"Surely, first generation geostatisticians can neither accept nor admit that each distance-weighted average-cum-kriged estimate has its own variance. Would future generations want to be wrong?"
Even if the weighted average has its own variance it has no meaning.
For two reasons:
First. In kriging methodology we are looking for the error variance equal to variance of the true value minus a weighted average. We are not looking for the variance of a single weighted average.To deal with this problem, we need to introduce a random model.
"In mathematical statistics, one-to-one correspondence between weighted averages and variances is sine qua non."
Second. Let us consider the simplest case.
The random variable V with finite (n = 2) set of its possible outcome values v_1 and v_2 with probabilities (respectively) p_1 and p_2 then:
The mean (weighted average) is
E{V} = p_1 * v_1 + p_2 * v_2
the variance is
D^2{V} = p_1 * ( v_1 )^2 + p_2 * ( v_2 )^2 - ( E{V} )^2
Any additional variance (degrees of freedom ) for this weighted average
has (have) no meaning.
Similarly, kriging is based on stationary random process model, however the classic kriging weights are not probabilities but try to play the role of a joint probabilities that depend only on distance beetwen the random variables, the input values are seen as possible outcome values of a random variable.
Any additional variance (degrees of freedom) for such distance-weighted average has (have) no meaning.
Best regards
tom andrew suslo
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