44 larger its weight gets smaller and smaller with

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– 44 – larger, its weight gets smaller and smaller. With this, in nonlinear fitting all datapoints are always included and their weights automatically adjust as the fit parameters home into their correct values. And you can’t get into the chasing-tail syndrome that can happen with the strict on-off inclusion. 12.1. Stetson’s sliding weight Stetson recommends using a sliding weight. To explain this, we review the ML concept of chi-square fitting. We define chi-square as χ 2 = M - 1 summationdisplay m =0 ( y m a · f ( x m )) 2 σ 2 m , (12.1a) and we minimize χ 2 by setting its derivative with respect to each parameter a n equal to zero: 2 da n = 2 M - 1 summationdisplay m =0 f n ( x m y m σ 2 m . (12.1b) Here Δ y m = ( y m a · f ( x m )). For each coefficient a n setting this to zero gives M - 1 summationdisplay m =0 f n ( x y m σ 2 m = 0 . (12.1c) Now we wish to modify this equation by introducing a weight w ( | Δ y m | ) that makes datapoints with large | Δ y m | contribute less, so it reads like this: M - 1 summationdisplay m =0 w ( | Δ y m | ) f n ( x m y m σ 2 m = 0 . (12.2) It’s clear that we need the following properties for w y m ) : 1. w y m ) = w ( | Δ y m | ) , meaning simply that it should depend on the absolute value of the residual and not bias the solution one way or the other. 2. w ( | Δ y m | ) 1 as | Δ y m |→ 0, meaning that datapoints with small residuals contribute their full weight. 3. w ( | Δ y m | ) 0 as | Δ y m |→∞ , meaning that datapoints with large residuals contribute nothing.
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– 45 – Stetson recommends w ( | Δ y m | ) = 1 1 + parenleftBig | Δ y | ασ parenrightBig β . (12.3) This function w ( | Δ y m | ) has the desired properties. Also, for all β it equals 0.5 for | Δ y m | = ασ . As β →∞ the cutoff gets steeper and steeper, so in this limit it becomes equivalent to a complete cutoff for | Δ y m | > ασ . Stetson recommends α = 2 to 2.5, β = 2 to 4 on the basis of years of experience. Stetson is a true expert and we should take his advice seriously; he provides a vibrant discussion to justify these choices in real life, including an interesting set of numerical experiments. However, for large M I see a problem with the choice α = 2 to 2.5. For large β , for which the cutoff is sharp, it seems to me that the cutoff should duplicate Chauvenet’s criterion. Referring to equation 6.5, this occurs by setting α = 2 erf - 1 parenleftbigg 1 1 2 M parenrightbigg (12.4) and I recommend making this change, at least for problems having reasonably large M ; this makes α larger than Stetson’s choice. I’m more of a purist than Stetson, probably because I’m a radio astronomer and often fit thousands of spectral datapoints that are, indeed, characterized mainly by Gaussian statistics. Stetson is an optical astronomer and probably sees a lot more departures from things like cosmic rays. Nevertheless, in a CCD image with millions of pixels, of which only a fraction are characterized by non-Gaussian problems such as cosmic ray hits, it seems to me only reasonable to increase α above Stetson’s recommended values by using equation 12.4.
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