Proposed Manuscript Chronology

[Jorge_Stolfi]

I'll take that bet.  ... For each pair i took the mean JSD minus the average of the internal group spreads to estimate the squared distance between centroids. I then scored all 84 triples on ": Y lies between X and Z when X and Z are farther apart than either is from Y. ... In random controls, 9 unaligned centroids in 8 or 50d never got down to 7 in 2,000.

Could you please give more details on these experiments?  Especially

• Which statistics do you compute for each page, and how many are there?
  
• How do you compute the distances between the centroids?
  
• How did you generate the random clouds?
  

Meanwhile, here is my attempt to reproduce your experiment, without the "intrinsic order"premise.

I created four roundish clouds in R^4, with centroids at the vertices of a regular tetrahedon.  Here are they in two different projections:
       
For each triple of clouds, I computed the ratio S = (dist(X,Y) + dist(Y,Z))/dist(X,Z), where dist is Euclidean distance in R^4 between cloud centroids, and X,Z are the two clouds in the triple with maximum dist(X,Z).  The ratio S can vary between 1 (when X,Y,Z are collinear in R^4) and 2 (when X,Y,Z are equidistant).

The ratio S of course came out the same for all triples, namely 2.0 exactly.

Then I repeated the test with the four centroids picked at random with uniform distribution within a cube of R^4:
   
The ratios S for the four distinct triples were 1.2744, 1.4229, 1.3460, 1.1910.  Are these numbers comparable to those in your "random" test?

Then I repeated the computation with the centroids in a regular tetrahedron, but stretched in an arbitrary direction of R^4 by a factor of 10:
   
The ratios S for the four distinct triples were 1.1810,  1.1013,  1.1013,  1.1810. 

Are those last values close enough to S = 1?

Note that in the second data set there still was no intrinsic order of the centroids.  The clouds are ordered along the direction of stretch, but that order is created by the stretching.  I could have obtained any of the 4!/2 = 3 possible orders by choosing different directions for the stretching.

The stretching is equivalent to picking the centroids in random order but with a strong correlation between their four R^4 data coordinates.  The same is expected to be the case when the points are VMS per-page statistics and the clouds are sections with same topic and text type.  It is undisputed that there are strong correlations between the frequencies of characters and digraphs over the sections.

All the best, --stolfi