Chronological Path Stability/Uncertainty Analysis

[Jorge_Stolfi]

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.

In that item I was referring to the pages of one cloud, not to two clouds.

Let's fix on sub-section H-1a,.  For each curve C, all pages p of cloud 1a projected onto the curve at positions t(p,C) very close together around some mean t value tbar©; say 0.05±0.01. (Isn't that so?) (Except for an outlier about 1/3 the way down the list; let's ignore that).  We can replace that curve C by a straight line segment which is tangent to the curve at that mean value tbar©,.  The projections of each page p on that line segment will be practically the same,and thus all within 0.05±0.01.

The fact that all pages p have projections tbar(p,C) = ~0.05±0.01 says that the extent of the cloud H-1a along the direction of that axis is about ±0.01 times some length L that is approximately the length of the whole curve.

For the plot in question you used dozens of curves; that is, you projected each page on dozens of different axes.  For every curve C'  (IIUC) the projections t(p,C') of all pages p were close together, only around some different value tbar(C',1a) that varied between ~0.00 and ~0.20 depending on C'.  Which means that, along all those directions in R^256, the extent of the cloud H-1a is equally small, namely about ±0.01 times L.

That is why I drew H-1a as a small smudge on my sketch.  Maybe H-1a is much longer in other directions of R^256, but that elongation does not show up in your plots.

And the same applies  to every other sub-section except 1.5a.  The fact that, for each curve C, the pages of the sub-section S project to a narrow range of t values tbar(C,S)±0.01 says that the extent of that sub-section S, along any of the tangent directions of those curves, is  ±0.01 times L.

Sub-section H-1.5a is a bit different, because for each curve C the projections t(p,C) of the points in that cloud are spread over a longer range, say tbar(C,1.5a)±0.03.  That is why I drew that section as a smudge that is elongated in the direction of the curves.

The distance of each cloud to the curve is very roughly proportional 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.

Ah, thanks.  Your nice plot clarifies things a lot:
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This plot shows that indeed the t positions shift by up to 0.20 depending on the curve C mainly because the curve itself shifts -- not because the direction of projection changes.

But another important information I get from this plot is that the set of all the curves is partitioned into two compact and well-separated families, F1 and F2.  Thus the theory of discontinuous projection that I had guessed to explain the split plot of B-4a and S-4c is unnecessary.  The two parts of each plot are just the t values for the curves of F1 and for those of F2.

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 ...

Do you mean that, for curves from F1, subsection H-1a projects around t = 0.05 and H-1b projects around t = 2.05; while for F2 curves, it is the reverse -- H-1a projects around t = 2.05 and H-1b projects around t = 0.05?

Or is the case that, for F1 curves, both subsections project near  t =0.05, while for F2 curves both project around t = 0.25?

If it is the latter, that is evidence that, in that sub-pace at least, the two sections are practically coincident.

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

You should consider at least possible (if not expected) that some (if not all) of the statistical differences between pages and sections are due to topic, style, etc -- not to time. Then You are not allowed to view links. Register or Login to view. and other "hard to place" pages will be no big mystery. 

Page You are not allowed to view links. Register or Login to view. explains that girls bathing on blue water pools should watch out for aardvarks.  They come out of their burrows (where they hide from aasvogels) on warm afternoons to mate and gorge on aaliis berries.  The text warns that they should not be confused with aardwolves.  It explains that there are four types of aardvark, the blue aardvark, the teal aardvark, the red aardvark, and the pink aardvark (the colors being purely theoretical, not actual).  Each type can be scared away by wearing a shower-cap of the proper color and reciting "by Aaron and Aalar, I command AAALL aardvarks to stay AAAWAY! Aaah! Aaah! Aaah!"   The page is an outlier only because of all those "aa".

Seriously, that page obviously does not belong to Herbal, A or B.  It should be put into Bio, even if it is an outlier there too.

Being an outlier means that it has some rather unusual digraph frequencies.  If you want to understand its puzzle, you should identify the anomalous digraphs, then look for the words that contain them.  You may find that its anomaly is due to a single "unusual' word, or maybe a few such words, that are used quite a number of times on that page, and nowhere else.  (I hereby propose the term "aardword" for that concept.    )

By the way, you should just ignore the pages that cannot be clearly classified into any of the major sections sections by their image contents, including f1r, f57v, f58r, f58v, f76r, f85r1, f85r2, f86v6, f86v5, f86v4, f86v3, You are not allowed to view links. Register or Login to view. from line 31 on, and of course f116v.  I guess that you want to include those pages because you see them as intermediate stages between the main sections.  But you should check that afterwards, if and when you think you figured out the order of the main sections.  Until then, including those pages in the analysis will only distract and confuse.

All the best, --stolfi