Flunking his PhD Thesis

Put it on paper; call it a PhD thesis; get it approved! Simple comme bonjour, n’est-ce pas! But his PhD supervisors at Université de Paris Sorbonne did not approve Professor Georges Matheron’s PhD thesis. On the contrary, they wanted to know what I had wanted to know since the Bre-X fraud! The title of Matheron’s thesis was “Les variables régionalisées et leur estimation: une application de la theory des fonctions aléatoires aux sciences de la nature”. How about that? Thank goodness French was my very first foreign language!

How did Matheron test for spatial dependence in sample spaces and sampling units? He never did! That is why Matheron got stuck on the very first page of his PhD thesis. He didn’t know in 1965 how to test for spatial dependence between measured values. His PhD supervisors had posted on his thesis two (2) sets of whole numbers with the same central value. One set was ordered and the other was randomly distributed.  Matheron’s PhD thesis added up to 301 pages of dense text and scores of symbols. But his PhD supervisors deemed it not enough to merit a PhD in his novel science of geostatistics! It’s rather silly that the Creator of Geostatistics and the Founder of Spatial Statistics did not know how to test for spatial dependence between sets of integers. But why didn’t he know?  Applying Fisher’s F-test and counting degrees of freedom have never been part and parcel of his novel science. His failure to test for spatial dependence was part and parcel of what he fondly called his new science of geostatistics when he took it to North America in June 1970.

Professor Georges Matheron came with his most gifted disciples. Neither knew how to 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. His disciples believed Matheron was teaching a new science. His PhD supervisors were aware that his new science was an invalid variant of applied statistics. Matheron’s thinking was alive in the eyes of his disciples. He had always taught that a distance-weighted average AKA a kriged estimate does not have a variance. How about that? Strip the variance off the distance-weighted average and call what’s left “a kriged estimate”. Good grief! Distance-weighted averages have variances but kriged estimates no longer do! D G Krige had not come all the way to Lawrence, Kansas. M David and A G Journel were busy writing textbooks about Matheron’s novel science.

A colloquium took place on campus at the University of Kansas, Lawrence in June 1970. D F Merriam, Chief of Geologic Research, Kansas Geological Survey, kept a record and Plenum Press put it in print. No list of visitors was kept. The event was useful to those who do work with applied statistics. Koch and Link, the authors of Statistical Analysis of Geological Data, talked about their work. Part I was published in 1970 and Part II came along in 1971. Both the famous Central Limit Theorem and the concept of Degrees of Freedom are still alive in Koch and Link’s work. I have had copies of both parts since the 1970s. I have used a data set in Sampling and Weighing of Bulk Solids. Tukey’s WSD test has also played a role in my work. Some Further Inputs describes what Professor Dr J W Tukey had seen in real time at Lawrence, Kansas. He wondered what would happen in two-dimensional sample spaces. Good grief! I was already working with three- dimensional sampling units and sample spaces. 

Marechal and Serra’s Random kriging and Matheron’s Random Functions and their Application in Geology had both been cooked up either at the Centre de Géostatistique or at Centre de Morphologie Mathématique. The variance had been stripped off the distance-weighted average and the concept of degrees of freedom was dismissed. Why did the geostatistical mind have distance-weighted averages morph into kriged estimates? The odd geostatistocrat may still remember the “famous Central Limit Theorem”.  All it would have taken is a passing grade in Statistics 101.

Matheron talked about Random Functions and their Application in Geology. He set the stage with a bizarre paradigm of Brownian motion along a straight line in deep time. It made counting degrees of freedom an exercise in extreme futility. Those who would have been tempted to count them would have scored a failing grade on Geostatistictics 101. Ranked high among vagaries in Matheron’s take on spatial dependence was his reference to the “quasistationarity” condition! Good grief!

Marechal and Serra talked about Random Kriging. Terms such as punctual kriging put into perspective what this new science of geostatistics was all about. Figure 10 did as little for Matheron’s new science as it would do for David’s 1977 Geostatistical Ore Reserve Estimation.   

Figure 10 – Grades of n samples belonging to

nine rectangles P of pattern surrounding x

A facsimile of Marechal and Serra’s Figure 10 is given in David’s first textbook as Fig. 203 on page 286 in Chapter 10 The Practice of Kriging. The National Research Council of Canada has given generous support to David’s imperfect thinking. It did so with its Grant NRC7035. NRC did not engage in statistical quality control in those days. NRC has changed its name and approves Markov chains. So much for SQC!

What's wrong with Matheron's 1965 PhD Thesis?

Once upon a time a young geologist in Algiers derived the degree of associative dependence between lead and silver grades of drill core samples. What he didn’t derive were length-weighted average lead and silver grades. Neither did he test for spatial dependence between metal grades of ordered core samples. The geologist did do a bit of applied statistics! He called his article Note statistique No1. In time, one of his dedicated disciples changed it to Note géostatistique No1. He did do so after the Internet was born! He is still the custodian of his master’s magnum opus. He may want to work with Matheron’s new science of geostatistics from the 1950s to eternity. Good grief! That’s long time! And it’s a headache already! The more so since Note géostatistique No28 shows krigeage in its title. Did Matheron ask Krige whether he wanted his name to become a genuine eponym?

Matheron was a master at working with mathematical symbols. He couldn’t possibly teach his students how to test for spatial dependence between mathematical symbols. What’s more, he didn’t even know in the 1950s how to test for spatial dependence between measured values in ordered sets. Neither did he know how to test for spatial dependence in his 1965 PhD thesis! As a matter of fact, Matheron has never tested for spatial dependence between measured values in ordered sets. 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. Degrees of freedom for both sets ought to be counted and taken into account. Matheron is remembered as Founder of Spatial Statistics and as Creator of Geostatistics. I couldn’t have cared less what his disciples called him. But why did he never test for spatial dependence by applying Fisher’s F-test? Why did he strip the variances of distance-weighted averages cum kriged estimates? Why did he assume spatial dependence between measured values in ordered sets?

Those who were to judge Matheron’s PhD Thesis on November 10, 1965 may well have asked him to put in plain words the nitty-gritty of his thesis.  Matheron had called it “LES VARIABLES RÉGIONALISÉES ET LEUR ESTIMATION”. His PhD supervisors were Professor Dr Swartz, President, Professor Dr Fortet and Professor Dr Caileux, Examinators. This team proposed a second thesis with the title “PROPOSITIONS DONNÉES PAR LA FACULTÉ”. Did Matheron’s supervisors ask him to jump hoops? And how far would Matheron jump to defend variance-deprived distance-weighted averages cum kriged estimates? The very first of 301 pages of Matheron’s 1965 thesis mesmerized me. Why had Matheron cooked up a pair of prime data sets? Why were both inserted under INTRODUCTION on the very first page? Why didn’t he show how to test for spatial dependence? Why is it that PhD candidate George Matheron did not know how to test for spatial dependence and how count degrees of freedom?

All it takes to test for spatial dependence is to compare observed F-values with tabulated F-values. Of course, degrees of freedom ought to be counted and be taken into account. I have applied Fisher’s F-test to verify spatial dependence in sample spaces and sampling units alike. I have done so ever since I worked on ASTM and ISO Standards. Geostatistical software converted Bre-X’s bogus grade and Busang’s barren rock into a massive phantom gold resource.  I unscrambled the Bre-X salting scam by proving that the intrinsic variance of gold was statistically identical to zero. Of course, it is of critical importance to have a good grasp of the properties of variances.

It became Matheron’s new science of geostatistics when the variance was stripped off the distance-weighted average and what was left was called a kriged estimate. Did Matheron really think that he had created a new science. Geostatistocrats still believe he did!  

Abuse of Statistics

My take on Abuse of Statistics was put in print in March 1992. It didn’t end up in CIM Bulletin but in CIM Forum. That’s where “Articles of a controversial nature” tend to end up. Merks and Merks in 1991 had shown how to test for spatial dependence by applying Fisher’s F-test to gold grades of ordered rounds mined from a drift. What a pity that geostatisticians in the 1990s didn’t test for spatial dependence between measured values in ordered sets. It is imperative in mineral exploration, mining and mineral processing that degrees of freedom be counted. On-stream analyzers measure and monitor metal grades of mill feed and tailing! That’s why confidence intervals and ranges for metal contents and grades are easy to derive.

So it came about that CIM Bulletin had decided to print in CIM Forum a technical brief on Abuse of Statistics in 1992. But why then had Armstrong and Champigny’s “A study on kriging small blocks” seen the light in CIM Bulletin of March 1989. Why was Abuse of Statistics published in CIM Forum? Why was placing distance-weighted averages AKA kriged estimates between measured values deemed sound science in CIM Bulletin? Why do geostatisticians not test for spatial dependence between measured values in ordered sets? Why are degrees of freedom ignored? Simple questions but still no answers! 

Dr W D Sinclair, Editor CIM Bulletin, brought up CIM Forum in his letter of September 21, 1992. He was aware that I have served on various standard committees since 1974. Dr F P Agterberg was his Associate Editor in those days. Both were scholars with the Geological Survey of Canada. They agreed that CIM Forum was a fitting format but that my article should pass rigorous scrutiny. Dr Agterberg wanted to know when H G Wells had said: “Statistical thinking will one day be as necessary as the ability to read and write”. Surely, one cannot be rigorous enough when entrusted with peer review for CIM Forum. Was it Agterberg who had approved Armstrong and Champigny’s study in 1989? Or was it perhaps David himself?

It was in Darrell Huff’s 1954 How to Lie with Statistics where H G Wells’s quote was printed ad verbatim. Huff had also referred to Disraeli’s famous lament: “There are three kinds of lies: lies, damned lies, and statistics”. But where had Huff found so much praise for statistical thinking? I had been reading Sherborne’s Another Kind of Life. I tried to contact Dr Sherborne and was pleased he did respond. He attributed Wells’s quote to Samuel Wilks’s 1954 presidential address to the American Statistical Association. It does have an extensive website. Wilks had strung together a rather rambling train of thought whereas Wells was much more frugal with words.

Dr Michael Sherborne has tracked Wells’s train of thought to page 204 of Wells’s book Mankind in the Making: “The great body of physical science, a great deal of the essential fact of financial science, and endless social and political problems are only accessible and only thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of great new complex world-wide Stats that are now developing, it is necessary to be able to compute, to think in averages and maxima, as it is now to be able to read and write.”

H G Wells

At the same time a budding geologist in Algiers did not know how to derive length-weighted average lead and silver grades determined in core samples of variable length. In fact, he even thought he was working with applied statistics. In time Professor Dr Georges Matheron stripped the variance off the distance-weighted average, called what was left a kriged estimate to honor D G Krige. Next, he praised what he had cooked up and decided to call it the new science of geostatistics. It is a fact that Matheron’s new science is as doomed as the dodo once was!