Equivalence to Numerical Least Squares for a Real Data Set

These simple formulae should of course be equivalent to the errors obtained from a numerical nonlinear least squares analysis. Put another way, such an analysis should have all the shortcomings of the one we performed in the previous section, with respect to the assumptions about the ideal nature of the errors. To demonstrate this, we therefore compared the above formulae with the results of a routine based on the Levenberg-Marquardt method, as described in Press et al. (1992). And to add a dose of ``realism'', we took the times {t<sub>i</sub>} from the Strømgren y observations of the star 4CVn obtained in the 1996 Delta Scuti campaign. In Table 9, we show the ratio of the errors computed with our formulae to those computed with the nonlinear least squares routine. The columns labeled freq, amp, and phase are the relevant ratios of the errors for the frequencies, amplitudes, and phases. The nonlinear analysis involved a simultaneous fit for the 6 largest amplitude frequencies in this data set. This is technically not a completely valid comparison with our formulae, since we considered fits for only one sinusoidal component. Furthermore, our formulae were derived assuming that no other sinusoidal signals were present, which is clearly not the case in this data set. By fitting the 6 largest frequencies simultaneously, we believe that we actually do a better job of removing the different frequencies' effects on each other, and therefore of simulating the conditions under which our expressions for the errors are valid. As can be seen from Table 9, the ratio of the errors calculated in these two ways is fairly close to 1.0, and in fact deviates no more than 12% from this value. First of all, this shows that the coverage in the 1996 campaign was sufficient so that our assumptions about the orthogonality of the sinusoids was valid. Second, it serves as a reminder that the simplified assumptions about the noise which we have made are also being made by the nonlinear least squares procedure. Therefore, the numerically obtained results should not be regarded in any sense as more realistic or ``better''. Thus, the errors computed with either method just provide a lower limit on the size of the true errors. We therefore urge extreme caution in their application and use.

Table: Comparison of analytical and numerical results

Mode Frequency	Ratio of Analytic to Numerical Errors
(c/d)	freq	amp	phase
8.59	0.912	1.001	0.934
7.37	0.887	0.962	0.925
5.05	0.924	0.953	0.924
6.12	0.886	0.951	0.907
5.85	0.916	0.948	0.904
5.53	0.904	0.957	0.919