Chronological Path Stability/Uncertainty Analysis

[Jorge_Stolfi]

Not at all, as i have explained with the multiple examples of paths between cities.  I will use some maps below to illustrate.

Your analogy is somewhat misleading, in a way that may be affecting many of your interpretations.  You say: 

• Kansas City and Springfield are slightly ambiguous, going from OKC -> Springfield -> Kansas City is roughly the same distance as OKC->Kansas City -> Springfield. Its clear they both occupy the same 2 slots, but the relative ordering is uncertain
  

But that is not what you measure when you compare two pages p,q!  In general, none of your curves goes through p or q, or even near them.  What you measure is the distance between the two points on your route that are closest in some sense to p and q.  

The two pages are Abilene and Albuquerque, and the projections on the route are Dallas and Oklahoma; and what you plot is the positions of these two cities along you route.

So you take several trips, starting from San Antonio or Houston, ending at Springfield or St. Louis, but mostly following the same route in between.  For each trip you ask Google Maps which are the best exits for Abilene and Albuquerque, and it always tell you Dallas and Oklahoma.  Depending in where you started, you will plot the two pages at different t values -- those of Dallas and Oklahoma -- but always in the same order and at the same distance apart.  

But that does not imply that Abilene and Albuquerque are close together, or that they are aligned in the N-S direction.  That apparent stability of the order and separation of Abilene and Albuquerque is an artifact of the set of routes you picked, not a property of the two pages.

Let v be the dominating direction of the current fitted curves.   In those plots, the projection of v run mostly left to right.  What would happen to the fitted curves if you stretched the data along a direction u in R^256 that is perpendicular to v?  Wouldn't the new fitted curves run mostly in the direction u? Wouldn't that completely scramble the t order of the sections?

The projections run mostly left to right because I have the same dimension on the X axis in each plot, MDS1. MDS1 captures the most obvious temporal gradient,

Beep! You are assuming the conclusion here...

That is not the temporal gradient.  It is just the direction in R^256 space along which the data has maximum extent,according to whatever criterion MDS uses.  

Basically, [the left-right diection] is because of the way that I've plotted them, it is not an artefact of the method. The line does indeed move in different directions perpendicular to this in certain regions, which is why i included the MDS5 and MDS6 charts.

Sorry, I was not asking for a different plot projection.  I was asking what happens if you stretch the data in R^256 space before applying MDS etc.   

That would change the direction of maximum extent of the data.  The new MDS1 could be almost perpendicular to the old one, leading to completely different ordering of the sections and pages. 

The longest mostly straight road trip between two state capitals in the US would probably be Sacramento to Augusta (Maine).  But if a passing black hole were to stretch the US (not just the map!) in the N-S direction by a factor of 3, the longest mostly straight trip would be something like New Orleans to Minneapolis.  Then the relative order of Denver and Miami, projected onto this new route, would be opposite to the old one...

A vertical white streak cannot exist across all color swabs, and they are not plotting artefacts..

Please look again at You are not allowed to view links. Register or Login to view..  Those white vertical streaks, across each color swab, are real, quite obvious, and obviously a bug.  From what I understand, as the curves vary, the t-value of each page should vary smoothly over its range -- and the t-values of different pages should not change in absolute lockstep.  

The streaks may not be a plotting bug, as I guessed before (they are not aligned from section to section), but they must be the result of a spurious quantization of the t-values occurring somewhere.

I tested both the lower portion [of f84v] and the lower circular together. Including the circular does reduce how much of an outlier this is, but not by enough to resolve it's anomalous status.

By "lower circular" you mean the mini-parag at bottom right, beyond the pool?

Many of the astro/cosmo sections exhibit word construction that seems somewhere in between pharma and Herbal B.

Here is another Free Advice (you can heed it without registering, but only for this month, hurry!) You should consider only text that consists of full parags.  Excluding labels, titles, radial lines, circular text, etc. And mini-parags like those on f84v, f67r2, f68v3.  

Which means excluding half of the Astro/Cosmo pages and all of Zodiac, unfortunately.  But those non-parag texts are expected to be anomalous, and the position of a page could shift a lot simply because it has more or fewer labels compared to normal text.

As i mentioned, the missing bifolia from Quire 8/H would likely greatly have aided in understanding more about this transition if we had them.

They could...  Or they could have broken the current ordering... 

If this were just chol, I would agree with you, but it is not. It is ALL vocabulary. When arranged in the way my chronology , the total vocabulary retained from the start falls gradually each step, such that by the end (bio) almost nothing remains.

Bio is indeed the only section where it would make sense to use page similarity (specifically, lexeme sharing) to infer the correct reading order.   Because it is the only section where one may expect a single continuous discourse that extend across page boundaries.

But, even there, the computed ordering may not be the reading order.  In a novel where the story mostly switches back and forth between two locations with mostly different characters, ordering the pages by lexeme sharing would put all the pages of location A together before those of location B, or the other way around.  

the gradient separates known topical sections.

Beep! I still see no "gradient" between the sections.  

I see 4-5 well-separated clusters with are not linearly arranged in any of the 2D projections I have seen, including those I made 20+ years ago.  

Those clusters can always be ordered by projecting along some axis, straight or curved; but there is no justification for interpreting that order as chronological.

Even because that concept may not be well defined.  For each section, there is the time when the Scribe put it to vellum, and the time when the Author created the draft.  There may be also the times when one or more earlier drafts were written. And, if each section is a transcription/translation/adaptation of an older ur-book by a different Ur-Author, there would be the dates when those ur-books were written.  These various dates may well be in different orders...

All the best, --stolfi