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. Matheron may not have told Journel that he thought in 1954 he was a statistician. It took almost ten years to teach Journel how to assume, krige, and smooth with a lot of 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 profusely 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 lost 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 her 1979 Practical Geostatistics makes an easy read. Her cradle once rocked on the side of the Channel where Sir R A Fisher was knighted. Clark confessed 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. She slashed for "mathematical convenience" the factor 2 in df₀=2(n-1) degrees of freedom for ordered sets, cooked up her silly semi- variogram, and scolded 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 do 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 to measure the risk of playing all sorts of games of chance.

No real data in 1954 - Casting dice in 1978

The set of three (3) stopes is given on the same page. Each set of four units within its stope was put together with a six-sided unbiased die such that each unit 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 is required to test whether or not a given die is unbiased. The question of whether Journel's die was biased may have been 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 σ²_k=0 is given 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 study implies that requirement of functional independence may be violated a little but not a lot All that geostatistical gobbledycook is cooked 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 of the few leaders at institutes of higher learning who did bother to 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 ask why the variance of Agterberg’s distance-weighted average point grade is still 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, Clark's 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 Mark Corey to do. Visit NRCan's Emeritus Scientist in the privacy of 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 to work with my method and several ISO Technical Committees approved it. So, we 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 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 and ignore the true variance of the set of nine measured values. David did sense 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 average point grades
each derived from the same set of nine holes

Counting degrees of freedom for his set of nine holes would have been a foolproof test to find out whether David really understood statistics. What he looked at in this 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 testing for spatial dependence and counting degrees of freedom.

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 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. 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 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 cooked up so much poppycock?

Agterberg did 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. But I saw 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!

Warming up a little or a lot?

Our world is not getting hot any time soon. Where it’s getting hot is under the collars of those who thought up global warming. Al Gore and the UN Intergovernmental Panel on Climate Change were awarded the 2007 Nobel Peace Prize for thinking up global warming. The almost US president and his UN think tank do think it’s getting warmer. So, it’s got to be so! They are telling tall tales and crafting cool books. A few worked up a frenzy worrying it doesn’t get warm soon enough. Others upped the odds by predicting it does so at an alarming rate. That sort of scare works so long as nobody knows the standard rate of warming for a little planet like ours.

Al Gore’s Inconvenient Truth is a tour de force in child psychology. It shows a polar bear on an ice floe drifting at some cool spot somewhere in our world. It’s scene that pulls a child’s heartstring just as much as does a puppy under a Christmas tree. Our offspring may be around long enough to witness that what goes up must come down if only because the sun is beyond control of church and state. And that’s just as well. Nobody worries much that the sun itself is running out of hydrogen at an astounding rate. The good news is that its hydrogen will last some 10 billion years. The bad news is that it's numerically a lot less than IMF’s trillion dollars.

Apollo’s Blue Marble photograph

NASA satellites transmit more than stunning photographs. Massive sets of temperatures have been transmitted ever since this famous photograph was shot in December 1972. It took me aback that annual temperatures in the lower troposphere display spatial dependence. Until I was told that long term cycles in ocean currents do impact lower troposphere temperatures. That’s why it makes scientific sense to verify spatial dependence in our own sample space of time. By inverse logic, it is a scientific fraud to assume spatial dependence without proof.

Bad luck had it awhile back that a NASA satellite failed to deploy. This one was to measure carbon dioxide concentrations in the troposphere. Geoscientists might have found out some 30 years later how carbon dioxide concentrations drive the greenhouse effect. That’s why patience is as much a virtue in the study of climate change as is a good grasp of statistics. Geostatistical data analysis is a catch-22 in the sense that interpolation between measured values creates an appearance of spatial dependence where it doesn’t exist.

Much of the USA and most of Canada is missing in the Apollo photograph. The USA may not have felt like showing a lot below the 49^th Parallel in those days. Canada’s vastness stretches from the Atlantic ocean to the Pacific ocean, and winds up into the arctic where Northern Lights shimmer when the sun takes leave during long winters. Canada’s vastness twists and turns into a multitude of different climate zones. Environment Canada (EC) manages a treasure trove for those who take the study of climate change seriously.

EC’s Adjusted Historical Canadian Climate Data Base gives temperatures for a large number of locations dating back to the 1930s. I was given permission to access EC's database. I downloaded temperatures for the international airports at Calgary, Alberta, at Ottawa and Toronto, Ontario, and at Vancouver and Victoria, British Columbia. I also downloaded temperatures for Coral Harbour, Territory of Nunavut. I did so at different times and for different reasons. Excel 2007 spreadsheet templates give the statistics for each set, a plot of the annual means, and a chart with the sampling variogram. The most relevant statistics are summarized below.

Summary of statistics for six locations in Canada

For the Toronto Lester B Pearson International Airport the difference of 2.30 centigrade between the first annual mean of 6.00 in 1940 and the last annual mean of 8.30 centigrade in 2008 is statistically significant at 95% probability. Higher annual temperatures do account for the observed difference of 2.30 centigrade. Observed differences at other locations are not significant. Randomly distributed variations in measured temperatures account for those observed differences.

Annual means at Toronto Lester B Pearson International Airport

A plot of annual differences in a chart shows a distinct trend toward higher temperatures. Such a trend also indicates spatial dependence between annual temperatures in the ordered set. A sampling variogram is a chart in which the variance terms of the ordered set are plotted against the variance of the set and the lower limits of its asymmetric 95% and 99% confidence ranges. It shows where orderliness at the selected location in our sample space of time dissipates into randomness.

Toronto Lester B Pearson International Airport

The sampling variogram for annual temperatures measured at the Toronto Lester B Pearson International Airport displays a significant degree of spatial dependence. The question is then why geostatisticians like to assume spatial dependence rather than verify it by applying Fisher’s F-test to the variance of the set and the variance terms of the ordered set. The mining industry is pleased to krige and smooth from here to eternity. Professional engineers and geoscientists with provincial securities commissions, too, do krige and smooth with the best. But here's the cinch! NASA and NOAA are not about to krige and smooth because the mining industry says so. What should the Harper Government do when the rules of statistics are rigged? Goverments do not sort out abuse of statistics. Good grief! It's the Great Lake Study where sound statistics will resurface. Come cold or warm water!

Sorting out junk statistics

It takes a long time to sort out junk statistics. What will kill Matheronian geostatistics is that its proponents are drifting into the study of climate change. That’s why I've put together a sort of standardized rant. It pays to work on various ISO Technical Committees! Here’s what I like to tell those who abuse statistics with reckless abandon...

... My story is about what was once hailed as Matheron's new science of geostatistics. Matheronian geostatistics plays a role not only in reserve and resource estimation for the world’s mining industry but even more so in the study of climate change. Real statistics turned into geostatistics under the leadership of Professor Dr Georges Matheron, a French geologist and a self-made wizard of odd statistics. My 20-year battle against the geostatocracy and its army of degrees of freedom fighters is chronicled on my website. I have brought my concerns to the attention of the Federal Government of Canada, the Provincial Government of British Columbia, the Ontario Securities Commission, the US Securities and Exchange Commission, and the US Senate Committee on Transportation, Science, & Technology.

Dr Frederik P Agterberg, Past President, International Association for Mathematical Geosciences, called Professor Dr Georges Matheron (1930-2000) the Creator of Spatial Statistics. Agterberg ranked him on a par with giants of mathematical statistics such as Sir Ronald A Fisher (1890-1962) and Professor Dr J W Tukey (1915-2000). Agterberg was wrong! Matheron failed to derive the variance of his length-weighted average in 1954 and in 1960.

Agterberg's distance-weighted average point grade

Agterberg himself failed to derive the variance of his distance-weighted average in his 1970 Autocorrelation Functions in Geology and again in his 1974 Geomathematics. Agterberg’s problem is that as few as a pair of measured values, determined in samples selected at positions with different coordinates in a finite sampling unit or sample space, gives an infinite set of zero- dimensional, variance-deprived distance-weighted average point grades. Infinite sets of kriged estimates and zero kriging variances are the very reasons why the world's mining industry welcomed geostatistics with reckless abandon. Geostatistics converted Bre-X’s bogus grades and Busang’s barren rock into a massive phantom gold resource. I applied analysis of variance and proved the intrinsic variance of Busang's gold to be statistically identical to zero. How many mineral inventories in annual reports are bound to shrink during mining?

Lord Kelvin (William Thomson 1824-1907) once said, “…when you can measure what you are speaking about, and express it in numbers, you know something about it, but when you cannot express it in numbers your knowledge is of the meagre and unsatisfactory kind…” Lord Kelvin knew more about degrees Kelvin and degrees Celsius than about degrees of freedom. Lord Kelvin and Sir Ronald A Fisher (1890-1960) were marginal contemporaries. Lord Kelvin would have wondered about the wisdom behind assumed spatial dependence between measured values in ordered sets. Sir Ronald A Fisher could have verified spatial dependence by applying his F-test to the variance of a set of measured values and the first variance term of the ordered set.

Not all scientists need to know as much about Fisher's F-test as do geoscientists. All too few know how to verify spatial dependence by applying Fisher’s F-test, and how to derive sampling variograms that show where orderliness in our own sample space of time dissipates into randomness. So much concern about climate change! So little concern about sound sampling practices and proven statistical methods! I make a clear and concise case against geostatistics on my blog and on my website. Surely, sound sampling practices and proven statistical methods ought to be taught at all universities on this planet and be implemented in all international standards. I’m working hard to make it happen. What will ... do about it?