MTH
lsfit_2008.pdf

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errors equation 8.10 is appropriate. Use your head! 8.4. Datapoints with known relative but unknown absolute dispersions Here the σ m are all different. The m th row of the equation-of-condition matrix X and the m th element of the data vector Y get divided by their corresponding σ m . The equation embodied in each row of the matrix equation 2.2 remains unchanged, but the different rows are weighted differently with respect to each other. Consider two measurements with intrinsic measurement uncertainties ( σ 1 , σ 2 ); suppose σ 1 < σ 2 . After being divided by their respective σ m ’s, all of the numbers in row 1 are larger than those in row 2. In all subsequent matrix operations, these larger numbers contribute more to all of the matrix-element products and sums. Thus, the measurement with smaller uncertainty has more influence on the final result, as it should. Suppose that the above two measurements were taken under identical conditions except that

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– 29 – measurement 1 received more integration time than measurement 2; we have σ 1 σ 2 = parenleftBig τ 1 τ 2 parenrightBig - 1 / 2 , so the rows of X χ are weighted as τ 1 / 2 . This means that during the computation of [ α χ ] = X T χ · X χ , the self-products of row 1 are weighted as τ 1 . This means that each datapoint is weighted as τ , which is exactly what you’d expect! Note that this is also exactly the same weighting scheme used in a weighted average, in which the weights are proportional to parenleftBig 1 σ m parenrightBig 2 . We conclude that the weighting scheme of the first two steps in section 8.2 agrees with common sense. Suppose you don’t know the intrinsic measurement dispersion σ m , but you do know the relative dispersion of the various measurements. For example, this would be the case if the datapoints were taken under identical conditions except for integration time; then σ m τ - 1 / 2 . In this case, multiply each row by its weight w 1 σ m and proceed as above. (The factors 1 σ m in the equations of condition become 1 σ 2 m in the normal equations.) 8.5. Persnickety Diatribe on Choosing σ m 8.5.1. Choosing and correcting σ m In the previous section, equation 8.10 taught us that—formally, at least—the variances in the derived fit parameters (or their uncertainties, which are the square roots) depend only on the adopted uncertainties σ m and not on the actual variance of the datapoints . Are you bothered by the fact that the variances of the derived parameters s a are independent of the data residuals? You should be: it is obvious that the residuals should affect s a . Formally, s a depends only on the adopted uncertainties σ m , which are chosen beforehand by you—you’re supposed be such a good experimentalist that you really do know the intrinsic uncertainty in your measured values. Moreover, you are assuming that there are no other sources of uncertainty—such as “cosmic scatter” or an inappropriate model to which you are fitting the data.
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