Comments on: What Do You Mean By Close?

By: pat

pat — Mon, 18 May 2009 22:10:20 +0000

Yes… I accidentally did set up the integral incorrectly at first to answer the random-point to random-point question. It does work out to be the distance between centroids. As such, it’s not terribly tough to calculate. But, I’m liking the idea of toying with non-symmetric distance.

It can’t be uphill both ways….

By: John Armstrong

John Armstrong — Mon, 18 May 2009 21:58:01 +0000

If you’re allowing random points in the starting state, why restrict to the nearest point in the ending state? Why not take the average distance from a random point in the starting state to a random point in the ending state?

This has two advantages. First, the resulting function is symmetric. Second, the integrals should be easier to set up and work out.

I’m not sure, but a first guess would be that this works out to be controlled by the distances between the centroids. Obvious problems with that would be extremely nonconvex states, but I don’t think we have any that are that bad.

By: pat

pat — Mon, 18 May 2009 21:35:14 +0000

That’d be my guess, too. When I started trying to write down an integral for this notion of closeness, I accidentally started with the symmetric version you mention here.

I haven’t done much with metrics and measure in a long time and nothing beyond undergraduate-level metrics at all. I am interested in toying with them now though to see what breaks if any of the requirements are violated. I know some of the things that go wrong if the triangle inequality doesn’t hold. But, I never messed with symmetry before.

By: Michael Lugo

Michael Lugo — Mon, 18 May 2009 21:12:01 +0000

That’s an interesting way of reinterpreting my question. One way to do something like your distance, but symmetric, would be to say that the distance between state A and state B is the average distance between a random point in state A and a random point in state B.

In this case, determining the answer to the question “which two states are closest together?” is much more difficult. But it’s clear that one wants two small states which border each other; my guess is Connecticut-Rhode Island.