Projection, Projection, Projection
So I was a bit curious as to how projections other than PCA might work on the senate data.
Random Projection: So this looks pretty easy, just multiply by a random matrix! But how could that possibly work? In fact it sounds stupid! Well according to some complicated math that I don't really understand: it could actually work pretty well:
2007-20082009
Well it sort of works but not really. There is a price you pay for just being random!
Principal Component Analysis(PCA): So "random projection" is actually a random LINEAR projection and PCA is in some sense the BEST linear projection so we would expect PCA to be better right???
2007-20082009
Yeah, it actually is better. But it also requires much more computation- so there is a trade off there.
Multi-dimensional Scaling(MDS): This is in some sense the BEST NON-LINEAR projection. Unfortunately we can't compute it exactly but MATLAB has a function to approximate it. I actually have no idea how good of a job the function does, but its code is several pages long, so it has to be good ... right?
2007-20082009
CONCLUSIONS?
So what "value-add" do you get from this that you wouldn't get just from following politics? After all some things that jump out in the visualizations are pretty well known:
(1) Senator Collins and Snowe of Maine are the most liberal Republicans.
(2) Ben Nelson of Nebraska is an awfully conservative Democrat.
etc.
But here is one thing that I noticed that I thought was kind of interesting:
Democrats seem to vote together more frequently than Republicans.
This is a bit counter intuitive since Democrats are considered to be more ideologically diverse than Republicans. Perhaps it is just a function of being in the majority: the Majority Leader only schedules votes on bills on which most Democrats agree.
Still it is not the kind of thing that would be easy to notice without quantitative data.
Random Projection: So this looks pretty easy, just multiply by a random matrix! But how could that possibly work? In fact it sounds stupid! Well according to some complicated math that I don't really understand: it could actually work pretty well:
2007-20082009
Well it sort of works but not really. There is a price you pay for just being random!
Principal Component Analysis(PCA): So "random projection" is actually a random LINEAR projection and PCA is in some sense the BEST linear projection so we would expect PCA to be better right???
2007-20082009
Yeah, it actually is better. But it also requires much more computation- so there is a trade off there.
Multi-dimensional Scaling(MDS): This is in some sense the BEST NON-LINEAR projection. Unfortunately we can't compute it exactly but MATLAB has a function to approximate it. I actually have no idea how good of a job the function does, but its code is several pages long, so it has to be good ... right?
2007-20082009
CONCLUSIONS?
So what "value-add" do you get from this that you wouldn't get just from following politics? After all some things that jump out in the visualizations are pretty well known:
(1) Senator Collins and Snowe of Maine are the most liberal Republicans.
(2) Ben Nelson of Nebraska is an awfully conservative Democrat.
etc.
But here is one thing that I noticed that I thought was kind of interesting:
Democrats seem to vote together more frequently than Republicans.
This is a bit counter intuitive since Democrats are considered to be more ideologically diverse than Republicans. Perhaps it is just a function of being in the majority: the Majority Leader only schedules votes on bills on which most Democrats agree.
Still it is not the kind of thing that would be easy to notice without quantitative data.