## 5 Discussion

Table 3: Empirical quantile angles \(Q_{p}\) for the principal component (PC) and anchored principal component (aPC) for \(d=3\) and various \(n\). aPC) and value of MAD (or aMAD), it is possible to find the \(d\) value using Table 4 and then find an approximately close angle \(Q_{0.95}(PC[d])\) (or \(Q_{0.95}(aPC[d])\)) in Table 2. Thus, one must choose between \(d=5\) or \(d=10\) in order to subsequently detect the corresponding \(Q_{0.99}\) or \(Q_{0.975}\). For cases of very noisy data (that usually correspond to a large MAD and therefore at small values \(n\) in the demagnetization process are often rejected), we add here approximations to the leading order in n (see [5]):  
\[\mathrm{rms(MAD)}\sim\frac{72.5}{\mathrm{d}\sqrt{\mathrm{n}}}\qquad\mathrm{ rms(aMAD)}\sim\frac{44.4}{\mathrm{d}\sqrt{\mathrm{n}}} \tag{5}\]  
where rms angles are expressed in degrees. We have formulated above how to localize the direction of the ancient magnetic field within the intersection of several confidential cones with a common apex and axes that (generally speaking) do not coincide. According to the method, this localization is by nature probabilistic. Comparing the traditional a95 method with the new process described here, we see in Figure 1 that a possible benefit of the new method may be the size of the confidence region. To statistically test this property, we would need a sufficiently large data collection of lava flows, and process it in two ways:  
1. By the traditional a95 method;
2. By a new method proposed here that accounts for the accuracy of individual samples. For comparison, it is necessary to agree on exactly how the size of the confidence region will be estimated from its shape on the sphere of directions. Processing a new large collection of lava flows would be a very significant effort. A statistical comparison of the two methods on original samples of erupted rocks is beyond the scope of this article but will be the content of a future publication. We present here only the basic ideas of a comparative statistical analysis. Area is an obvious parameter for characterizing the size of a region on the surface of a directional sphere. A computational procedure will be required. By the set of confidence cones corresponding to the individual specimens in the sample, one can determine the area corresponding to their intersection on the sphere, this will clarify the compatibility of the specimens in the sample. The numerical implementation of this procedure utilizes a quasi-uniform grid on the surface of the sphere. The appropriate estimation of the confidence domain for the standard a95 approach is straightforward, even analytically. Note, however, that in the standard approach the question of compatibility with the Fisher distribution is usually solved visually, although a formal test exists. Additionally, the traditional method ignores the accuracies of individual directions and, consequently, in practice, it is plausible that a95 might be applied to incompatible specimens. This must be taken into account when comparing the statistical efficiency of the two methods. \begin{table}
\begin{tabular}{c c c c c c} \hline \(n\) & \(C_{MAD}\) & \(C_{aMAD}\) & \(n\) & \(C_{MAD}\) & \(C_{aMAD}\) \\ \hline
3 & 7.69 & 6.00 & 10 & 2.54 & 4.14 \\
4 & 3.90 & 5.00 & 11 & 2.51 & 4.12 \\
5 & 3.18 & 4.63 & 12 & 2.48 & 4.11 \\
6 & 2.88 & 4.43 & 13 & 2.46 & 4.08 \\
7 & 2.71 & 4.31 & 14 & 2.44 & 4.08 \\
8 & 2.63 & 4.24 & 15 & 2.43 & 4.06 \\
9 & 2.57 & 4.18 & 16 & 2.43 & 4.05 \\ \hline \end{tabular}
\end{table} 
Table 4: Recommended \(C_{MAD}\) and \(C_{aMAD}\) scaling factors to convert MAD and aMAD angles into \(\alpha_{95}\) estimates. When comparing the new approach with the traditional one, we must also keep in mind the semantic difference in the meaning of the indicators. While the new method indicates the area where the common direction for all specimens lies, the traditional method a95 indicates only the area where the Fisher distribution axis falls.