Use and misuse of statistics

“Statistical thinking will one day be as necessary for

efficient citizenship as the ability to read and write.”

H G Wells (1866-1946)

Wells was a prolific writer with a keen sense of rights and wrongs in his life and time. What had inspired him to praise statistical thinking were the works of Karl Pearson (1857-1936), and of Sir Ronald Aylmer Fisher (1890-1962). Pearson worked with large data sets whereas Fisher worked with small data sets. That was what inspired Fisher to add degrees of freedom to Pearson’s chi-square distribution. Thus was born a feud between giants of statistics. Degrees of freedom converted probability theory into applied statistics, and sampling theory into sampling practice. Fisher and Pearson were both outstanding statisticians. They inspired H G Wells and scores of statisticians. Applied statistics shall stand the test of time until our sun bloats into a red giant and Van Gogh’s Sun Flowers burn to a crisp.

Why did geoscientists get into geostatistical thinking? All it took was a young French geologist who went to work at a mine in Algeria in 1954. He measured associative dependence between lead and silver grades of drill-core samples. But he did not count degrees of freedom. So, he did not know whether his correlation coefficient was significant at 95%, 99% or 99.9% probability. What is more, his drill-core samples varied in length. As a result, the number of degrees of freedom is a positive irrational rather than a positive integer. He did not know how to test for spatial dependence by applying Fisher’s F-test to the variance of the set of measured values and the first variance term of the ordered set. His first paper was not peer reviewed. Nobody asked him to report primary data and give references. As luck would have it, he was without peers. Professor Dr Georges Matheron and his magnum opus were accepted on face value. His students thought of him as “creator of geostatistics”. Dr Frederik P Agterberg in his eulogy called him “founder of spatial statistics”. Yet, between 1954 and 2000 Professor Dr Georges Matheron did not teach his disciples how to test for spatial dependence and how to count degrees of freedom.

My son and I wrote Precision Estimates for Ore Reserves. It was based on applied statistics as it had been developed by Fisher and Pearson and was praised by Wells. We did test for spatial dependence by applying Fisher’s F-test to the variance of a set of measured values and the first variance term of the ordered set. We had studied David’s 1977 Geostatistical Ore Reserve Estimation. Professor Dr Michel David did not show how to test for spatial dependence and how to count degrees of freedom. Our paper was submitted to CIM Bulletin on September 28, 1989. We did not criticize geostatistics nor did we refer to it. CIM Bulletin rejected it but Erzmetall praised and published it in October 1991.

Bre-X Minerals was selling stock and getting ready to drill at Busang. The internet would not be ready for a while. The mining industry liked unbiased confidence limits for masses of metals contained in mined ores and mineral concentrates. What it did not like in the 1990s and still does not like in 2013 are unbiased confidence limits for masses of metals contained in reserves. I had sent to CIM Bulletin on September 21, 1992, an article on Abuse of Statistics. The Editor advised that articles of a controversial nature can be published in CIM Forum. I was asked to cite a specific reference for the quotation in which H G Wells spoke so highly about statistical thinking. I had found it long ago in Darrell Huff’s How to lie with statistics. Penguin Books published the first edition in 1954 when young Matheron was working with statistics in Algeria. Matheron and Agterberg would have been pleased had Wells praised geostatistical thinking.

Geostatistics messed up the study of climate change. Spatial dependence in our sample space of time may or may not dissipate into randomness. Sampling variogram shows whether, where and when it does. High school students ought to be taught how to construct sampling variograms. It would have made H G Wells smile. 

Setting new standards?

The Bre-X fraud inspired the Toronto Stock Exchange (TSE) and the Ontario Securities Commission (OSC) to set up a task force. Its objective was to take a close look at National Instrument 43-101. The Members of the task force are given in this Interim Report. Mr Morley P Carscallen, OSC’s Vice Chair, and Mr John W Carson, TSE’s Senior Vice-President, took on this task in April 1997. It is a fact that Bre-X’s bogus gold grades and Busang’s barren rock were made to look by hook and by crook like a gold resource. But who were the crooks? And who set the hook for Bre-X’s shareholders? OSC’s own qualified persons have yet to grasp the fact that geostatistics is a scientific fraud! Perhaps ironically, it was geostatistical software that made Bre-X’s bogus grades and Busang’s barren rock to look like massive gold resource!

I have put on paper why geostatistics is a scientific fraud. A few simple steps were all it took to cook it up! The first step was to strip the variance off the distance-weighted average. The second step was to call what was left a kriged estimate to honor D G Krige and his work. Matheron taught his disciples how to work with infinite sets of kriged estimates and zero kriging variances. What a shame that such a simple scientific fraud underpinned what was called a new science. Matheron himself never got into counting degrees of freedom. Neither did Stanford’s Journel, UBC’s Sinclair, and similarly gifted scholars.  

Young Dr A J Sinclair took to geostatistics in the 1970s. He may well have thought that Matheron had a fresh take on applied statistics. In those days Sinclair was entrusted with teaching UBC’s students all about Earth Sciences. CIM Bulletin asked Sinclair in 1990 to review Precision Estimates for Ore Reserves. My son and I had shown how to test for spatial dependence between a set of gold grades determined in ordered rounds in a drift. Given that interleaved bulk samples had not been selected, it was impossible to estimate the intrinsic variance of gold. Professor Dr A J Sinclair, PEng, PGeo rejected our article. We were pleased that it was praised by and published in Erzmetall, October 1991.

What a surprise that David’s peers wanted to praise his 1977 Geostatistical Ore Reserve Estimation! Why would his peers want to praise infinite sets of simulated values? The stage for an international forum was set at McGill University on June 3-5, 1993. It was called Geostatistics for the Next Century. What is so striking in retrospect is the fact that Bre-X Minerals was already drilling in Borneo when David was praised by his peers! Nobody was interested in the properties of variances in 1993! Yet, the additive property of variances in a measurement chain played a key role in unscrambling the Bre-X fraud.

 Measurement variance included

 Measurement variance subtracted

The Mining Standard Task Force released its Final Report in January 1999. Why had MSTF not pointed out that geostatistical software had convert Bre-X’s bogus grades and Busang’s barren rock so slickly into a massive phantom gold resource? MSTF’s Final Report was made public in January 1999. On a positive note, Dr A J Sinclair no longer graces National Instrument 43-101. On a negative note, Setting New Standards still didn’t explain at all how the Bre-X fraud could have been nipped in the bud. So it was that the Mining Standard Task Force ended up as a farce. The properties of variances were nowhere to be found. Sinclair still teaches students at UBC's Department of Earth and Ocean Sciences how to assume spatial dependence, krige, smooth, and rig the rules of applied statistics with impunity. So much for scientific integrity!

I have set up several sources of information on my website. Under Correspondence are listed all sorts of letters in a context of source and time. Academic freedom to teach a scientific fraud makes no sense at all. The fact that "geostatistics has flourished in the scientific literature for more than four decades" does not imply that spatial dependence between measured values in ordered sets may be assumed. Neither does it imply that degrees of freedom need not be counted.

To krige or not to krige?

Not only is it a verb with a touch of a noun but it is also a true eponym. Matheron had written in 1960 what he himself had called Krigeage d’un panneau rectangulaire par sa périphérie. Nowadays it is posted as Note géostatistique No 28. An anthology of Matheron’s life and time, and of his creation of geostatistics, is posted on a massive website. Danie G Krige had put together a Preface to David’s 1977 Geostatistical Ore Reserve Estimation. References to Krige pop up on many pages. Journel’s 1978 Mining Geostatistics, too, refers not only to D G Krige but also to the zero kriging variance. 

Geostatistical software made Bre-X’s bogus grades and Busang’s barren rock look like a massive gold resource. So why had geostatistics been hailed as a new science in the 1970s. The Bre-X scam was well on its way when geostatisticians got together to praise David’s 1977 Geostatistical Ore Reserve Estimation. He was praised at a celebration called Geostatistics for the Next Century at Montreal on June 3-5, 1993. My take on The Properties of Variances clashed with the celebrations at McGill University. What applied statistics did do is prove that the intrinsic variability of Bre-X’s gold was statistically identical to zero. How about that? The geostatocracy is still poised in 2012 to assume, krige, smooth, and rig the rules of applied statistics with impunity.  

David’s 1977 textbook displayed his tenuous grasp of applied statistics. The author points on page 33 of Chapter 2 to what he calls “the famous central limit theorem”. On page 286 in Figure 203 he shows how to derive a set of sixteen (16) “famous central limit theorems” from the same set of nine (9) holes. Next, he points out on this page, “Writing all the necessary covariances for that system of equations is a good test to find out whether one really understands geostatistics”. Counting degrees of freedom would have  shown that the author of the first textbook on geostatistics did grasp applied statistics.

It is simple to verify spatial dependence between measured values in an ordered set by applying Fisher’s F-test to the variance of the set and the first variance term of the ordered set. The F-test requires that degrees of freedom be counted. Stanford’s Journel claims that spatial dependence between measured values may be assumed. For crying out loud! He did so in his letter to JMG’s Editor. Now how’s that for a nouveau science! Surely, spatial dependence in sample spaces should be proved beyond reasonable doubt. It took but two steps to go from goofy geostatistics to a genuine fraud. The first step was to strip the variance off the distance-weighted average. The second step was to call a kriged estimate what had once been a distance-weighted average with a variance. Now that’s simple comme bonjour, n’est ce pas? Kriging is a stacked game of chance. Thou shall not krige when scientific integrity matters!

Mineral Inventory Studies of Precious Metal Deposits in British Columbia is one work of geostatistical fiction that I have kept on file. The study that peeked my interest most of all was Ordinary Block Kriging with Geological Control, A Practical Approach to Estimating Mineral Inventory, Nickel Plate Mine, Hedley, British Columbia. I did so simply because primary data are given. The authors of this study were A J Sinclair et al. It hit the spotlight on June 3-5, 1993 when “Geostatistics for the Next Century” was hailed for no reason whatsoever!   

Dr A J Sinclair, Professor Emeritus (Geological Engineering), was 2000-2001 recipient of a distinguished lecturer award. Sinclair talked about “Geology and data analysis: essential components of high quality resource/reserve estimation”.  He talked across the country in both official languages. His paper on Ordinary Block Kriging with Geological Control, A Practical Approach to Estimating Mineral Inventory, Nickel Plate Mine, Hedley, BC was presented when David was praised at McGill in June 1993. I applied Fisher F-test to test for spatial dependence.

 

Fisher's F-test for spatial dependence

The set of production data didn’t display a significant degree of spatial dependence. Neither did the set of ordinary block kriging data. Bartlett’s chi-squared test would have shown significant discrepancies not only between variances of sets but also between first variance terms of ordered sets. 

    

95% Confidence limits for arithmetic means

The central values in this table are arithmetic means. Confidence intervals and ranges are derived in Excel spreadsheet files. Shortly, a link to both files will be be posted.

  

The Society for Mining, Metallurgy, and Exploration published in Volume 308 Transactions 2000 a reviewed paper entitled Borehole statistics with spreadsheet software. The paper shows how to fingerprint boreholes. Its reviewer expected it would “stir up a hornets’ nest” but it never did! This paper underpins a report in which confidence limits for a large gold reserve had been derived. It was submitted to Barrick Gold early in 1998. 

Metrology in mining and metallurgy

Trans Tech Publication printed Volume 4 in its series on Bulk Materials Handling in 1985. It was called Sampling and Weighing of Bulk Solids. An unauthorized translation into Mandarin surfaced in November 1989. We do have a Canadian copyright on Metrology in Mining and Metallurgy. This text will also deal in detail with mineral exploration. It will do so because the Bre-X fraud was by far the worst salting scam I have ever unscrambled. I did it for Barrick Gold Corporation several months before Bre-X’s boss salter wound up in the Kalimantan jungle. That’s but one reason why I have registered the Canadian copyright for Metrology in Mining and Metallurgy.  

What has put my work on the map was the interleaved sampling protocol for mineral concentrates. The same protocol underpins the design of a mechanical sampling system to determine trace elements in cathode copper. I know how to derive 95% confidence limits for metal grades and contents of reserves and of proven parts of resources. Page 120 of my textbook in Section 4.5 Propagation of Variances gives the variance of a general function as defined in probability theory. One would expect a scholar with a PhD in epidemiology and biostatistics to be familiar with the properties of variances and the concept of degrees of freedom. I had given a pair of copies of Sampling and Weighing of Bulk Solids to Dr Martha Piper and she gave both to Professor Dr Alastair J Sinclair, PEng, PGeo. Dr Piper could have but did not give a copy to Dr M Klawe, her Dean of Science in those days. UBC’s library in 2011 finally put a copy of my book on one of its shelves. Why it took much too long would make a story in itself at this stage.

UBC’s Department of Geological Sciences took a liking to Matheron’s new science of geostatistics. It came about when Professor Dr Alastair J Sinclair, PEng, PGeo thought that his students stood to benefit more from Matheronian geostatistics than from applied statistics. Dr Piper might have been aware that one-to-one correspondence between functions and variances is sine qua non in applied statistics. All it would have taken in those days was a brief call to Professor Dr Nathan Divinsky.

A peculiar event took place at the Department of Geological Sciences on November 22, 1989. That’s when Dr A J Sinclair greeted those who took my short course on Sampling Precious Metal Deposits, Metrology - A New Look.  CIM Bulletin had earlier entrusted Sinclair with the review of Precision Estimates for Ore Reserves. He had initialed his review with AJS:131 on November 15, 1989. What may have troubled Professor Dr Alastair J Sinclair, PEng, PGeo was that Matheron’s science of geostatistics had left us cold. It may explain why he hopped in and out of Room 330A like a jack-in-the-box. But he could have asked the odd question during my talk. For Sir R A Fisher’s sake!

One-to-one correspondence between functions and variances was as far beyond the grasps of David and of Sinclair in 1989 as it was beyond Matheron’s grasp in 1952. The question is why variances of distance-weighted averages are still missing in 2012. It is true that the distance-weighted average itself was never lost but had merely morphed into a kriged estimate. But its variance had vanished when Matheron and his disciples had cooked up geostatistics.

The attachment to my letter of November 30, 1994 to Mr John Drury, CIM Ad Hoc Reserve Definitions Committee, shows how to derive the variance of a mass of metal in crushed ore or insitu ore.

ISO/DIS 13543-Determination of Mass of Contained Metal in the Lot

 

Borehole statistics with spreadsheet software
SME Volume 308, Transactions 2000

It had come about that the new science of geostatistics called for a mind-numbing step. Matheron and his minions stripped the variance off the distance-weighted average and called what was left a kriged estimate. The miracle of that stripped variance was embraced at UBC with as much zeal as it was at Stanford. Professor Dr A J Journel was asked why Fisher’s F-test was not applied to test for spatial dependence between measured values in ordered sets. His reply has graced my website since 2003. Journel seems to encourage those who assume, krige, smooth and rig the rules of applied statistics with impunity!

From human error to scientific fraud

Such reads the caption that these days graces my website. A few changes have been made since it was posted in 2003. What pleased me most was that loads of facsimiles and scores of snail mails could be whittled down to links. It didn’t take Merks and Merks long to figure out why geostatistics is an invalid variant of applied statistics. All it took was a close look at geostatistics when CIM Bulletin did reject Precision Estimates for Ore Reserves. We did so since it was praised by and published in Erzmetall 44 (1991) Nr 10. It was easy  to find out what was wrong with geostatistics. It matters not at all that the distance-weighted average is called a kriged estimate. What does matter is that it did somehow shed its variance.  Geostatistocrats have not yet put into plain words why each and every kriged estimate has lost its variance.  

Matheron’s new science of geostatistics has made landfall on this continent in 1970. A geostatistics colloquium in North America took place on campus at The University of Kansas, Lawrence on 7-9 June 1970. Its proceedings were recorded by Daniel F Merriam and published by Plenum Press, New York-London, 1970. A Maréchal and J Serra had graduated at the Centre de Morphologie Mathématique at Fontainebleau, France. They had come to shed light on Random Kriging. The authors point to Punctual Kriging in Figure 10. It shows how to derive a set of sixteen (16) grades from a set of nine (9) grades. It looked a bit of a slight of hand but it seemed to make sense to Professor Dr Michel David. So he posted  Maréchal and Serra’s Figure 10 on page 286 in Chapter 10 The Practice of Kriging of his 1977 textbook.

Figure 10 – Grades of n samples belonging to
nine rectangles P of pattern surrounding x

Figure 203 – Pattern showing all points within B,

which are estimated from the same nine holes

Why geostatistics is but a bogus variant of applied statistics is simple comme bonjour! Functions do have variances. No ifs or buts! That’s why one-to-one correspondence between functions and variances is sine qua non in applied statistics. Degrees of freedom are positive integers when all measured values in the set have the same weight. Degrees of freedom are positive irrationals when all measured values in the set have variable weights.

The power of applied statistics has served me well throughout my career. It did because so much of applied statistics is intuitive. For example, any set of measured values has a central value, a variance, a standard deviation and a coefficient of variation. The central value is either its arithmetic mean or some weighted average. Numbers of measured values in sets define confidence limits for central values. Testing for spatial dependence between measured values in ordered sets shows where orderliness in sample spaces or sampling units dissipates into randomness. Never did it make any sense in my work to assume spatial dependence between measured values in ordered sets.  What does make sense is testing for spatial dependence, skewness and kurtosis.

The central limit theorem defines the relationship between a set of measured values and its central value. Even David did refer to “the famous central limit theorem”. Yet, he didn’t deem it famous enough to add to his Index. Testing for spatial dependence between measured values in sample spaces and sampling units plays a key role in scores of applications in a wide range of disciplines. Participation in several standard committees served to make applied statistics indispensable in so many ways. I do have but a few simple questions at this stage. Why did Professor Dr Georges Matheron (1930-2000) cook up such a silly variant of applied statistics? Why was Matheron’s work deemed beyond peer review! Why didn’t anybody point out to him that all functions do have variances? Why doesn’t the mining industry care about unbiased confidence limits for metal contents and grades of reserves and resources?

Today I woke up as a certified octogenarian. I took a ride on my stationary bike and got nowhere. Yet I felt good. But I am still sick and tired of those who play games with other people’s money.  All I want to do at this stage of my life is show how to work with sound statistics and how to get rid of bogus science.

UBC still stuck with geostatistics

 Professor Dr Alastair J Sinclair has been teaching earth sciences at the University of British Columbia since 1964. It was but a dozen years after Matheron tried his hand at applied statistics. Young Georges Matheron in 1952 was an up-and-coming geologist in Algiers. He had a penchant for applied statistics in those days. For example, he knew how to test for associative dependence between lead and silver grades in drill core samples of variable lengths. What he did not know was how to derive variances of length-weighted average lead and silver grades. Perhaps ironically, young Matheron at that time thought he was working with applied statistics. Yet he didn’t know how to test for spatial dependence in sample spaces and sampling units by applying Fisher’s F-test to the variance of the set and the first variance term of the ordered set.  Neither did he derive length-weighted average lead and silver grades for his data set.

Professor Dr Georges Matheron
Abuser of applied statistics
Creator of geostatistics
Founder of spatial statistics

Professor Dr Georges Matheron was not at all into sharing primary data with his students. Even for his 1965 PhD Thesis he saw fit to cook up a funny pair of minuscule primary data sets.

 Matheron's primary data sets for 1965 PhD Thesis

So I have decided to show how to test for spatial dependence between Matheron’s make-believe primary data. All I did was apply Fisher's F-test to the variance of the set and the first variance terms of the ordered set and compare the observed F-values with tabulated F-values.

Stats for Matheron's 1965 PhD Thesis

Matheron’s magnum opus is posted on a massive website. Its webmaster made a few minor changes to suggest that Matheron had applied geostatistics somewhat sooner than he had done in real time.

It is ironic to the extreme that geostatistics was hailed as a new science when Matheron and his disciples brought it to campus at the University of Kansas in June 1970. Matheron’s own tour-de-force at this colloquium was to invoke Brownian motion along a straight line. He did so to infer that his random functions are continuous between measured values. The study on Random kriging by A Marechal and J Serra at the Centre de Morphology Mathematique seemed successful under Matheron’s supervision. Figure 10 in this 1970 study metamorphosed in Figure 203 on page 286 in Chapter 10 The Practice of Kriging in Professor Dr Michel David’s 1977 Geostatistical Ore Reserve Estimation.

David’s 1977 textbook and Gy’s 1979 Sampling of Particulate Materials, Theory and Practice, stand side-by-side on a shelf in my office. One time soon I’ll use them to prove how the French sampling school has messed up statistical thinking. And all it really took was to ignore one-to-one correspondence between functions and variances, to assume spatial dependence between measured values in ordered sets, and to pay no attention to counting degrees of freedom.

Dr Alastair J Sinclair, PEng, PGeo

UBC Emeritus Professor

Professor Dr Alastair J Sinclair described in Applied Mineral Inventory Estimation how his “exciting and invigorating career” took off when he was exposed to Matheron’s ideas, and how he had “the good fortune to work with Journel, Huijbregts and Deraisme”. Good grief! Those were Matheron’s earliest students who took his musings for dogma, and who didn’t have a clue that the variance of the distance-weighted average cum kriged estimate had vanished into thin air on Matheron’s watch. Sinclair’s list of those who he was “fortunate to have worked with at various times” reads like a Who’s who in the world's  geostatistical fraternity. Sinclair credits all of them to have contributed to his education. For once I do agree! I’m all in favor of giving credit where credit is due. But to give credit to everybody who has taught Professor Dr Alastair j Sinclair, PEng, PGeo how to apply a flawed variant of applied statistics is a bit over the top. Some geostatistocrats on Sinclair’s list know that each and every distance-weighted average cum kriged estimate does have its own variance. No ifs or buts! Whether Al likes it or not!

I wrote one more letter to Dr Martha C Piper, President, The University of British Columbia. I pointed out that H G Wells (1866-1946) had predicted, “Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write”. I mentioned that statistical thinking served me well indeed as a consultant, a lecturer, an author and a publisher, and as a global citizen of sorts on IMO and ISO Technical Committees such as TC69-The application of statistical methods.

Professor Dr Nathan Divinsky was charged in 1949 with the teaching of mathematics to UBC students. He retired as a professor in the mathematics department in 1991. I met a few of his former students who enjoyed his teaching and appreciated the power of applied statistics. Once upon a time I called him to ask whether statistical inferences are possible without degrees of freedom. I’ll always remember that he said, “But without degrees of freedom statistical inferences are impossible”. Dr Nathan Divinsky passed away at 86. He was married for eleven years to former Prime Minister Kim Campbell. Who would dare doubt such a short, crisp and to the point response by a Professor of Mathematics? May he rest in peace!

McGill toils with Markov chains

McGill University claims to be at the cutting edge of defining ore deposits with Markov chains. The National Post on August 15, 2005 published an article with the caption “It’s mining by the numbers”. McGill’s Professor Dr Roussos Dimitrakopoulos pointed out that, “Uncertainty means probabilistic models, and there are a gazillion types of them”. He has yet to show how to select the least biased model. He has a $3.5 million budget to put Markov chains to work. I’ll call him Dr RD for short. I do respect his blatant chutzpah! Dr RD cited a study by the World Bank that alleged 73% of North American mines had failed. What he didn’t point out is that geostatistical software is to blame!

It is a bit of a mystery when, where and why Dr RD made up his mind to travel all the way back to Markov chains. Stringing Markov chains overnight on a fast computer seems to somehow pin down ore deposits. But Dr RD didn't know that Markov chains cannot possibly give unbiased confidence limits for metal contents and grades of ore deposits! Markov and his chains may have made some sense before Fisher and Pearson feuded about degrees of freedom for the chi-square distribution. Why is it that counting degrees of freedom is still baffling the most gifted geostatistical gurus?

What’s more, Dr RD’s grasp of the properties of variances was already flawed in June 1993. At that time he was in a rush to get Geostatistics for the Next Century going. I had submitted by registered mail on March 10, 1993 an abstract for The Properties of Variances. I received an unsigned letter dated March 31, 1993. As luck would have it “a number of potential participants and their very interesting abstracts couldn’t be accommodated”. It so happened that I was one of those! All I wanted to do was show  how to derive unbiased confidence limits for metal contents and grades of ore reserves. We had shown how to do it in 1990. Professor Dr Michel David blew a fuse because we had applied “our own method”. Whose method had he expected? Given geostatistical peer review at CIM Bulletin in the 1990s I had asked JASA’s Editor for a courtesy review of The Properties of Variances. It passed JASA’s litmus test! A copy of The Properties of Variances is posted on my website. Peruse the properties of variances, count degrees of freedom, and derive confidence limits for mineral inventories. It is simple comme bon jour! I did it for Barrick Gold in 1998.

Professor Dr Michel David and his 1977 Geostatistical Ore Reserve Estimation were honored at Montreal, Quebec on June 3-7, 1993. Geostatistical scholars had come to praise the author of the first textbook. It deals with Matheron’s new science in mind numbing detail. David brought up “the famous central limit theorem “ on page 33 in Chapter 2 Contribution of Distributions to Mineral Reserves Problems. Chapter 10 The Practice of Kriging shows how to derive sixteen (16) famous central limit theorems from the same nine (9) holes. David pointed out on page 286, “Writing all the necessary covariances for that system of equations is a good test to find out whether one really understands geostatistics!” Good grief! Counting degrees of freedom for his system of equations would have been a good test to find out whether David did grasp applied statistics. David's Index does not refer to Markov chains. But who would want to bring up Markov chains at David’s bash?

Lost: variance of kriged estimate

Found: zero kriging variance 

It was none other than Stanford’s Professor Dr Andre G Journel who did! He had put forward a paper to shore up his own vision. It was called “Modeling Uncertainty, Some Conceptual Thoughts”. He had embellished his thoughts with prettified statements such as stochastic simulation, random models, Bayes’ updating, likelihood functions, sequential simulation and non-Gaussian models. That’s what preoccupied the mind of Matheron’s most gifted disciple in June 1993. It may well have turned off some of those who had come to praise David’s 1977 Geostatistical Ore Reserve Estimation! 

Every Spring quarter Emeritus Professor Dr A G Journel teaches an advanced PhD level seminar. What he does not teach is that each and every distance-weighted average AKA kriged estimate does have its own variance in applied statistics. What he ought to study is Dr Isobel Clark’s 1979 Practical Geostatistics. She derived the variance of a distance-weighted average AKA kriged estimate. Alas, Dr Clark didn’t test for spatial dependence between hypothetical uranium concentrations in her ordered set. Neither did she know that degrees of freedom for her set are positive irrationals rather than positive integers.