Teaching junk statistics at UBC

The stage was set in 1964 to teach junk statistics at the University of British Columbia. It was the year Professor Dr Alastair J Sinclair took on his duty to teach earth sciences to UBC’s students. It was but a few years after Matheron dabbled at his own kind of unreal statistics and fumbled real variances. Just the same, Matheron’s junk statistics was hailed as new science on campus at the University of Kansas in June 1970. His tour de force at that time was to call on Brownian motion to infer the continuity of his famed stationary random function. UK’s campus was a fitting venue because that’s where Agterberg failed for the first time to derive the real variance of his distance-weighted average point grade. Here’s why it gives too rich an abundance of data in mineral exploration. As few as a pair of measured values, when determined in samples taken at positions with different coordinates in a finite sample space, gives an infinite set of Agterberg’s point grades, a zero voodoo variance, and not a single degree of freedom. How about that? Some kind of perpetual motion in mineral exploration!

Sinclair details in Applied Mineral Inventory Estimation how his “exciting and invigorating career” took off when he was exposed to Matheron’s ideas, and how he had had “the good fortune to work with Journel, Huijbregts and Deraisme.” Those were Matheron’s earliest students who took his musings for dogma, and who didn’t have a clue which variances were lost on Matheron’s watch. Sinclair’s list of folks he was “fortunate to have worked with at various times” reads like a Who’s Who in the geostatistical fraternity. He credits all of them to have contributed to his education. I’m all in favor of giving credit where credit is due. But to give credit to everybody who taught him junk statistics is over the top. Some geostatistocrats on Sinclair’s list now know each weighted average has its own variance. And the odd one might even know why! One cannot help wonder how the cream of Matheron’s crop saw fit to make junk statistics look so good to Sinclair starting in 1972. So much so that Sinclair felt compelled to write his own textbook. Of course, all of that spelled bad news for UBC’s students.

When I met Sinclair at his UBC office in August 1992, I talked about real statistics. I showed how to count degrees of freedom for the set of nine holes in Figure 203 of David’s 1977 Geostatistical Ore Reserve Estimation. In Sinclair’s world, the concept of degrees of freedom breaks down in matters of spatial dependence. But it’s alive and well in my world. Sinclair did not see much of a difference between Matheron’s surreal geostatistics and Fisher’s real statistics. In fact, he knew as much about real statistics in August 1992 as he did in September 1989. That’s when CIM Bulletin entrusted Sinclair and David with the review of Precision Estimates for Ore Reserves. David blew a fuse because our paper was “without a single reference to 20 years of work in geostatistical ore reserve estimation.” And we didn’t even know we had written a geostatistical paper! So, we were baffled when Dr L R Fyffe, Editor, CIM’s Geology Division wrote on November 23, 1989, “Both reviewers recommend publication with major revisions.”

But big troubles were looming in the esoteric universe of those who infer, krige, smooth, and rig the rules of real statistics with reckless abandon. When I was working on Sampling and Weighing of Bulk Solids in the early 1980s, I studied David’s 1977 Geostatistical Ore Reserve Estimation. I found way too many symbols and far too few measured values. Brownian motion, too, played some kind of cameo role in this work of geostatistical fiction. The author confessed his work is "not for professional statisticians." In fact, he even predicted, “…statisticians will find many unqualified statements…” What David didn’t predict was he would deny anything was wrong in surreal geostatistics.

So what were we to do? Spice our paper with symbols? Scrap measured values? Delete Fisher’s F-test for spatial dependence? Call David to the task? Ask him to put in plain words his “good test to find out whether one really understands geostatistics” on page 286 of his 1977 textbook? Or try to pacify CIM Bulletin’s keepers of Matheron’s tablets with a few tidbits of token stuff? So we huffed and puffed a lot and added but a few references to works of geostatistical scholars such as Dagbert, David, Journel and Huijbregts. Our marginally revised paper was rejected on February 7, 1990. My son completed his PhD in computing science. I resolved to raise a stink. I did it then. And I still do now! Sinclair is but one reason. Bre-X’s phantom gold resource is another!

On November 23, 1989, CIM Bulletin’s editor wrote “Both reviewers recommend publication with major revisions.” Sinclair started some charade of sorts on November 22, 1989, at 08:30AM. He welcomed those who attended my short course on Sampling Precious Metal Deposits: Metrology-A New Look. The venue was Room 330A at UBC’s Department of Geological Sciences. The course was sponsored by its Mineral Deposits Research Unit. Sinclair didn’t have time to listen and moved about a lot. In fact, he popped in and out of Room 330A like a Jack-in-the-Box. Sinclair didn’t ask any questions. Was it because the paper he rejected was part of my notes? Did he worry others might ask questions? Did he worry I would talk too much about real statistics and too little about Matheronian geostatistics?

Dr J A McDonald, Interim Director, Mineral Deposits Research Unit, on February 21, 1990, wrote, “We certainly were pleased with the response to your course and have elected to maintain the theme with a 5-day course to be held April 23-27, 1990, entitled Geostatistics for the Mining Industry, New Concepts, New Tools.” How about that for cruel and unusual punishment? Sinclair was in damage control mode. So much more has happened in our stand-off on real statistics ever since I met Sinclair in his UBC Office in August 1992. Much of it will stay untold for some time to come.

Dr Alastair J Sinclair, PEng, PGeo, has striking credentials. He is a former Member of the Discipline Committee of the Association of Professional Engineers and Geoscientists of British Columbia with its Code of Ethics to protect the public at large. He was CIM’s Distinguished Lecturer for 2000-2001. He taught a short course at the UBC Robson Square Campus, Vancouver, BC, on May 15-16, 2008. What he didn’t teach was that each distance-weighted average has its own variance. He didn’t teach how to verify spatial dependence by applying analysis of variance and how to count degrees of freedom. Neither did he teach how to derive unbiased confidence interval and ranges for metal grades and contents of mineral inventories. Sadly, Sinclair is still teaching junk statistics!

Teaching junk science by consensus

The Centre de Géosciences/Géostatistique deserves praise for posting to its Online Library a treasure trove of writings. A great deal came from the seminal work of Professor Dr Georges Matheron (1930-2000). Most of it merits long overdue scrutiny and review. The problem is not so much that Matheron put a few spurious findings on paper but that his students took it for doctrine. The Online Library has made it easy to pinpoint what Matheron did wrong and when he did so.

Matheron derived the length-weighted average grade of a set of metal grades determined in core samples with variable lengths. He did so in his Rectificatif of January 13, 1955, to Formule des Minerais Connexes of November 25, 1954 (see Note Statistique No 1). What he didn’t derive was the variance of this length-weighted average grade. Neither did he show how to test for spatial dependence between grades of ordered core samples by applying analysis of variance. He didn’t report primary data sets because of his penchant for working with symbols rather than with real measured values.

Matheron concocted the honorific eponym krigeage in his 1960 Krigeage d’un Panneau Rectangulaire par sa Périphérie. In this Note géostatistique No 28, Matheron derived k*, his “estimateur”, and a precursor to kriged estimate or kriged estimator. In real statistics, Matheron’s k* is in fact the length-weighted average grade of a single block. In this case, too, he didn’t derive var(k*), the variance of his “estimateur”. Sadly, kriging became a curse of sorts for Professor D G Krige.

Matheron’s Stationary Random Function seemed not to have troubled those who were at the first geostatistics colloquium in the USA in 1970. Matheron even called on Brownian motion to infer by hook or by crook the continuity of his Riemann integral. He didn’t explain what Brownian motion and mineral deposits have in common. Matheron, unlike John von Neumann in 1941 and Anders Hald in 1952, didn’t work ever in his life with Riemann sums. On the contrary, he would rather infer spatial dependence than apply Fisher’s F-test to the variance of a set and the first variance term of the ordered set.

It is to Matheron’s credit that it was not him who lost variances of all weighted averages. It was Dr Frederik P Agterberg who failed to derive the variance of his distance-weighted average. He did derive the distance-weighted average grade of a set of five (5) points at positions with different coordinates but failed to derive the variance of this central value. What he didn’t point out was that as few as two such points define an infinite set of distance-weighted averages. He fumbled the variance of his central value for the first time in his 1970 colloquium paper and once again in his 1974 Geomathematics.

Matheron’s length-weighted average grade was reborn as an honorific kriged estimate or estimator. But then Agterberg’s distance-weighted average grade was honored in the same way! And here’s the clincher! An infinite set of Agterberg's zero-dimensional point grades fits along any borehole, and within any ore block, sampling unit or sample space. That’s why distance-weighted average point grades without variances became the heart and soul of geostatistics. Matheron’s seminal work merely set the stage for Agterberg’s giant step into the abyss of mineral reserve and resource estimation with confidence but without confidence intervals and ranges.

The above figure is a facsimile of Fig. 203 on page 286 of David's 1977 Geostatistical Ore Reserve Estimation. It shows the infinite set of "estimated" values within B derived from the same set of nine (9) holes.

The more geostatistocrats tinkered with real statistics, the more flawed geostatistics grew. It’s a scientific fraud to derive confidence limits from pseudo kriging variances. It’s as silly to talk about confidence without limits as it is to infer spatial dependence within or between boreholes. To discount degrees of freedom would make no sense at all in real statistics. To count degrees of freedom makes no sense in geostatistics. That’s the very reason why geostatistics does not give unbiased confidence limits for metal contents and grades of mineral reserves or mineral resources.

Professor Dr Roussos Dimitrakopoulos is a catch of sorts for the Department of Mining, Metals and Materials Engineering at McGill University. I don’t know why! I told him in 1993 that weighted averages have variances because one-to-one correspondence between functions and variances is sine qua non in statistics. This basic rule is still beyond his grasp in 2008. All the same, he is Editor-in-Chief, Journal of Mathematical Geosciences. Agterberg, President, International Association for Mathematical Geosciences, left his fingerprints when he failed to derive the variance of his distance-weighted average point grade. Dimitrakopoulos talks about “gazillion types” of probabilistic models. What he doesn’t talk about is that the odds to select the least biased subset of some infinite set of kriged estimates are immeasurable. The problem is not so much he himself believes it but the world’s mining industry believes it. The more so because he does all of that with voodoo variances.

Mining engineers, mine geologists, resource analysts, and project managers were invited to a course on Applied Risk Assessment for Ore Reserves and Mine Planning at McGill University. The same course deals with Strategic Risk Quantification and Management for Ore Reserves and Mine Planning and with Conditional Simulation for the Mining Industry. That’s a lot of buzz for a bundle of bucks! Too bad that voodoo variances underpin all that risk assessment and quantification stuff! That’s why one should come with a buddy. For it’s more difficult to baffle a few birds of a feather than a single sitting duck. Dimitrakopoulos should explain why Agterberg’s distance-weighted average point grade aborted its variance during its rebirth as an honorific kriged estimate on Matheron’s watch.