# How to use/interpret/visualise a distance matrix with unknown coordinates?

How would you go about using a distance matrix where points have no coordinates, i.e. what useful statistics can be made and how to visualise it in some useful way?

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You'll probably get better answers on the stats stackexchange. – underdark Feb 16 '12 at 11:39
Thanks, I considered that but wanted to try it here first and not duplicate the question. Will wait a little longer and then try the stats stack. – Vladtn Feb 16 '12 at 12:22
Are the coordinates unknown because the destinations are anonymous, or do you know where they are but don't know how to get coords for them? If it's the latter case, a little more explanation might get you traction in the GIS stack. – Scro Feb 16 '12 at 14:06
The destinations are anonymised indeed... – Vladtn Feb 16 '12 at 14:15

One class of solutions uses Multidimensional scaling. This addresses exactly your question: given a set of distances (often obtained among points in a high dimensional space), find an embedding in one, two, or three dimensions that preserves the distances as closely as possible.

This figure is an MDS rendering of distances among all 183 top-ten Hollywood movie stars from 1932 through 2006. "Distances" were based on data about co-starring in movies (but had nothing to do with time or location). Each point represents a star. Especially notable stars are named. Points are connected with a Euclidean minimum spanning tree to highlight close connections. (It was drawn with a GIS, showing how we can apply spatial methods of analysis to non-spatial relationships).

MDS is found in many commercial statistical packages. It is also freely available in add-ons to R.

You could also take any automatic procedure for drawing an abstract graph and use it for this purpose. This is somewhat more specialized than MDS and so is more likely to be found in commercial or research software dedicated to visualization of graphs. I do know that Mathematica provides several graph-embedding methods: see this page for a discussion of some graph drawing algorithms (starting near the middle).

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Thanks for the very useful answer. Could you say where the image comes from? – Vladtn Feb 16 '12 at 17:10
It comes from me :-). It was created five years ago using some home-grown MDS software (I would use an R-based solution today) to determine the point coordinates; ArcView 3 was used to compute the EMST and produce the map. – whuber Feb 16 '12 at 17:13
Awesome! that's what I hoped. So just a quick question: how are the distances computed? In your case co-starting sounds binary, but can accumulate, so am I right to thing that your distance matrix is between actors and that the values is the number of times they acted together? – Vladtn Feb 16 '12 at 17:16
I tried a bunch of distances. Here, "The “distance” between two stars in the map depends primarily on the overlap of their careers: when one career (that is, period on the list) coincides with a portion of another, then the two stars are nearly coincident. No overlap puts them at a distance of one unit apart. Distance is increased by 0.04 units per year of difference in their careers (measured as the mean year in which each star appears) and by 0.001 times the difference in mean ranks on the list. (This latter helps assure that any two stars will be separated, at least by a tiny amount.)" – whuber Feb 16 '12 at 17:19
I have registered just to vote up. Nice job dude. However, some problem lies in this method. The MDS is mathmatically optimal for this problem in the sense of MSE. However sometimes there are very dense clusters in the result which is "right" mathematically but not "proper" for visualization. Do you have any idea on this? – SolessChong Aug 7 '13 at 10:54