The world is flat again!

Ancient people used to believe that the Earth was flat and it was not until roughly 500 years ago that we acknowledged the concept of our Mother-ship being a sphere. The technological tsunami that has been well on its way since the World Wide Web kicked into high gear in the mid Nineties managed to flatten the world again. As we keep surfing the top of the wave – and the Internet – we finally have a tool that has great potential for countries and people of any ethnical and religious background to participate in business globally without having to ever leave their houses.
There is not a week that goes by where businesses call or e-mail me offering their products and engineering services – from more than 8000 Kilometers away. Not only does our technology finally offer opportunities to those of us who used to suffer from centuries of colonialism and its ugly repercussions, it also has proven itself by making business more transparent and thus fairer in general. The times are gone by when manufacturers tried to manipulate the quotation and buying process by controlling if not limiting information. Now we just need a browser and a good search engine and wealth of information opens itself up to us. The Bulk Blog and the Bulk Online Forum are just the case in point for this phenomenon.
This type of trading makes the whole quotation and buying experience much fairer and provides unique business opportunities for producers that are sincere and genuine in what they offer to the market. The emergence of social marketing that at its root is basing a company’s success to think about inviting customers to buy instead of pushing themselves onto the customer and actively having to sell products or services. What that means is that now a consumer has so many choices that what a seller offers had better be providing value to them. It should (I feel it is a must) be offered for customers to take at their free will without manipulation. The Web lowers the smoke screen that conventional sales and distribution outlets used to benefit from and raises the bar for us all to be honest. Consultants are probably the only ones who could argue that there income are threatened by the free information made available on the Net. On the other hand, they also have a great chance to use the technology to their benefit. Even the Web cannot exist without relationships that are based on trust and commitment management. There still need to be people who need to work on answering Web based requests.
Of course then there is the rate of speed at which one can acquire knowledge and do business. It is fascinating to me that nowadays I have the opportunity to not travel to a job site any more because you I log on to the machinery controllers and “be there” without being there. Machinery troubleshooting is possible from wherever we have access to the Net. Even better, you are even alerted by the machine with an SMS or e-mail that it is on trouble. Can you imagine how much time, money and natural resources you – and therefore your customer – do not need to spend anymore? Plant managers have an opportunity assessing life cycle costs for product ranging from a small blower to high value assets such as kilns – phenomenal.
I have trouble nowadays thinking back to a time when we did not have technological marvels such as E-Mail, Skype, Twitter, customer relationship databases, Intranets, Microsoft Roundtable and so many more. It still comes with disadvantages such as leaving poor folks out who do not have the money to participate. The Internet has changed our lives forever and it has brought us humans closer together than ever before in history. It also will provide well for entities that use the technology in a way that is beneficial for all the stake and share holders – including the customer as well as our environment. How flat can the Earth get?

Who wrote bogus stats, when, where, and why?

Professor Dr Roussos Dimitrakopoulos came up all the way from Down Under to chair a Forum on Geostatistics for the Next Century at McGill University on June 3-5, 1993. His task was to honor Professor Dr Michel David for writing the very first textbook on Matheron’s new science of geostatistics. What David didn’t know was how to test for spatial dependence and how to count degrees of freedom. He wrote his first textbook against all odds since he didn’t even know that functions do have variances. I have written a bit about the properties of variances. So, I send by registered mail an abstract to that futuristic forum at McGill University. Some person at McGill’s Conference Office encouraged me in an unsigned letter of March 31, 1993, to submit my abstract to another event. I’ll dig up more bits and pieces about the properties of variances.
Dimitrakopoulos likes McGill a lot. In fact, he settled down in La Belle Province after the Bre-X fraud was no longer on his mind. In a candid interview with the National Post on August 15, 2005, he clarified the intricacies behind his valuations of mining projects. Here’s what he said, “You drill a few holes, you think you understand something but what you know is very, very little. Uncertainty means probabilistic models, and there are a gazillion types of them.” How about that? Some mining investors might wonder how RD selects the least biased probabilistic model. Peter Ravenscroft, a senior executive with Rio Tinto and an expert at geostatistics himself, thinks what RD does is kind of cool and gave him a stack of dough.
Professor Dr Roussos Dimitrakopoulos was present at APCOM 2009 in Vancouver, British Columbia. The first line of his abstract reads, “Conventional approaches to estimating reserves and optimizing mine planning and production forecasting result in single, often biased forecasts.” I wonder what would have happened if Stochastic Mine Planning Optimization: New Concepts, Applications, and Monetary Value in an Ever Uncertain Market, had been applied to Bre-X’s exploration data. Financial institutions should demand that the International Organization for Standardization set up a Technical Committee on Reserve and Resource Estimation. It’s long past due. Matheron thought he was a statistician in 1954. Yet, his Note Statistique No 1 shows he didn’t know how to test for spatial dependence between metal grades in ordered core samples. Neither did he know how to derive the variances of length-weighted average lead and silver grades determined in core samples of variable lengths. So much for Matheron’s new science of geostatistics!

Dr Frederik P Agterberg wrote in 2000 that Matheron was the Founder of Spatial Statistics. Matheron thought he was a statistician in 1954 when he wrote his Note Statistique No 1. He didn’t write about spatial dependence between metal grades of ordered core sections with variable length. He did derive length-weighted average lead and silver grades but didn’t derive the variances of these central values. In 1907 he stirred up “Brownian motion on a straight line.” He did so because he liked Riemann integrals better than Riemann sums. He wrote in his 1978 Foreword to Mining Geostatistics why he proposed the name geostatistics in the 1960s. Professor Georges Matheron would have been shocked had he read in his obituary that he was the Founder of Spatial Statistics. Agterberg invited me on October 1, 2004, to present my views at the next IAMG annual meeting in Toronto. I happen to know a lot about IAMG events where geostatistocrats talk bafflegab. I would rather make my case against bogus stats at APCOM 2009.

Dr Michel David wrote a few words of caution in his 1977 Geostatistical Ore Reserve Estimation. First, he wrote, “…statisticians will find many unqualified statements…” Then, he blew the sales of his work by writing, “This is not a book for professional statisticians.” But he was indeed right. David did prove it when he wrote his test for geostatistical proficiency. He took M&S’s set of nine (9) measured values and “estimated” the same set of sixteen (16) what he came to call “…points…” He wrote on page 286 of his textbook, “Writing all the necessary covariances for that system of equations is a good test to find out whether one really understands geostatistics.” Why did the author of the very first textbook on geostatistics fail to derive the variance of each of this sixteen (16) functionally dependent values? Why didn’t he count degrees of freedom? If M&S’s set of nine (9) measured values were evenly spaced, the set and the ordered set would give df=n-1=8 and dfo=2(n-1)=16 respectively. Why is the geostatocracy still asleep at the switch? Why is Bre-X’s massive phantom gold resource all but forgotten?

A Marechal and J Serra wrote Random kriging in 1970 to celebrate the first krige and smooth bash in North America. M&S toiled under Matheron’s tutelage at his Center de Morphology Mathematique, Fontainebleau, France. So, why did M&S set out to simplify Matheron’s kriging equations with their own random kriging procedure? Under Punctual Kriging in Random kriging they show how to get a set of sixteen (16) functionally dependent values from a set of nine (9) measured values. M&S didn’t show how to derive a variance of a functionally dependent value. Neither did they show 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. What Matheron never taught M&S was how to count degrees of freedom. In his own 1970 Random functions and their applications in geology Matheron wrote, “Let us denote a Brownian motion on a straight line.” In Matheron’s mind it somehow seemed to replace Riemann sums with Riemann integrals. Matheron never explained what Brownian motion and ore deposits have in common. M&S put Random kriging “within the geostatistical framework of the French school.” Go figure why!

Dr Isobel Clark is the author of Practical Geostatistics. She wrote on the first page of Chapter 5 Kriging, “It would seem sensible to use a weighted average of the sample values, with the ‘closer’ sample values having more weight.” On the same page she wrote, “The arithmetic mean is simply a special case where all the weights are identical.” She wrote in her Preface that Journel and others at Fontainebleau taught her all she knows about the theory of the Theory of Regionalized Variables.” She transposed for “mathematical convenience” the factor two (2) in dfo=2(n-1), the number of degrees of freedom for an ordered set of n measured values. That’s how Clark’s semi-variogram was born. Why did Fisher’s F-test for spatial dependence between hypothetical uranium data fail to make Clark’s grade in her 1979 Practical Geostatistics? And why does nobody care?

Statistically dysfunctional geoscientists write all sorts of things that are bound to hound them in time. Read what Stanford`s Journel wrote to the Editor of the Journal for Mathematical Geology. What he did was set the stage for conditional simulation on Stanford stationary. Take note of when he wrote it. And read what JMG’s Editor wrote to me. So, my feeling that geostatistics is invalid might be correct. How about that? He also wrote that different “flavors” of geostatistics may fail at different times. Now that’s kind of cool. I do know which flavor failed in the Bre-X fraud. It was the flavor of assuming continued gold mineralization between salted boreholes. The odd geostatistician might be taught how to test for spatial dependence and how to count degrees of freedom. Most are doomed to assume, krige, smooth, and rig the rules of statistics.

Influence of electrostatic charge on pneumatic conveying

In pneumatic conveying, the air molecules pass the material particles and when they touch, the combination of the 2 exchange electrons until electrostatic equilibration is reached.

When the air and material part from each other, the exchanged electrical charges remain also separated and leave the particles electro statically charged.

In the example of pvc-resin, the air becomes positively charged and the pvc-resin becomes negatively charged. (see tribo electric series)

This phenomenon is called the tribo-electro effect.

The electrostatic charge of a particle increases with the number of contacts between air molecules and the particles until the maximum possible charge exchange is reached.

The maximum possible charge exchange is limited by the ratio between product particles and air particles. (In fact this is the Solid Loading Ratio).

In the example of PVC-resin, the pvc particles collect electrons up to a maximum.

This maximum absorption of electrons depends on the availability of air molecules, supplying these electrons.

A low SLR implies that there are more air molecules available per product particle, resulting in a maximum charged voltage.

Electrostatic charging is therefore stronger at low SLR’s

Continue reading Influence of electrostatic charge on pneumatic conveying

Degrees of freedom fighters struck at Stanford

It`s a strange but annual ritual of sorts. Degrees of freedom fighters assume, krige, smooth, and rig the rules of statistics. Today`s fighters call that mathematical statistics. This year the stage was set at Stanford Campus on 23-28 August. Once upon a time IAMG stood for International Association for Mathematical Geology. A few years ago IAMG morphed into International Association for Mathematical Geosciences. Its present mission is to promote, worldwide, the advancement of mathematics, statistics and informatics in the Geosciences. This latest variant of IAMG talks about statistics without counting degrees of freedom.

The famous feud between Pearson (1857-1936) and Fisher (1890-1962) came about because of degrees of freedom. Fisher added degrees of freedom to Pearson’s chi-square distribution, and was knighted in 1952. Fisher’s F-test is applied to verify spatial dependence in sampling units and sample spaces alike. Pearson’s coefficient of variation, too, stood the test of time. Meanwhile in Algiers, young Matheron didn’t count degrees of freedom. In fact, he didn’t have a clue what degrees of freedom were all about. IAMG’s most advanced thinkers still do not count degrees of freedom.

The very first textbook about Matheron’s new science of geostatistics was David’s 1977 Geostatistical Ore Reserve Estimation. Table 1.IV Copper grades Prince Lyell in Chapter 1 Elementary Statistical Theory and Applications gives a chi-square distribution with 13 degrees of freedom. David’s Index lists neither Chi-square distribution nor Degrees of freedom. What the author did list are Best linear unbiased estimator, Brownian motion, and Bull’s eye shot.

Figure 203 on page 286 of David’s first textbook takes the cake for boldness. The same figure saw the light as Figure 10 in Marechal and Sierra’s 1970 Random Kriging. It is printed in Proceedings of a Colloquium on Geostatistics held on campus at the University of Kansas, Lawrence on 7-9 June 1970.

Fig. 203. Pattern showing all the point within B,
which are estimated from the same nine holes.

David derived the covariances of his set of sixteen “samples”, each of which was “estimated” from the same nine holes.What he didn’t do was count degrees of freedom. His set of nine (9) holes gives df=n-1=9-1=8 degrees of freedom. The ordered set gives dfo=2(n-1)=2(9-1)=16 degrees of freedom. The number of degrees of freedom is a positive integer for evenly spaced holes but becomes a positive irrational for unevenly spaced holes.

A set of sixteen (16) functionally dependent values does not give a single degree of freedom. What David did not know either is that every functionally dependent value does have its own variance. He did know that his set of nine (9) holes gives an infinite set of functionally dependent values. David called them simulated values but statistically dysfunctional thinkers call them kriged estimates. The question is then why kriging variances of sets of kriged estimates became the building blocks of Matheronian geostatstics.

Dr Jef Caers chairs IAMG 2009. He is Associate Professor, Energy Resources Engineering, with Stanford University. His 1993 MS in Mining Engineering and Geophysics and his 1997 PhD in Engineering were obtained with the Katholieke Universiteit, Leuven, Belgium. He speaks French fluently. This is why he should belatedly review Matheron’s 1954 Note Statistique No 1 to assess if anything else but degrees of freedom and primary data went missing. Some scholar at Stanford Earth Sciences should know all about associative dependence, functional dependence and spatial dependence. I think Dr Jef Caers may be that scholar!

Casting dice and tossing coins at Stanford

Behind Stanford`s motto “The wind of freedom blows” is a rich history. It was President Gerhard Casper on October 5, 1995 who put a score of fine points to it. Who could possibly object to the freedom to teach and be taught sound sciences? When President Casper spoke in 1995 the freedom to assume spatial dependence between measured values in ordered sets had been entrenched in geostatistics since 1978. Herbert Hoover, Thirty-First President and Stanford`s very first mining engineer, would have been shocked. Who would put a mine stope together by casting dice? How could geostatistics have converted Bre-X’s bogus grades and Busang’s barren rock into a massive phantom gold resource?

Here’s what I have been trying to bring to the attention of Dr J L Hennessy, Stanford’s President. Geostatistics ignores the concept of degrees of freedom and violates one-to-one correspondence between functions and variances. Agterberg’s distance-weighted average does not have a variance. Neither does David’s distance-weighted average. I pointed out that it took the Papacy 300 years to right a wrong. I did so the last time I wrote to Stanford’s President on February 13, 2008. I wrote that I thought Stanford could right a wrong much faster. He could have asked a Stanford statistician whether or not the geostatocracy has the freedom to assume spatial dependence between measured values in ordered sets. What I wrote in 2008 didn’t hit Dr Hennessy’s list of things to do.

I do appreciate my own freedom and am a stickler for degrees of freedom. So, I looked at Stanford’s statistical scholars and warmed to what I read about Professor Dr Persi Diaconis. He looked like the kind of scholar who would take seriously my crusade against the geostatocracy and its army of degrees of freedom fighters. Stanford Report of June 7, 2004, pointed out, “Persi Diaconis has spent much of his life turning scams inside out.” Now there’s a professional scam buster of sorts. It became even better than I thought it would be when I read what Professor Dr Persi Warren Diaconis had done. He left home at 14, hit the road with Dai Vernon, the famous Ottawa-born slight-of-hand magician, and got Vernon’s magic touch.

When I was searching Stanford’s website for a genuine statistician, I found out that Dr Diaconis doesn’t respond to email. I took a chance and did send him an email anyway on February 23, 2009. That was more than year after my last email to Stanford’s President. Diaconis is indeed true to his word and did not respond to my email. I had suggested that Stanford should give real statistics a fighting chance. So, I decided to call Diaconis but nobody picked up the phone. I called between March 26 and April 22, 2009, and did so between 13:00 and 16:00 PST. I called sixteen times and the line was busy twice. I could have but decided not leave a message.

Diaconis knows how to toss a coin. So much so that the same side of the coin comes up ten times in a row. He designed a mechanical coin tossing contraption that gives the same odds. What he did do was defy the Central Limit Theorem. Coins and dice played cameo roles when I taught sampling theory and practice in places are far apart as Greenland and Tasmania, and as Finland and the Philippines. I put in plain words how to tamper with the outcomes of tossing coins and casting dice. What I didn’t show is how to test for bias. A Stanford student should cast the same die often enough to infer absence of bias within acceptable bias detection limits. The catch-22 is that abrasion is bound to cause a bias before acceptable bias limits are obtained.

I taught sampling theory and practice on the basis of a binomial sampling unit that consists of 90% white beans and 10% of the same but red-dyed beans.

Each participant would take a small increment and a large increment, and count white and red beans in each. This simple sampling experiment made it easy to explain Visman’s sampling theory and practice, and his composition and distribution components of the sampling variance. Visman’s work proved that the most effective method to estimate the variance of the stochastic variable of interest in a sampling unit or a sampling space is to partition the set of primary increments into a pair of interleaved subsets. Of course, one pair of subsets gives but one degree of freedom. That’s why SQC programs should be implemented on a routine basis. The interleaved sampling protocol has been incorporated in several ISO standards.

The wind of freedom blows at Stanford University. What geostatistocrats have blown is the concept of degrees of freedom. Agterberg blew the variance of the distance-weighted average. Journel blew Fisher’s F-test for spatial dependence. Once upon a time Herbert Hoover wrote, “It should be stated at the outset that it is utterly impossible to accurately value any mine, owing to the many speculative factors involved. The best that can be done is to state that the value lies between certain limits, and that the various stages above the minimum given represent various degrees of risk.” Hoover’s 1909 Principles of Mining Valuation, Organization and Administration still make sense. Why then is the world’s mining industry hooked on assuming, kriging, smoothing, and rigging the rules of real statistics?

Teaching junk statistics at Stanford

Stanford University is Professor Dr Andre G Journel`s world. He has put down deep roots at Stanford since 1978. Journel teaches the same flaky stats that Professor Dr Georges Matheron taught him between 1969 and 1978. Journel was Matheron`s most gifted student. Matheron taught him all of the ins and outs of his novel science of geostatistics. What Matheron may not have told Journel is that he himself thought in 1954 he was a statistician. It took some ten years to teach Journel how to assume, krige, and smooth with confidence and pride. Journel was Mining Project Engineer at the Centre de Morphology Mathematique from 1969 to 1973, and Maitre de Recherches at the Centre de Geostatistique from 1973 to 1978. Not surprisingly, he worked as much with symbols as Matheron did in his magnum opus. What Matheron failed to show his star disciple is how to test for spatial dependence between ordered sets of measured values in sample spaces and sampling units. Matheron and Journel never found the variance of Agterberg’s distance-weighted average point grade.

Journel is the lead author of Mining Geostatistics. When the ink had dried in 1978 he took his book to Stanford’s students and taught them all about assuming, kriging and smoothing. My copy is a “1981 reprint with corrections.” Matheron’s Foreword makes a deeply dense read. In contrast, Dr Isobel Clark’s Preface to 1979 Practical Geostatistics is an easy read. Her cradle once rocked on the side of the Channel where Sir R A Fisher was knighted. Clark confessed that it was Journel who taught her all she knows about the Theory of Regionalized Variables. Clark messed up degrees of freedom for ordered sets of measured values. What she did do is slash for “mathematical convenience” the factor 2 in df₀=2(n-1) degrees of freedom for ordered sets, cook up her silly semi- variogram, and scold the poor souls who “sloppily call it a variogram”. Clearly, Clark and Journel disagreed about semi-variograms and variograms. Neither knew how to test for spatial dependence, how to chart sampling variograms, or how to count degrees of freedom.
Matheron’s 1978 Foreword to Mining Geostatistics went off on a tangent just as much as did his 1954 Note statistique No 1. He beat around the bush about geologists who “stress structure” and statisticians who “stress randomness.” Matheron’s point of view flies in the face of Visman’s sampling theory with its composition and distribution variances. Matheron predicted, “The user of Mining Geostatistics will come across nothing more than variances and covariances, vectors and matrices”. Matrices and vectors do indeed abound from cover to cover but so do pseudo variances and pseudo covariances. What all those so called “variances” and “covariances” in Mining Geostatistics have in common with genuine variances and covariances are squared dimensions. The concept of degrees of freedom, too, failed to make the grade in Matheronian geostatistics. And that’s what will kill the kriging game!
I came across a genuine variance in a numerical example on page 63 of Mining Geostatistics. The authors divided a stope into four equal units, and assigned to each unit a grade equal to the outcome of a cast of “an unbiased six-sided die.” Now that does indeed give a genuine variance. Casting an unbiased die a large number of times gives a uniform probability distribution with a population mean of μ=3.5 and a population variance of σ²=2.917. The authors deserve praise for giving correct values, and for pointing out that the die ought to be unbiased. Surely, Stanford’s students ought to be taught how toproperly assess the risk of playing all sorts of games of chance.

No real data in 1954 – Casting dice in 1978

A set of three (3) stopes is presented on the same page. Each set of four units within a stope was put together with a six-sided unbiased die such that each has the same mean of 3.5. That sort of applied research is time-consuming but of critical importance when teaching all of the intricacies of geostatistics. A touch of classical statistics would be required to test whether or not a given die is indeed unbiased. The question of whether or not Jourel’s die was biased was solved by assuming it was unbiased. Fisher’s F-test shows that the variances of the sets and the first variance terms of ordered sets are statistically identical. Read what Journel said about “Fischerian (sic) statistics” in October 1992. How’s that for creative thinking and writing?
The zero kriging variance of σ²k=0 can be found on page 308, Chapter V The Estimation of in situ resources in Mining Geostatistics. Another unique feature of Matheronian geostatistics is one-to-one correspondence between zero kriging variances and infinite sets of kriged estimates. Even the OCS might find it a bit of a stretch to report a 95% confidence interval of zero ounces of gold for a mineral inventory with 9.9 million ounces. Armstrong and Champigny solved this Catch-22 with a strict caution against over smoothing. They did so in A Study on Kriging Small Blocks, CIM Bulletin, March 1989. The authors suggest that the requirement of functional independence may be violated a little but not a lot All that geostatistical gobbledycook is dished up because one-to-one correspondence between distance-weighted averages and variances became null and void in Agterberg’s 1974 Geomathematics.
On a positive note, Dr John L Hennessy, Stanford’s President, is but one leader at an institute of higher learning who did respond to my letters.

On August 23-28, 2009, IAMG’s Annual Conference will be held at Stanford University. What a wonderful opportunity for Stanford’s President to peek around the corner and find out why the variance of Agterberg’s distance-weighted average point grade went missing. Or he might ask Professor Dr Persi Diaconis to pose a few questions on his behalf. Diaconis is Stanford’s Mary V Sunseri Professor of Statistics and Mathematics. He’ll know all about the Central Limit Theorem and its role in sampling theory and practice.

Geostatistics continues to evolve as a discipline

That’s what Mark Corey wrote when Canada’s Minister of Natural Resources asked him to respond to my message. Mark Corey is Director General Mapping Services Branch and Assistant Deputy Minister, Earth Sciences Sector. He is the chief mapmaker for NRCan so to speak. I was ticked off big time when he called geostatistics a discipline. But I told myself it could have been worse. He could have called it a scientific discipline. He is also one of several experts behind NRCan’s 2008 “bulletproof” climate report. He testified at the Senate Committee for Energy, Environment and Natural Resources. I wish I could have asked him a few questions.
What I wanted him to tell me in plain words is why each and every distance-weighted average point grade doesn’t have its own variance. Dr Frits P Agterberg thought his distance-weighted average point grade didn’t have a variance in the early 1970s. Agterberg was wrong then. He’s wrong now. It’s high time for NRCan’s Emeritus Scientist to explain why his distance-weighted average point grade still doesn’t have a variance in 2009!
None of the five (5) points in the next picture have anything to do with pixels on a map. Each point stands for some sort of hypothetical uranium concentration that was measured in some way in samples selected in this sample space at positions with known Easting and Northing coordinates. I didn’t make it up but Dr Isobel Clark did in her 1979 Practical Geostatistics. She worried whether or not the Central Limit Theorem would hold so she didn’t derive it. Clark’s figure would have been a dead ringer for Agterberg’s 1970 and 1974 figures if it were not for her hypothetical uranium concentrations.

Fig. 1.1. Hypothetical sampling and estimation situation
Fig. 4.1. Hypothetical sampling and estimation situation – a uranium deposit

I want to prove Clark’s set of hypothetical uranium concentrations does not display a significant degree of spatial dependence. So, let’s take a systematic walk that visits each point only once and covers the shortest possible distance. Clark’s selected position is not equidistant to each of her hypothetical uranium concentrations. That’s why the number of degrees of freedom is not a positive integer but a positive irrational. Applying Fisher’s F-test to var(x)= 4,480, the variance of the set, and var1(x)=3.640, the first variance term of the ordered set, gives an observed F-value of F = 4,480/3,640 = 1.23. This observed F-value does not exceed the tabulated F-value of F0.05;4;4.90 = 6.38 at 95% probability. Therefore, the distance-weighted average hypothetical uranium concentration of 371 ppm is not an unbiased estimate.
Clark didn’t need Agterberg’s approval to derive confidence limits and ranges for this point grade. Neither did I and came up with a 95% confidence interval of 95% CI = +/-111 ppm or 95% CI = +/-29.8%rel, and a 95% confidence range with a lower limit of 95% CRL=261 ppm and an upper limit of 95% CRU=482 ppm.
Here’s what I would want NRCan’s Mark Corey to do. Visit your Emeritus Scientist in his ivory tower and borrow his 1974 Geomathematics. Go to Chapter 6 Probability and Statistics and look at Fisher’s F-test in Section 6.13. That will be all. At least for now!

Not quite fit for professional statisticians

Professor Dr Michel David said so himself. He pointed out his textbook is not for professional statisticians. He was talking about his very first textbook. I bought a copy of Geostatistical Ore Reserve Estimation, and worked my way through it. David was dead on when he predicted, “…statisticians will find many unqualified statements here.” All I really wanted to know is how David derived unbiased confidence limits for metal grades and contents of ore deposits. But he didn’t do it! Why would the author of the very first textbook on geostatistics fail to show how to derive unbiased confidence limits?

I had derived unbiased confidence limits for metal grades and contents of concentrate shipments. Mines and smelters want to know the risks associated with trading mineral concentrates. Metal traders were keen on my method and several ISO Technical Committees approved it. So, we I put together an analogous method, called it Precision Estimates for Ore Reserves and submitted it for review to CIM Bulletin. I still don’t know why our paper ended up on David’s desk. What I do know is that David blew a fuse when he saw we didn’t even refer to geostatistics let alone work with it.

In Section 10.2.3.3. Combination of point and random kriging, David refers to Maréchal and Serra’s Random kriging. These authors were with the Centre de Morphology Mathematique when they presented it at the celebrated Geostatistics colloquium on campus at the University of Kansas, Lawrence on June 7-9, 1970. In a section called Punctual Kriging these authors showed nine measured grades and sixteen functionally dependent grades.

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

M&S’s Figure 10 morphed into Figure 203 on page 286 of David’s 1977 book. On the same page David claimed, “Writing all the necessary covariances for that system of equations is a good test to find out whether one really understands geostatistics.” What David didn’t do was take a systematic walk that visits each hole only once and covers the shortest distance. But neither did Agterberg in 1970. Nor did M&S take a systematic hike on campus at that time.

Fig. 203. Pattern showing all the points within B,
which are estimated from the same nine holes

Each of David’s sixteen (16) points within B is in fact a distance-weighted average point grade. It makes no sense at all to derive the false covariance of a set of functionally dependent values but ignore the true variance of the set of nine (9) measured values. David sensed something was amiss. In Section 12.2 Conditional Simulations of Chapter 12 Orebody Modelling he confessed , “There is an infinite set of simulated values which will have these properties.”

Infinite set of distance-weighted averages point grades
derived from the same set of nine (9) holes

Counting degrees of freedom for his set of nine (9) holes would have been a foolproof test to find out whether David grasped statistics. What he saw in the black hole were Agterberg’s distance-weighted average point grades. Each is a zero-dimensional point grade. And each one of them lost its variance on Agterberg’s watch. Dr F P Agterberg, Emeritus Scientist with Natural Resources Canada, did approve Abuse of Statistics but wasn’t himself into counting degrees of freedom and testing for spatial dependence.

David’s 1977 Geostatistical Ore Reserve Estimation and Journel and Huijbregts’s Mining Geostatistics rank among the worst textbooks I’ve ever read. Until David’s 1988 Handbook of Applied Advanced Geostatistical Ore Reserve Estimation came along. His work was founded by the Natural Science and Engineering Research Council of Canada with Grant No 7035. What a waste!

Geostatistical data analysis – Quo Vadis?

More than 20 years ago G M Philip and D F Watson posed the question Matheronian Geostatistics – Quo Vadis? Philip and Watson’s question was published in Mathematical Geology, Vol 18, No 1, 1986. The text consisted of 21 pages and the list of references counted 86 works. Sir R A Fisher’s 1959 Statistical methods and scientific inference is on the list. Fisher’s ubiquitous F-test is applied to test for spatial dependence in sampling units and sample spaces alike. To assume spatial dependence appealed more to Matheron than to verify it by applying a sort of test cooked up by some kind of knight across the Channel. So, counting degrees of freedom failed to make Matheron’s list of things to do. He worked mostly with symbols and rarely with real data. Shortsighted statistical thinking still runs rampant at the Centre de Géostatistique, 5 Rue St Honoré, Fontainebleau, France.

Matheron’s edifice groupe de réflexion statistique nouveau

Matheron’s rebuttal in his Letter to the Editor was called Philipian/Watsonian High (Flying) Philosophy. It was published in Mathematical Geology, Vol 18, No 5, 1986. It shed a bright light on Matheron’s mind when he ranted, “But all of this is clear now: geostatistics is just a dastardly conspiracy organized with diabolic cunning, by a secret order of one-dimensional Jesuits.”

Assume, krige, smooth, and be happy

Here’s what Matheron’s new science of geostatistics is all about. Matheron in 1954, in his very first Note Statistique No 1, failed to derive variances of weighted average lead and silver grades of ordered core samples of variable length and density. In his 1960 Note Geostatistique No 28 Matheron coined his honorific krigeage eponym. What he didn’t do was test for spatial dependence between ordered block grades. In his 1970 Random functions and their applications in geology Matheron brought to light a likeness of some sort between ore deposits and Brownian motion along a straight line. That’s why Matheron and his flock took to working with Riemann integrals rather than with Riemann sums. It explains why counting degrees of freedom failed to make the grade in Matheronian geostatistics. Blatant disrespect for Fisher’s F-test, for degrees of freedom, and for the Central Limit Theorem, prove my point. Professor Dr Georges Matheron was a self-made wizard of odd statistics.

Once upon a time I was an accidental reader of A Sampling Manual and Reference Guide for Environment Canada Inspectors. It was also called The Inspector’s Field Sampling Manual. I read the First edition. I thought about reading a Second edition and got headache. Will that be crafted by the most gifted geostatistocrat in Canada? Or will some lowest bidder put it together? But who put Geostatistical data analysis in the First edition? Did Environment Canada, too, have a geostatistically qualified Emeritus Scientist on board?

Section 2.1.2 Sampling Approaches points to random sampling, systematic sampling and judgement (sic) sampling. EC’s inspectors are also taught, “Systematic samples taken at regular time intervals can be used for geostatistical data analysis, to produce site maps showing analyte locations and concentrations. Geostatistical data analysis is a repetitive process, showing how patterns of analyte change or remain stable over distances and time spans.” Close but not quite close enough for EC’s average inspector. What a pity that meaningful examples are missing as much in EC’s First edition as they are in Matheron’s magnum opus.

One example points to shellfish samples taken at 1-km intervals along a shore. What EC’s Inspectors are not taught is how to test for spatial dependence between ordered shellfish counts. A sampling variogram would give much more valuable information than a simple test for spatial dependence. EC’s inspectors should not even think about charting semi-variograms. The status quo is unacceptable if Environment Canada wants to study climate change. So, what’s EC’s brass waiting for? Students at Canadian Universities may want to explore EC’s National Climate Data and Information Archive. Many student’s do not even know why geostatistical data analysis is a scientific fraud. It’s time to call a scientific inquiry!

Junk statistics at Natural Resources Canada

Dr Frits P Agterberg is Emeritus Scientist with the Geological Survey of Canada. In the early 1990s he was but one of many scientists with NRCan’s precursor. Several scientists are members of Canadian Advisory Committees to the International Organization for Standardization. I have never met Agterberg at any such event. I derived a method that gives confidence limits for metal contents and grades of mined ores and mineral concentrates. It was approved as ISO DIS 13543, Determination of Mass of Contained Metal in the Lot. What I wanted to do was apply the same method to in-situ ores and coals. Cominco’s geologists taught me a bit about kriging and smoothing but I didn’t get the gist of it. A mining company gave me a set of gold assays for ordered rounds in a drift to play with.

My son and I studied a few geostat books. We found David’s 1977 Geostatistical Ore Reserve Estimation to be short on statistics and long on geostat drivel. David did pay tribute to the ‘famous’ Central Limit Theorem but didn’t take to working with it. Neither did he take to counting degrees of freedom. Degrees of freedom played a cameo role when he pointed to an earlier work of geologists. David didn’t derive confidence limits for metal grades and contents of ore deposits. He was right on cue when he wrote “…statisticians will find many unqualified statements…” David didn’t write he would throw a temper tantrum if any one dared to.

David was but one of a score of geostat thinkers who thrashed Precision Estimates for Ore Reserves and made a mockery of peer review in the process. But did they ever know how to stake out their own turf! They coined all sorts of terms and never stopped nattering nonsense neologisms among themselves. I’m not gifted in verbal discourse. It took me a while to grasp that kriged estimates, kriged estimators, estimated values, and simulated values, are birds of a feather in geostat speak. What did blow my mind were infinite sets of kriged estimates, zero kriging variances, and a dreadful disrespect for degrees of freedom. Who could have made up so much poppycock?

Agterberg made up quite a bit of it. He cooked up a distance-weighted average point grade that didn’t have a variance. He failed to put in plain words why his function lost its variance. Neither did he ever tell me why his zero-dimensional distance-weighted average point grade didn’t have a variance. It led me to guess that this lost variance wasn’t his proudest feat. What is still beyond Agterberg’s grasp in 2009 is one-to-one correspondence between functions and variances.

Matheron’s new science of geostatistics drifted across the Channel and the Atlantic Ocean and made landfall on the North American continent in 1970. The mining industry was gung-ho to swallow least biased subsets of infinite sets of kriged estimates with hook, line and sinker. Kriging and smoothing sounded so soothing. How its practitioners could beat the odds of selecting least biased subsets of infinite sets of kriged estimates troubled but a few. The list of those who couldn’t care less would stack a Mining Hall of Shame. I got to the bottom of Matheron’s odd statistics long before the Bre-X fraud. But no one cared!

I took my time to find out who lost what, when and where. It was Agterberg who brought to light a typical geologic prediction problem in 1970. I took a look and saw a distance-weighted average. He found a typical kriging problem in 1974. All I saw was the same distance-weighted average.

Typical geologic prediction problem
Typical kriging problem

Here are a few of Agterberg’s real problems. He didn’t know how to test for spatial dependence between his ordered point grades. He didn’t know how to derive the variance of his distance-weighted average point grade. He didn’t know how to count degrees of freedom either for the set or for the ordered set. Yet, Agterberg does point to degrees of freedom on pages 174, 190 and 254 of his 1974 Geomathematics.

What’s more, Agterberg didn’t take to door-to-door sales walks. Such a walk would visit each point only once and cover the shortest possible distance between all points. He could then have applied Fisher’s F-test to the variance of the set and the first variance term of the ordered set. Agterberg himself did apply Fisher’s F-test on page 187 of his 1974 Geomathematics. And he does refer to Sir Ronald A Fisher‘s work on nine (9) pages!

Agterberg does not refer to Dr Jan Visman’s work. Visman was a Dutch coal mining engineer who worked with the Dutch State Mines during the war. His PhD thesis proved the variance of the primary sample selection stage to be the sum of the composition variance and the distribution variance. Visman immigrated to Canada in 1951 and worked with the Department of Mines and Technical Surveys in Ottawa. He wrote Towards a Common Basis for the Sampling of Materials (Research Report R 93, July 1962). The advent of ash analyzers for coal and on-stream analyzers for slurry flows led to a fundamental understanding of what spatial dependence between measured values in ordered sets is all about. Yet, spatial dependence in sampling units and sample spaces stayed as profound a mystery to the geostatocracy as were the properties of variances.

Agterberg prefers oral criticism. Once upon a time he did reply in writing. On October 11, 2004, he called me …an iconoclast with respect to spatial statistics and kriging.” He insisted, “By now this approach is well established in mathematical statistics.” He got it all wrong again. Kriging and mathematical statistics have as little in common as alchemy and chemistry. What is ringing kriging’s bell is climate change. That’s were Agterberg’s zero-dimensional distance-weighted average will always have a variance. Whether he likes it or not!

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