On the other hand if your data dont follow gaussian

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On the other hand, if your data don’t follow Gaussian statistics as their intrinsic pdf, then you should think twice before using least squares! (Like, maybe you should try the median fitting discussed in § 13.) You may wish to relax Chauvenet’s criterion by increasing the Δ x beyond which you discard points. This is being conservative and, in the presence of some non-Gaussian statistics, not a bad idea. But think about why you are doing this before you do it. Maybe the intrinsic statistics aren’t Gaussian? You should never make Chauvenet’s criterion more stringent by decreasing the Δ x beyond which you discard points. This rule hardly needs elaboration: it means you are discarding datapoints that follow the assumed pdf! Most statistics books (e.g. Taylor, BR) harp on the purity aspect. One extreme: don’t throw out any datum without examining it from all aspects to see if discarding it is justified. The other extreme: apply Chauvenet’s criterion, but do it only once and certainly not repeatedly. Being real-life astronomers, our approach is different. There do exist outliers. They increase the calculated value of σ . When you discard them, you are left with a more nearly perfect approximation to Gaussian statistics and the new σ calculated therefrom will be smaller than when including the outliers. Because the original σ was too large, there may be points that should have been discarded that weren’t. So our approach is: repeatedly apply Chauvenet’s criterion until it converges. If it doesn’t converge, or if it discards an inordinately large number of datapoints, you’ve got real problems and need to look at the situation from a global perspective.
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– 20 – Many observers use the 3 σ criterion: discard any points with residuals exceeding 3 σ . This is definitely not a good idea: the limit 3 σ is Chauvenet’s criterion for M = 185 datapoints. Very often M exceeds this, often by a lot. To apply Chauvenet’s criterion it’s most convenient to calculate the inverse error function. For this, you have two choices. One (for sissies like myself), you can use inverf.pro from my area heiles/idl/gen . But the real he-man will want to learn about using a root-finding algorithm such as Newton’s method (NR § 9.4 and 9.6) together with the error function; both procedures exist in IDL as newton and errorf . You at least ought to skim lightly some of NR’s chapter 9 about root finding, because some day you’ll need it. 7. NONLINEAR LEAST SQUARES The least-squares formulation requires that the data values depend linearly on the unknown coefficients. For example, in equation 0.1, the unknown coefficients A and B enter linearly. Suppose you have a nonlinear dependence, such as wanting to solve for A and B with equations of condition that look like sin( At m ) + Bt m = y m . (7.1) What do you do here? You linearize the process, using the following procedure.
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