Proposed Manuscript Chronology

[Jorge_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.

And I respectfully insist that this does not make sense.  Again, what the plots show is five fairly well-separated roundish clouds of points, originally in a space of dimension d much greater than 4 but projected onto some 2D subspace thereof.  

Again, there will be directions in that d-space where the centroids appear in the order 1-2-3-4-5, but other directions where they appear in the order 3-5-2-4-1 or any other order.  

You may use some criterion like PCA to chose a particular direction in d-space, and that will define an order for the centroids, namely by projection on an axis parallel to that direction.  But that order is not meaningful.   You can see this if you consider a linear transformation of the data that is not an isometry: that can completely change the PCA direction, and hence the apparent order of the clouds.

Once you have picked a direction, you can always fit a quadratic (or higher polynomial) curve to the centroids as a function of position along that direction.  But that "horseshoe" is no more significant than the chosen direction itself.  Again, it depends totally on the coordinate system you use, and on how you pick the direction that determines the independent variable of the fit.

Think of picking any three countries in Europe and plotting the coordinates of their major cities as points on a 2D plot. You get three clouds of points.  If you pick a direction by PCA or any other method, you will get an ordering of those three countries, and a "Guttman horseshoe" from that data.  But obviously that order will not be significant.

All the best, --stolfi