I had this idea a while back, but I only got a chance to implement it the other day.
It is a fairly obvious idea - probably one of the first things you would think of doing. Indeed,
Keith Poole et. al. appear to do something pretty similar.
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http://voteview.com/senate110.htm
But I had to do some manual prepocessing before it was suitable for Matlab.
Yeah, I was pleasantly surprised by just how well basic PCA works on this, it is pretty amazing!
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Like consider one bill being split into two bills, on which everyone would vote the same as they would with the one big bill; you ideally wouldn't really want this change to affect your PCA results, because nobody's ideology really changed, you just changed how you chunked bills together. But in point of fact it would affect the results, wouldn't it?
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i.e. if you had two very tight distinct clusters of senators they should still be there even after the duplication.
So that idea might be useful if a duplicate vote can cause the political visualization to change DRAMATICALLY but I don't think that can happen.
This is a lot easier to see with something like MDS which directly uses a metric because duplicating a vote obviously doesn't change the distance between two senators so the result doesn't change. PCA seems a bit harder to think about (but a lot easier to implement!)
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