Proposed Manuscript Chronology

[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.