Matrix report worth its weight in gold

Same time thirteen years ago some of Bre-X’s test results for gold landed on my desk. I had not asked for Bre-X’s data. But I had agreed to and signed a three-year confidentiality agreement with Barrick Gold Corporation. I did so on December 16, 1996. It was the very same confidentiality agreement that Barrick Gold Corporation and Bre-X Minerals had signed a few days earlier. The first set of Bre-X data were transmitted by facsimile on December 17, 1996. I didn’t know then that my life would never be the same. Bre-X‘s infamous phantom gold resource is but part of a tangled tale with as many twists and turns as Matheron took to create his new science of geostatistics. It’s a tale that taught me a lot more about the mining industry than I cared to know.
I sorted out the Bre-X fraud faster than Bre-X’s salting squad took to cook it up. I think Barrick liked what I did. At least Barrick did when I applied statistics to prove that Bre-X was a salting scam. So much so that I signed on July 4, 1997 a Consulting Services Agreement with Barrick Goldstrike Mines Inc. I submitted on August 18, 1997 my report on Statistical Quality and Grade Control . Geostatisticians on Barrick’s staff didn’t think much of it. I had applied Fisher’s F-test to verify spatial dependence between gold grades of ordered core sections from a single borehole by applyingit to the variance of the set and the first variance term of the ordered set. I had done the same thing with Bre-X’s salted boreholes. Stanford’s Journel would have assumed rather than verified spatial dependence. But then, Matheron’s most gifted disciple never signed a Consulting Services Agreement with Barrick Goldstrike Mines Inc.
When I was working with Bre-X’s test results my closest contact was a staff mining engineer at Barrick Gold Corporation in Toronto. We got along great because he knew plenty about sampling and assaying. So, he knew why Bre-X’s bogus grades and Busang’s barren rock added up to a geostatistically engineered gold resource. He also knew how to test for spatial dependence, and why geostatistics should not be applied in reserve and resource estimation. And he asked me whether I wanted to take a look at a large set of borehole data for a real gold deposit. Guess what? So, I did agree to and signed on October 22, 1997 a confidentiality agreement with Barrick Gold Corporation. I submitted my report on Confidence Limits for Gold Contents and Grades on February 9, 1998. When I called my contact to find out what he thought of my report, he said, “It’s worth its weight in gold”. I didn’t ask him to put it in writing. His word was good enough for me!

Worth its Weight in Gold

Geologists, mining engineers and mineral process engineers rarely agree on metal grades of in-situ ores, mined ores and mill feed. I witnessed many such rituals. Top brass wants high mineral inventories in glossy annual report and geostatisticians always deliver. Barrick’s geologists may find confidence limits for gold contents and grades of mineral inventories a bit much of a commitment. Shareholders do want a measure for risk.
Another year passed by, Christmas 1999 came along, and the Confidentiality Agreement between Barrick Gold Corporation and Bre-X Minerals expired. I liked to talk about the Bre-X fraud. Barrick engaged lawyers who wanted to come to Vancouver and tell me not to talk. I called on a friend and the visit to Vancouver was cancelled. All I have done since Christmas 1996 is show why geostatistics is a scientific fraud.
What Barrick asked me ten year later blew my mind. Barrick wanted consulting services. I’m not about to describe the required services but it had nothing to do with confidence limits for gold contents and grades of in-situ ore. I agreed to and signed on March 20, 2007 a Consulting Services Agreement for services to be provided at Barrick Technology Centre, Vancouver, BC. My contact had a lot of practical experience but stood to gain from a touch of real statistics. Before we could get going he was needed at Barrick’s Bulyanhulu gold mine in northwest Tanzania. Long before Barrick acquired Placer-Dome and its former Bulyanhulu gold deposit I knew Placer-Dome had born geostatisticians on board.
A Munk Debates on scientific fraud makes no sense whatsoever. Who would dare make a case for scientific fraud? Yet, a scientific fraud underpins the geostatistical practice of reserve and resource estimation all over the world. Blatantly biased, shameless self-serving peer review is all it took. But that’s another story. I have called it Behind Bre-X, The Whistleblower’s Story.

Who wants more Munk Debates

Who wouldn’t! Debates beat apathy. The Munk Debates is cool. The more so since climate change was the theme for the Fourth Munk Debates. Climate change, just like continental drift, has been around for a few billion years. It took geologists from 1915 to 1950 to slow down to continental drift and call it plate tectonics. So, it’s about time to debate climate change. And why not call it weather dynamics? I work with metrology, the science of measurement. I took a crack at testing whether or not annual temperatures at several locations in Canada have changed significantly as a function of time. The average temperature of 6.57 centigrade in 2007 at Ottawa International Airport was significantly higher than the average temperature of 4.79 centigrade in 1939. Similarly, the average temperature of 8.30 centigrade in 2007 at Toronto International Airport was significantly higher than the average temperature of 6.04 centigrade in 1939. Average temperatures didn't change at international airports in Calgary, Vancouver and Victoria. Neither did the average temperatures in Coral Harbour and Iqaluit change significantly during the test period under examination.


Some grasp of statistics is required to apply Fisher’s F-test and verify spatial dependence between annual temperatures in ordered sets. Weather dynamics do change from day to day, from week to week, and from month to month. Such short-term changes in temperatures do not merit a Munk Debates. What does merit a Munk Debates is the question whether or not geostatistics is a scientific fraud.
Here’s in a nutshell my take on the Fourth Munk Debates. Elizabeth May is Leader of the Green Party of Canada. She is a gifted and confident speaker. She knows a lot of environmental stuff. She doesn’t know much about temperatures recorded by Environment Canada. Given that the Leader of the Green Party does speak a lot in public, she should know where temperatures went up or down, since when, and by how much.
George Monbiot was her partner in the Fourth Munk Debates. He is a superb scribe with the Guardian newspaper where his penchant for hyperboles runs rampant. How to measure climate change as a function of space and time is far beyond his grasp. Monbiot says cool things such as, “Canada is a cultured, peaceful nation, which every so often allows a band of Neanderthals to trample over it.” He doesn’t know Sir Ronald A Fisher ‘s work is trampled over by a tribe of statistically dysfunctional geoscientists bred in France, Great Britain, and elsewhere on this planet. The May/Monbiot side debated The Case For Climate Change.
Lord Nigel Lawson and Bjorn Lomborg debated The Case Against Climate Change. Lord Lawson is in a class apart when it comes to a life of public service in the United Kingdom of Great Britain. His work has done much to cool down global warming to climate change. He is the author of An Appeal to Reason, A Cool Look at Global Warming. He is the Chairman of Oxford Investment Partners, and of Central Europe Trust. As such, he knows all about mining conglomerates and mineral inventories in annual reports. He is bound to remember the Bre-X fraud. He may be unaware that geostatistical software converted Bre-X’s bogus grades and Busang’s barren rock into a huge phantom gold resource. Neither may Lord Lawson remember the cast of characters behind the Bre-X fraud.
Bjorn Lomborg’s claim to fame is based on The Skeptical Environmentalist and on Cool It. He is adjunct professor at the Copenhagen Business School. He also set up the Copenhagen Consensus Center to bring together those who set priorities for the world. I had brought to his attention in August 2008 that junk statistics underpins Matheron’s new science of geostatistics. I wanted to know whether he applies geostatistical data analysis. Environment Canada points to geostatistical data analysis in its handbook for inspectors. The skeptical environmentalist did not respond to my message.
The Merks and Merks team wants to debate The Case Against Geostatistics. Dr Frits P Agterberg, Emeritus Scientist with Natural Resources Canada, and Dr Roussos Dimitrakopoulos, Professor with McGill University, are highly qualified to debate The Case For Geostatistics. Both are serving in key positions with IAMG (International Association for Mathematical Geosciences). Once upon a time, IAMG stood for International Association for Mathematical Geology. Nowadays, our world needs more mathematical statistics.

Chatting with NRCan's Emeritus Scientist

Dr Frits P Agterberg is Emeritus Scientist with Natural Resources Canada. He wrote a textbook on Geomathematics and scores of papers on a wide range of geological topics. He is the nimblest of geostatistical minds on this planet. His gift to assume spatial dependence between measured values in ordered sets is second to none but Stanford’s Journel. I called him on November 4, 2009, at NRCan in Ottawa but he was away from his Office. I caught him at home when I called his residence at 09:10 AM PDST. I asked him to explain why his zero-dimensional distance-weighted average point grade does not have a variance.

He hummed and huffed but didn’t speak to the matter of the missing variance. All I wanted to know is why the variance of his distance-weighted average went missing. I pointed out that the Central Limit Theorem pops up if all of his measured points are equidistant to his selected point. NRCan’s Emeritus Scientist beats around the bush with the best. His textbook does refer to the Central Limit Theorem in Chapter 6 Probability and Statistics and Chapter 7 Frequency Distributions of Independent Random Variables but not in Chapter 10 Stationary Random Variables and Kriging. NRCan’s Emeritus Scientist has yet to give a clear and concise explanation why the Central Limit Theorem doesn’t apply to his distance-weighted average point grade.

I included Agterberg’s problems in my talk about Metrology in Mineral Exploration. I wanted to make a case at APCOM 2009 that distance-weighted average point grades do have variances. Nobody was ready for my show-and-tell but I got a gift. It was Clark’s Practical Geostatistics 2000. I found out that semi-variograms are still alive and below par. Here’s Clark’s problem. Her set of five (5) hypothetical uranium data doesn’t display a significant degree of spatial dependence. Thus, the concentration at the selected coordinates is not necessarily an unbiased estimate. Let’s find out what happens when coordinates are selected beyond her sample space.

Who expects the distance-weighted average point grade to converge on zero? And who expects it to converge on the arithmetic mean? It's a good test to find who is geostatistically gifted and who is not. I would rather test for spatial dependence between measured values in ordered sets and chart sampling variograms that show where spatial dependence dissipates into randomness. Come hell, high water, global cooling, polar warming, or another Bre-X.

My first APCOM affair was just as cluttered with geostat drivel as are all of IAMG’s shindigs. McGill’s Professor Dr Roussos Dimitrakopoulos sought to shed light on stochastic mine planning optimization. He is Editor-in-Chief, Journal for Mathematical Geosciences. That’s why all his work passes his own litmus test for scientific integrity with flying colors. Somehow, it may have slipped his mind how geostatistical software converted Bre-X’s bogus grades and Busang’s barren rock so smoothly into a massive phantom gold resource. But then, the geostatocracy has worked long and hard to ensure mining professionals never get a grasp of classical statistics.

It brings me back to my chat with NRCan’s Emeritus Scientist. I brought to his attention that a good test to verify McGill's stochastic mine planning optimization would be to apply it to Bre-X’s data. Agterberg saw it differently because Bre-X's data was “no real data”. No real data? But mining investors thought Bre-X was real! Didn't Gemcom’s software convert Bre-X’s bogus grades and Busang’s barren rock into a massive phantom gold resource? And wasn't the battle to take over Bre-X Minerals a really bizarre affair?

This was my second chat with NRCan’s Emeritus Scientist after we had found out in 1989 that geostatistics is a scientific fraud. It brought back an odd dialogue in 1992 with Dr W D Sinclair, Editor, CIM Bulletin, and Dr F P Agterberg, Associate Editor. We talked about a technical brief on Abuse of Statistics. I'll keep that tangled tale for some other place and time!

Spatial dependence in mineral exploration

Some twenty years ago my son and I submitted to CIM Bulletin a paper on Precision Estimates for Ore Reserves. David, CIM Bulletin's reviewer, blew a fuse because we didn’t refer to “twenty years of geostatistical literature”. We did study David’s 1977 Geostatistical Ore Reserve Estimation and Clark’s 1979 Practical Geostatistics. Neither author showed how to test for spatial dependence. So, we showed how to test for spatial dependence between gold assays determined in bulk samples taken from twelve (12) rounds in a drift. CIM Bulletin was but one of several journals to reject our paper. Yet, the very same paper was praised by and published in Erzmetall 44, October 1991. We could not show how to estimate the intrinsic variance of gold because but a single bulk sample was taken from each round.
It was easy to estimate the intrinsic variance of gold in Bre-X’s phantom resource. Bre-X’s quality control program was based on selecting and testing duplicate test portions of every tenth crushed and salted core sample. The set of duplicate gold assays for Bre-X’s bonanza borehole BSSE198 gave enough degrees of freedom to estimate the analytical variance with a high degree of precision. Fisher’s F-test proved that the analytical variance and the first variance term of the ordered set are statistically identical. Hence, the intrinsic variance of gold in BSSE198 was statistically identical to zero. Plenty of placer gold was present in crushed and salted core samples but Bre-X’s bonanza borehole BSSE198 was barren.


When APCOM 2009 asked for abstracts, I talked to my son about presenting one more paper on our home turf. His talk about EMF at some school of mines in Nantes, France, took him too far away from Vancouver to attend APCOM 2009. Our abstract was based on a bulk sampling program at the Cerattepe project in Turkey where core recovery was poor. So, I advised my client to implement an interleaved bulk sampling program in order to derive unbiased confidence limits for in-situ gold and silver. Our abstract was accepted and Metrology in Mineral Exploration was approved.
I spoke to a small group on Thursday, October 8, 2009, at 15:30. I showed how to unscramble the Bre-X fraud, and how to derive the statistics for Cerattepe's bulk sampling program.

Spatial dependence is significant at 99.9% probability
Lag of 4.30 m at 95% probability is defined for gold

Spatial dependence is significant at 99.9% probability
Lag of 4.09 m at 95% probability is defined for silver

I explained how to correct those sampling variogram for the extraneous measurement variance estimated from pairs of interleaved primary samples, and how to derive 95% confidence limits for in-situ masses of gold and silver.

I asked my audience why the variance of Agterberg’s distance-weighted average point grade is still missing. I didn't get any response. Not a single question was asked. There was but a pinch of polite applause. My soul mate got an anonymous note together with the second coming of Clark’s 1979 Practical Geostatistics on DVD. Which APCOM 2009 sponsor ignored my question but did hand my spouse that anonymous note? Was it Gemcom? Or did Geovariances do it?
I was tickled pink with that priceless gift. In her first coming Clark cooked up a semi-variogram, berated those who "sloppily" call it a variogram. Yet, Clark praised Journal and his buddies for teaching her all she knows about “the theory of the Theory of Regionalized Variables.” Journel may well have taught Clark how to assume spatial dependence between measured values in ordered sets. He might even have cautioned Clark, too, not to become “too encumbered with Fischerian [sic!] statistics”. But what did Professor Dr William V Harper teach Dr Isobel Clark between 1979 and 2000? Sadly, Clark’s learning curve simply flat lined! She still doesn’t test for spatial dependence in sampling units and sample spaces. She still scolds those who work with variograms rather than with her own sacred semi-variogram. There's still no progress!
Statistics or geostatistics? Sampling error or nugget effect? Clark talked about those questions at WCSB4 in Cape Town on 21-23 October 2009. Sampling error adds a nice touch of Gy’ological thinking to Clark’s repertoire. Testing for spatial dependence failed to make her grade. Why did she take the factor two (2) out of degrees of freedom for ordered sets. Why does she deem too sloppy sampling variograms that show where orderliness in sample spaces or sampling units dissipates into randomness. Clark and Harper are ready to take Gy's sampling theory to sampling practices in mineral exploration, mining, processing, smelting and refining? Why does Harper not recognize that geostatistics is a scientific fraud? Strip the variance of the distance-weighted average, assume spatial dependence between measured values, interpolate by kriging, smooth the least biased subset of some infinite set of distance-weighted averages, and rig the rules of real statistics with impunity.

Who wrote bogus stats, when, where, and why

Professor Dr Roussos Dimitrakopoulos came up all the way from Down Under to chair a Forum on Geostatistics for the Next Century at McGill University on June 3-5, 1993. His task was to honor Professor Dr Michel David for writing the very first textbook on Matheron’s new science of geostatistics. David didn’t know how to test for spatial dependence and how to count degrees of freedom. He wrote his first textbook against all odds since he didn’t even know that functions do have variances. I have written quite a bit about the properties of variances. So, I send by registered mail an abstract to that futuristic forum at McGill University. Some person at McGill’s Conference Office encouraged me in an unsigned letter of March 31, 1993, to submit my abstract to another event. I'll have to dig up more bits and pieces about genuine variances.
Dimitrakopoulos likes McGill a lot. In fact, he settled down in La Belle Province after the Bre-X fraud was no longer on his mind. In a candid interview with the National Post on August 15, 2005, he clarified the intricacies behind his valuations of mining projects. Here’s what he said, “You drill a few holes, you think you understand something but what you know is very, very little. Uncertainty means probabilistic models, and there are a gazillion types of them.” How about that? Some mining investors might wonder how RD selects the least biased probabilistic model. Peter Ravenscroft, a senior executive with Rio Tinto and an expert at geostatistics, thinks what RD does is kind of cool and gave him a stack of dough.
Professor Dr Roussos Dimitrakopoulos was present at APCOM 2009 in Vancouver, British Columbia. The first line of his abstract reads, “Conventional approaches to estimating reserves and optimizing mine planning and production forecasting result in single, often biased forecasts.” I wonder what would have happened if Stochastic Mine Planning Optimization: New Concepts, Applications, and Monetary Value in an Ever Uncertain Market, had been applied to Bre-X’s exploration data. I also wonder why regulators and financial institutions do not insist the International Organization for Standardization set up a Technical Committee on Reserve and Resource Estimation. It's long past due! Matheron thought he was a statistician in 1954. Yet, his Note Statistique No 1 shows he didn't know how to test for spatial dependence between metal grades in ordered core samples. Neither did he know how to derive variances of length-weighted average lead and silver grades determined in core samples of variable lengths. So much for Matheron's new science of geostatistics!

Dr Frederik P Agterberg wrote in 2000 that Matheron was the Founder of Spatial Statistics. Matheron thought he was a statistician in 1954 when he wrote his Note Statistique No 1. He didn't write about spatial dependence between metal grades of ordered core sections with variable length. He did derive length-weighted average lead and silver grades but didn't derive the variances of these central values. In 1907 he stirred up "Brownian motion on a straight line." He did so because he liked Riemann integrals better than Riemann sums. He wrote in his 1978 Foreword to Mining Geostatistics why he proposed the name geostatistics in the 1960s. Professor Georges Matheron would have been shocked had he read in his obituary that he was the Founder of Spatial Statistics. Agterberg invited me on October 1, 2004, to present my views at the next IAMG annual meeting in Toronto. I happen to know a lot about IAMG events where geostatistocrats talk bafflegab. I would rather make my case against bogus stats at APCOM 2009.

Dr Michel David wrote a few words of caution in his 1977 Geostatistical Ore Reserve Estimation. First, he wrote, "...statisticians will find many unqualified statements..." Then, he blew the sales of his work by writing, "This is not a book for professional statisticians." But he was indeed right. David did prove it when he wrote his test for geostatistical proficiency. He took M&S's set of nine (9) measured values and "estimated" the same set of sixteen (16) what he came to call "...points..." He wrote on page 286 of his textbook, "Writing all the necessary covariances for that system of equations is a good test to find out whether one really understands geostatistics." Why did the author of the very first textbook on geostatistics fail to derive the variance of each of this sixteen (16) functionally dependent values? Why didn't he count degrees of freedom? If M&S's set of nine (9) measured values were evenly spaced, the set and the ordered set would give df=n-1=8 and dfo=2(n-1)=16 respectively. Why is the geostatocracy still asleep at the switch? Why is Bre-X's massive phantom gold resource all but forgotten?

A Marechal and J Serra wrote Random kriging in 1970 to celebrate the first krige and smooth bash in North America. M&S toiled under Matheron's tutelage at his Center de Morphology Mathematique, Fontainebleau, France. So, why did M&S set out to simplify Matheron's kriging equations with their own random kriging procedure? Under Punctual Kriging in Random kriging they show how to get a set of sixteen (16) functionally dependent values from a set of nine (9) measured values. M&S didn't show how to derive a variance of a functionally dependent value. Neither did they show how to test for spatial dependence by applying Fisher's F-test to the variance of the set of measured values and the first variance term of the ordered set. What Matheron never taught M&S was how to count degrees of freedom. In his own 1970 Random functions and their applications in geology Matheron wrote, "Let us denote a Brownian motion on a straight line." In Matheron's mind it somehow seemed to replace Riemann sums with Riemann integrals. Matheron never explained what Brownian motion and ore deposits have in common. M&S put Random kriging "within the geostatistical framework of the French school." Go figure why!

Dr Isobel Clark is the author of Practical Geostatistics. She wrote on the first page of Chapter 5 Kriging, "It would seem sensible to use a weighted average of the sample values, with the 'closer' sample values having more weight." On the same page she wrote, "The arithmetic mean is simply a special case where all the weights are identical." She wrote in her Preface that Journel and others at Fontainebleau taught her all she knows about the theory of the Theory of Regionalized Variables." She transposed for "mathematical convenience" the factor two (2) in dfo=2(n-1), the number of degrees of freedom for an ordered set of n measured values. That's how Clark's semi-variogram was born. Why did Fisher's F-test for spatial dependence between hypothetical uranium data fail to make Clark's grade in her 1979 Practical Geostatistics? And why does nobody care?

Statistically dysfunctional geoscientists write all sorts of things that are bound to hound them in time. Read what Stanford’s Journel wrote to the Editor of the Journal for Mathematical Geology. What he did was set the stage for conditional simulation on Stanford stationary. Take note of when he wrote it. And read what JMG’s Editor wrote to me. So, my feeling that geostatistics is invalid might be correct. How about that? He also wrote that different “flavors” of geostatistics may fail at different times. Now that’s kind of cool. I do know which flavor failed in the Bre-X fraud. It was the flavor of assuming continued gold mineralization between salted boreholes. The odd geostatistician might be taught how to test for spatial dependence and how to count degrees of freedom. Most are doomed to assume, krige, smooth, and rig the rules of statistics.

Degrees of freedom fighters struck at Stanford

It’s a strange but annual ritual of sorts. Degrees of freedom fighters assume, krige, smooth, and rig the rules of statistics. Today's fighters call that mathematical statistics. This year the stage was set at Stanford Campus on 23-28 August. Once upon a time IAMG stood for International Association for Mathematical Geology. A few years ago IAMG morphed into International Association for Mathematical Geosciences. Its present mission is to promote, worldwide, the advancement of mathematics, statistics and informatics in the Geosciences. This latest variant of IAMG talks about statistics without degrees of freedom.

The famous feud between Pearson (1857-1936) and Fisher (1890-1962) came about because of degrees of freedom. Fisher added degrees of freedom to Pearson’s chi-square distribution, and was knighted in 1952. Fisher's F-test is applied to verify spatial dependence in sampling units and sample spaces alike. Pearson’s coefficient of variation, too, stood the test of time. Meanwhile in Algiers, young Matheron didn’t count degrees of freedom. In fact, he didn’t have a clue what degrees of freedom were all about. IAMG’s most advanced thinkers still do not count degrees of freedom.

The very first textbook about Matheron’s new science of geostatistics was David’s 1977 Geostatistical Ore Reserve Estimation. Table 1.IV Copper grades Prince Lyell in Chapter 1 Elementary Statistical Theory and Applications gives a chi-square distribution with 13 degrees of freedom. David’s Index lists neither Chi-square distribution nor Degrees of freedom. What the author did list are Best linear unbiased estimator, Brownian motion, and Bull’s eye shot.

Figure 203 on page 286 of David’s first textbook takes the cake for boldness. The same figure saw the light as Figure 10 in Marechal and Sierra’s 1970 Random Kriging. It is printed in Proceedings of a Colloquium on Geostatistics held on campus at the University of Kansas, Lawrence on 7-9 June 1970.

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

which are estimated from the same nine holes.

David derived the covariances of his set of sixteen "samples", each of which was "estimated" from the same nine holes.What he didn't do was count degrees of freedom. His set of nine (9) holes gives df=n-1=9-1=8 degrees of freedom. The ordered set gives dfo=2(n-1)=2(9-1)=16 degrees of freedom. The number of degrees of freedom is a positive integer for evenly spaced holes but becomes a positive irrational for unevenly spaced holes.

A set of sixteen (16) functionally dependent values does not give a single degree of freedom. What David did not know either is that every functionally dependent value does have its own variance. He did know that his set of nine (9) holes gives an infinite set of functionally dependent values. David called them simulated values but statistically dysfunctional thinkers call them kriged estimates. The question is then why kriging variances of sets of kriged estimates became the building blocks of Matheronian geostatstics.

Dr Jef Caers chairs IAMG 2009. He is Associate Professor, Energy Resources Engineering, with Stanford University. His 1993 MS in Mining Engineering and Geophysics and his 1997 PhD in Engineering were obtained with the Katholieke Universiteit, Leuven, Belgium. He speaks French fluently. This is why he should belatedly review Matheron’s 1954 Note Statistique No 1 to assess if anything else but degrees of freedom and primary data went missing. Some scholar at Stanford Earth Sciences should know all about associative dependence, functional dependence and spatial dependence. I think Dr Jef Caers may be that scholar!

Casting dice and tossing coins at Stanford

Behind Stanford’s motto “The wind of freedom blows” is a rich history. It was President Gerhard Casper on October 5, 1995 who put a score of fine points to it. Who could possibly object to the freedom to teach and be taught sound sciences? When President Casper spoke in 1995 the freedom to assume spatial dependence between measured values in ordered sets had been entrenched in geostatistics since 1978. Herbert Hoover, Thirty-First President and Stanford’s very first mining engineer, would have been shocked. Who would put a mine stope together by casting dice? How could geostatistics have converted Bre-X’s bogus grades and Busang’s barren rock into a massive phantom gold resource?

Here’s what I have been trying to bring to the attention of Dr J L Hennessy, Stanford’s President. Geostatistics ignores the concept of degrees of freedom and violates one-to-one correspondence between functions and variances. Agterberg’s distance-weighted average does not have a variance. Neither does David’s distance-weighted average. I pointed out that it took the Papacy 300 years to right a wrong. I did so the last time I wrote to Stanford’s President on February 13, 2008. I wrote that I thought Stanford could right a wrong much faster. He could have asked a Stanford statistician whether or not the geostatocracy has the freedom to assume spatial dependence between measured values in ordered sets. What I wrote in 2008 didn’t hit Dr Hennessy’s list of things to do.

I do appreciate my own freedom and am a stickler for degrees of freedom. So, I looked at Stanford’s statistical scholars and warmed to what I read about Professor Dr Persi Diaconis. He looked like the kind of scholar who would take seriously my crusade against the geostatocracy and its army of degrees of freedom fighters. Stanford Report of June 7, 2004, pointed out, “Persi Diaconis has spent much of his life turning scams inside out.” Now there’s a professional scam buster of sorts. It became even better than I thought it would be when I read what Professor Dr Persi Warren Diaconis had done. He left home at 14, hit the road with Dai Vernon, the famous Ottawa-born slight-of-hand magician, and got Vernon’s magic touch.

When I was searching Stanford’s website for a genuine statistician, I found out that Dr Diaconis doesn’t respond to email. I took a chance and did send him an email anyway on February 23, 2009. That was more than year after my last email to Stanford’s President. Diaconis is indeed true to his word and did not respond to my email. I had suggested that Stanford should give real statistics a fighting chance. So, I decided to call Diaconis but nobody picked up the phone. I called between March 26 and April 22, 2009, and did so between 13:00 and 16:00 PST. I called sixteen times and the line was busy twice. I could have but decided not leave a message.

Diaconis knows how to toss a coin. So much so that he can make the same side of a coin come up ten times in a row. He designed a mechanical coin tossing contraption that gives the same odds. What he did do was defy the Central Limit Theorem. Coins and dice played cameo roles when I taught sampling theory and practice in places are far apart as Greenland and Tasmania, and as Finland and the Philippines. I put in plain words how to tamper with the outcomes of tossing coins and casting dice. What I didn’t show is how to test for bias. A Stanford student should cast the same die often enough to infer absence of bias within acceptable bias detection limits. The catch-22 is that abrasion is bound to cause a bias before acceptable bias limits are obtained.

I taught sampling theory and practice on the basis of a binomial sampling unit that consists of 90% white beans and 10% of the same but red-dyed beans.

Each participant would take a small increment and a large increment, and count white and red beans in each. This simple sampling experiment made it easy to explain Visman’s sampling theory and practice, and his composition and distribution components of the sampling variance. Visman’s work proved that the most effective method to estimate the variance of the stochastic variable of interest in a sampling unit or a sampling space is to partition the set of primary increments into a pair of interleaved subsets. Of course, one pair of subsets gives but one degree of freedom. That’s why SQC programs should be implemented on a routine basis. The interleaved sampling protocol has been incorporated in several ISO standards.

The wind of freedom blows at Stanford University. What geostatistocrats have blown is the concept of degrees of freedom. Agterberg blew the variance of the distance-weighted average. Journel blew Fisher’s F-test for spatial dependence. Once upon a time Herbert Hoover wrote, “It should be stated at the outset that it is utterly impossible to accurately value any mine, owing to the many speculative factors involved. The best that can be done is to state that the value lies between certain limits, and that the various stages above the minimum given represent various degrees of risk.” Hoover’s 1909 Principles of Mining Valuation, Organization and Administration still make sense. Why then is the world’s mining industry hooked on assuming, kriging, smoothing, and rigging the rules of real statistics?

Teaching junk statistics at Stanford

Stanford University is Professor Dr Andre G Journel’s world. He has put down deep roots at Stanford since 1978. Journel teaches the same flaky stats that Professor Dr Georges Matheron taught him between 1969 and 1978. Journel was Matheron’s most gifted student. Matheron taught him all of the ins and outs of his novel science of geostatistics. Matheron may not have told Journel that he thought in 1954 he was a statistician. It took almost ten years to teach Journel how to assume, krige, and smooth with a lot of confidence and pride. Journel was Mining Project Engineer at the Centre de Morphology Mathematique from 1969 to 1973, and Maitre de Recherches at the Centre de Geostatistique from 1973 to 1978. Not surprisingly, he worked as profusely with symbols as Matheron did in his magnum opus. What Matheron failed to show his star disciple is how to test for spatial dependence between ordered sets of measured values in sample spaces and sampling units. Matheron and Journel never found the lost variance of Agterberg's distance-weighted average point grade.

Journel is the lead author of Mining Geostatistics. When the ink had dried in 1978 he took his book to Stanford’s students and taught them all about assuming, kriging and smoothing. My copy is a “1981 reprint with corrections.” Matheron’s Foreword makes a deeply dense read. In contrast, Dr Isobel Clark’s Preface to her 1979 Practical Geostatistics makes an easy read. Her cradle once rocked on the side of the Channel where Sir R A Fisher was knighted. Clark confessed it was Journel who taught her all she knows about the Theory of Regionalized Variables. Clark messed up degrees of freedom for ordered sets of measured values. She slashed for "mathematical convenience" the factor 2 in df₀=2(n-1) degrees of freedom for ordered sets, cooked up her silly semi- variogram, and scolded the poor souls who “sloppily call it a variogram”. Clearly, Clark and Journel disagreed about semi-variograms and variograms. Neither knew how to test for spatial dependence, how to chart sampling variograms, or how to count degrees of freedom.
Matheron’s 1978 Foreword to Mining Geostatistics went off on a tangent just as much as did his 1954 Note statistique No 1. He beat around the bush about geologists who “stress structure” and statisticians who “stress randomness.” Matheron’s point of view flies in the face of Visman’s sampling theory with its composition and distribution variances. Matheron predicted, “The user of Mining Geostatistics will come across nothing more than variances and covariances, vectors and matrices”. Matrices and vectors do indeed abound from cover to cover but so do pseudo variances and pseudo covariances. What all those so called “variances” and “covariances” in Mining Geostatistics do have in common with genuine variances and covariances are squared dimensions. The concept of degrees of freedom, too, failed to make the grade in Matheronian geostatistics. And that’s what will kill the kriging game!
I came across a genuine variance in a numerical example on page 63 of Mining Geostatistics. The authors divided a stope into four equal units, and assigned to each unit a grade equal to the outcome of a cast of “an unbiased six-sided die.” Now that does indeed give a genuine variance. Casting an unbiased die a large number of times gives a uniform probability distribution with a population mean of μ=3.5 and a population variance of σ²=2.917. The authors deserve praise for giving correct values, and for pointing out that the die ought to be unbiased. Surely, Stanford’s students ought to be taught how to measure the risk of playing all sorts of games of chance.

No real data in 1954 - Casting dice in 1978

The set of three (3) stopes is given on the same page. Each set of four units within its stope was put together with a six-sided unbiased die such that each unit has the same mean of 3.5. That sort of applied research is time-consuming but of critical importance when teaching all of the intricacies of geostatistics. A touch of classical statistics is required to test whether or not a given die is unbiased. The question of whether Journel's die was biased may have been solved by assuming it was unbiased. Fisher’s F-test shows that the variances of the sets and the first variance terms of ordered sets are statistically identical. Read what Journel said about “Fischerian (sic) statistics” in October 1992. How’s that for creative thinking and writing?
The zero kriging variance of σ²_k=0 is given on page 308, Chapter V The Estimation of in situ resources in Mining Geostatistics. Another unique feature of Matheronian geostatistics is one-to-one correspondence between zero kriging variances and infinite sets of kriged estimates. Even the OCS might find it a bit of a stretch to report a 95% confidence interval of zero ounces of gold for a mineral inventory with 9.9 million ounces.
Armstrong and Champigny solved this Catch-22 with a strict caution against over-smoothing. They did so in A Study on Kriging Small Blocks, CIM Bulletin, March 1989. The study implies that requirement of functional independence may be violated a little but not a lot All that geostatistical gobbledycook is cooked up because one-to-one correspondence between distance-weighted averages and variances became null and void in Agterberg's 1974 Geomathematics.
On a positive note, Dr John L Hennessy, Stanford’s President, is but one of the few leaders at institutes of higher learning who did bother to respond to my letters.

On August 23-28, 2009, IAMG’s Annual Conference will be held at Stanford University. What a wonderful opportunity for Stanford's President to peek around the corner and ask why the variance of Agterberg’s distance-weighted average point grade is still missing. Or he might ask Professor Dr Persi Diaconis to pose a few questions on his behalf. Diaconis is Stanford's Mary V Sunseri Professor of Statistics and Mathematics. He’ll know all about the Central Limit Theorem and its role in sampling theory and practice.

Geostatistics continues to evolve as a discipline

That's what Mark Corey wrote when Canada's Minister of Natural Resources asked him to respond to my message. Mark Corey is Director General Mapping Services Branch and Assistant Deputy Minister, Earth Sciences Sector. He is the chief mapmaker for NRCan so to speak. I was ticked off big time when he called geostatistics a discipline. But I told myself it could have been worse. He could have called it a scientific discipline. He is also one of several experts behind NRCan's 2008 "bulletproof" climate report. He testified at the Senate Committee for Energy, Environment and Natural Resources. I wish I could have asked him a few questions.
What I wanted him to tell me in plain words is why each and every distance-weighted average point grade doesn't have its own variance. Dr Frits P Agterberg thought his distance-weighted average point grade didn't have a variance in the early 1970s. Agterberg was wrong then. He's wrong now. It's high time for NRCan's Emeritus Scientist to explain why his distance-weighted average point grade still doesn't have a variance in 2009!
None of the five (5) points in the next picture have anything to do with pixels on a map. Each point stands for some sort of hypothetical uranium concentration that was measured in some way in samples selected in this sample space at positions with known Easting and Northing coordinates. I didn't make it up but Dr Isobel Clark did in her 1979 Practical Geostatistics. She worried whether or not the Central Limit Theorem would hold so she didn't derive it. Clark's figure would have been a dead ringer for Agterberg's 1970 and 1974 figures if it were not for her hypothetical uranium concentrations.

Fig. 1.1. Hypothetical sampling and estimation situation
Fig. 4.1. Hypothetical sampling and estimation situation - a uranium deposit

I want to prove Clark's set of hypothetical uranium concentrations does not display a significant degree of spatial dependence. So, let's take a systematic walk that visits each point only once and covers the shortest possible distance. Clark's selected position is not equidistant to each of her hypothetical uranium concentrations. That's why the number of degrees of freedom is not a positive integer but a positive irrational. Applying Fisher's F-test to var(x) = 4,480, the variance of the set, and var1(x) = 3.640, the first variance term of the ordered set, gives an observed F-value of F = 4,480/3,640 = 1.23. This observed F-value does not exceed the tabulated F-value of F0.05;4;4.90 = 6.38 at 95% probability. Therefore, Clark's distance-weighted average hypothetical uranium concentration of 371 ppm is not an unbiased estimate.
Clark didn't need Agterberg's approval to derive confidence limits and ranges for this point grade. Neither did I and came up with a 95% confidence interval of 95% CI = +/-111 ppm or 95% CI = +/-29.8%rel, and a 95% confidence range with a lower limit of 95% CRL=261 ppm and an upper limit of 95% CRU=482 ppm.
Here's what I would want Mark Corey to do. Visit NRCan's Emeritus Scientist in the privacy of his ivory tower and borrow his 1974 Geomathematics. Go to Chapter 6 Probability and Statistics and look at Fisher's F-test in Section 6.13. That will be all. At least for now!

Not quite fit for professional statisticians

Professor Dr Michel David said so himself. He pointed out his textbook is not for professional statisticians. He was talking about his very first textbook. I bought a copy of Geostatistical Ore Reserve Estimation, and worked my way through it. David was dead on when he predicted, "…statisticians will find many unqualified statements here.” All I really wanted to know is how David derived unbiased confidence limits for metal grades and contents of ore deposits. But he didn't do it! Why would the author of the very first textbook on geostatistics fail to show how to derive unbiased confidence limits?

I had derived unbiased confidence limits for metal grades and contents of concentrate shipments. Mines and smelters want to know the risks associated with trading mineral concentrates. Metal traders were keen to work with my method and several ISO Technical Committees approved it. So, we put together an analogous method, called it Precision Estimates for Ore Reserves, and submitted it for review to CIM Bulletin. I still don’t know why our paper ended up on David’s desk. What I do know is that David blew a fuse when he saw we didn’t even refer to geostatistics let alone work with it.

In Section 10.2.3.3. Combination of point and random kriging, David refers to Maréchal and Serra’s Random kriging. These authors were with the Centre de Morphology Mathematique when they presented it at the celebrated Geostatistics colloquium on campus at the University of Kansas, Lawrence on June 7-9, 1970. In a section called Punctual Kriging these authors showed nine measured grades and sixteen functionally dependent grades.

Figure 10 - Grades of n samples belonging to
nine rectangles P of pattern surrounding x

M&S’s Figure 10 morphed into Figure 203 on page 286 of David’s 1977 book. On the same page David claimed, “Writing all the necessary covariances for that system of equations is a good test to find out whether one really understands geostatistics.” What David didn't do was take a systematic walk that visits each hole only once and covers the shortest distance. But neither did Agterberg in 1970. Nor did M&S take a systematic hike on campus at that time.

Fig. 203. Pattern showing all the points within B,
which are estimated from the same nine holes

Each of David's sixteen points within B is in fact a distance-weighted average point grade. It makes no sense at all to derive the false covariance of a set of functionally dependent values and ignore the true variance of the set of nine measured values. David did sense something was amiss. In Section 12.2 Conditional Simulations of Chapter 12 Orebody Modelling he confessed , “There is an infinite set of simulated values which will have these properties.”

Infinite set of distance-weighted average point grades
each derived from the same set of nine holes

Counting degrees of freedom for his set of nine holes would have been a foolproof test to find out whether David really understood statistics. What he looked at in this black hole were Agterberg's distance-weighted average point grades. Each is a zero- dimensional point grade. And each one of them lost its variance on Agterberg's watch. Dr F P Agterberg, Emeritus Scientist with Natural Resources Canada, did approve Abuse of Statistics but wasn't himself into testing for spatial dependence and counting degrees of freedom.

David's 1977 Geostatistical Ore Reserve Estimation and Journel and Huijbregts's Mining Geostatistics rank among the worst textbooks I've ever read. Until David's 1988 Handbook of Applied Advanced Geostatistical Ore Reserve Estimation came along. His work was founded by the Natural Science and Engineering Research Council of Canada with Grant No 7035. What a waste!

Geostatistical data analysis - Quo Vadis

More than 20 years ago G M Philip and D F Watson posed the question Matheronian Geostatistics - Quo Vadis? Philip and Watson’s question was published in Mathematical Geology, Vol 18, No 1, 1986. The text consisted of 21 pages and the list of references counted 86 works. Sir R A Fisher’s 1959 Statistical methods and scientific inference is on the list. Fisher’s ubiquitous F-test is applied to test for spatial dependence in sampling units and sample spaces alike. To assume spatial dependence appealed more to Matheron than to verify it by applying a sort of test cooked up by some kind of knight across the Channel. So, counting degrees of freedom failed to make Matheron’s list of things to do. He worked mostly with symbols and rarely with real data. Shortsighted thinking still runs rampant at the Centre de Géostatistique, 5 Rue St Honoré, Fontainebleau, France.

Matheron’s edifice groupe de réflexion statistique nouveau

Matheron’s rebuttal in his Letter to the Editor was called Philipian/Watsonian High (Flying) Philosophy. It was published in Mathematical Geology, Vol 18, No 5, 1986. It shed a bright light on Matheron’s mind when he ranted, “But all of this is clear now: geostatistics is just a dastardly conspiracy organized with diabolic cunning, by a secret order of one-dimensional Jesuits.”

Assume, krige, smooth, and be happy

Here’s what Matheron’s new science of geostatistics is all about. Matheron in 1954, in his very first Note Statistique No 1, failed to derive variances of weighted average lead and silver grades of ordered core samples of variable length and density. In his 1960 Note Geostatistique No 28 Matheron coined his honorific krigeage eponym. What he didn’t do was test for spatial dependence between ordered block grades. In his 1970 Random functions and their applications in geology Matheron brought to light a likeness of some sort between ore deposits and Brownian motion. That’s why Matheron and his flock took to working with Riemann integrals rather than with Riemann sums. It explains why counting degrees of freedom failed to make the grade in Matheronian geostatistics. Blatant disrespect for Fisher’s F-test, for degrees of freedom, and for the Central Limit Theorem, prove my point. Professor Dr Georges Matheron was a self-made wizard of odd statistics.

Once upon a time I was an accidental reader of A Sampling Manual and Reference Guide for Environment Canada Inspectors. It was also called The Inspector’s Field Sampling Manual. I read the First edition. I thought about reading a Second edition and got headache. Will that be crafted by the most gifted geostatistocrat in Canada? Or will some lowest bidder put it together? But who put Geostatistical data analysis in the First edition? Did Environment Canada, too, have a geostatistically qualified Emeritus Scientist on board?

Section 2.1.2 Sampling Approaches points to random sampling, systematic sampling and judgement (sic) sampling. EC’s inspectors are also taught, “Systematic samples taken at regular time intervals can be used for geostatistical data analysis, to produce site maps showing analyte locations and concentrations. Geostatistical data analysis is a repetitive process, showing how patterns of analyte change or remain stable over distances and time spans.” Close but not quite close enough for EC’s average inspector. What a pity that meaningful examples are missing as much in EC's First edition as they are in Matheron's magnum opus.

One example points to shellfish samples taken at 1-km intervals along a shore. What EC’s Inspectors are not taught is how to test for spatial dependence between ordered shellfish counts. A sampling variogram would give much more valuable information than a simple test for spatial dependence. EC's inspectors should not even think about charting semi-variograms. The status quo is unacceptable if Environment Canada wants to study climate change. So, what's EC's brass waiting for? Students at Canadian Universities may want to explore EC's National Climate Data and Information Archive. Many student's do not even know why geostatistical data analysis is a scientific fraud. It's time to call a scientific inquiry!