Chronological Path Stability/Uncertainty Analysis

[Jorge_Stolfi]

I think you might be misunderstanding the plots. The plots are showing where certain sections may appear across many runs. They are not saying that these sections are shuffled in together in some runs. On the contrary, the sections are grouped together and the entire section shifts.

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?

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.
   
2. 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.  
   
3. 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.
   
4. 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.
   
5. 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.
   
6. A similar situation may explain why clouds 3c and 4c seem to have complementary ranges of t values.  Specifically, I conjecture that the cloud 3c+4c is elongated in the direction perpendicular to the curve.  Your cloud 3c is the half of that joint cloud that is closer to the curve, and it projects to a continuous range of t values as the curves move.  Your could 4c is the other half of that joint cloud; being farther away from the curves, its t values jump discontinuously between ~0.70 and ~0.90 as the curves move.
   
7. 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.
   

What do you think of this model?

All the best, --stolfi