Proposed Manuscript Chronology

[pjburkshire]

Novels have reading order.  I don't think of reference books as having a reading order.  Most people don't sit down and read a cookbook.  They go directly to the page with the topic of interest.

[DG97EEB]

Novels have reading order.  I don't think of reference books as having a reading order.  Most people don't sit down and read a cookbook.  They go directly to the page with the topic of interest.

Agreed... CoReMa might be a better source than REN as a result... Thousands of historical recipe texts..

[Rafal]

These are nice results and they agree with other non statistical observations.

For example people observed that nymphs (naked women) in biological sections are better drawn than in Zodiac. Also Zodiac drawing seems to improve from earliest Zodiac signs to latest. It suggests the artist was gaining exeperience and indicates drawing order which may be as well generally creation order.

The results to some point agree with binding order. Although they also seem to indicate that bifolios are very important. If I understood correctly the text over one bifolio is similar to itself and different from other bifolios. People noticed it some time ago and called it BAAFU. Prescott Currier once introduced LAAFU (Line as a functional unit) and here by analogy we have Bifolio as a functional unit.

This is a curious result. In case of Herbal each page is one plant. We have 4 plants on bifolio. Why are descriptions of these plants more similar to each other than to other plants? If we consider some autocitation or constant mutation of the text it could be explained. But I guess it would be hard to explain for natural text.

I believe it would be nice to compare these results to natural texts. Like others noticed, two types of texts should be considered:

- chronological story
- encyclopedia style where each page is separate entry, like a herbal

I wonder if you could restore the text order from such data in both cases and if you would have similar effects like bifolio clusters.

[npcompl33t]

Thank you very much for this thread! As far as I know, Guttman and Kendall were never mentioned in the context of Voynich studies. It’s great to know that Rene’s findings fit into a pattern that also shows up in the archeological data of Flinders Petrie.

I have always been puzzled by that V-shape of PCA1 vs PCA2, but I think I can now partially understand why this happens.

A detail I am not sure about is that the first plot in post #1 shows Biological as the extreme B-side subset, while the last post shows part of Stars as the extreme. I guess this depends on the shape of the green line in the second plot (MDS1, MDS2): that green curve seems to me to be a little weird at both extremes, so I am not fully convinced of the stars-bio-stars chronology.

Yes, it greatly depends on the curve's path. 

Think of it like this: if you did PCA on a spring / slinky / spiral, one dimension , looking at it from the top, would show a circle. the 2nd might show a series of descending lines. A spiral does have a single path connecting all the disjoint points across multiple dimensions, but because PCA is squashing them all down into the ones with the highest separation, no pair of PCA dimensions recovers the path. To recover the path, you must trace a higher dimensional curve through multiple dimensions. Examining the order in the first 2 dimensions alone will not produce the correct order, and in fact, points that appear very close to each other may actually be on opposite ends of the path.

Another example is how dinosaur fossils appear squashed completely flat, making it hard to reconstruct their actual skeleton. Just because two bones are next to each other in the fossil, doesn't mean they were next to each other on the 3d dinosaur.

If you look at the Guttman Horseshoe studies i shared in the OP, they are all discussing this exact confound. Points near each other on the PCA curve may actually be misleadingly far apart, but that distance is captured across several of the higher order dimensions. This is why performing the curve correction is necessary to recover the true sequence.

The green curve is essentially tracing that spiral. It appears odd in the plots, because those plots are still only showing the top down and profile dimensions, making some points appear closer than they actually are, and distoriting the overall sequence. In a sense this is showing the 'squashed dinosaur', and the portions of the curve that appear odd are portions that are moving in directions that are perpendicular to the ones being shown on the 2 MDS dimensions. The green curve is connecting them through the unseen dimensions, but it also still appears squashed on the MDS1 and MDS2 for the same reason.

In this case, the things that separate Stars and Bio, are not particularly things that separate the other sections, which is why that differentiation is not as visible.

EDIT: 
This shows what  i'm talking about. I have isolated the dimensions that appear to separate bio and stars. The green line in these dimensions is the same path shown in the OP. It is still possible that a different path could be drawn that connects the points differently through the higher dimensions. An alternate initialization may connect the points better and adjust the final ordering, swapping stars and Bio. This is just the best curve i was able to draw. But looking at a few dimensions in isolation may warp or skew the order.

   


Additional work can help to narrow down the exact sequencing.  If we can identify the exact mechanical changes that are occurring, perhaps we can reconstruct the specific sequence here with more certainty. Using alternative methods and seeing which sequences result in the most cross-method agreement would also help us understand which sequence is best supported.


EDIT 2: 

Here is an alternate initialization that traces a different path through stars and bio changing the ordering. This fit is *slightly* worse, but only by a few percentage points. It captures 0.1% more pooled bifolio variation, and the proposed curve has about 2% lower held out pooled error. The proposed version fits Stars-S better, while this version fits Bio/Stars T better. I think this shows that this particular ordering sequence is very sensitive and we probably need additional work to fully resolve the exact ordering of these 3 sections.

   


The differences between these two curves imply more than just an alternative ordering though. Note under the proposed, the spread of Stars T and Bio along the curve is much tighter, while in the alternative, they are much more dispersed along the line.

This means the different curves would actually have pretty significant impacts on the mechanical interpretation of these sections:

• Proposed: Bio represents a comparatively stable production phase, with its internal differences mostly unrelated to progression.
  
• Alternative: Bio contains a stronger internal progression that the sequence follows.
  

So the alternative attributes more of Bio’s internal variation to the proposed chronological progression, potentially suggesting a more extended or more internally changing production phase, compared to the proposed curve.

More importantly, the alternative curve dramatically changes the initial sequence, swapping the 2 Herbal A sections. This seemed like a less reliable ordering of the Herbal A sections, which is why i ultimately favored the proposed curve.

[Jorge_Stolfi]

This is what is known as a Guttman horseshoe. It is a You are not allowed to view links. Register or Login to view. You are not allowed to view links. Register or Login to view.that can happen with both PCA and MDS.

This "Guttman effect" is a fancy name for the banal observation that, if you try to fit a simple arc of curve to a bunch of random points in 2 or more dimensions, the result will usually be a curved arc rather than a straight line.

There is no reason to assume that the different sections of the VMS were created by a single process (natural or mechanic) whose parameters varied gradually along a single dimension (like time).  Therefore, there is no justification for trying to fit a single curve to the clouds of points obtained from each section.  They are expected to have different statistics because of different subjects, formats, and styles.   

If cloud B happens to lie somewhat between clouds A and C in a particular projection, it does not follow that it was created after A and before C,or vice-versa.  It does not even follow that its topic is somehow "intermediate" between those of A and C.

All the best, --stolfi

[npcompl33t]

This is what is known as a Guttman horseshoe. It is a You are not allowed to view links. Register or Login to view. You are not allowed to view links. Register or Login to view.that can happen with both PCA and MDS.

This "Guttman effect" is a fancy name for the banal observation that, if you try to fit a simple arc of curve to a bunch of random points in 2 or more dimensions, the result will usually be a curved arc rather than a straight line.

There is no reason to assume that the different sections of the VMS were created by a single process (natural or mechanic) whose parameters varied gradually along a single dimension (like time).  Therefore, there is no justification for trying to fit a single curve to the clouds of points obtained from each section.  They are expected to have different statistics because of different subjects, formats, and styles.   

If cloud B happens to lie somewhat between clouds A and C in a particular projection, it does not follow that it was created after A and before C,or vice-versa.  It does not even follow that its topic is somehow "intermediate" between those of A and C.

All the best, --stolfi

Respectfully, this statement is not accurate. The Guttman effect by definition can only occur when a strong, single underlying linear gradient or sequential ordering is forced into a lower-dimensional space using linear dimension reduction techniques. Its very appearance does in fact mean there is some single underlying linear gradient in the data. You can argue that gradient does not represent time, but you can't claim that such a gradient does not exist. It very clearly does, and is a prequisite for the horseshoe shape to appear.

Similarly if the Guttman effect is present, then fitting a curve is not only justified, it is the documented correct approach to recover the underlying linear gradient or sequential ordering.

[RadioFM]

I think this result was to be expected. Let's see if I'm getting this right:

You're calculating the transition probabilities on a page by page basis. Torsten and others have shown in the past that neighbouring pages read similarly (albeit with some exceptions, likely due to misbinding).

The Jensen-Shannon Divergence as I understand it is symmetrical, so when taking pairwise distances between the probability distributions from each page, you'll have a diagonal full of zeros (each distribution has a distance 0 to itself) and smaller numbers the more similar the distribution is. Therefore, the vector of Jensen-Shannon Divergence of any Herbal A folio will have small numbers (small distances -- similar glyph groupings -- similar glyph transitions) on the coordinates respective to other Herbal A folios, a little bit larger values on Cosmo coordinates (Cosmo reads somewhat similar to Herbal A), then Pharma, and so on in increasing magnitude.

For example you would have the first 3 rows of the pairwise Jensen-Shannon Divergence matrix looking like this (allow me to re-order and group folia):

   Herbal A            ···   Herbal B  ···                     Pharma  ···            Cosmo   ···             Bio ···                   Stars ··· 
[   0    0.05   0.04   ··· 0.29  0.55  0.66  0.77  ···  0.25  0.29  0.23 ··· 0.11  0.18  0.15  ··· 0.80  0.82  0.86 ··· 0.70  0.76 ··· ]
[ 0.05   0      0.02   ··· 0.27  0.53  0.68  0.71  ···  0.29  0.31  0.28 ··· 0.07  0.10  0.08  ··· 0.83  0.83  0.84 ··· 0.74  0.71 ··· ]
[ 0.04  0.2      0      ··· 0.31  0.47  0.63  0.69  ···  0.24  0.26  0.27 ··· 0.09  0.13  0.07  ··· 0.79  0.81  0.85 ··· 0.68  0.77 ··· ]

As you can imagine, in an N~230 (1 per folio?) dimensional space, each folio will have lower values on those coordinates mapping to its own section. Naturally the points cluster for each section with some in-betweens, probably consistent with RZ's extended languages, Torsten's self-citation theory and so on. So it's not surprising to find clusters of sections in some projection, in fact we expect PCA or MDS to do just that as best it can.

The horseshoe curve is IMO the algorithm trying to keep the triangle inequality in the N-space (not of J-S Divergence but of the matrix norm in the MDS), telling you: "Hey, Pharma is different than Herbal A and Bio / Stars, but it's not exactly a smooth segway/intermediate all N=230 dimensions considered, you know?"

I'm not entirely getting the "polynomial correction" step. You can flatten any polynomic function into a line by compositing it with another function afterwards. Are you basically interpolating the 3rd PC dimension with a 4 degree polynomial and then mapping/flattening that to a line? Because in that case I'd argue that'd be skewing the data just to fit a line, not particularly insightful

Your results don't look wrong, btw -- on the contrary, they are consistent with René's, MarcoP's, Torsten's and others'. I just think you're basically retreading the same path, except for the line fit step, which I'm not quite understanding what you did

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PS I got some questions to better understand your methodology:

1. Glyph transition: Are you using a Markov Model (conditioned on the last glyph only?)
   
2. Glyph transition matrix: How do you merge/combine the folio and/or line level transitions into a single transition matrix per folio?
   
3. By "[environmental] gradient" you mean the gradient (1st order derivatives) on the R^N space of the Jensen-Shannon distances, correct?
   

Thanks

[npcompl33t]

I think this result was to be expected. Let's see if I'm getting this right:



The horseshoe curve is IMO the algorithm trying to keep the triangle inequality in the N-space (not of J-S Divergence but of the matrix norm in the MDS), telling you: "Hey, Pharma is different than Herbal A and Bio / Stars, but it's not exactly a smooth segway all N=230 pairwise Jensen-Shannon Divergence distances considered, you know?"

I'm not entirely getting the "polynomial correction" step. You can flatten any polynomic function into a line by compositing it with another function afterwards. Are you basically interpolating the 3rd PC dimension with a 4 degree polynomial and then mapping/flattening that to a line? Because in that case I'd argue that'd be skewing the data just to fit a line, not particularly insightful

Your results don't look wrong, btw -- on the contrary, they're are consistent with René's, MarcoP's, Torsten's and others'. I just think you're basically retreading the same path, except for the line fit step, which I'm not quite understanding what you did

I agree that it is not unusual to find clusters corresponding to questions. This actually asks a different section. Can the relationships between those clusters and individual bifolia be explained by a continuous progression.

Your summary is close, the distance matrix rows are not being used directly as features, it is instead operating directly on the underlying pairwise representations, but otherwise you have the right idea.  Distance metrics, in particular where features tend to arise and then disappear entirely, will cause this type of effect, and is typically referred to as distance saturation. 

The horseshoe also does not automatically arise from having clusters or satisfying a triangle inequality. 

The example is like this, if you were to distribute points along a spiral path, these points will have a sequential ordering. Yet measuring distance along any one axis does not actually recover the correct ordering of these points. Measuring the distance in between the individual points shows that more distance exists than any single axis can explain, and individually you cannot recover the ordering of a points separated on this spiral from pairwise distances between these points. SOme points that sit close to one another, may actually be on different loops of the spiral, considerably separated by the path distance. PCA is going to squash all these distances down to primary axis distances. It would show that two points on different spiral loops are actually closer than their true path distance.

The curve is attempting to find the continuous path through these points, that explains all of the intermediate distances. Reconstructing the full path through the spiral requires flattening it out, and that is what the curve is doing. It is not strictly a polynomial curve, its a parameterized curve that uses cubic splines with eleven knots and a smoothing penalty. The polynomial representation relates to identifying the specific shape of the w pattern in the 1st vs 3rd dimension plot, as opposed to the more common s shape. 

The final straightening step is indeed just a change of coordinates, it is not evidence itself. The evidence is whether the fitted path describes other observations. In this case, the dispersion of the individual pages of the same bifolium is reduced with curve. 

In the spiral example, lets say as you move along the spiral, you create 4 points that are slightly randomly offset from the spiral rather than a single point on the spiral. You would expect a line that minimizes the overall variation of these points to be a better approximation of the full spiral. This is why i reported the fact the curve reduces the bifolio level spread.

This does not necessarily prove that the curve or the MDS1 dimension reflect chronology. However, time is a known mechanism that can result in the horseshoe shape, as with the flinders petrie example. In that example, the raw measurements produce the same horseshoe, and it was proven that by 'flattening the spiral', you can recover the chronology of the pots. This does not prove the voynich horseshoe gradient represents a chronology. It is a theory to explain the gradient, based on the bifolio/production level dispersion of points, combined with the fact time can cause the horseshoe shape. 

Does that make sense?

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Response to your other questions:

1. Yes, first order transition probabilities, conditioned on the preceding glyph. P(a|b), the probability of the next glyph being b, given the current glyph a. This is a first-order Markov representation of the text.
   
2. I don't average the transition matrixes. I first calculate the JS divergence between the full matrixes, then average the distances. Averaging the matrices would blend different observations together obscuring contrast. I did this because i found that Early A, using full lines and word transitions separately resulted in different features being used. After inspecting these features, i realized they were capturing meaningful information the other lacked. The low text per page in Herbal A meant it was critical to capture as much distinguishing information as possible, so I elected to combine them in the above way. Doing so made my Herbal A clusters more stable than using either in isolation. The later sections that have more text per page were largely unchanged across either representation.
   
3. No, gradient here does not mean a mathematical derivative. It means an ordered progression in composition. "environmental gradient" is used in ecological ordination, where this type of pattern often appears. for example, species composition changing along a temperature gradient. "latent transition-profile gradient" would probably be a more accurate term. The glyph-transition profiles change along an inferred direction or curved path. The reconstructed sequence assigns positions along that path using normalized arc length. Interpreting this progression as production chronology is a further hypothesis, nothing in the math itself directly establishes the underlying cause of the gradient is time, that is my hypothesis. 
   

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EDIT 2: Realized i never addressed this, but the horseshoe shape appears when data has one dominant ordering, where items close together on it are similar, and similarity falls off with separation. Noticing it is a legitimate reason to go looking for that ordering / attempting to trace a path. That is the motivation and reasoning for drawing the curve. Its not just randomly tracing curves through the space with no justification. Based on the appearance of the Guttman Effect, there should be an underlying dominant order. What that ordering represents is up for debate, I am proposing time.

[Jorge_Stolfi]

This "Guttman effect" is a fancy name for the banal observation that, if you try to fit a simple arc of curve to a bunch of random points in 2 or more dimensions, the result will usually be a curved arc rather than a straight line.

Respectfully, this statement is not accurate. The Guttman effect by definition can only occur when a strong, single underlying linear gradient or sequential ordering is forced into a lower-dimensional space using linear dimension reduction techniques.

Sorry, I was referring to the "Guttman horseshoe".

Its very appearance does in fact mean there is some single underlying linear gradient in the data. You can argue that gradient does not represent time, but you can't claim that such a gradient does not exist. It very clearly does, and is a prequisite for the horseshoe shape to appear.

I still take issue to that.

I can't tell whether you are referring to statistics in general or to the VMS in particular.  I will assume the later for concreteness.

Those scatterplots do not look like a single snake-like cloud -- which would justify interpreting the data as the result of some stochastic process that depends non-linearly on one parameter that varies continuously over some interval. 

What I see in them is about 5 distinct clouds that look like they are mostly pairwise separable in n-dimensional space.  Some clouds are a bit elongated in those projections, but not in directions that point towards other clouds. 

If they are projected along some particular directions, the clouds may seem to be in a certain linear order.  But that is likely to happen if you take any n+1 clouds of points in d-space, for d >= n, with completely independent d-dimensional random distributions.  Like, if you take four random points in 3-space to be the centroids of four point clouds, for any order of the numbers 1..4 there will be directions in 3-space along which those centroids will seem to occur in that order.  And if you pick the main PCA direction, the deviation of the centroids from a straight line in that direction will be smaller than their extent in that direction.

So the "gradient" in that VMS data, and the ordering of the clouds, is an imposed assumption.  Not a property of that data.

All the best, --stolfi

[npcompl33t]

I can't tell whether you are referring to statistics in general or to the VMS in particular.  I will assume the later for concreteness.

Those scatterplots do not look like a single snake-like cloud -- which would justify interpreting the data as the result of some stochastic process that depends non-linearly on one parameter that varies continuously over some interval. 

What I see in them is about 5 distinct clouds that look like they are mostly pairwise separable in n-dimensional space.  Some clouds are a bit elongated in those projections, but not in directions that point towards other clouds. 

If they are projected along some particular directions, the clouds may seem to be in a certain linear order.  But that is likely to happen if you take any n+1 clouds of points in d-space, for d >= n, with completely independent d-dimensional random distributions.  Like, if you take four random points in 3-space to be the centroids of four point clouds, for any order of the numbers 1..4 there will be directions in 3-space along which those centroids will seem to occur in that order.  And if you pick the main PCA direction, the deviation of the centroids from a straight line in that direction will be smaller than their extent in that direction.

So the "gradient" in that VMS data, and the ordering of the clouds, is an imposed assumption.  Not a property of that data.

All the best, --stolfi

Respectfully, again this is not accurate. if a guttman horseshoe is present, it means there is a dominant, sequential order underlying the data. Guttman Horseshoes appear (not just in the VMS) when a sequential ordering dominates the other features in the data. 

A snake is not necessary in the 3rd dimension. As i state in my OP, A guttman curve can take a more complex shape than an S in MDS3.

It can take a higher order polynomial shape than an S, as is You are not allowed to view links. Register or Login to view. I will paste the relevant figure below.

   

Here , 3 is the snake pattern you are describing. However, a guttman horeshsoe can take a higher polynomial appearance, it does not have to be a snake. Polynomial 4 , the purple line in the above chart, is a w. This same shape IS seen in MDS 3. As seen in the right portion of the next image.

   

This is a textbook guttman horseshoe, through and through. This means there is a dominant sequential order underlying the data. 

You can take issue with my hypothesis that this dominant sequential ordering is time, but a dominant sequential ordering is justified and evident.