Wednesday, May 30, 2012

McGill toils with Markov chains

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

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

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

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

Lost: variance of kriged estimate
Found: zero kriging variance 

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

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

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