Sunday, October 31, 2010

Geostatistics in a nutshell

Assume, krige, smooth, and rig the rules of applied statistics. That’s all! It’s simple to assume spatial dependence between measured values in ordered sets. And to krige or not to krige is never a question. Not since Matheron himself in the 1960s cooked up the krigeage eponym. It took on a life of its own when the kriged estimate and the kriging variance became the heart and soul of Matheron’s new science of geostatistics. Smoothing sounds so soothing! Yet, the smoothing stage should not be taken lightly. In fact, the kriging variance of the least biased subset of some infinite set of kriged estimates should not be “over-smoothed”. Of course, one would not expect those who are taught to assume, krige and smooth to do too much of it with too few data. Matheron thought in 1954 that he worked with applied statistics. He was wrong! Journel in 1992 taught that spatial dependence between measured values in ordered sets may be assumed. Matheron’s most gifted disciple was wrong! It’s a piece of cake to assume spatial dependence between measured values in sampling units and sample spaces. To apply Fisher’s F-test to the variance of a set of measured values and the first variance term of the ordered set is much more intuitive. It was Journel who pointed out that one’s reading ought not to be "... too encumbered by classical Fischerian [sic!] statistics”.

To krige or not to krige has never been a dilemma in my work. Kriging between measured values in sampling units or sample spaces either enhances spatial dependence or gives a false positive. It was a cinch to prove that how the geostatocracy rigged the rules of applied statistics. Scores of textbooks show what went wrong, when and why. Geostatistics is all the rage with the world’s mining industry. It is true that the practice of assuming, kriging and smoothing does a lot with small sets of boreholes. But geostatistics, unlike applied statistics, does not give unbiased confidence limits for metal contents and metal grades of mineral inventories in annual reports. And that’s a fact!

Professor Dr Michel David wrote a few textbooks on geostatistics. My son and I studied his 1977 Geostatistical Ore Reserve Estimation. What we learned is that David did not derive confidence limits for metal contents and grades of ore reserves. So, we did it in our own paper on Precision Estimates for Ore Reserves. My son and I took at different times the same stats courses at Simon Fraser University. Ed earned his PhD in Computing Science and went to work at Big Blue in Toronto. Our paper was sent on September 28, 1989 to CIM Bulletin. David did reject it because we had shown “our own method”. Now whose method had he expected? He foretold in 1977 that “statisticians will find many unqualified statements”. Why then was he surprised when we did? Precision Estimates for Ore Reserves was praised by and published in Erzmetall 44, (1991).

The National Research Council of Canada was so taken with Matheron’s new science that it sponsored David’s 1977 Geostatistical Ore Reserve Estimation. Patronage played a role in Grant NRC7035. The very first page of his 1977 book shows an epiphany that had come to David. That’s where he wrote, “To our statistician readers, we apologize”. And so he should! In Section 2.1.1 The Standard Error of the Mean, the author pointed to what he then praised as “the famous Central Limit Theorem”. David was inspired by Figure 10 in Maréchal and Serra’s 1970 Random Kriging. M&S brought a bag of geostat stuff to the USA in 1970. It was made up at the Centre de Morphologie Mathématique, Fontainebleau, France. M&S had crafted it under Matheron’s guidance. So it came about that Random Kriging saw the light at the first krige-and-smooth shindig at the University of Kansas, Lawrence, USA on 7-9 June 1970. Dr Frederik P Agterberg, Dr Daniel F Merriam and a gathering of geostatistocrats were tickled pink. The central limit theorem had morphed into the kriged estimate. The few statisticians didn't question Brownian motion along a straight line.

Professor Dr Michel David in his 1977 textbook explained how M&S had derived sixteen (16) “famous Central Limit Theorems” from a set of nine (9) holes.

Chapter 10 Figure 203 page 286

The author clarified, “Pattern of all the points within B, which are estimated from the same nine holes”. Each and every one of David’s “estimated points” is a function of the same nine (9) holes. As such, each estimated point (otherwise known as Central Limit Theorem) does have its own variance. In fact, one-to-one correspondence between functions and variances is sine qua non in applied statistics. In short order, David’s estimated points morphed into kriged estimates. In Section Combination of Point and Random Kriging on the same page he suggested, “Writing all the necessary covariances for that system of equations might be a good test to find out whether one really understands geostatistics”. Here’s what I have already pointed out some twenty years ago, “Counting degrees of freedom for that system of equations is a good test to find out whether really understands applied statistics”.
Somehow it seems to make sense that geoscientists apply geostatistics. Some geoscientists call it “mathematical statistics” but did get rid of degrees of freedom. I work with Volk’s Applied Statistics for Engineers and with Visman’s sampling theory and practice. Applied statistics defines the sampling variogram, which, in turn, defines spatial dependence in sampling units and sample spaces alike.

Dr Frederik P Agterberg

He seems as ambitious and ambiguous today as he was in 1970. He ignores fundamental rules of applied statistics. He has yet to explain why he stripped the variance off his distance-weighted average. He did review and approve Abuse of Statistics. I asked Natural Resources Canada on September 21, 2010 for permission to interview Dr Frederik P Agterberg in writing. I have yet to receive a response to my request. It made me wonder whether or not he is still Emeritus Scientist with Natural Resources Canada.