Covariance in error analysis.
- From: glhansen@xxxxxxxxxxxxxxxxxxxxx (Gregory L. Hansen)
- Date: Tue, 27 Dec 2005 16:11:48 +0000 (UTC)
Everybody knows the rule, after making the usual assumptions, for finding
the uncertainty of a prediction from measured variables,
s^2 = \sum_ij (@f/@x_i)(@f/@x_j) s_ij^2
where the s_ij are the entries in the covariance matrix, and s_ij=0 for
i!=j when the measured parameters are uncorrelated. And it's common
knowledge that the best estimate given several measurements is a weighted
average,
<x> = (\sum_i x_i/s_ii^2) / (\sum 1/s_ii^2)
And plugging that into the first equation, the uncertainty is
s^2 = 1/(\sum 1/s_ii^2)
But I can't help noticing the lack of off-diagonal terms. What is the
best average when the {x_i} are correlated?
--
"When the fool walks through the street, in his lack of understanding he
calls everything foolish." -- Ecclesiastes 10:3, New American Bible
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