Saturday, August 15, 2009

Degrees of freedom fighters struck at Stanford

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

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

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

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

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

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

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

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