Chronological Path Stability/Uncertainty Analysis

[npcompl33t]

Indeed I am confused.  If each Golf sub-section is treated (or behaves) as a single object, why are the lines in H-1.5a staggered rather than aligned, as in H-1a and H-1b?

And what about those "anomalous" lines, like the one at the bottom of H-3b?

excellent questions. 1.5a is staggered because its pages more consistently fall into sequence across all of the model runs. In a sequence, the most easily identifiable ordered segments are those in the center, because they contain many overlapping sections before and after. That is why the 1.5a and 2a are so clearly offset. 1.5a in particular appears to be a period of high change, connecting 1a/1b to 2a, which is why i have named it 1.5a / Transitional Herbal A.


The anomalous lines are indeed anomalous. specifically one that jumps forward is f86v4. Looking at the MDS plot here shows what is happening -- this page is on the far end of all the other 3b sections, much closer to the 2a. Below i've highlighted it on the MDS chart.

   


It is extremely anomalous, appearing very far offset from its bifolia, the largest offset by far, across any bifolia, as shown below.

   


I absolutely think this needs more research to untangle f86v4, it could be key to discovering the ordering of these sections.

Your data is in R^256, but I think that this model in R^2 could reproduce your plot, including your "move together" information:

Your Golf sub-sections are the magenta smudges.  In this sketch, I tried to account for the following points:

1. The fact that all pages of each sub-section plot at about the same t value on each curve says that the diameter of each cloud (say, the RMS of the JS distance between pages in that cloud) is small compared to the overall span of the curve.
   

Not quite. It means that those two sections are estimated to fall within the same t slot, and are probably adjacent in the time. It doesn't mean that the differences are small, just that the ordering is ambiguous depending on how you want to draw the line. They can still have substantial distance between them, they are still placed sequentially, just in some cases the line goes left to 3a first and in others the line goes right to 3b first. 

Another example, Houston TX, Austin TX, and Dallas TX all form a triangle. if you start in a random city, and drive to another random city, then drive to the last, and only record your distances, it will be hard to determine the exact order without knowing which city you started in. Houston -> austin ->dallas is almost the same set of distances as houston->dallas->austin. IF you drive to 4 cities, and the 4th is Oklahoma city, i can infer dallas is probably the 3rd city, because it lies halfway in between austin and houston and oklahoma city. However it is still ambiguous if you start the path in austin or in houston.  

I show exactly how this is occuring with the Stars/Bio ordering iYou are not allowed to view links. Register or Login to view.

I tried to place each cloud so that its projection onto the curve (by Euclidean distance on that plot) is at the mean t value shown in your plot.  

I just finished up work adding this as a published view to my website for the clusters, i will share it below.

1. The distance of each cloud to the curve is very roughly proporional to the spread of its t values on the plot, assuming that most of this spread is due to change in the slope of the curve rather than the curve itself moving in the general t direction.
   

No, the spread of the cloud has nothing to do with the distance from each to the curve.  In my city example above, austin and houston would both have overlapping clouds. Some simulations have you starting in austin, some in houston.

1. The cloud pairs 1a+1b, 3c+4c, 3a+3b, 5a+6 were drawn as single clouds because they cannot be split based on the plot.  Maybe, as you say, they can be split in other dimensions; but those dimensions apparently do not affect their projection on any of the curves.
   

No, it has to do with the ambiguity as in my city example, austin and houston would be drawn as a single cloud pair. You can still clearly split one apart from the other in geographic space.

If the curves were straight line segments, the t value of the projection of a page would vary smoothly (indeed, linearly)  with the parameters of the curve.  But if the curves are not straight, the nearest t value may suffer a large jump for small changes in those parameters.  That is my theory for why the plot of 4a clusters at two distinct t values, ~0.67 and ~0.87.  In my sketch, as the curve changes from C1 to C2, at some moment the nearest points for cloud 4a jump from the right arm of C1 (at t = ~0.87) to the left arm of C2 (at t = ~0.67) without going through intermediate values.

This is only ture in the non ambiguous section. The distances between dallas and oklahoma city would be like 1.5a and 2a on the chart -- there is really only one ordering that makes sense with them, so they fall into distinct bands with linear increasing t bands. Ambiguous sections do not. The point of this analysis was to identify which sections are ambiguous, rather than simply saying the path is houston->austin->dallas->OKC then calling it a day. This analysis shows that really houston->austin or austin->houston are both valid paths.

[*]The anomalous page at the end of your section 3b woud be an outlier (not shown), located near 3b in my sketch but further away from the curve.

This is accurate, You are not allowed to view links. Register or Login to view. is quite far from the curve, the farthest such outlier by far. Its assignment to Hb is listed as tentative, the MDS plots above i think show why it was placed there and why it is ambiguous.