Unscrambling the French sampling school

My grandma taught me not to put all my eggs in one basket. She was a caring matriarch who told inspiring stories. She played card games but odds were beyond her grasp. She played for pennies but not with other people’s pennies. She didn’t have a PhD in anything. I took her word and never put all my eggs in one basket.

Dr Pierre Gy (1924-...) and Professor Dr Georges Matheron (1930-2000) put the French sampling school on the world map. Matheron never put core samples from a single hole in one basket so to speak. But Gy did put a set of primary increments taken from a sampling unit in one basket. So he didn’t even get a single degree of freedom. The interleaved sampling protocol is described in several ISO Standard Methods. It is also described in Chapter 6 Spatial Dependence in Material Sampling of a textbook on Approaches in Material Sampling. Dr Bastiaan Geelhoed edited the text. IOS Press published the book in 2010.

Matheron marched to a new low when he sampled in situ ores. So he didn't put in one basket a set of core samples from a single borehole. But he failed to derive measures for precision, to test for spatial dependence between grades of ordered core sections, and to count degrees of freedom. Quelle dommage! Matheron thought that Gy knew a lot about sampling theory and sampling practice. Gy’s L’Échantillonage des Minerais en Vrac was printed in two parts and on 656 pages. Tome 1 is dated January 15, 1967, and Tome 2 hit the shelves on September 15, 1971.

Gy's sampling slide rule

Gy pioneered a slide rule of sorts to simplify the sampling of mined ores. His sampling constant C is a function of c, the mineralogical composition factor, of l, the liberation factor, of f, the particle shape factor, and of g, the size range factor. Hence, Gy’s sampling “constant” is a function of a set of four (4) stochastic variables. As such, Gy's constant C does have its own variance.

Some sampling constant!

Matheron wrote a three-page Synopsis to Gy’s Tome 1 Theory Generale. He praised Gy’s work for defining, “... accuracy and precision, bias and random error, etc...” Gy, in turn, praised Matheron’s 1965 PhD thesis. Gy did refer to Visman’s 1947 PhD Thesis and to his 1962 Towards a common basis for the sampling of materials. Gy didn’t mention Sir R A Fisher, Anders Hald, Carl Pearson, or William Volk. Why then did Gy deserve Matheron’s praise?

Dr Pierre M Gy is a chemical engineer with a deterministic take on sampling. He is the most prolific author of works on sampling. He sent me a copy of his 1979 Sampling of Particulate Materials, Theory and Practice. It was marked Christmas 1979 and signed underneath. Gy pointed to degrees of freedom in Chapter 14. His Index does not list degrees of freedom between “degenerate splitting processes” and “degree of representativeness”. Another odd entry in this Index is “SF = Student-Fisher”. Student’s t-test proves or disproves bias between paired data. Fisher’s F-test proves or disproves whether two variances are statistically identical or differ significantly. Both statistical tests demand that degrees of freedom be counted!

Matheron praised Gy’s work in 1967 and Gy, in turn, praised Matheron’s work in 1979. Here’s what Gy wrote literally:

"The sampling of compact solids and more specifically mineral deposits
is covered by the science known as ”Geostatistics”. The fundamentals
of this science, established by Krige, Sichel, deWijs were developed by
Matheron and his team (references in appendix). Worked out in France,
Matheron’s theories are slowly but steadily gaining acceptance in
English speaking countries around the world thanks to
an increasing teaching and to technical textbooks such as
Michel David’s “Geostatistical Ore Reserve Estimation” (1977)".

Now that's a nice little tit-for-tat between scholars who created the French sampling school! Matheron and his disciples cooked up quite a variant of applied statistics! Thank goodness, his magnum opus is posted on CdG’s website. I scanned his 1965 PhD Thesis for degrés de fidelité but didn’t find any at all in 301 pages of dense probability theory. But I did find two sets of numerical data. Matheron’s Set A looks a lot less variable than Set B but both sets have the same central value. So, I applied Fisher’s F-test to the variances of the sets and the first variance terms of the ordered sets.

Data sets in Matheron's PhD thesis

I have pasted Matheron’s A- and B-sets on a truncated title page of his 1965 PhD Thesis. This title page and Fisher’s F-tests for his A- and B-sets are posted on my website. Matheron and Gy didn't know how to test for spatial dependence in sampling units and sample spaces. The root of the problem is these scholars didn’t grasp the properties of variances. But then, neither did my grandma!