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

[Jorge_Stolfi]

(in my opinion, the talk about much later painters, and thus irrelevant colors, is pure speculation).

But the claim that colors are later than the binding has at least two crumbs of evidence: 

• the green pigment over the page number on f42r, attested by Rene and others from microscope examinatons;
  
• the blue paint spots at the upper left corner of f40r, that look like they are continuous with the blue "hearts" of You are not allowed to view links. Register or Login to view. but are in fact not aligned and do not match their color.
  

Granted, both of these bits of evidence can be disputed.  But the opposite theory -- that the color paints are original -- has no evidence at all, not even historical precedent.

Note that this "color paints are original" is actually two claims:

C1. The color paints were applied while the VMS was still in possession of the Author, and under his supervision.

C2. The Scribe applied colored paints to each page (or bifolio) as soon as he finished drawing the outlines, before proceeding to the nexp page (bifolio).

C1 would be sufficient for one to consider colors when trying to identify the plants.   But even if C1 is assumed, C2 seems rather unlikely.  But one needs C2 in order to use colors as the clue for the order of creation of the bifolios.

An intermediate hypothesis would be 

C1.5.The Scribe created the bifolios in batches.  For each batch, the Scribe drew the outlines of figures and text on all pages, and then he applied color paints to all those pages.

With C1.5 one could use colors at least to guess the order of creation among sections.

All the best, --stolfi

[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.

[npcompl33t]

You are not allowed to view links. Register or Login to view.. You can play around with the sorting mechanisms, and view the individual page/bifolia level variation across all the runs from the above charts.

I believe this ordering is more robust than my original chronology, and it better communicates the uncertainty and ambiguity around specific page/bifolia ordering. 

I hope this level of testing and information around the stability helps to improve confidence in the overall broad chronology, while explicitly showing sections that exhibit more ambiguity or uncertainty.

[npcompl33t]

@jorge_stolfi -- this should help clear things up. This plots all of the ensemble lines at once. You can see they aren't randomly dispersed, they cluster around a few equilibria. Overlapping segments on the chart above mean that there are two possible equilibriums, most lines route through any one area in just 1 or 2 ways, despite individual random variance in the actual paths.

   

   


PS: As a data person that loves charts,  i think these are some really cool charts. And I think there is actually a ton of information in these the other charts missed, with respect to stable ordeerings. i think i probably need to spend some time studying these.


EDIT: Replaced charts with one that shows order as a color scale so you can see the directionality.

[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:
You are not allowed to view links. Register or Login to view.
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

[npcompl33t]

@Jorge_Stolfi I'm going to try to respond to all of your points and questions, but let me know if I miss anything. I appreciate your continued dialog about this.

First, a small correction, you refer to the above charts as showing 'dozens' of curves, they show 1,600 curves, so several orders of magnitude off.

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.

This is exactly correct, I'm glad the plots helped here. Collapsing to two curves F1 and F2 may be a bit of an oversimplification but I think for the purposes of our discussion it is fine. Just to be clear though, the saved fits contain combinations of:

• 1a before/after 1b
  
• bio 4a before/after stars 4c
  
• Herbal B 3a before/after Herbal B 3b
  

So really it can't be reduced to just two fixed global sequences.

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.

Your first example is correct: when H-1a projects around t-0.05, H-1b projects around t-2.05, and for the F2 curves it is the reverse. Neither F1 nor F2 shuffles H-1a and H-1b together. So F1 or F2 curve places 1a and 1b as occupying the same position.

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.

This is reasonable for locally straight projections away from endpoints. However,  projections onto a curved or endpoint limited path can complicate the straight-line approximation. Many projections land at curve bends, where there is no single tangent. That limits how literally we can apply the straight-line approximation.

For individual pages, within a single fit, the typical central 80% of positions spans approximately 10.5% of the curve for 1a and 11.0% for 1b. For pooled bifolia, it is lower, between 6-8% roughly.

1.5a is not unique elongated, its typical within-fit width is about 7% for pages and 6% for bifolia.

Ultimately the above plots only capture the 1st 3 dimensions of a 256d space, so its natural they don't show all or even the majority of the distances. I'll share the MDS5 plots below, as they capture another dimension specifically useful for evaluating the end orderings.

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".

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.    )

I specifically tested this, removing the 3 largest word types on an already text-light page only removes a few % points of the divergence. A handful of repeated words does not explain most of its difference. The differences come from different word construction, specifically use of the bigram ol (apart from the single ol words, removing those does not change this), along with eo, initial ch, and o-t. 

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. 

Topic can produce distinct clusters. What it cannot produce is a gradient. For example, there is no reason that arranging topics in the order i've listed would result in chol's counts gradually decreasing at each step. On a broad scale, the exact amount of 1a vocabulary retained from step to step declines gradually at each step. Topic cannot explain this. Style can, if it is a function of chronology or time.

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.

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.

These were all checked, and not every page was placed into a section. That is why there is an 'unclassified' label on the initial charts i shared. Each page was placed into a group only if it made sense, the relative certainty was established for every individual assignment. It was placed in 3b, because that is legitimately its best match, in the MDS plots i shared it genuinely looks like it is a part of that grouping.

Again, you are also only viewing the top 3 dimensions in a 256d space, you do not have the information to make that call from just the above plots alone. I've shared the full 256d MDS space if you want to take a deeper look. 

Ignoring pages that seem to cause contradictions is exactly what can lead to warped perspectives about the MS as a whole, I do not think sweeping away the difficult pages under the rug will lead to a better understanding of the MS.

[npcompl33t]

Here are the MDS1 vs 5 and MDS1 vs 6 plots. These show how the ambiguity related to the ending sections arises. They almost form a coherent circle of points. The exact ordering of these segments depends on where you enter the circle. This makes sense, if the points all form a ring, you can be reasonably sure of the ordering but starting at different points on the circle results in different sets of orders.

If Phileas Fogg began his journey in New York as opposed to London, suddenly it is on opposite sides of the path from San Francisco, even though the two are spatially adjacent. This ordering swap does not mean that there is no difference between San Francisco, New York, and London. It does not imply any kind of inconsistency with respect to there being a path connecting all of them. 


   

   

[Jorge_Stolfi]

@Jorge_Stolfi I'm going to try to respond to all of your points and questions, but let me know if I miss anything.

Thanks for the extensive reply and the new graphs!

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.

Just to be clear though, the saved fits contain combinations of:

• 1a before/after 1b
  
• bio 4a before/after stars 4c
  
• Herbal B 3a before/after Herbal B 3b
  

So really it can't be reduced to just two fixed global sequences.

So you mean that there are three or more families of curves, such that the curves of different families make different combinations of choices at each of the "river bifurcations" visible on those curve plots?

The existence of multiple families of fitted curves, that yield different t orders for the pages and sections, is an argument against your thesis -- that the chronological order of writing (COW) can be deduced from the oredering of the points in R^256 space.  Right?  

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 happento 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?

That brings up some moe fundamental questions I have about the general methodology and the Jensen-ShennongShannon distance.  Will ask them later.

when H-1a projects around t-0.05, H-1b projects around t-2.05, and for the F2 curves it is the reverse.

Ah, OK, so H-1a and H-1b do seem to be well-separated clouds in R^256 space.

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?) ...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 individual pages, within a single fit, the typical central 80% of positions spans approximately 10.5% of the curve for 1a and 11.0% for 1b. For pooled bifolia, it is lower, between 6-8% roughly.

Sorry, I am confused.  Do you mean that, for a single curve, the projections of the H-1a pages span a range of ~0.10 in t values? Not ~0.02, as I had assumed above?

In You are not allowed to view links. Register or Login to view. there are some white vertical streaks across all color swabs.  I had assumed that they were due to the use of a small number of curves.  That assumption influenced my misunderstanding that the t projections of the H-1a pages, for each single curve, spanned only ±0.01.  But you say you had thousands of curves, and now I see that those white bands are equally spaced in t.  So now I guess that they are plotting artifacts, perhaps due to something as banal as writing the data out to some intermediate file with too few decimal digits.  Is that so?

1.5a is not unique elongated, its typical within-fit width is about 7% for pages and 6% for bifolia.

All other color swabs in that plot are roughly rectangular.  With the new information, I interpret this fact a meaning that the t-ordering of the pages of those sections is drastically different for different curves, even within the same family.   Which would mean that the order of the projections of those pages along a direction u in R^256 space changes drastically even with small changes in u.  Which seems to imply that the pages of each section are not mostly aligned along some direction v, but are scattered randomly in a roundish cloud.  No?

But H-15a seems to be different: its swab is a parallelogram instead of a rectangle.  So I suppose that not only its pages project in the same order along any curve, but in fact that order is highly correlated with the order of those pages in your data file.  Which order is this?

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.  ... The page is an outlier only because of all those "aa". ... If you want to understand its puzzle, you should identify the anomalous digraphs, then look for the words that contain them.

I specifically tested this, removing the 3 largest word types on an already text-light page only removes a few % points of the divergence. A handful of repeated words does not explain most of its difference. The differences come from different word construction, specifically use of the bigram ol (apart from the single ol words, removing those does not change this), along with eo, initial ch, and o-t.

Is that page still an outlier if you delete all spaces (points, dashes, commas)?  The Scribe seems to have been inconsistent with spaces around some elements, such as ol, al, or, ar.

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.

[f84v] was placed in H-3b, because that is legitimately its best match, in the MDS plots i shared it genuinely looks like it is a part of that grouping.

It may be the best match, but it is still a very poor match.

But if the anomalies of You are not allowed to view links. Register or Login to view. are so scattered, then I retract my idea of an "aardword" effect and the proposal to move it to Bio.  Instead, I propose that You are not allowed to view links. Register or Login to view. is not really part of Bio, although it may be related to it (like an epilogue about the history of the section). It is a section by itself, another "unknown" section, like the next page f85r1.  The illustrations logically belonged to the previous page, but did not fit there so the Scribe spilled them over to the unrelated page You are not allowed to view links. Register or Login to view. (something that happens all the time on books and papers even today).

It may also be that only the lower half of You are not allowed to view links. Register or Login to view. is anomalous and is not part of Bio proper (like the lower half of You are not allowed to view links. Register or Login to view. does not seem to be part of the Starred Parags section proper).  That half seems to be a single long parag, twice as long as the average parags of other Bio pages.

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.

Topic can produce distinct clusters. What it cannot produce is a gradient. For example, there is no reason that arranging topics in the order i've listed would result in chol's counts gradually decreasing at each step.

If you listed the pages in order of mean t-value along some curve, and you fitted that curve to a bunch of statistics, and those statistics happen to be correlated with the frequency of chol, then the frequency of chol will tend to vary gradually along that ordering.  

It is again like sorting books by height, and observing that the number of words per page (along with many other book properties) varies gradually along the resulting ordering.

On a broad scale, the exact amount of 1a vocabulary retained from step to step declines gradually at each step. Topic cannot explain this. Style can, if it is a function of chronology or time.

But, again, what page ordering are you assuming when you say "step to step"?  The order of mean t-position along the fitted curves?

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. ...  including those pages in the analysis will only distract and confuse.

Ignoring pages that seem to cause contradictions is exactly what can lead to warped perspectives about the MS as a whole, I do not think sweeping away the difficult pages under the rug will lead to a better understanding of the MS.

Well, I think that in fact it is generally better to ignore outliers until one understand the inliers.

• If an outlier was generated by the same process that generated the inliers (like sampling from a distribution with a fat tail), ignoring the outlier will not make much difference to understanding the process;
  
• But if an outlier was generated by a glitch or external interference in the process (like the Author accidentally including a love letter in his draft), including it in the analysis will only distract, distort the fitted curves and PCA directions, create bogus correlations, etc.
  

All the best, --stolfi

[npcompl33t]

The existence of multiple families of fitted curves, that yield different t orders for the pages and sections, is an argument against your thesis -- that the chronological order of writing (COW) can be deduced from the oredering of the points in R^256 space.  Right?  

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

   

In the above example, lets say we are taking a road trip, with a general NE direction, and can only measure the distance we travel and the distance between each point. We can be relatively sure that:

• Austin and Houston are both at the beginning of the trip, but Austin->Houston->Dallas is roughly identical to Houston->Austin->Dallas. We can't say for sure which the path stars in, but its clear it must start in one of these and the other must be 2nd.
  
• Dallas and Oklahoma City are adjacent points on the path, occupying positions 2 and 3
  
• 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
  
• We can be certain that St Luis is the final point, the most NE.
  

We can be relatively certain as to the structure of the path while still being uncertain about relative order of certain pieces. Rather than present a single path and claim it is the only option, this analysis gives us a way to understand where those ambiguities fall. It is fully in agreement with my overall theory, and I am being extremely transparent about which specific components my methodology cannot resolve the ordering of. 

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 happento 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, so viewing it this way makes it easy to consistently follow the paths. Basically, this 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. None of this 'scrambles' the order. The only uncertainty is injected when relative distances between certain points in certain regions create equally viable paths, as in my city example. 

 Do you mean that, for a single curve, the projections of the H-1a pages span a range of ~0.10 in t values? Not ~0.02, as I had assumed above?

yes, this is correct.  

In You are not allowed to view links. Register or Login to view. there are some white vertical streaks across all color swabs.  I had assumed that they were due to the use of a small number of curves.  That assumption influenced my misunderstanding that the t projections of the H-1a pages, for each single curve, spanned only ±0.01.  But you say you had thousands of curves, and now I see that those white bands are equally spaced in t.  So now I guess that they are plotting artifacts, perhaps due to something as banal as writing the data out to some intermediate file with too few decimal digits.  Is that so?

A vertical white streak cannot exist across all color swabs, and they are not plotting artefacts. This chart is similar in theory to a bifurcation diagram. A vertical white streak means that whatever page does occupy that, is preferred across all of the curve variants. For whatever reason, the paths all exhibit some sort of stability or attractor in that area. Thus these are the most confidently assigned positions. In my map /city example, Dallas -> OKC conceptually could resemble the 1.5a and 2a sections. Regardless of the different Austin/Houston and later Kansas City orders, all paths converge and agree on a particular stretch. So the vertical white bands mean that there is mostly agreement across all generated curves that whatever color is present is in the correct

All other color swabs in that plot are roughly rectangular.  With the new information, I interpret this fact a meaning that the t-ordering of the pages of those sections is drastically different for different curves, even within the same family.   Which would mean that the order of the projections of those pages along a direction u in R^256 space changes drastically even with small changes in u.  Which seems to imply that the pages of each section are not mostly aligned along some direction v, but are scattered randomly in a roundish cloud.  No?

No, this again can be seen in my city example. Consider Houston and Austin 1a and 1b. Regardless of which comes first, they both occupy the exact same region of t on the curve. Neither can appear after Dallas. So their possible positions on t are identical, and their distribution appears rectangular.

But H-15a seems to be different: its swab is a parallelogram instead of a rectangle.  So I suppose that not only its pages project in the same order along any curve, but in fact that order is highly correlated with the order of those pages in your data file.  Which order is this?

This is similar to the stretch between dallas and OKC. If you consider each page to be a small city in between dallas and OKC, virtually every possible path has to pass through these point in the same order. So they end up appearing staggered along the T axis. Does that make sense?

But if the anomalies of You are not allowed to view links. Register or Login to view. are so scattered, then I retract my idea of an "aardword" effect and the proposal to move it to Bio.  Instead, I propose that You are not allowed to view links. Register or Login to view. is not really part of Bio, although it may be related to it (like an epilogue about the history of the section). It is a section by itself, another "unknown" section, like the next page f85r1.  The illustrations logically belonged to the previous page, but did not fit there so the Scribe spilled them over to the unrelated page You are not allowed to view links. Register or Login to view. (something that happens all the time on books and papers even today).

It may also be that only the lower half of You are not allowed to view links. Register or Login to view. is anomalous and is not part of Bio proper (like the lower half of You are not allowed to view links. Register or Login to view. does not seem to be part of the Starred Parags section proper).  That half seems to be a single long parag, twice as long as the average parags of other Bio pages

I tested both the lower portion 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. Many of the astro/cosmo sections exhibit word construction that seems somewhere in between pharma and Herbal B. It's clear that something transformative is happening to the language during this time, I'm not claiming to have a full understanding of what is going on here, your explanation could be correct. 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. 

If you listed the pages in order of mean t-value along some curve, and you fitted that curve to a bunch of statistics, and those statistics happen to be correlated with the frequency of chol, then the frequency of chol will tend to vary gradually along that ordering.  

It is again like sorting books by height, and observing that the number of words per page (along with many other book properties) varies gradually along the resulting 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. 

The book size example completely falls apart here, and the gradient separates known topical sections. Why do some Herbal A pages vastly prefer forms with eo in them? Why does chol drop by over 50% within the same topic? This should be the same topic, same language?