11 A detailed investigation of the effects of hole-cutting

The considerations of the previous section indicate the possibility that a selective application of the Mk II and Mk III auto-folders might be beneficial. But the only possible objective justification for using the active hole-cutting process of the Mk III auto-folder selectively is that the transition regions of the selected rotation curves are intrinsically noisier than the transition regions of the non-selected rotation curves. This raises the question of how we can determine which rotation curves have intrinsically noisy transition regions.

   
11.1 A working hypothesis

A consideration of the relative strengths of the B and C peaks in each of the diagrams of Fig. 5 provides prima facie evidence for the idea that C-peak rotation curves have intrinsically noisy transition regions (since hole-cutting has increased the strength of the C peak), whilst B-peak rotation curves have relatively quiet transition regions (since hole-cutting has decreased the strength of the B peak).

Since rotation curves with intrinsically noisy transition regions will necessarily be subject to more inaccurate folding than other rotation curves then, by the observations of Appendix D, it might be expected that C-peak rotation curves will be associated with correspondingly greater changes in $(R_{{\rm min}},R_{{\rm opt}})$ correlations through hole-cutting than B-peak rotation curves.

Generalizing, this leads to the working hypothesis that an analysis of changes in $(R_{{\rm min}},R_{{\rm opt}})$ correlations through the hole-cutting process as a function of $\ln A$ might reveal significant non-uniformities indicating corresponding non-uniformities in the distribution of rotation curves with intrinsically noisy transition regions. This eventuality would then provide the required objective rationale of when to employ hole-cutting as an active component of the folding process, and when not to. The following subsections describe this analysis.

11.2 Plan of analysis

We require a detailed assessment of the effects of the hole-cutting strategy on $(R_{{\rm min}},R_{{\rm opt}})$ correlations as a function of $\ln A$. Since the potential objective is to determine for what values of $\ln A$ the Mk III auto-folder should be used and since, in practice, these values of $\ln A$ will be obtained via the prior use of the Mk II auto-folder (cf. Sect. 9), it follows that the $R_{{\rm min}}$ values used in the proposed analysis should likewise be drawn from Mk II auto-folder solutions.

Comparing the Mk II and Mk III auto-folder solutions of Fig. 5, we were able to identify an approximate partition of the range $2.2 \leq \ln A \leq 6$ in which to study the effects of the hole-cutting strategy and, with a little experimentation, were able to refine this into the four cases, $2.2 < \ln A \leq 4.2$, $4.2 < \ln A \leq 4.5$, $4.5 < \ln A \leq 4.8$ and $4.8 < \ln A \leq 6$.

It is to be emphasized that, although the foregoing ranges were identified by numerical experimentation, the changes in the behaviour of the $(R_{{\rm min}},R_{{\rm opt}})$ correlations between these ranges are so strong, and the sample sizes so large, that the ordinary processes of random statistical fluctuation as a source of the variations can be ruled out with virtual certainty; consequently, considerable reliance can be placed on qualitative deductions made from the results summarized in Table 1 which lists the pre-hole cutting and the post-hole cutting values of the indices of determination, R², for each of the four $\ln A$ ranges:

   

Table 1: Effects of hole-cutting on coefficient of determination

	R2 before	R2 after
$\ln A$ range	hole-cutting	hole-cutting	N
(2.2, 4.2]	$20.7\%$	$26.3\%$	348
(4.2, 4.5]	$~6.4\%$	$34.6\%$	150
(4.5, 4.8]	$~1.9\%$	$~8.4\%$	141
(4.8, 6.0]	$17.8\%$	$40.2\%$	224

It is clear that there are two distinct modes of behaviour for changes in the $(R_{{\rm min}},R_{{\rm opt}})$ correlation through the hole-cutting process: there are very strong changes in $\ln A$ ranges (4.2, 4.5] and (4.8, 6.0], and relatively weak changes in $\ln A$ ranges (2.2, 4.2] and (4.5, 4.8]Recalling our working hypothesis (cf. Sect. 11.1) that significant change in the $(R_{{\rm min}},R_{{\rm opt}})$ correlation is potentially indicative of rotation curves with intrinsically noisy transition regions, then Table 1 provides an objective rationale for employing the Mk III auto-folder on the sub-intervals $4.2 < \ln A \leq 4.5$ and $4.8< \ln A \leq 6.0$, but not on the remaining intervals.