Here’s how to in a nutshell. The most brazen lie of all was to deny that weighted averages do have variances. The stage for this lie was set at the French Geological Survey in Algeria on November 25, 1954. It came about when a novice in geology with a knack for probability theory put together his very first research paper. The author had called his paper Formule des Minerais Connexes. He had set out to prove associative dependence between lead and silver in lead ore. He worked with symbols on the first four pages. Handwritten on page 5 are arithmetic mean grades of 0.45% lead and 100 g/t silver, variances of 1.82 for lead and 1.46 for silver, and a correlation coefficient of 0.85. He had worked with symbols until page 5 and did omit his set of primary data. Neither did he refer to any of his peers. Those peculiar practices would remain this author’s modus operandi for life.
The budding author was to be the renowned Professor Dr Georges Matheron, the founder of spatial statistics and the creator of geostatistics. What young Matheron had derived in his 1954 paper were arithmetic mean lead and silver grades of drill core samples. But he had not taken into account that his core samples varied in lengths. So he did derive length-weighted average lead and silver grades and appended a correction to his 1954 paper on January 13, 1955. What he had not done is derive the variances of his length-weighted average lead and silver grades. Neither did he test for, or even talk about, spatial dependence between metal grades of ordered core samples. Matheron’s first paper showed that testing for spatial dependence was beyond his grasp in 1954.
Why was Formule des Minerais Connexes marked Note statistique No 1? Matheron had not derived variances to compute confidence limits for arithmetic mean lead and silver grades but applied correlation-regression analyis. Statisticians do know that the central limit theorem underpins sampling theory and practice. So why didn’t young Matheron derive confidence limits? Surely, he was familiar with this theorem, wasn’t he? Or was it because he thought he was some sort of genius at probability theory? That would explain why he worked mostly with symbols and rarely with real data. Had he worked with real data, he would still have cooked up odd statistics because the variances of his central values went missing. That’s why he was but a self-made wizard of odd statistics. It was Matheron who called the weighted average a kriged estimate as a tribute to the first mining engineer who took to working with weighted averages. Matheron never bothered to differentiate area-, count-, density-, distance-, length-, mass- and volume-weighted averages. But thenn neither did any of his disciples.
Matheron’s followers, unlike real statisticians, didn’t take to counting degrees of freedom. Statisticians do know why and when degrees of freedom should be counted. Geostatisticians don’t know much about degrees of freedom but they do know how to blame others when good grades go bad. They always blame mine planners, grade control engineers, or assayers whenever predicted grades fail to pan out. They claim over-smoothing causes kriging variances of kriged estimates to rise and fall. Kriging variances rise and fall because they are pseudo variances that have but squared dimensions in common with true variances. Of course, Matheron’s odd new science is never to blame for bad grades or bad statistics.
It is a fact that Matheron fumbled the variance of his length-weighted average in 1954. Several years before the Bre-X fraud I derived the variance of a length- and density-weighted average metal grade. The following example is based on core samples from an ore deposit in Canada. The mine itself is no longer as Canadian as it once was. The Excel template with the set of primary data and its derived statistics are posted on a popular but wicked website.
My website was set up early in the Millennium. I loved to send emails with links to my reviews of Matheron’s new science of geostatistics. The students at the Centre de Géostatistique (CDG) in Fontainebleau, France, ranked on high on my list of those who ought to pass Statistics 101. I was pleased when PDF files of Matheron’s work were posted with CDG’s online library. But I was surprised to find out that Matheron’s first paper was no longer listed as Note statistique No 1 in the column marked Reference but as Note géostatistique No 1. Just the same, the PDF file of this paper and its appended correction are still marked Note statistique No 1. On October 27, 2008, five out of six of Matheron’s 1954 papers were still marked Note statistique Nrs 2 to 6.
What was going on? Was the birth date of Matheron’s new science of geostatistics under review? Who reviewed it? And why? Why not retype the whole paper? Why not add the variances of length-weighted average lead and silver grades? And how about testing for spatial dependence between metal grades of ordered core samples? Where have all of Matheron’s sets of primary data gone? And what has happened to his old Underwood typewriter? I have so many questions but hear nothing but silence!
Matheron himself moved from odd statistics to geostatistics in 1959 when he went without a glitch from Note statistique no 19to Note géostatistique no 20. Check it out before geostat revisionists strike again. I admit to having paraphrased Darrell Huff’s How to lie with statistics. But I couldn’t have made up that this delightful little work was published for the first time in 1954. That’s precisely when young Matheron was setting the stage for his new science of geostatistics in North Africa. Matheron, the creator of geostatistics, never read Huff’s work. But then, Huff didn’t read Matheron’s first paper either. Thank goodness Darrell Huff’s How to lie with statistics is still in print!
The choice between a turbo compressor or a positive displacement pump (blower or screw compressor with internal compression) air mover for a pneumatic conveying system can be evaluated by the influence of the pump characteristics on the pneumatic conveying parameters.
A positive displacement pump displaces a constant volume of air, irrespective of the pressure at the inlet.
A turbo compressor compresses the air adiabatically and is therefore best compared with a screw compressor with internal compression.
A turbo compressor transfers impulse to the air in its impeller.
The more mass flow of air, the more power is consumed.
Therefore, the turbo is kept at a constant mass flow of air by regulated throttle at the inlet, which keeps the pressure ratio constant.
The application of controllable diffusers at the exit of the impeller makes it regulate to control the airflow between approx. 50% to 100% without efficiency reduction.
A turbo compressor always operates at its design point and therefore always consumes a constant (full) power over the full pressure range of the system.
The screw compressor with internal compression consumes less energy at partial pressure, but still requires energy for the internal compression.
The energy consumption of a blower is proportional to the system pressure drop.
In pneumatic conveying systems there are pressure systems and vacuum systems to recognize.
In a pressure system, a positive displacement pump and a turbo compressor deliver a constant mass flow of air, as the inlet conditions of the air are constant (atmospheric).
In a pneumatic conveying system with fluctuating pressure, the turbo compressor has the disadvantage of consuming high energy per ton at lower pressures as the power demand of the turbo is constant.
Applying a turbo compressor for sewage aeration with a constant pressure (water height) is a good choice.
In pneumatic vacuum conveying, the influence of the pump characteristics is much more complex.
Is our world going green? It may be a long while before we know. That’s because scores of geoscientists have gone nuts and work with junk statistics. In Canada, too, geoscientists would rather infer than test for spatial dependence in sampling units and sample spaces. The more so since it’s all in The Inspector’s Field Sampling Manual. Nobody should have to read it. Not even EC’s own inspectors. I had to in the early 2000s because Environment Canada had taken a client of mine to court. It was about my statistical analysis of test results determined in interleaved primary samples. So I worked my way through EC’s manual and found all sorts of sampling methods. What I didn’t find was the interleaved sampling method. I had put this method on my list of smart statistics long before global warming got hot.
Here’s what I did find out when I struggled with EC’s manual. Inspectors are 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 analytes change or remain stable over distances or time spans.”
Geostatistics already rubbed me the wrong way long before it converted Bre-X’s bogus grades and Busang’s barren rock into a massive phantom gold resource. In fact, Matheron’s new science of geostatistics has been a thorn in my side for some twenty years. That sort of junk statistics still runs rampant in the Journal for Mathematical Sciences. Just the same, EC’s field inspectors read under Systematic (Stratified) Sampling , “1) shellfish samples taken at 1-km intervals along a shore, 2) water samples taken from varying depths in the water column.”Numerical examples are missing as much in A Sampling Manual and Reference Guide for Environment Canada Inspectors as they were throughout Matheron’s seminal work. Not all of EC’s geoscientists know as little about testing for spatial dependence in sampling units and sample spaces as do those who cooked up The Inspector’s Field Sampling Manual.
In his letter of October 15, 1992, to Dr R Ehlich, Editor, Journal for Mathematical Geology, Stanford’s Professor Dr A G Journel claimed , “The very reason for geostatistics or spatial statistics in general is the acceptance (a decision rather) that spatially distributed data should be considered a priori as dependent one to another, unless proven otherwise.” He believed that my anger“arises fro [sic] a misreading of geostatistical theory, or a reading too encumbered by classical ‘Fischerian’ [sic] statistics.” JMG’s Editor advised me in his letter of October 26, 1992, “Your feeling that geostatistics is invalid might be correct.”
Each and every geoscientist on this planet ought to know how to test for spatial dependence and how to chart sampling variograms that show where spatial dependence in our own sample space of time dissipates into randomness. Following is an Excel spreadsheet template that shows how to apply Fisher’s F-test. Geoscientists should figure out why Excel’s FINV-function requires the number of degrees of freedom both for the set and for the ordered set.
Of course, it’s easy to become a geostatistically smart geoscientist. All it takes is to infer spatial dependence between measured values, interpolate by kriging, select the least biased subset of some infinite set of kriged estimates, smooth its kriging variance to perfection, and rig the rules of real statistics with impunity. All but a few of those who have gone nuts and work with junk statistics have written books about geostatistics!
A poster in my office reads, “Metrology, the Science of Measurement.” It’s a bit faded because I’ve had it for so long. Standards Council of Canada had it printed for educational purposes. I got my poster with a set of slides about international units of measure. Most of them have since been redefined. The famous platinum-iridium artifact that has so long defined the International Unit of Mass is about to bite the dust. A sphere of pure silicon will take its place. The famous Central Limit Theorem has stood the test of time since Abraham de Moivre (1667-1754) brought to the world The Doctrine of Chances. De Moivre’s work underpins sampling theory and sampling practice. His work is bound to stand the test of time until our planet runs out of it.
The science of measurement has always played a key role in my work. That’s why I put together Sampling and Weighing of Bulk Solids after I had completed my assignment with Cominco Ltd. I was pleased to see it in print in 1985. What pleased me even more was that ISO Technical Committee 183–Copper, lead, zinc and nickel ores and concentrates approved an ISO standard method based on deriving confidence intervals and ranges for metal contents of concentrate and ore shipments.
Several years later I got a slim paperback the cover of which I didn’t recognize. What I did recognize inside of it were my own charts and graphs embedded between Chinese characters. A friend of mine told me it was a Mandarin translation printed on rice paper. My book is protected by copyright but I have yet to be paid a single yuan. Teaching innovative sampling practices and sound statistical methods ranks much higher on my list of things to do than becoming a small c capitalist.
Sampling and Weighing of Bulk Solids
Mandarin translation, November 1989
My son and I were pleased when Precision Estimates for Ore Reserves was praised by Erzmetall and published in its October 1991 issue. The more so since peer reviewers in Canada, the USA and Britain did reject that very paper. One of CIM Bulletin’s reviewers spotted a lack of references to geostatistical literature. The other was ticked off because we were not “…relying on the abundant geostatistical literature…” We had found out that geostatisticians do not explain how to derive confidence intervals and ranges for metal contents of in-situ ore. So we did in our paper and submitted it to CIM Bulletin on September 28, 1989.
Both of us had taken statistics courses at the same university but at different times. Ed leads the Eclipse Modeling Framework project and coleads of the Eclipse Modeling project. He is a coauthor of the authoritative book “EMF: Eclipse Modeling Framework” which is nearing completion of a second edition. He is an elected member of the Eclipse Foundation Board of Directors and has been recognized by the Eclipse Community Awards as Top Ambassador and Top Committer. Ed is currently interested in all aspects of Eclipse modeling and its application and is well recognized for his dedication to the Eclipse community, posting literally thousands of newsgroup answers each year. He spent 16 years at IBM, achieving the level of Senior Technical Staff Member after completing his Ph.D. at Simon Fraser University. He has started his own small company, Macro Modeling, is a partner of itemis AG, and serves on Skyway Software’s Board of Advisors. His experience in modeling technology spans 25 years.
I was proud to have his pre-IBM credentials printed on the backside of Part 1– Precision and Bias for Mass Measurement Techniques. I shall convert all Lotus 1-2-3 files into Excel files and post them on my website. Some time ago Dr W E Sharp, the Editor-in-Chief for what was recently renamed the Journal of Mathematical Geosciences, wanted Dr Ed Merks to review papers on computer applications. What Sharp also asked me was to write a paper on testing for spatial dependence by applying Fisher’s F-test. I did but we couldn’t agree on degrees of freedom for ordered sets.
Metrology in Mining and Metallurgy
First part and also the last one
After Part 1 was completed in 1992 I went to work on Part 2– Precision and Bias for Ore Reserves. It was coming along nicely until Barrick Gold asked me in December 1996 to look at Bre-X’s test results for gold in crushed core and Lakefield’s test results for gold in library core. The hypothesis that 2.9 m crushed core and 0.1 m library core were once part of the same 3.0 m whole core proved to be highly improbable. CIM’s statistically dysfunctional but otherwise qualified persons were not at all keen to know how Bre-X’s salting scam could have been avoided altogether. Surely, life after Bre-X couldn’t have been any more bizarre. But that’s another story altogether!
The ISO copyright office in Geneva, Switzerland, suggests that it holds the copyright to ISO/FDIS 12745:2007(E)–Precision and bias of mass measurement techniques. Yet, this ISO standard is an ad verbatim copy of Part 1–Precision and bias for mass measurement techniques. Part 1 is supposed to be protected by Canadian copyright. So what gives? Didn’t ISO have to ask permission to reprint? What’s this world coming to when ISO violated Canadian copyright in 2007 just as much as China did in 1992?
What Ed and I have decided to do is put together a paper on Metrology in Mineral Exploration. I want to present it at APCOM 2009 in Vancouver, BC. Home sweet home! Maybe I’ll talk Ed into coming home for a while. I’ll have to post an abstract before the deadline. By the way, APCOM stands for Applications of Computers and Operations Research in the Mineral Industry. Acronym talk does make a lot of sense, doesn’t it?
The cause is bigger than the individual and the product is more than the sum of the parts
I have had the privilege to be part of a crack engineering team that gelled and matured over a several year period and then continued to get better and better with time and with each challenge.As is natural, each team member revealed particular strengths.The individual strengths tended to be diverse and cumulatively greatly broadened the capability.When inspired by leadership, towards a common cause that is deemed bigger than the individual, the individual strengths dovetail, not only filling the gaps but with strengthened bonds produce a powerful force.This is not a case for promoting specialization, quite the contrary.Individual strengths and leanings happen naturally.Indeed, individuals that are trained and inspired to be well rounded and complete don’t lose their strengths and leanings but achieve greater versatility and productivity and through a broader understanding of all functions are able to enhance the performance of others by their support.
Recently I had the privilege of witnessing such exemplary teamwork in action at the Victor Diamond Mining project, in Northern Ontario, Canada.I was there for Dos Santos International, starting-up and commissioning our three DSI Snake Sandwich High-Angle Conveyors.At the morning launch meetings as at the evening recap, the enthusiasm and sense of purpose was contagious.Clearly the cause was larger than the individual and this sense was shared by all team members from management to labor of the participating companies; the owner, the EPCM, the installation contractor and the various suppliers.Assignments, both planned and unexpected were embraced with enthusiasm and performed with pride and purpose.It’s no wonder that the Victor project is an example of success, coming in ahead of schedule and under budget.
The accomplishments, product of the teamwork, are the more impressive when the size, location and schedule of the Victor Project are considered.The following stats are taken from “E&MJ Dec. 2005” and “Canadian Business Dec. 2005”:
Project cost, (US Dollars) $ 982 million
Project life is 17 years
Productive mine life is 12 years, based on only one of 16 pipes (grading 22.3 carats/100 t), 6 million carats. Exploration continues on others in order to extend the mine life
Mine will produce annual revenue of (US Dollars) $ 117 million
The Victor kimberlite has a surface area of 15 hectors
Mine is located in James Bay Lowlands of northern Ontario, 90 km west of the coastal community of Attawapiskat
Mine is accessible only by air, and supplied by ice roads during 2 to 2½ months in the winter
Environmental permits were approved in late October, 2005
Construction began in early 2006
Mine production began in early 2008, nearly a full year ahead of schedule
Project manpower grew to more than 800 during construction and settled to 380 for the productive mine life
I was privileged and honored to be a part of (if only as a supplier and observer) this very successful and exemplary project and team. Joseph A. Dos Santos, PE
In the bulk-online forum are several pneumatic conveying questions posed, using the descriptions dense phase conveying and dilute phase conveying.
It seemed then that there was not a general understanding about the definition of the two conveying regimes.
After the discussion on the Forum it became clear that the definition was related to the so called Zenz-diagram.
The Zenz diagram is widely accepted as a description of pneumatic conveying with explanatory properties.
Since the calculation of a Zenz diagram is now possible by an extensive computer program, it is also possible to investigate how the diagram is formed.
The calculation approach is described in the Bulkblog article “Pneumatic Conveying, Performance and Calculations!”. By varying the air flow at constant capacity, the resulting partial pressure drops were calculated and combined into a table.
The summation of the partial pressure drops results in the total pressure drop of the system under the chosen conditions.
Dividing the calculated pressure drops by the total length, the pressure drop per meter is derived.
This procedure could also be differentiated to partial pressure drops over partial lengths.
Then it can be checked whether one part of the conveying pipeline is in f.i. dense phase, while another part of the conveying pipeline is dilute phase. This not executed for this article.
The curve in the Zenz – diagram represents pneumatic conveying as the pressure drop per unit of length as a function of the air flow (or air velocity).
For this curve the solids flow rate and the pipeline are kept constant.
For a cement conveying pipe line, this curve is calculated.
The calculation curves are given below:
From 0.8 m3/sec to 2.0 m3/sec, the pressure drop decreases.
This can be explained as the stronger influence of the decreasing loading ratio, opposed to the
weaker influence of the increasing velocity, which would increase the pressure drop per meter.
In addition, the residence time of the particles becomes shorter with increasing velocity and the required pressure drop for keeping the particles in suspension decreases.
From 2.0 m3/sec to 6.0 m3/sec, the pressure drop increases.
This can be explained as the weaker influence of the decreasing loading ratio and the decreasing pressure drop for keeping the particles in suspension, opposed to the stronger influence of the increasing velocity, which increases the pressure drop per meter.
The lowest pressure drop per meter occurs at 2.0 m3/sec.
Left of this point of the lowest pressure drop per meter, the pneumatic conveying is considered: dense phase and on the right of this point, the pneumatic conveying is considered: dilute phase.
As can be read from the calculation table, the loading ratio (mu) is higher on the left part of the curve than on the right part of the curve.
Regarding the energy consumption per ton conveyed, the lowest value occurs at 0.9 m3/sec.
This can be explained as follows:
The energy consumption per ton is depending on the required power for the air flow.
(solids flow rate is kept constant)
This required power is determined as a function of (pressure * flow ).
It appears that the minimum in pressure drop does not coincide with the lowest power demand of the air flow.
As soon as the decreasing airflow (causing lower power demand) is compensated by the increasing pressure drop, the lowest energy consumption per conveyed ton is reached.
The calculation for an air flow of 0.8 m3/sec indicated the beginning of sedimentation in the pipeline, due to the velocities becoming too low.
From this calculation, it can be concluded that a pneumatic conveying design for the lowest possible energy demand, is also a design, using the lowest possible air flow (or velocity).
The lowest possible velocities are also favorable for particle degradation and component’s wear.
Contribution of partial pressure drops to the total pressure drop
To investigate the physical background of the shape of the Zenz diagram, a cement pressure conveying installation is assumed and calculated, whereby the partial pressure drops are noticed.
The installation is described by:
Horizontal conveying length=71 m
Vertical conveying length=28m
Number of bends=2
Pipe diameter=243 mm (10”)
Capacity basis for Zenz diagram=200tons/hr
The compressor airflow is varied from 0.5 m3/sec to 3.0 m3/sec
The calculation results are presented in the following table.
Lord Kelvin (William Thomson, 1824-1907) was a brilliant scientist and an innovative engineer. His honorific name is forever linked to the absolute temperature of zero degrees Kelvin. His work often called for all sorts of variables to be measured. Here’s what he 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’s view struck a chord with me because of the Dutch truism,“Meten is weten.” It translates into something like, “To measure is to know.” It may have messed up a perfect rhyme but didn’t impact good sense. It’s a leitmotif in my life!
Lord Kelvin knew all about degrees Kelvin and degrees Celsius. But he couldn’t have been conversant with degrees of freedom because Sir Ronald A Fisher (1890-1960) was hardly his contemporary. Lord Kelvin might have wondered why geoscientists would rather assume than measure spatial dependence. Sir Ronald A Fisher could have verified spatial dependence by applying his ubiquitous F-test to the variance of a set of measured values and the first variance term of the ordered set. He may not have had time to apply that variant of his F-test because of his conflict with Karl Pearson (1857-1936). It was Fisher in 1928 who added degrees of freedom to Pearson’s chi-square distribution.
Not all students need to know as much about Fisher’s F-test as do those who study geosciences. The question is why geostatistically gifted geoscientists would rather assume spatial dependence than measure it. How do they figure out where orderliness in our own sample space of time dissipates into randomness? Sampling variograms, unlike semi-variograms, cannot be derived without counting degrees of freedom. So much concern about climate change and global warming. So little concern about sound sampling practices and proven statistical methods!
I derived sampling variograms for the set that underpinsA 2000-Year Global Temperature Reconstruction based on Non-Tree Ring Proxies. I downloaded the data that covers Year 16 to Year 1980, and derived corrected and uncorrected sampling variograms. The corrected sampling variogram takes into account the loss of degrees of freedom during reiteration. I transmitted both to Dr Craig Loehle, the author of this fascinating study. Excel spreadsheet templates on my website show how to derive uncorrected and corrected sampling variograms.
Uncorrected sampling variogram
Spatial dependence in this uncorrected sampling variogram dissipates into randomness at a lag of 394 years. The variance of the set gives 95% CI = +/-1 centrigrade between consecutive years. The first variance term of the ordered set gives 95% CI = +/-0.1 centrigrade between consecutive years.
Corrected sampling variogram
Spatial dependence in the corrected sampling variogram dissipates into randomness at a lag of 294 years. It is possible to derive 95% confidence intervals anywhere within this lag.
Sampling variograms are part of my story about the junk statistics behind what was once called Matheron’s new science of geostatistics. I want to explain its role in mineral reserve and resource estimation in the mining industry but even more so in measuring climate change and global warming. Classical statistics turned into junk statistics under the guidance of Professor Dr Georges Matheron (1930-2000), a French probabilist who turned into a self-made wizard of odd statistics. A brief history of Matheronian geostatistics is posted on my blog. My 20-year campaign against the geostatocracy and its army of degrees of freedom fighters is chronicled on my website. Agterberg ranked Matheron 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 fumbled the variance of the length-weighted average grade of core samples of variable lengths in 1954. Agterberg himself fumbled the variance of his own distance-weighted average point grade in his 1970 Autocorrelation Functions in Geology and again in his 1974 Geomathematics.
Agterberg seems to believe it’s too late to reunite his distance-weighted average point grade and its long-lost variance. I disagree because it’s never too late to right a wrong. What he did do was change the International Association of Mathematical Geologyinto the International Association for Mathematical Geosciences. Of course, geoscientists do bring in more dollars and cents than did geologists alone. I’m trying to made a clear and concise case that sound sampling practices and proven statistical methods ought to be taught at all universities on this planet. Time will tell whether or not such institutions of higher learning agree that functions do have variances, and that Agterberg’s distance-weighted average point grade is no exception!
Not a word from CRIRSCO’s Chairman. I just want to know whether or not functions do have variances at Rio Tinto’s operations. Surely, Weatherstone wouldn’t toss a coin to make up his mind, would he? My functions do have variances. I work with central values such as arithmetic means and all sorts of weighted averages. It would be off the wall if the variance were stripped off any of those functions. But that’s exactly what had come to pass in Agterberg’s work. I’ve tried to find out what fate befell the variance of the distance-weighted average. I did find out who lost what and when. And it was not pretty in the early 1990s! When Matheron’s seminal work was posted on the web it became bizarre. The geostatistocrats turned silent and resolved to protect their turf and evade the question. They do know what’s true and what’s false. And I know scientific truth will prevail in the end.
Agterberg talked about his distance-weighted average point grade for the first time during a geostatistics colloquium on campus at The University of Kansas in June 1970. He did so in his paper on Autocorrelation functions in geology. The caption under Figure 1 states; “Geologic prediction problem: values are known for five irregularly spaced Points P1 –P5. Value at P0 is unknown and to be predicted from five unknown values.”
Agterberg’s 1970 Figure 1 and 1974 Figure 64
Agterberg’s 1970 sample space became Figure 64 in Chapter 10. Stationary Random Variables and Kriging of his 1974 Geomathematics. Now his caption states, “Typical kriging problem, values are known at five points. Problem is to estimate value at point P0 from the known values at P1 –P5”. Agterberg seemed to imply his 1970 geologic prediction problem and his 1974 typical kriging problem do differ in some way. Yet, he applied the same function to derive his predicted value as well as his estimated value. His symbols suggest a matrix notation in both his paper and textbook.
The following function sums the products of weighting factors and measured values to obtain Agterberg’s distance-weighted average point grade.
Agterberg’s distance-weighted average
Agterberg’s distance-weighted average point grade is a function of his set of measured values. That’s why the central value of this set of measured values does have a variance in classical statistics. Agterberg did work with the Central Limit Theorem in a few chapters of his 1974 Geomathematics. Why then is this theorem nowhere to be found in Chapter 10 Stationary Random Variables and Kriging? All the more so because this theorem can be brought back to the work of Abraham de Moivre (1667-1754).
David mentioned the “famous” Central Limit Theorem in his 1977 Geostatistical Ore Reserve Estimation. He didn’t deem it quite famous enough to either work with it or to list it in his Index. Neither did he grasp why thecentral limit theorem is the quintessence of sampling theory and practice. Agterberg may have fumbled the variance of the distance-weighted average point grade because he fell in with the self-made masters of junk statistics. What a pity he didn’t talk with Dr Jan Visman before completing his 1974 opus.
The next function gives the variance of Agterberg’s distance-weighted average point grade. As such it defines the Central Limit Theorem as it applies to Agterberg’s central value. I should point out that this central value is in fact the zero-dimensional point grade for Agterberg’s selected position P0.
Agterberg’s long-lost variance
Agterberg worked with symbols rather than measured values. Otherwise, Fisher’s F-test could have been applied to test for spatial dependence in the sample space defined by his set. This test verifies whether var(x), the variance of a set, and var1(x), the first variance term of the ordered set, are statistically identical or differ significantly. The above function shows the first variance term of the ordered set. In Section 12.2 Conditional Simulation of his 1977 work, David brought up some infinite set of simulated values. What he talked about was Agterberg’s infinite set of zero-dimensional, distance-weighted average point grades. I do miss some ISO Standard on Mineral Reserve and Resource Estimation where a word means what it says, and where text, context and symbols make for an unambiguous read.
But I digress as we tend to do in our family. Do CRIRSCO’s Chairman and his Crirsconians know that our sun will have bloated to a red giant and scorched Van Gogh’s Sunflowersto a crisp long before Agterberg’s infinite set of zero-dimensional point grades is tallied? And I don’t want to get going on the immeasurable odds of selecting the least biased subset of some infinite set. Weatherstone should contact the International Association of Mathematical Geosciences and request IAMG’s President to bring back together his distance-weighted average and its long-lost variance. That’s all. At least for now!
Niall Weatherstone of Rio Tinto and Larry Smith of Vale Inco have been asked to study a geostatistical factoid and a statistical fact. I asked them to do so by email on July 8, 2008. Next time they chat I want them to discuss whether or not geostatistics is an invalid variant of classical statistics. I’ve asked Weatherstone to transmit my question to all members of his team. CRIRSCO’s Chairman has yet to confirm whether he did or not. I just want to bring to the attention of his Crirsconians my ironclad case against the junk science of geostatistics.
Not all Crirsconians assume, krige, and smooth quite as much as do Parker and Rendu. The problem is nobody grasps how to derive unbiased confidence intervals and ranges for contents and grades of reserves and resources. Otherwise, Weatherstone would have blown his horn when he talked to Smith. A few geostatistical authors referred per chance to statistical facts. Nobody has responded to my questions about geostatistical factoids. The great debate between Shurtz and Parker got nowhere because the question of why kriging variances “drop off” was never raised. So I’ll take my turn at explaining the rise and fall of kriging variances.
In the 1990s I didn’t geostat speak quite as well as did those who assume, krige and smooth. I did assume Matheron knew what he was writing about but he wasn’t. Bre-X proved it makes no sense to infer gold mineralization between salted boreholes. The Bre-X fraud taught me more about assuming, kriging, and smoothing than I wanted to know. And I wasn’t taught to blather with confidence about confidence without limits. It reminds me of another story I’ll have to blog about some other day. It’s easy to take off on a tangent because I have so many factoids and facts to pick and choose from.
Functions have variances is a statistical fact I’ve quoted to Weatherstone and Smith. Not all functions have variances I cited as a geostatistical factoid. Factoid and fact are mutually exclusive but not equiprobable. One-to-one correspondence between functions and variances is a condition sine qua non in classical statistics. Therefore, factoid and fact have as much in common as do a stuffed dodo and a soaring eagle. My opinion on the role of classical statistics in reserve and resource estimation is necessarily biased.
The very function that should never have been stripped off its variance is the distance-weighted average. For this central value is in fact a zero-dimensional point grade. All the same, its variance was stripped off twice on Agterberg’s watch. David did refer to “the famous central limit theorem.” What he didn’t mention is the central limit theorem defines not only the variance of the arithmetic mean of a set of measured values with equal weights but also the variance of the weighted average of a set of measured values with variable weights. It doesn’t matter that a weighted average is called an honorific kriged estimate. What does matter is that the kriged estimate had been stripped off its variance.
Two or more test results for samples taken at positions with different coordinates in a finite sample space give an infinite set of distance-weighted average point grades. The catch is that not a single distance-weighted average point grade in an infinite set has its own variance. So, Matheron’s disciples had no choice but to contrive the surreal kriging variance of some subset of an infinite set of kriged estimates. That set the stage for a mad scramble to write the very first textbook on a fatally flawed variant of classical statistics.
Step-out drilling at Busang’s South East Zone produced nine (9) salted holes on SEZ-44 and eleven (11) salted holes on SEZ-49. Interpolation by kriging gave three (3) lines with nine (9) kriged holes each. Following is the YX plot for Bre-X’s salted and kriged holes.
Fisher’s F-test is applied to verify spatial dependence. The test is based on comparing the observed F-value between the variance of a set and the first variance of the ordered set with tabulated F-values at different probability levels and with applicable degrees of freedom. Neither set of salted holes displays a significant degree of spatial dependence. By contrast, the observed F-values for sets of kriged holes seem to imply a high degree of spatial dependence.
If I didn’t know kriged holes were functions of salted holes, then I would infer a high degree of spatial dependence between kriged holes but randomness between salted holes. Surely, it’s divine to create order where chaos rules! But do Crirsconians ever wonder about Excel functions such CHIINV, FINV, and TINV? Wouldn’t Weatherstone want to have a metallurgist with a good grasp of classical statistics on his team?
High variances give low degrees of precision. I like to work with confidence intervals in relative percentages because it easy to compare precision estimates at a glance. SEZ-44 gives 95% CI= ±23.5%rel whereas SEZ-49 gives 95% CI= ±26.4%rel. By contrast, low variances give high degrees of precision. Three (3) lines of kriged holes give confidence intervals of 95% CI= ±0.8%rel to 95% CI= ±1.6%rel. Crirsconians should know not only how to verify spatial dependence by applying Fisher’s F-test but also how to count degrees of freedom. Kriging variances cannot help but going up and down as yoyos!
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