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Dear Dr van den Boogaart, To each its own variance may seem a bit much but one-to-one correspondence between functionally dependent values and variance is inviolable in mathematical statistics. Just the same, Ive reformulated the proposition to recognize your criticism. Heres a more carefully crafted TRUE or FALSE proposition: A functionally dependent value of two or more measured values ( ) has a variance. You may wish to fill in the blanks between the brackets with a statement such as of a stochastic variable, or of a stochastic variable in a sample space or sampling unit, if it would assist you in formulating a mathematically rigorous proposition. You may insert "random" rather than "stochastic" without confusing me. You need not insert with variable weights because that matter came up on ai-geostats late last year. Heres a link to the formula whose numerator was approved but whose denominator did raise some geostatistical eyebrows simply because degrees of freedom become positive irrationals when weights are variable. http://ai-geostats.jrc.it/documents/JW_Merks/Lost%20Variance.pdf
Perhaps you should not insert determined in samples selected at positions with different coordinates in a sample space because that proposition applies only to the variance of the distance-weighted average. You ask me why the variance question is so important. Thats a very good question indeed and once more heres why! The problem is that geostatistics violates the requirement of functional independence and ignores the concept of degrees of freedom in mathematical statistics. Therefore, the VARIANCE of the distance-weighted average cannot possibly be replaced with the PSEUDO VARIANCE of a SET of degrees-of-freedom and variance-deprived functionally dependent weighted averages. After all, it causes kriging covariances to rise and kriging variances to fall, a phenomenon that caused Armstrong and Champigny to caution against oversmoothing. Thats why the variance question is so important! http://www.geostatscam.com/Adobe/CBull98903.pfd
By the way, it was not me but you who brought up Cauchy in your email of July 20, 2006, under 1) on the fifth page of your 12-page response. What a pity you couldnt find time to count functionally dependent and independent holes in Fig 203 on page 286 of Davids seminal textbook. I guess the first textbook on geostatistics has become somewhat dated with so many new textbooks flooding the market. Kind regards, Jan W Merks |
