Alfred Inselberg, the inventor of parallel coordinates, sent along this fascinating handout with a bunch of color graphs illustrating the power of the parallel-coordinates idea.
Here’s a cool picture, along with Inselberg’s caption:
In the background is a dataset with 32 variables and 2 categories. On the left is the plot of the first two variables in the original order, on the right are the best two variables after classification. The algorithms discovers the best 9 variables (features) needed to describe the classification rule, with 4% error, and orders them according to their predictive power.
A couple more below:
In 3-D a surface s is represented by two linked planar regions s¯123 , s¯231′ . They consist of the pairs of points representing all its tangent planes. In N-dimensions a hypersurface is represented by (N −1) regions as the hypercube above.
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Developable surfaces are represented by curves. Note the two dualities cusp ↔ inflection point and bitangent (tangent at two points) plane ↔ crossing point. Three such curves represent the corresponding hypersurface in 4-D and so on.
Cool stuff. He has a book on this coming out in the fall.
Andrew,
One thing I've come to realize after reading many of your papers and blog postings, is that while you seem to pay incredible attention to graphs, I can rarely understand them. You love plotting tons of information in a panel of graphs, and invent new graph types to illustrate your points in clever ways. But I always give up before I can understand them. I'm sure they all illustrate your point beautifully provided I spend the effort to try to understand them. But isn't great writing easy on the reader, and shouldn't great graphs be the same?
This post is an example: if it isn't obvious what is going on from reading the axes I think the graph basically fails. I'm not sure if this speaks poorly of me as a reader, or of the graph maker? I wonder if other readers have the same problem? Or perhaps you're trying to define new graph types that one day I will be able to instinctively understand.
Best
Joe
Glad to see the signal amidst the noise (for once!)
Joe:
These aren't graphs that I'm making. They're graphs from Al Inselberg's presentation. And they're not supposed to be self-explanatory; he explains them during his talk. My point in posting them here was to show how cool they look; this coolness may be a motivation for people to look into these ideas further.
Regarding your larger point, there is a place for simple graphs and a place for more complicated graphs. In Red State, Blue State, for example, we start with simple pictures and build up to more elaborate analyses and graphs.
"A picture is worth a thousand words" — but how would we say this with a picture? We really need
both.
Here the pictures are almost self-explanatory.
The first picture shows a projection(left) from 32-D to 2-D where the two data categories are mixed, and the two categories separated(right).
Third picture shows a developable surface with a
line of cusps (left) which is mapped into two
inflection points (one on each curve). The interscection indicates that there is a plane
tangent at TWO lines (bitangent) of the surface. One can see this by squinting a bit. The analogue of such a surface in 4-D is represented
by 3 such curves etc. Such (and more) features may occur in multidimensional data or problems and can now be detected.
Interesting details, i.e. complexity, errors and more are left for the talk. Still I loved Andrew's Red/Blue state example. Any chance of knowing Joe's and JDL's full names?
This looks like a visual/manual Support Vector Machine. (SVM)