Sound statistics or goofy geostatistics?

It took a while to post on my website much of what I know about sampling and applied statistics in mineral exploration, mining, mineral processing, smelting and refining. My webmaster and I have done so in the format of downloadable PDF files. Cash flow is used to derive unbiased confidence limits for content and grade of a reserve, and of the proven part of a resource.

The Canadian Institute of Mining, Metallurgy and Petroleum in 2007 made me a Life Member. I am the most irate Life Member of that iconic institution. Every year CIMMP asks its Life Members for donations. It wants to teach students all about mineral exploration, mining, mineral processing, smelting and refining. Sounds familiar, doesn’t it! But I have never donated a penny. Not as long as UBC’s Emeritus Professor Dr Alastair J Sinclair, PEng is teaching students to assume spatial dependence between measured values in ordered sets. Why doesn’t he grasp that each distance-weighted average does have its own variance? Did he ever peruse Clark’s 1979 Practical Geostatistics? She derived the variance of her distance-weighted average hypothetical uranium concentration. Alas, what she didn’t do was test for spatial dependence between measured values in her wacky sample space. So it seems that Clark’s take on applied statistics is imperfect.

Practical geostatistics

The odd mining mogul seems to pine for moral integrity but I pine for scientific integrity. Scores of scientists pay attention to my take on geostatistics. On November 14, 1990 I mailed the first of several letters to Professor Dr Robert Ehrlich, Editor of the Journal of Mathematical Geology. I did so by snail mail and enclosed copies of Precision Estimates for Ore Reserves and of Sampling and Weighing of Bulk Solids. On October 26, 1992 JMG’s Editor wrote, “Your feeling that geostatistics is invalid might be correct”. Attached to his letter was what Stanford’s Professor Dr A G Journel had written “a bit reluctantly”. I have also posted the letter on my website under Correspondence. It seemed to Journel that “my anger arises fro a misreading of geostatistical theory, or a reading too encumbered by classical “Fischerian” statistics”.

The next paragraph shows another ad verbatim example of goofy geostatistical thinking:

1 – Data and degrees of freedom

"The very reason for geostatistics or spatial statistics in general is the acceptance (a decision rather) that spatially distributed data should be considered a priori as dependent one to another, unless proven otherwise. It is that spatial dependence which allows differentiated local interpolation and mapping in general. Were the data independent one from another then only global statistics can be retrieved. In presence of dependence the classical notion of degrees of freedom vanishes; n spatially dependent data do not provide n degrees of freedom”.

Another stunning farce was Geostatistics for the Next Century. Geostatisticians from far and wide had flocked to McGill, Montreal, Canada on June 3 to 5, 1993. They had come to honor Professor Dr Michel David for his contribution to Matheron’s new science of geostatistics. I tried to get on the program but my paper on The Properties of Variances didn’t arouse any interest.

Way too many years I have been exposed to geostat drivel. It has made me a perfect cynic. But I do have friends. A true friend is a precious gift. Sometimes it’s tough to find out who your true friends are. Dr.-Ing Reinhard Wohlbier is a true friend. Trans Tech Publications in 1985 printed my textbook on Sampling and Weighing of Bulk Solids. Thanks to Reinhard it has now been posted on my website as a PDF file. Various papers I have put together for Trans Tech Publications through the years are about to be posted on my website. Reinhard has formidable knowledge of the handling of materials in bulk. I wish Reinhard and Ute well.

Todd Higden, too, is a true friend. He is Creative Director of Frontline Multimedia. He has set up most of my first take on Geostatscam.com. My website went online in September 2005. Todd has posted scores of downloadable PDF files in 2012. It takes less than ten minutes at legal speed to drive to Todd’s Office. I have more papers to scan and to post on my website. I want my readers to know the difference between sound statistics and surreal geostatistics. Todd does not need to know much about either but he'll get to know a little!

My partner for life and my son are far more than true friends. Hennie has never been awarded a PhD in Psychology for putting up with her driven hubby. In contrast, Ed was awarded a PhD in Computing Science at Simon Fraser University in 1992. He was also awarded the Dean’s Medal in 1986 and in 1992. My son and I did put together Precision Estimates for Ore Reserves. It was thrashed by our geostatistical peers at CIM Bulletin. In spite of that Erzmetall praised it for “splendid preparation” and published it in October 1991. We have also put together Precision and Bias for Mass Measurement Techniques. ISO did like it. So much so that it became ISO 12745:2008. Nowadays Ed leads the top-level Eclipse Modeling Project as well as the Eclipse Modeling Framework subproject. He has put on the same page his blog, my blog and the bulk-online blog. Now that’s cool!

A tale of two papers

CIM Bulletin approved and published in 1999 a paper called Simulation models for mineral processing plants. In contrast, CIM Bulletin did reject in 1990 what Merks and Merks had called Precision estimates for ore reserves. Professor Dr Michel David (1945-2000) decided to reject our paper because we had applied our own method and had given too few references to the geostatistical literature. Emeritus Professor Dr Alastair J Sinclair, PEng, PGeo found the variance of a general function a bit dated!

Variance of a general function

Dr Sinclair frowned on functions whose roots are traceable to applied statistics. He is Emeritus Professor at the University of British Columbia. He may still be teaching students that working with variance-deprived distance weighted averages AKA kriged estimates does make sense in Matheronian geostatistics.

Applied statistics has always played a key role in my teaching. The published paper was based on process simulation with the pseudo-random number generator of the standard uniform distribution. The variance of the general function as defined by Volk in his Applied Statistics for Engineers was of critical importance. This function made it simple to derive confidence limits for metal contents of mined ores and mineral concentrates. So I was delighted that the Metallurgical Society of the Canadian Institute of Mining had approved my paper. The more so since MetSoc had not found any errors. That’s how it came to pass that CIM Bulletin did publish this paper in September 1999.

A few years later I did spot a mistake not only in Simulation models for mineral processing plants but also in my book on Sampling and Weighing of Bulk Solids. The number of degrees of freedom for the first variance term of measured values in an ordered set is df=2(n-1) rather than df=2n-1. I also found out that the number of degrees of freedom for sets of measured values with variable weights are no longer positive integers but become positive irrationals. Both my book and my paper have been corrected.

Merks and Merks’s Precision estimates for ore reserves was the first paper on this topic that my son and I had put together. When Gy had sent me a copy of his 1979 Sampling of particulate materials, Theory and practice, I found out about David’s 1977 Geostatistical ore reserve estimation. It struck me as odd for any author to predict “…statisticians will find many unqualified statements…” Why hadn’t he asked a real statistician to peruse his work? And why had geostatistics been hailed as a new science in the 1970s? What I decided to do at that time was to keep David’s 1977 opus for scrutiny. The time for scrutiny came about in the late 1990s!

One would expect those who ignore degrees of freedom not to be entrusted with the works of those who do count degrees of freedom. CIM Bulletin did trust geostatistical peer review but I called it a blatantly biased and shamelessly self-serving sham. Let me briefly explain why! We had submitted our paper on September 28, 1989. We did expect the Geology Division of CIM to review our paper in an unbiased manner. I was tickled pink when the Editor of CIM Bulletin wrote on November 23, 1989 that both reviewers recommended publication with major revisions. But I turned red when I read what “mayor revisions” were necessary. So who had asked for major revisions? Professor Dr Michel David and Professor Dr Alastair J Sinclair had been entrusted with the task to protect the central tenets of Matheron’s new science of geostatistics.

David was in a tiff when he wrote “the authors had presented their own method”. Good grief! Whose method had David expected? What the author of the first textbook on geostatistics did expect most of all were scores of references to the geostatistical literature. We did find what David himself had predicted that statisticians would find. He wrote:"...statisticians will find many unqualified statements here". We did indeed! What David also felt is that we should have made reference to Gy’s 1979 Sampling of Particulate Materials, Theory and Practice. We should have pointed out that Gy's sampling constant does have its own variance whether his followers like it or not! Too few scientists and engineers are grasping what problems the French sampling school has caused!

Goofing with Gy's sampling errors

Searching for sound sampling practices always deserves praise. Sampling experts in South Africa have tried to put into plain words what sound sampling practices are all about. What a shame that they took a shine to Pierre Gy’s sampling errors. So much so that they have decided to put together a study of Gy’s sampling errors. Gy had grasped but little of what he had come to call “sampling errors”. What a pity that sampling experts in South Africa, too, have failed to come to grips with Gy’s sampling errors as much as had Gy himself.

What irked me is that sampling experts in South Africa praised Gy’s unpublished tale on “Minimum mass of a sample needed to represent a mineral lot”. Gy didn’t refer to it in his 1979 Sampling of Particulate Materials. But why didn’t he refer to Visman’s 1947 PhD Thesis anymore? And why did he take to praising David’s 1977 Geostatistical Ore Reserve Estimation? Famous French sampling scholars such as Professor Dr G Matheron, Dr P M Gy and Professor Dr M David have left no doubt that the properties of variances were far beyond their shared grasp. That’s why they never were familiar with one-to-one correspondence between functions and variances. And that’s why degrees of freedom have become such a burden!

Part 1 and Part 2 of this 2007 review of Pierre Gy’s sampling errors were strung together by R C A Minnitt, P M Rice and C Spangenbergs. All that is necessary to get goofy statistics is to pay no attention to those who do grasp applied statistics. I have sent an email to Professors F Cawood and R Minnitt. I spelled out why mining students ought not to be taught to assume spatial dependence between measured values in ordered sets, to interpolate by kriging, and to ignore the rules of applied statistics with impunity. Added to my email was a link to pre-read copies of Volk’s textbook on Applied Statistics for Engineers for the benefit of those who are studying mining engineering at the Witwatersrand University.

It feels at times as if I grew up with Volk’s Applied Statistics for Engineers under my pillow. I bought my first copy when a friend told me he liked it a lot. I did so before moving to Canada in 1969. Volk’s Chapter Seven Analysis of Variance deals in rich detail with all of the properties of variances. Section 7.3 Confidence range of variances shows how to derive lower and upper confidence limits for an observed variance as a function of the number of degrees of freedom. The odd inquisitive student may wonder why statisticians do count degrees of freedom. Those who are against counting degrees of freedom are bound to score passing grades wherever Gy’s sampling errors are taught.

Volk’s 1980 reprint is still as sound as was his 1958 work. I have lost two copies while I was teaching sampling and statistics around the world. Initially, I taught coal sampling for the McGraw-Hill Seminar Center. After Trans Tech Publications had published Sampling and Weighing of Bulk Solids in 1985 I drifted into mineral exploration and mining. Showing why geostatistics is a scientific fraud ranks high on my list of things to do on this planet. Volk’s tattered 1958 textbook remains my favorite. I would have written a Wiki page about William Volk and his work if I had known him as well as I did Dr Jan Visman. But Wiki is not what it used to be when those who assume, krige and smooth, do rig the rules of applied statistics.

The above table gives arithmetic mean assay results (g/t Au) for each of four (4) sets of samples. The variances in the second line would be obtained if infinite sets of test portions were selected and assayed. However, this sampling tree experiment was based on selecting and assaying finite sets of thirty (30) test portions which give 29 degrees of freedom. Excel’s FINV function has been applied to derive lower and upper limits of 95% confidence ranges in (g/t)².