Proposed Manuscript Chronology

[Jorge_Stolfi]

Another example, here is a 3D plot of the You are not allowed to view links. Register or Login to view.. 
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You can see the horseshoe appearing. This means there is a dominant sequential ordering underlying the data. This sequence is not directly captured by any MDS dimension. It can be recovered though, which is what the plot on the right is.

That is a good example ... of hallucination.

There is no "gradient" or "sequential ordering" or "horseshoe" in fig1.  There are two widely separated clouds of points.  Each cloud has a funny shape which does not seem to be quite well modeled by a with a multivariate Gaussian distribution. Each cloud is fairly elongated and slightly bent.  But the main axes of the two clouds do not seem to be aligned.  

You can take that whole dataset as a single cloud, compute some major axis of variation (by PCA or whatever) then sort the points along that axis.  But that sorting will not mean anything besides the fact that the point were sorted along that axis. You can fit a parabola to those points, using the projection on that axis as independent variable; but that curve does not mean anything.  The ordering is not in the data; it is an imposed and arbitrary assumption. 

The principal direction itself is not a meaningful property of the data, because it can drastically change if the data coordinates are redefined, even by a linear transformation.

All the best, --stolfi