Agterberg's way

Here’s what Agterberg wrote to me, “It seems that you are an iconoclast with respect to spatial statistics including kriging.” He did so in his reply to my email of October 7, 2004, on the subject of The Silence of the Pundits. That’s not quite what I had written to him. I didn’t bring up spatial statistics or kriging. It seemed as if Agterberg’s tribute to Matheron had become his new reality. All I had asked were questions about the distance-weighted average. I didn’t know in 2004 that Agterberg himself had derived this distance-weighted average point grade first in his 1970 Autocorrelation Functions in Geology and once more in his 1974 Geomathematics. What kept me spellbound in this Millennium was Matheron’s mind-numbing opus after it was posted on the website of the Centre de Géosciences. Since December 12, 2008, all I get to look at is “Not found.” I was used to Matheron’s prose and symbols but did miss his primary data. I wish his collected works were posted for posterity. It is such stunning stuff.

Agterberg brought up a friend of mine with similar criticisms who had “orally presented his views at IAMG meetings.” Agterberg thought I might wish to do the same. Good grief! What I do is put my thoughts in writing. I did so with The Properties of Variances in 1993. I wanted to bring the properties of variances within the grasp of geostatistical thinkers. Many had gathered at McGill to celebrate Geostatistics for the Next Century. It sounded somewhat premature but geostatistics was growing in leaps and bounds in those heady days. The properties of real variances were rather late in coming and the Bre-X fraud was just around the corner. As luck would have it, the properties of variances didn’t quite suit the tribute to David’s work with its infinite sets of simulated values and zero pseudo variances. That sort of science fiction still underpins McGill’s curriculum for budding geoscientists. McGill University is a source of goofy geosciences.

Philip and Watson’s Matheronian Geostatistics: Quo Vadis? (MG, Vol 18, No 1, 1986) made Matheron fit to be tied up. His rebuttal took the form of a Letter to the Editor (MG, Vol 18, No 5) on the subject of Philipian/Watsonian High (Flying) Philosophy. Agterberg’s way is oral criticisms but I really liked Matheron’s written rebuttal. On the other hand, Matheron’s temper tantrum driven tirade might have boggled the odd geostatistical mind. I wrote about voodoo statistics in the 1990s but it failed to trigger another mind numbing tirade.

Matheron was called the Founder of Spatial Statistics and the Creator of Geostatistics. Why did his ramblings merit twin epitaphs? The more so since Berry and Marble’s 1968 Spatial Analysis, a Reader in Statistical Geography, makes no mention of Matheron’s work. Chapter 8 Fourier Analysis in Geology in Section IV Analysis of Spatial Distributions refers to Agterberg’s Methods of Trend-Surface Analysis. Agterberg talked about it at a 1964 symposium with Applications of Statistics in its lengthy title. Just the same, Matheron did dismiss trend surface analysis at the 1970 geostatistics colloquium. Why did the masterminds not see eye-to-eye on spatial statistics when Matheron brought his new science to the USA?

All that gibberish troubled me even more when I read Agterberg’s response to my questions of October 11, 2004. On September 23, 2004, I had posed the same questions to the Councilors of the International Association for Mathematical Geology, and to the Editor and his Associate and Assistant Editors of the Journal for Mathematical Geology.

Who lost the variance of a single distance-weighted average?

Who found the variance of a set of distance-weighted averages?

Only one Assistant Editor responded by pondering, “If geostatistics is not furthering a certain problem, a different type of mathematics may solve it.” Now there’s one partially open JMG mind at work! It didn’t tempt me into giving oral criticisms at any IAMG meeting.

Here’s what I wrote on October 12th in response to Agterberg’s Aberdeen message of October 11, 2004. “I just want to know when and on whose watch the variance of the single distance-weighted average vanished, and when and under whose tutelage the kriging variance and covariance of a set of kriged estimates became the cornerstones of geostatistics, spatial statistics, kriging, smoothing, or any other popular computation that violates the requirement of functional independence and the concept of degrees of freedom”. His way was not to respond.

Agterberg had failed to derive the variance of his distance-weighted average point grade first in 1970 and again in 1974. What he did do was make a sham of scientific integrity when he was IAMG’s President. He did call it the International Association of Mathematical Geosciences. Agterberg’s way was to stay silent. It’s the wrong way in science. The right way would be to revise Geomathematics.

Agterberg's tribute

It’s high time to try and read Agterberg’s state of mind in his tribute to the life and times of Professor Dr George Matheron. It taught me so much more about his way of thinking than I had learned when we talked in the early 1990s. Neither could I have found out what I needed to know had the Centre de Géosciences (CG) not posted Matheron’s works on its website. When I looked at CG’s spiced up website for the first time I found out that he wrote his Note statistique No 1 in 1954. So, it seems safe to assume Matheron thought he was working with statistics. His thoughts are accessible again since CG’s website is back online.

Agterberg said in his tribute that Matheron “commenced work on regionalized random variables inspired by De Wijs and Krige.” Let’s take a look at Matheron’s very first paper and try to figure out what he did in his Formule des Minerais Connexes. He tested for associative dependence between lead and silver grades in lead ore. He derived length-weighted average lead and silver grades of core samples that varied in lengths. What he didn’t do was derive variances of length-weighted average lead and silver grades. Neither did he test for spatial dependence between metal grades of ordered core samples. He didn’t give his primary data but scribbled a few stats in this 1954 paper. He didn’t refer to De Wijs or to Krige. In fact, Matheron rarely referred to the works of others.

Where’s the Central Limit Theorem?

Matheron was a master at working with symbols. Yet, he wouldn’t have made the grade in statistics because the Central Limit Theorem was beyond his grasp. The Founder of Spatial Statistics did indeed have a long way to go in 1954. So, he penned nothing but Notes Statistique until 1959. That's when he tucked Note géostatisque No 20 tightly behind Note statistique No19. So, why did he switch from stats to geostats? It took quite a while to explain but here’s what Matheron said in 1978. He did it because “geologists stress structure” and “statisticians stress randomness.” That sort of drivel does stand the test of time in Matheron’s Foreword to Mining Geostatistics just as much as Journel’s mad zero kriging variance does in Section V.A. Theory of Kriging.

What did D G Krige do that so inspired young Matheron? In 1954 Krige had looked at, “A statistical approach to some mine valuation problems on the Witwatersrand.” It still reads like real statistics, doesn’t it? In 1960 he did reflect, “On the departure of ore value distributions from the lognormal model in South African gold mines.” Isn't that the nasty reality at gold mines? So, Krige did indeed work with statistics in those days. He may since have had some epiphany because he cooked up in 1976, “A review of the development of geostatistics.” This is why Krige was highly qualified to put a preface to David’s 1977 Geostatistical Ore Reserve Estimation with its infinite set of simulated values in Section 12.2 Conditional Simulations.

Why did H J De Wijs wind up in Agterberg’s tribute to Matheron? Agterberg had found out in 1958 that De Wijs worked with formulas that “differed drastically from those used by mathematical statisticians.” Agterberg preferred “the conventional method of serial correlation.” Why would Agterberg talk about mathematical statistics and serial correlation in 1958 when he himself had stripped the variance of his own distance-weighted average point grade in 1970 and in 1974? Agterberg ought to explain why in 2009!

De Wijs brought vector analysis without confidence limits to mining engineering at the Technical University of Delft in the Netherlands when he left Bolivia after the Second World War. Jan Visman worked at the Dutch coal mines during the war and surfaced with tuberculosis, a novel approach to sampling theory and practice, and a huge set of test results determined in samples taken from heterogeneous sampling units of coal. So much information, in fact, that he was encouraged to write his PhD thesis on this subject. And that’s exactly what he did! He continued to work as a mining engineer at the Dutch State Mines. When he found out that the Dutch Government was thinking of closing its coal mines he migrated to Canada in 1951. He worked briefly in Ottawa until 1955, and moved to Alberta where his formidable expertise was put to work in the coal industry.

Going, going, gone in geostatistics

Visman’s sampling experiment with pairs of small and large increments is described in ASTM D2234-Collection of a Gross Sample of Coal, Annex A1. Test Method for Determining the Variance Components of a Coal. Visman’s sampling theory has been quoted in a range of works. Following are some surprising references to Visman’s work, and to the lack thereof after Gy's work was widely accepted for no apparent reason.

Gy’s 1967 L’Échantillonnage des Minerais en Vrac, Tome 1 … two

Gy’s 1973 L’Échantillonnage des Minerais en Vrac, Tome 2 … eight

David’s 1977 Geostatistical Ore Reserve Estimation … two

Journel & Huijbregts’s 1978 Mining Geostatistics … zero

Clark’s 1979 Practical Geostatistics … zero

Gy’s 1979 Sampling Particulate Materials, Theory Practice … zero

Visman's sampling theory is based on the additive property of variances. None of the above works deals with the additive property of variances in a measurement hierarchy.

How to measure what we speak about

NASA satellites have been measuring lower troposphere global temperatures since 1979. At that time I went around the world at a snail’s pace. Lord Kelvin’s thoughts about how to measure what we speak about were much on my mind in those days. I thought a lot of metrology in general, and of sampling and statistics in detail. I was to visit all of Cominco’s operations around the world. My task was to assess the sampling and weighing of a wide range of materials. Of course, it couldn’t possibly have crossed my mind that I would look in 2008 at the statistics for 30 years of lower troposphere global temperatures.

My job with Cominco did have its perks. When I was at the Black Angel mine in Greenland, I saw Wegener’s sledge on a glacier above the Banana ore zone. I knew how geologists had struggled with Wegener’s continental drift, and how they slowed it down to plate techtonics.

Southeast Coast of Greenland

I knew geologists were struggling with Matheron’s new science of geostatistics. I travelled around the world with a bag of red and white beans, a HP41 calculator and a little printer to make the Central Limit Theorem come alive during workshops on sampling and statistics. I lost my bag of beans because it was confiscated at customs in Australia.

On-stream analyzers that measure metal grades of slurry flows at mineral processing plants ranked high on my list of tools to work with. The fact that the printed list of measured values was just peeled of the printer at the end of a shift rubbed me the wrong way. I got into the habit of asking who did what with measured values. It was not much at that time because on-stream analyzers were as rare as weather satellites. Daily sheets made up a monthly pile, and that was the end of it. I entered the odd set in my HP41 to derive the arithmetic mean and its confidence limits for a single shift. But that was too tedious a task. That’s why spreadsheet software ranked high on my list of stuff to work with.

I met a metallurgist who tried to put to work Box and Jenkins 1976 Time series analysis. So, he did have a few questions. I explained what Visman’s sampling theory had taught me. First of all, the variance terms of an ordered set of measured values give a sampling variogram. Secondly, the lag of a sampling variogram shows where orderliness in a sample space or a sampling unit dissipates into randomness. The problem is that Time series analysis doesn’t work with sampling variograms. So, the metallurgist got rid of his Box and Jenkins and I took his Time series analysis. Box and Jenkins referred to M S Bartlett, R A Fisher, A Hald, and J W Tukey but not to F P Agterberg or G Matheron. Box and Jenkins provide interesting data sets. I’ve got to look at the statistics for Wölfer’s Yearly Sunspot Numbers for the period from 1770 to 1869.

Sunspots

Visman’s sampling theory did come alive while I was working with Cominco. So much so that I decided to put together Sampling and Weighing of Bulk Solids. The interleaved sampling protocol plays a key role in deriving confidence limits for the mass of metal contained in a concentrate shipment. So, I was pleased that ISO Technical Committee 183 approved ISO/DIS 13543–Determination of Mass of Contained Metal in the Lot. I was already thinking about measuring the mass of metal contained in an ore deposit! But CIM’s geostatistical thinkers had different thoughts. For example, CIM’s Geological Society rejected Precision Estimates for Ore Reserves. In contrast, CIM’s Metallurgical Society approved Simulation Models for Mineral Processing Plants.

In other words, testing for spatial dependence is acceptable when applied to an ordered set of metal grades in a slurry flow. Testing for spatial dependence is unacceptable when applied to metal grades of ordered rounds in a drift. So I talked to Dr W D Sinclair, Editor, CIM Bulletin. He was but one of a few who would listen to my objection against such ambiguity. In fact, I put together a technical brief and called it Abuse of Statistics. I mailed it on July 2, 1992, and asked it be reviewed by a statistician. A few weeks later Sinclair called and said Dr F P Agterberg, his Associate Editor, was on the line with a question. What Agterberg wanted to know is when and where Wells did praise statistical thinking. That was all!

H G Wells

I didn’t know when or where Wells said it! I didn’t even know whether he said it or not! What I did know was that Darrell Huff thought he had said it. In fact, he did quote it in How to Lie with Statistics. I didn’t know much about Agterberg in 1992. What I did know then was that David in his 1977 Geostatistical Ore Reserve Estimation referred to Agterberg’s 1974 Geomathematics. And I found out that Agterberg didn’t trust statisticians when he reviewed Abuse of Statistics.

F P Agterberg

Agterberg , CIM Bulletin’s Associate Editor in 1992, was a leading scholar with the Geological Survey of Canada. Yet, he didn’t know that functions do have variances. It does explain why he fumbled the variance of his own distance-weighted average zero-dimensional point grade first in 1970, and again in 1974. He could have told me in 1992 that this variance was gone but chose not to. Agterberg was the President of the International Association for Mathematical Geology when it was recreated as the International Association for Mathematical Geosciences. He is presently IAMG’s Past President. He still denies that his zero-dimensional distance-weighted average point grade does have a variance. Agterberg was wrong in 1970, in 1974, and in 1992. And he is still wrong in 2009. That's bad news for geoscientists!