# 114 how small is small when looking at w n just

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11.4. How Small is “Small”? When looking at w n , just exactly how small is “small”? 11.4.1. Strictly Speaking. . . Strictly speaking, this is governed by numerical machine accuracy. NR suggests measuring this by the ratio of the largest w n to the smallest. This ratio is the condition number . Single precision 32-bit floats have accuracy 10 - 6 , so if the condition number is larger than 10 6 you certainly have problems. 11.4.2. Practically Speaking. . . Practically speaking it’s not machine accuracy that counts. Rather, it’s the accuracy of your data and the ability of those uncertain data to define the parameters. In any real problem, plot the vector w . If the values span a large range, then the data with small w n might not be defining the associated vectors well enough. If so, then practically speaking, these vectors are degenerate. How to tell what constitutes “small w n ”? As far as I know, and as far NR says, it’s an art.

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– 43 – 11.5. Doing SVD in IDL 11.5.1. IDL’s SVD routine la svd IDL’S la_svd procedure 6 provides the SVD decomposition. Thus for equation 11.4, the IDL SVD decomposition is given by la svd , X , w , U , V ; (11.9) the notation is identical to that in equation 11.4. 11.5.2. My routine lsfit svd Implementing a least-squares SVD fit requires the ability to modify the weights. I’ve written an IDL routine lsfit_svd that makes the above process of dealing with the weights easy. When used without additional inputs, it returns the standard least-squares results such as the derived coefficients and the covariance matrix; it also returns [ w ], U , and V . It allows you to input those matrices and, also, the bracketleftbig 1 w bracketrightbig matrix so that you can tailor the weights to your heart’s content. 12. REJECTING BAD DATAPOINTS II: STETSON’S METHOD PLUS CHAUVENET’S CRITERION Chauvenet’s criterion is an on-off deal: either you include the datapoint or you don’t. This makes sense from a philosophical point of view: either a datapoint is good or not, so you should either include it or exclude it. However, when doing a nonlinear fit this presents a problem. As you iterate, the solution changes, and a given datapoint can change from being “bad” to “good”. Or vice-versa. You can imagine being in a situation in which the iteration oscillates between two solutions, one including a particular datapoint and the other excluding it; the solution never converges, it just keeps chasing its tail. Enter Stetson’s beautiful technique 7 . Stetson reasons that we shouldn’t have an on-off criterion. Rather, it should relieve a datapoint of its influence adiabatically: as its residual gets larger and 6 Old versions of IDL (before 5.6) had a significantly worse algorithm called svdc . Don’t use this unless you have to. 7 Stetson is one of those anomalies, a true expert on fitting. He invented many of the stellar photometry routines used in daophote , all of which use least-squares techniques. He provides a lively, engaging discussion of many fascinating and instructive aspects in his website: .
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