SME approved applied statistics

SME did so at the turn of the century. Its acronym stands for The Society for Mining, Metallurgy, and Exploration, Inc. Not only did it approve Process simulation with spreadsheet software and put it in print in Minerals & Metallurgical Processing, Vol 16, No 2, May 1999. It also approved Borehole statistics with spreadsheet software and published it in Volume 308 of Transactions 2000. I was tickled pink that SME’s reviewers approved applied statistics. The more so since SME’s reviewers thought my work would stir up a hornet’s nest! But not a single hornet stirred! As a matter of fact, Stanford’s Professor Dr A J Journel didn’t put up opposition against applied statistics! He may well remember what he wrote on October 15, 1992 to the Editor of the Journal of Mathematical Geology? Here are but a few paragraphs of Professor Journel’s musings!

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.

It is not correct to state categorically that kriging, or for that matter any other interpolation algorithm, does not add any information to the system. It does through the implicit or explicit model of correlation. Indeed, change the variogram model yet keep the same data, the kriging estimates change. This correlation/covariance/variogram model can be borrowed from another field or outcrop, it is then genuine new information. Or, it can be inferred from the same data used for kriging. In the latter case, new information is introduced through aspects of the sample bivariate (two-point) distribution. The important question is, of course, how representative of the unsampled area is that bivariate information, i.e., how appropriate is the prior “decision” of stationarity.

 What a world of difference between SME’s reviews and Professor Dr A J Journel’s shamelessly self-serving drivel. Merks and Merks 1991 Precision estimates for ore reserves was praised by and published in Erzmetall 44 (1991) Nr 10. David, the author of 1977 Geostatistical Ore Reserve Estimation, deemed our paper deficient in references to the geostatistical literature. Professor Dr A J Sinclair, PEng, PGeo, was the second reviewer. He objected to scores of equations incorporated in several ISO Standards after I had been active on several ISO Committees since 1974. Sinclair was thinking in the 1970s that Professor G Matheron was onto something! He didn’t notice that the variance had been stripped off the distance-weighted average AKA kriged estimate. UBC’s President didn’t notice either. Yet, I have delivered to the Office of Dr Martha Piper a pair copies of Sampling and Weighing of Bulk Solids

Geostatisticians always praise geostatistical theory and practice. Stripping the variance off the distance-weighted average and calling what’s left a kriged estimate are the heart and soul of Matheron’s new science. I saw an opportunity to talk about applied statistics come along when Geostatistics for the Next Century made an early landing at McGill University. The stage was set at McGill’s Conference Centre in August 1993. I had submitted by registered mail my view on Properties of Variances. I was told on behalf of the Chair, that all seats were taken. Somebody did wish me success with my research!

Professor Dr A G Journel, Geology and Environmental Sciences, Stanford University, talked about stochastic simulation and Bayes’ Theorem but didn’t yet drift into Markov chains. He didn’t talk about confidence limits for contents and grades of mineral resources. Meanwhile, Bre-X Minerals was drilling and salting in the Kalimantan jungle. Stanford’s Dr A G Journel got into teaching more of the same. Stanford’s Journel taught only those with a PhD in geostatistics! For teaching out loud!

From stripping variances to stringing Markov chains

Did Professor Georges Matheron (1930-2000) make fundamental contributions to science? That’s what Dr F P Agterberg wrote  in Matheron’s eulogy! Here’s what Professor Matheron ought to have known but didn’t! He did not know how to apply Fisher’s F-test to the variance of a set and the first variance term of the ordered set. He did not know how to count degrees of freedom! He did not apply Fisher’s F-test when his PhD supervisors asked him to show how to test for spatial dependence. He didn’t test for spatial dependence by applying Fisher’s F-test to the variance of a set and the first variance term of the ordered set! He did not compare his observed F-value with tabulated F-values at 95%, 99% and 99.9% probability. He did not take degrees of degrees of freedom into account.  Study http://www.geostatscam.com/ to find out why Matheron’s work did not include Fisher’s F-test!

 Geostatistocrats remember Matheron either as the Creator of Geostatistics or as the Founder of Spatial Statistics. Yet Matheron didn’t grasp in 1965 how to test for spatial dependence! He didn’t even know how to test for spatial dependence when his PhD supervisors had given him trivial data sets. One was a perfectly ordered set and the other was shuffled into randomness. Matheron’s PhD thesis rambled on for a whopping 301 pages of dense text and countless symbols. Yet, Matheron flunked his PhD thesis. The problem was not so much that he didn’t know how to test for spatial dependence. The real problem was that Matheron taught his disciples that spatial dependence between measured values in ordered sets may be assumed. Good grief! What made matters worse is that Matheron didn’t count degrees of freedom the same way as did Sir R A Fisher. So how did Matheron and his disciples study the dynamics of spatial dependence? They assumed spatial dependence.  Of course, it was doomed to fail without Fishers’ F-test!  

Matheron and a few of his students visited the University of Kansas, Lawrence on June 7-9, 1970. They had come to represent the Centre de Morphologie Mathematique, Paris, France. Matheron talked about Random functions and their application in geology. He invoked Brownian motion along a straight line to solve a few estimation problems.  Marechal and Serra spoke about Random Kriging. The caption under Figure 10 explicates:  Grades of n samples belonging to nine rectangles P of pattern surrounding x.  This figure was reborn as Figure 203 on page 286 of David’s 1977 Geostatistical Ore Reserve Estimation. Its author prevaricated about “an infinite set of simulated values”. Stanford’s Journel in his 1978 Mining Geostatistics derived the zero kriging variance

Professor Dr Michel David refers in Chapter 2 to what he called the famous central limit theorem. Yet, he didn’t refer to this theorem in his Index. In Figure 203 on page 286 David shows M&S’s set of sixteen (16) Central Limit Theorems each of which is a function of the same set of nine (9) measured values! Surely, stripping variances off functions does not make any sense whatsoever. Why does the geostatocracy work with kriging variances that tend towards zero? Why call geostatistics a new science when all that has been done is strip the variance off the distance-weighted average AKA kriged estimate? Why string Markov chains between measured values when ordered sets do not display a significant degree of spatial dependence? Why strip the variance off the distance-weighted average grade? Why not report unbiased confidence limits for metal contents and grades of mineral inventories? Surely, stringing Markov chains cannot possibly give unbiased confidence limits for metal contents and grades of mineral inventories!

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From human error to scientific fraud

The caption on my website reads: “Geostatistics: From human error to scientific fraud.” Much of it is still the way it was when I posted it in August 2005. Those who wish to grasp geosciences ought to take a close look at what it took Professor G Matheron to cook up his new science of geostatistics. Here’s what he did do in a nutshell!  He stripped the variance off the distance-weighted average and called what was left a kriged estimate. Infinite set of kriged estimates and zero kriging variances became the heart and soul of Matheronian geostatistics. Matheron’s disciples quickly went to work with infinite sets of kriged estimates and zero kriging variances. It was not so much a shame that Matheron goofed! What was a shame is that so many of his disciples goofed along! Fisher’s F-test and degrees of freedom never played a role in Matheron’s novel science of geostatistics. It is simple comme bon jour to find out that geostatistocrats neither took to testing for spatial dependence nor to counting degrees of freedom! 

Matheron_Stats

Professor G Matheron defied his PhD supervisors and flunked

What made Matheron’s new science of geostatistics click? Who knows! Stripping the variance off the distance-weighted average and calling what’s left “a kriged estimate” made no sense at all. All the same, infinite sets of kriged estimates and zero kriging variances became the heart and soul of geostatistics. So what did young Matheron do with applied statistics in 1954? He tested for associative dependence between lead and silver grades of drill core samples. The degree of associative dependence between lead and silver grades turned out to be ρ=0.85. What he didn’t do was test for spatial dependence between grades of ordered core samples. Neither did he derive weighting factors to take into account that core samples varied in density and length. Matheron marked it Note Statistique No 1 but made a correction. Sets of primary data were not as prominent in Matheron’s work as were his superstrings of symbols. The custodian of Matheron’s magnum opus thought it fitting to change Note Statistique No 1 to Note Geostatistique No 1. Perhaps a touch of deception but Matheron’s novel science was not going anywhere fast in deep time. Are all of Matheron’s sets of measured data archived? Was D C Krige tickled pink when his name turned into a genuine eponym? Many questions have yet to be answered!

 It took chutzpa to go the USA in June 1970 and peddle geostatistics as a new science. The more so since Professor Matheron would rather flunk his PhD thesis than reunite the distance-weighted average and its variance! So here’s how he flunked! His PhD supervisors had asked him to show how to test for spatial dependence between sets of randomized and ordered whole numbers. Yet, Matheron saw fit to ignore this request! His PhD thesis was called Regionalized variables and their estimation”. It added up to 301 pages of dense text with two (2) sets of whole numbers on the first page followed by scores of symbols on the next 300. What his PhD supervisors had asked him to do is to test both sets for spatial dependence. Since nobody had been inspired to do so I took to testing each set for spatial dependence.

MS_Fig10

M&S derive 16 kriged estimates

 Matheron brought along a pair of his disciples. Matheron talked about Brownian motion along a straight line. Marechal and Serra talked about Random Kriging and showed how to derive 16 kriged estimates from 9 measured grades. David pointed to M&S’s Figure 10 on page 286 in his 1977 . David wondered how to make infinite sets of simulated values smaller. Good grief! Matheron’s new science was a gift to those who ignore the power of Fisher’s F-test and the concept of degrees of freedom. So he  got the world stuck with infinite sets of variance-deprived kriged estimates and zero kriging variances! That’s what Matheron’s gift to mankind has been in a nutshell. Incredibly, he was hailed not only as the Founder of Geostatistics but also as the Creator of Spatial Statistics. Professor G Matheron passed away the 7th of August 2000.

A study on kriging small blocks

Margaret Armstrong and Normand Champigny called on but a few facts to get their small block study going in the 1980s. Following are two (2) facts that underpin their study:

Mine planners often insist on kriging very small blocks
Kriged estimates of very small blocks are over-smoothed

These geostatistical scholars had found out that kriged block grade estimates and measured grades no longer display associative dependence when variogram ranges are less than half the spacing between samples. Good grief! I couldn’t have thought that up even if I were a crafty kriger or a cunning smoother! Surely, geologists and mining engineers didn’t expect kriging to create random numbers! Yet, CIM Bulletin put in print what the authors thought about the rise and fall of kriging variances. Who were the peers who reviewed Armstrong and Champigny’s study? Didn’t they know why the kriging variance rises up to a maximum and then drops off? Who was the Editor of CIM Bulletin in 1989? What did she or he think of the rise and fall of kriging variances? But why did P I Brooker think in 1986 that kriging variances are robust?

Armstrong_Figure_2

Figure 2. Kriging variance as a function of the variogram range

After CIM Bulletin of March 1989 had landed on my desk it took but little time to figure out what was wrong with Armstrong and Champigny’s study. I didn’t find out that Matheron had flunked his PhD thesis in 1965 until his magnum opus was posted on a massive website. His disciples in 1970 stripped the variance off the distance-weighted average, and called what was left a kriged estimate! Infinite sets of kriged estimates and zero kriging variances made no sense in any scientific discipline but Matheron’s new science of geostatistics. Geostatistocrats kept kriging and smoothing simply because blatantly biased and shamelessly self-serving peer reviews made Matheron’s new science go too far. CIM Bulletin should have never approved and published A study on kriging small blocks. One-to-one correspondence between functions and variances is sine qua non in mathematical statistics! It would be a walk in the park to prove in a court of law that Matheron’s new science of geostatistics is an invalid variant of mathematical statistics.

Armstrong and Champigny studied kriging small blocks at the Centre de Géostatistique, Fontainebleau, France. Armstrong got what Matheron had failed to get in 1965. She was awarded a PhD in geostatistics at Matheron’s centre early in this century. She had already been awarded a Master’s degree in mathematical statistics at the University of Queensland, Australia. Why then didn’t she know how to test for spatial dependence between measured values in ordered sets by applying Fisher’s F-test to the variance of the set and the first variance term of the ordered set? Why hadn’t she mastered how to count degrees of freedom? Why did she allude to the fact that mine planners are often tempted to krige very small blocks. Does she deserve credit for asking how small is “too small”? I was pleased to note that Normand Champigny had been awarded a Diploma of Geostatistics. What a pity he didn’t know how to test for spatial dependence, and why degrees of freedom ought to be counted.

CIM Bulletin reviewed A study on kriging small blocks and published it under Ore Reserve Estimation in March 1989. All I want to know who approved this study for CIM Bulletin. Geostatistical peer review at CIM Bulletin was a blatantly biased and shamelessly self-serving sham. It got a lot worse after the Bre-X fraud. What has made peer review at CIM Bulletin a farce is the fact that Markov chains are strung to define stochastic mining plans!

Scores of geostatistocrats have been taught how to krige and smooth at the Centre de Géostatistique in Fontainebleau, France. Yet, Matheron himself did not know how to test for spatial dependence in 1965. What he did do in 1970 was think up Brownian motion along a straight line. He talked about his vision at The University of Kansas, Lawrence in June 1970.  Maréchal and Serra, in turn, had thought up Random Kriging. They derived sixteen (16) distance-weighted averages AKA kriged estimates from the same set of nine (9) measured values. David’s 1977 Geostatistical Ore Reserve Estimation on page 286 in Chapter 10 The Practice of Kriging shows the very same set that Maréchal and Serra’s Random Kriging had worked with in 1970.

MS_Fig10

Michel David’s 1977 M&S picture

Geostatistical software converted Bre-X’s bogus grades and Busang’s barren rock into a massive phantom gold resource. Applied statistics proved the intrinsic variance of Bre-X’s gold to be statistically identical to zero. It did so several months before Bre-X’s boss salter passed away, and before Munk’s Golden Phoenix was put in print.

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