Born krigers do more with less

Teaching statistics to born krigers takes a long time. What krigers do best is fit curves. Think what happens when a curve is fitted through a set of measured values. Most of all there is much pride and joy. A perfect curve is indeed a thing of beauty. Look at the Fourier transform of Wölfer annual sunspot counts from 1700 to 1987. Isn’t it as stunning as the original plot? It does put into perspective the power of mathematics when applied to an ordered set of measured values in our own sample space of time. A perfect fit is of less interest in my work than the statistics behind ordered sets of measured values. For example, I applied mathematical statistics to derive the statistics of Wölfer annual sunspot counts for the period from 1749 to 1924. I work with spreadsheet software because it is such a powerful tool to show and tell. Several Excel files are posted on my website under Statistics for geoscientists.

Wiki’s Kriging doesn’t test for spatial dependence between measured values in an ordered set. Wiki’s keepers of Krige’s grail didn’t even try. Here’s what they wrote about kriging, “The theory behind interpolation and extrapolation by Kriging was developed by the French mathematician Georges Matheron based on the Master’s thesis of Daniel Gerhardus Krige.” It’s short and crisp but not to the point.

Krige’s 1951 Master thesis brings up ‘knowledge of mathematical statistics’, ‘careful statistical analysis’, ‘science of statistics’, ‘modern statistical basis’, ‘application of statistics’, and so on. It does read like a thesis on statistics, doesn’t it? Nowhere did Krige bring up ‘geostatistics’. A 2003 Tribute to Krige alluded to “…his pioneering work in the application of mathematical statistics…” The same tribute alluded to Krige’s 1952 paper in which he “introduced, inter alias, the basic geostatistical concepts of ‘support’, ‘spatial structure’, ‘selective mining units’, and ‘grade-tonnage curves’. Did it take Krige one year and a bit of inter alias to switch from real statistics to a pinch of between-the-lines geostatistics? Not quite! He was a committed geostatistician when he wrote the Preface to David 1977 Geostatistical Ore Reserve Estimation. But when did Krige really take to kriging?

Matheron’s Note Statistique No 1 saw the light of day in North Africa on November 25, 1954. He coined the first krige-inspired eponym in his 1960 Krigeage d’un panneau rectangulaire par sa périphérie. Matheron didn’t refer to Krige’s 1951 Master thesis. Neither did he much refer to anyone’s work but his own. In those early days Matheron himself dawdled between statistics and geostatistics. But he was not much of a statistician even though he thought he was one.

It makes sense to compare Wiki’s Kriging with Krige’s teachings. Look at tFigure 1 in Wiki’s Kriging. The graph didn’t irk me quite as much as did the confidence intervals for measured values. Once upon a time I tried to get the set of measured values that underpin Figure 1 but its caretaker(s?) didn’t respond. So, I waited until it was time to take a stand against junk statistics on Wikipedia.

Figure 1

Example of one-dimensional data interpolation by kriging, with confidence intervals.
Squares indicate the location of the data.
The kriging interpolation is in red.
The confidence intervals are in green.

I enlarged Figure 1 and measured X- and Y-coordinates for all points in mm. I tested for spatial dependence by applying Fisher’s F-test to the variance of the set and the first variance term for the ordered set. I applied weighting factors because of unevenly spaced measured values. That’s why degrees of freedom become irrational numbers.

Given that the observed value of F=var1(x)/var(x)=1,504/1,408=1.07 is below the tabulated value of F0.05;dfo;df=6.04, it follows that the ordered set of measured values does not display a significant degree of spatial dependence. Hence, measured values in the ordered set are randomly distributed within this sample space. Therefore, interpolation between measured values makes as much sense as extrapolation beyond the set. As a matter of fact, it does give junk statistics of the worst kind wherever and whenever randomness rules. I do not know how Wiki’s Kriging caretakers cooked up the confidence intervals in Figure 1. Next, I applied plain vanilla statistics and plotted confidence intervals in this bar chart.

The bars in this graph, unlike the measured values in Figure 1, are evenly spaced. I’ll show in another block that interpolation between measured values in an ordered set does indeed give the same sort of junk statistics as did Bre-X and the kriging game.



Working with Wikipedia

Wikipedia is a wonderful source of information for all of us while we are doing our time in this universe. Wiki is reliable as a rule and tries to do right when in doubt. For example, under Geostatistics Wiki points out, “This article is in need of attention from an expert on the subject. WikiProject Geography or the Geography Portal may be able to help recruit one”. No kidding! Wiki’s expert would have to be some kind of jack-of- all-sciences. So many disciplines do have a role to play in geography.

Geologists and mining engineers got stuck with geostatistics when Matheron goofed but thought he had dug up a new science. They were taught not work with the Central Limit Theorem and to infer ore between widely spaced boreholes. To infer ore between step-out boreholes at a spacing of 200-m worked well indeed in the Bre-X case. On the other hand, to infer spatial dependence between closely spaced pixels makes sense. When I tested for spatial dependence between gold grades of ordered rounds in a drift, Journel called me “too encumbered” with Fisher’s statistics. It’s not surprising then that geoscientists at Stanford are taught to assume, krige and smooth voodoo variances. Geoscientists with a passion for order tend to do curve-fitting. Too many are led to believe that geostatistics is good for geoscientists. I know that geoscientists would enjoy working with real statistics just as much as Sir Ronald A Fisher once did.

I tried to add applied statistics to Wiki’s Geostatistics when it was still called Kriging. I did so when I was a new Wikipedian in 2005. I knew then that geostatistics is an invalid variant of applied statistics. My son and I had known why since the early 1990s. What I didn’t know in 2005 is who stripped the variance off the distance-weighted average. But I do know now who did and when! What I do not know is why. I’ll continue to explain my case against geostatistics in concise terms and with significant symbols. I do so not only as a member of several ISO Technical Committees but also as a blogger, as a webmaster, and, last but not least, as a Wikipedian.

Most Wikipedians have a strong need to leave a better informed world than we found. I’m no exception. I hold an edge in always having worked with applied statistics and grasped Visman’s sampling theory and practice. I know that geostatistics converted Bre-X’s bogus grades and Busang’s barren rock into a massive phantom gold resource. What I also know is that bogus assays for three to five salted boreholes would have been enough to nip this mind-boggling fraud in the bud. The world’s mining industry doesn’t want to know is what I would have done!

Neither does Pierre-Jean Lafleur want to know. He is a Professional Engineer and a reserve and resource expert with Watts, Griffis, and McOuat Limited. He doesn’t believe I called the Bre-X fraud several months before the boss salter vanished. Lafleur wrote, “The information he provides is unclear, and most likely untrue”. So he wiped it off Wiki’s Bre-X Minerals. Neither may he believe it was not I who put my name on that Wiki subject. But what I did do when my name came up with the wrong context was add the facts and a few links to subjects such as spatial dependence and sampling variogram.

Lafleur deserves some praise because he doesn’t work under a nom-de-plume. Too many Wikipedians work anonymously. When scientists and engineers want to be taken seriously on Wikipedia they should stand up and be counted. Wikipedia should not allow Wikipedians who hide behind pseudonyms to delete indisputable scientific facts. Examples in my discipline of sampling and statistics are the Central Limit Theorem, functional dependence, spatial dependence and degrees of freedom.

Look and see which stats derive from Matheron’s Formule des Minerais Connexes. What a pity that the seminal work of the Creator of Geostatistics and the Founder of Spatial Statistics is no longer posted on the web. In fact, Matheron was a self-made wizard of odd statistics. Here’s a link to Matheron’s correction of his very first paper. All I did was use Matheron’s corrected lead and silver grades and the variances of those grades. Enter a different number of core samples and see how the Central Limit Theorem impacts 95% confidence limits. Play with real statistics and find out what geostatisticians are missing.

Professional engineers and geoscientists claim to be guided by codes of ethics that protect the public at large. Provincial securities commissions in Canada employ reserve and resource experts to set the rules. But should foxes run henhouses? That’s a good-enough reason why a National Securities and Exchange Commission should turn provincial fiefdoms into branch offices. Reserve and resource experts in branch offices should then be asked to testify under oath and explain why the Central Limit Theorem and degrees of freedom are null and void in geostatistics.

Working with applied statistics is fun. And it’s kind of cool for our planet! Wikipedians should read what the International Association for Standardization is all about. ISO may violate the odd copyright, and ignores priority once in a while. And the UN is not perfect either. Only Wikipedia can bring scientific integrity to the world.

Agterberg’s way

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

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

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

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

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

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

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

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

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

Agterberg’s tribute

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

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

Where’s the Central Limit Theorem?

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

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

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

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

Going, going, gone in geostatistics

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

Gy’s 1967 L’Échantillonnage des Minerais en Vrac, Tome 1 two
Gy’s 1973 L’Échantillonnage des Minerais en Vrac, Tome 2eight
David’s 1977 Geostatistical Ore Reserve Estimation two
Journel & Huijbregts’s 1978 Mining Geostatisticszero
Clark’s 1979 Practical Geostatisticszero
Gy’s 1979 Sampling Particulate Materials, Theory Practicezero

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

How to measure what we speak about

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

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

Southeast Coast of Greenland

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

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

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

Sunspots

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

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

H G Wells

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

F P Agterberg

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

A Case For Free Advice – It Helps a Customer as Well as Yourself

When I browse through this and other forums I am often fascinated at how much free advice is made available by the many experts.  At first one might think that this unpaid consulting at its best.  I am a firm believer in helping people out with as much information as you can provide.  Why?  It reminds me of my own sour grape experience as an adolescent trying to install a new cylinder and piston on my moped.  I had bought the – I might add expensive – parts at my up to then favorite motor cycle dealer.  Since I had been a gearhead all my live the mechanical part of the job was no problem with the exception to the ignition timing.  The store owner would not provide me with that value insisting that I let him do that at his shop, which was 10 miles away from home.  The experience turn me off so bad that I took my chances, figured it out myself and never went back to the guy again.  How many people do you think I told about the bad experience? 

This is where suppliers and subject matter experts can make a huge difference in the lives of others.  Anything affects everything and who knows when the one whom you helped may be able to help you one day?  If you are interested read Joseph Jaworski’s book about “Synchronicity”, it is a great book.  While there is always a fine line between free advice and giving proprietary information away, what do you have to lose?  There is always a great step between obtaining advice and successfully implementing the solutions.  Usually the basic technical details may be provided in the initial request fro help, but I have rarely seen people provide either all the site details or assumptions made.  Kudos to anyone that can figure out his own issues based on the advice given to him.   You as the advice giver will always glean something new from a request for help. 

Energy consumption per ton of a pneumatic conveying system.

 

Pneumatic conveying installations suffer from the image of being not energy efficient.

Usually, this bad image is explained by the statement that a pneumatic conveying system is based on high velocities. High velocities are usually synonymous for high energy demand.

 

The definition of the efficiency of a pneumatic conveying system is (in case of an electric drive):

 

Total efficiency = Electric energy / tons       (in kWh/ton)

 

This Total efficiency can be divided in 4 partial efficiencies.

 

1) Drive efficiency = Mechanical energy / Electric energy

 2) Compressor efficiency = Thermal compressing energy / Mechanical energy

 3) Thermo dynamic conveying efficiency = Thermal compressing energy / Thermal expansion energy

 4) Pneumatic conveying efficiency = Thermal expansion energy / tons

  Continue reading Energy consumption per ton of a pneumatic conveying system.

How to work with real statistics

Lorne Gunter called on skeptics to unite. He did so in the National Post. His story was about scientists who don’t warm up to “the orthodoxy on global warming”. What a shame but a few got this call because it came on Monday, October 20, 2008. The timing couldn’t have been worse. It was another Monday when Wall Street and Bay Street watchers saw stock indices move straight south. Global warming isn’t of as much concern as are shrinking stock portfolios. It may explain why Lorne’s tale was told on a Monday. Sandra Rubin’s story, too, ran on a Monday. NP’s head honchos run their own stories mostly in weekend editions. NP’s very first edition was printed on October 27, 1998. At that time, it was Lord Black’s pride and joy. At this time, Lord Black is doing time and NP’s kingpins are still timing things their own way.

Lorne need not have urged skeptics to unite since they did so long ago. Skeptics do hold a dim view of pseudo scientists who play games with scientific integrity. I may well have been a born skeptic. I was taught more than I could grasp about heaven and hell from a pulpit in a Dutch village. Nowadays I teach how to test for spatial dependence in sampling units and sample spaces. Stanford’s Journel taught in 1992 that spatial dependence between measured values may be assumed. I never thought much of Journel’s thinking. Neither did JMG’s Editor. All I thought about at that time was to rid the world of Matheron’s junk statistics. Come hell or high water! And I still do!

The National Post brought to light on November 7th that President-Elect Barrack Obama is set to “Stop global warming”. It brought back that off the wall “Stop continental drift” slogan. Geologists slowed down continental drift by calling it plate tectonics. Plates are still moving, and earthquakes, magma flows and tsunamis are tagging along. The National Post on November 10 claimed that climate change, too, is on some kind of yes-we-can list. Surely, geoscientists should study climate change. What the study of global warming has done so far is set the stage for a constant belief bias.

Lorne’s story about skeptics and global warming came about because of the work of Professor Dr John R Christie. More than 300,000 daily temperature readings around the globe with NASA’s eight weather satellites over 30 years gave Christy and his coauthor a massive data set to work with. It was marked “Lower Troposphere Global Temperature: 1979-2008.” The authors had drawn a trend line thru a see-saw plot. It was the shape of this trend line that piqued my interest. What I wanted to do was test for spatial dependence between measured values and determine where orderliness in our own sample space of 30 years dissipates into randomness. So I asked Lorne and he did sent me the whole set that underpins the plot in his story!

The first step in the statistical analysis is to verify spatial dependence between observed temperatures in this sample space of time by applying Fisher’s F-test to the variance of the set and the first variance of the ordered set.

The observed value of F=6.27 exceeds the tabulated value of F0.001;df;dfo=1.32 at 99.9% probability by a margin of magnitude. Hence, monthly temperatures display an extraordinary high degree of spatial dependence. The probability that this inference is false is much less than 0.1%.

The second step is to verify whether or not the weighted average difference of 0.063 centigrade is statistically identical to zero. Since the first set and the last one have different degrees of freedom than intermediate sets, Student’s t-test is applied with a month-weighted average variance. Such weighted variances are called pooled variances in applied statistics.

The observed value of t=4.245 exceeds the tabulated value of t0.001;dfo=3.674. Hence, the probability is less than 0.1% that this weighted average difference of 0.063 centigrade is statistically identical to zero. Alternatively, this probability of 99.9% points to a statistically significant but small change of 0.063 centigrade during this 30-year period. Detection limits that take into account Type I risk only and the combined Type I and II risks are of critical importance in risk analysis and control. In this case, the Type I risk is ±0.031 centigrade, and the combined Type I risk and Type II risk is ±0.056 centigrade.

The third step is to verify whether or not the variances of ordered temperatures in centigrade constitute a homogeneous set.

Bartlett’s chi square test shows that the observed χ2-value of 22.979 falls between 42.557 at 5% probability and 17.708 at 95% probability. Hence, the set of variances for this 30-year period is homogeneous.

Sir Ronald A Fisher was knighted in 1953 for his work with analysis of variance. Dr F P Agterberg fumbled the variance of his distance-weighted average point grade in 1970 and in 1974. NASA started to measure Lower Troposphere Temperatures in 1979. I showed how to test for spatial dependence between metal grades in ordered sets for the first time in 1985. So why would any geoscientist assume spatial dependence between measured values in ordered sets? Agterberg is the President of the International Association for Mathematical Geosciences. He should explain why his distance-weighted average point grade does not have a variance.

Reverse Auction for Process Equipment – Hopefully a Thing of The Past

Do you remember the times when some customers actually thought that reverse auctions where the thing to do for just about anything?  It probably started out pretty well for items such as commodities.  I used to be in utter disbelieve when a few customers started to also include capital equipment.  It ended up as a lot of unpaid consulting on our part since part if not all of the system design information would almost always end up on the web. 

More confounding was trying to figure out the question what if anything the customer was supposed to get out of this deal.  Sure, the thought of a vendor having to compete with his own bid must have been appealing to buyers and accountants.  A paper towel is pretty much a paper towel, but a bulk conveying system is just too specialized to be able to obtain comparable quotes.  Almost every system even including primary air movers is the culmination of years worth of detail laden OEM manufacturer experience.  Without proper communication the chance of getting such an endeavor successfully up and running were slim to none.  I am glad that these auctions are mostly history.  How about you? 

How to lie with geostatistics

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 19 to 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!

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