Weighted error

Abandon error weighting entirely and revert to a pair of simple y-on-x least-... 

Beyond error-weighted least-squares isochrons... 

Robust and resistant isochrons can have very different characteristics than traditional least-squares or error-weighted least squares regressions. Some methods ignore analytical errors entirely, others infer them from the observed scatter of the data, and still others make use of analytical errors only to the extent that they are validated by their observed scatter. 

Figure 9. Comparison between performance of error-weighted least-squares isochron (dashed lines) with a robust isochron (solid line) for a dirty data-set at secular equilibrium). Both age and errors are greatly affected by inclusion of the two outliers (stippled error ellipses), whereas the robust isochron (Theil 1950 Vugrinovich 1981) is unchanged. Errors are 95% confidence.

Within each of the two classes in Table V, the first two sets of limits ((a), (b), (d), (e)) use the constant and variable weights, respectively, and assume B and A are exactly known (model-1 in Table IV). The remaining limits involve estimated parameters, based on the design x and the equations of Table III. Method (c) utilizes the parameters of Model-1 and constant weight method (f) uses Model-3 and variable y-errors (weight). 

A general goal is to minimize the sums of the squares of the N error weighted residuals (Titterington and Halliday, 1979 Ludwig, 1998)... 

Ballentine CJ, Hall CM (1999) Determining paleotemperature and other variables by using an error-weighted, non-linear inversion of noble gas concentrations in water. Geochim Cosmochim Acta 63 2315-2336... 

The lead-alpha dates of zircon in the Thiel Mountains Porphyry appeared to be confirmed by Eastin (1970) who obtained a whole-rock Rb-Sr date of 632 102 Ma and an initial Sr/ Sr ratio of 0.7086 0.0059 for specimens of the Porphyry. This date and the initial Sr/ Sr was derived by an error-weighted regression procedure applied to the analytical data in Appendix 8.5.3. The same data were used by Faure et al. (1977) to calculate a date of 660 79 Ma and an initial ratio of 0.7069. The large uncertainty ( 102 x 10 year) of the date calculated by Eastin (1970) arose because the data points do not constrain the slope and intercept of a straight line well enough to calculate a geologically useful crystallization date of the Thiel Mountains Porphyry. 

There is considerable interest in the possibility of absolute type measurements which may be made independently of the weight of the sample. This eradicates a possible source of error (weight) and increases sample throughput. 

We already know from the optimal arrangement of fragments of the density of states by means of the multi-histogram reweighting method, described in Section 4.5.2, that the optimal error weights are given by the reciprocal variance. Here, we have... 

Since the optimized, error-weighted recursion is much more powerful than the standard recursion in that it provides a smoother and faster convergence in the recursive process of estimating the multicanonical weights, it should generally be favored, even more so as the additional implementation effort is minimal. 

Figure 7.32. Electron mobility data and two function fits on log mobility scale. Dotted curve is for unity weighting factor. Solid curve is for relative error weighting factor.

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