Wednesday, August 27, 2008

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. After Matheron's seminal wrok 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 fals. And I know scientific truth always prevails 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 well 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 yearn for 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 ask its President to bring back together his distance-weighted average and its long-lost variance. That’s all. At least for now!

Thursday, August 07, 2008

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!