subset of an infinite set of kriged estimates. That set the stage for a mad scramble to write the very first textbook on a fatally flawed variant of classical statistics.

Step-out drilling at Busang’s South East Zone produced nine (9) salted holes on SEZ-44 and eleven (11) salted holes on SEZ-49. Interpolation by kriging gave three (3) lines with nine (9) kriged holes each. Following is the YX plot for Bre-X’s salted and kriged holes.

Fisher’s F-test is applied to verify spatial dependence. The test is based on comparing the observed F-value between the variance of a set and the first variance of the ordered set with tabulated F-values at different probability levels and with applicable degrees of freedom. Neither set of salted holes displays a significant degree of spatial dependence. By contrast, the observed F-values for sets of kriged holes seem to imply a high degree of spatial dependence.

If I didn’t know kriged holes were functions of salted holes, then I would infer a high degree of spatial dependence between kriged holes but randomness between salted holes. Surely, it’s divine to create order where chaos rules! But do Crirsconians ever wonder about Excel functions such CHIINV, FINV, and TINV? Wouldn’t Weatherstone want to have a metallurgist with a good grasp of classical statistics on his team?

High variances give low degrees of precision. I like to work with confidence intervals in relative percentages because it easy to compare precision estimates at a glance. SEZ-44 gives 95% CI= ±23.5%rel whereas SEZ-49 gives 95% CI= ±26.4%rel. By contrast, low variances give high degrees of precision. Three (3) lines of kriged holes give confidence intervals of 95% CI= ±0.8%rel to 95% CI= ±1.6%rel. Crirsconians should know not only how to verify spatial dependence by applying Fisher’s F-test but also how to count degrees of freedom. Kriging variances just cannot help but going up and down as yoyos!