Rebranding Professor Dr Georges Matheron

Dr Frederik P Agterberg tried to do so when he sang the praises of Professor Dr Georges Matheron and called him the Founder of Spatial Statistics. The keepers of Matheron’s magnum opus at his own Centre de Géostatistique didn’t quite see eye to eye with Agterberg’s rebranding. Matheron’s disciples were taught to hold him in the highest regard as the Creator of Geostatistics. It was Matheron himself who called geostatistics a new science in the early 1960s. Here’s in a nutshell what had inspired Matheron so much in his most creative of days. He taught that, “geologists stress structure and statisticians stress randomness”. I liked that a lot. I would have liked it even more had Matheron shown how to test for absence or presence of structure. All it would have taken is to apply Fisher’s F-test to the variance of a set of measured values and the first variance term of the ordered set. He would have had to count the number of degrees of freedom for each set. That was a bit of a problem. Matheron and his following never got around to counting degrees of freedom.

On a positive note, Matheron did test for associative dependence between lead and silver grades of drill core samples. This test may well have been the very reason why Matheron thought he was a statistician. His 1954 Formule des Minerais Connexes is indeed marked Note Statistique No 1. In his Rectificatif à la Note Statistique No 1, Matheron derived weighted average lead and silver grades. What he failed to derive were variances of weighted average lead and silver grades. So, I am quite pleased that the Centre de Géostatistique has posted so much of Matheron’s work. On the negative side, its webmaster saw fit to predate the evolution of Matheron’s new science of geostatistics. That’s why his very first paper did end up as Note Géostatistique No 1. Providentially, his 1954 Formule des Minerais Connexes and its Rectificatif are still marked Note Statistique No 1.

So it was that Matheron didn’t take to working with the Central Limit Theorem. David did recall the famous Central Limit Theorem in his 1977 Geostatistical Ore Reserve Estimation. He didn’t much work with it either. A critical subject that failed to make Matheron’s list of things to teach is one-to-one correspondence between functions and variances. Yet, it is a condition sine qua non in mathematical statistics. It is no wonder then that the properties of variances are beyond the grasp of the geostatistical fraternity. I have never thought much of Professor Dr Georges Matheron’s thinking. Whenever I do think of Matheron, I remember him as a self-made wizard of odd statistics.

Professor Dr Michel David took a shine to Matheron’s new science of geostatistics. David did so while he was teaching at l’École Polytechnique, University de Montréal, Québec, Canada. And he did predict that ‘statisticians would find many unqualified statements’ in his 1977 Geostatistical Ore Reserve Estimation. He didn’t predict he couldn’t care less if someone pointed out what was wrong and why. Some twenty years ago I did but few cared. So, I’ll just keep doing it again and again! Chapter 10 The Practice of Kriging shows how to do more with fewer boreholes by paying no attention at all to the rules of mathematical statistics.

Fig. 203. Pattern showing all the points within B,

which are estimated from the same nine holes.

David borrowed the above figure from Maréchal and Serra’s 1970 Random Kriging. Both were scholars at Matheron’s Centre de Morphology Mathématique. Here’s word for word what I have come to call David’s test for geostatistical acuity. “Writing all the necessary covariances for that system of equations is a good test to find out whether one really understands geostatistics”. I have pointed out that a good test to find out whether one really understands mathematical statistics is to count the number of degrees of freedom for David’s system of equations. The correct count is zero! That’s how Matheron and his timid minions took reserve and resource estimation into a dead-end street.

But even more bad science pops up in Chapter 12 Orebody Modelling. In Section 12.2 Conditional Simulation, David wrote about the infinite set of simulated values. He wonders how to make infinite sets smaller and get models closer to reality. In Section 12.2.1 Using a Simulated Model, he wrote about some pudding proof and a posteriori proved simulations. But nobody cries out loud in the face of such blatant nonsense. What are the odds to win when playing 649?

So, why then did Agterberg try to rebrand Matheron the Founder of Spatial Statistics after he had passed away? Now that’s a long story. The short of it is that there are many more geoscientists than geologists on our little planet. Remember global warming? And how to assume spatial dependence between measured values in ordered sets? That’s what way too many geoscientists are taught. Stay tuned for real statistics. And tune out to surreal geostatistics.

One more message to CIM's President

CIM stands for Canadian Institute of Mining, Metallurgy and Petroleum. Once upon a time I was a proud CIM Member. Today I am the accidental CIM Life Member. My first message to CIM’s President was snail mailed on March 20, 1992. CIM’s President was William E Stanley of The Coopers & Lybrand Group in Vancouver. He was the first of many whom I had told why geostatistics is an invalid variant of applied statistics. We met, he listened to my story, and I wrote him a letter. CIM Bulletin of March 1989 had published Armstrong and Champigny’s A Study on kriging Small Blocks. Both authors were geostatistical scholars at the Centre de Géostatistique, France. They thought up the study since, “The kriging variance rises up to a maximum and then drops off.” What they also found out is that “…mine planners are often tempted to kriging very small blocks.” How about that? Smoothing is good but smoothing very small blocks is bad. That sort of a pass-the-buck study did pass David’s peer review with red flags blazing.

Early in 1990 we found out that Precision Estimates for Ore Reserves was rejected. Our paper showed how to test for spatial dependence between gold grades of ordered rounds in a drift. David’s 1977 textbook didn’t show how to test for spatial dependence, or how to count degrees of freedom. Neither did his work show how to derive unbiased confidence limits for metal contents and grades of in-situ ores. So, I put Geostatistics or Voodoo Science on paper, and The Northern Miner printed it on March 20, 1992. Champigny was no longer a geostatistical scholar at the Centre de Géostatistique in France but a Senior Consultant with The Coopers & Lybrand Group in Toronto. He never lost his passion for kriging and smoothing. As a matter of fact, he rounded up a team of anonymous ore reserve practitioners to stand on guard against the rise and fall of kriging variances. What he and his team did prove was that the properties of variances were far beyond their grasp. The Northern Miner put Champigny’s rambling tale in print on May 18, 1992. Armstrong went beyond the pale and lectured on scientific integrity in De Geostatisticis of July 1992.

Following is the text of my emessage of January 13, 2010, to Michael J Allen, CIM’s President, Vice President, Engineering, with Teck Corporation, Member of APEGBC and SME, and a CIM Fellow:

About twenty years ago I reported to CIM that geostatistics is an invalid variant of applied statistics. Geostatistocrats with CIM Bulletin promptly put up a spirited battle to salvage the new science of geostatistics. And a fine job they did! Matheron's madness of surreal geostatistics even survived the Bre-X fraud. Statistics turned into geostatistics under the guidance of Professor Dr Georges Matheron, a French probabilist who became a self-made wizard of odd statistics in the 1950s. A brief history of my 20-year campaign against the geostatocracy and its army of degrees of freedom fighters is chronicled on my website.

Dr Frederik P Agterberg, Past President, International Association for Mathematical Geosciences formerly know as International Association for Mathematical Geology, called Matheron (1930-2000) the Founder of Spatial Statistics. Agterberg ranked Matheron on a par with giants of real statistics such as Sir Ronald A Fisher (1890-1962) and Professor Dr J W Tukey (1915-2000). Agterberg was wrong! Matheron fumbled the variance of the length-weighted average in 1954. Agterberg himself fumbled the variance of the distance-weighted average first in his 1970 Autocorrelation Functions in Geology and once more in his 1974 Geomathematics. Agterberg is Emeritus Scientist with Natural Resources Canada. He ought to but has yet to explain why his distance-weighted average point grade does not have a variance. After all, Gemcom's geostatistical software converted Bre-X's bogus grades and Busang's barren rock into a massive phantom gold resource. I applied Fisher's F-test to prove that the intrinsic variance of Bre-X's phantom gold resource was statistically identical to zero. Duplicate test results for gold by cyanide leaching determined in a few boreholes would have been enough to unravel the Bre-X fraud in a timely manner.

I make a clear and concise case for real statistics. Test for spatial dependence by applying Fisher's F-test to the variance of a set of measured values and the first variance term of the ordered set. Chart a sampling variogram to show where spatial dependence in a sample space (or in a sampling unit) dissipates into randomness. We applied Fisher's F-test in Precision Estimates for Ore Reserves. And we did it again in our APCOM 2009 paper entitled Metrology in Mineral Exploration.

Geostatisticians assume spatial dependence between measured values in ordered sets, interpolate by kriging, smooth some kind of least biased subset of an infinite set of Agterberg's zero- dimensional and variance-deprived distance-weighted average point grades AKA kriged estimates or kriged estimators, and rig the rules of real statistics with reckless abandon. I urge CIM to investigate whether or not geostatistics is a scientific fraud. I do so as a CIM Life Member. Please do not assume that CIM need not resolve this matter.

To strip or not to strip?

CIM Bulletin approved Abuse of Statistics for publication. Dr Frits Agterberg wanted to know when and where Wells spoke so highly about statistical thinking. I wasn’t about when Wells said what he did. What I do know is that Darrell Huff said Wells did. That’s good enough for me. Huff did so in his 1954 How to Lie with Statistics. It was the very same year that young Matheron didn’t know how to test for spatial dependence between metal grades of ordered core samples, how to derive the variance of the set of metal grades, and how to derive the variance of the central value of the set. Huff never found out what Matheron did wrong. But then, neither did Matheron himself! Agterberg, Armstrong, David, Journel and scores of geostatistocrats never broke rank with Matheron.

I want to move fast forward to the present. Michael J Allan, CIM President in 2010, writes under President’s Notes about A time of renewal. Let’s see what else he wrote. “Our work in providing standard reserve and resource definitions that are used by the country’s securities regulators is an example of the ongoing technical contributions CIM makes to the industry at large”. For heaven’s sake! Geostatistics is as alive and flawed as it was in the 1970s. So it seems that CIM is not about to kill the incredible kriging machine. Surely, infinite sets of kriged estimates and zero kriging variances set the stage for boundless krige and smooth fests. APEGBC ‘s Code of Ethics is not written to rule against scientific fraud. What will kill the kriging machine is the study of climate dynamics on our little planet. No ifs and buts!