Comment by codezero | Hacker News Reader

Comment by codezero | Hacker News Reader

This is interesting because I was thinking about a somewhat related sort of statistical measure (trying to determine average times over repeated measurements of a wall clock) and happened upon this: https://en.wikipedia.org/wiki/Mean_of_circular_quantities

I think this particular trend could benefit from some insights therein, they would help avoid the issue of hovering around the center of the 2d coordinate plane. If you use 3d coordinates on the sphere, your center of gravity will appear on the unit sphere and the radius can be used to determine how strongly it favors that particular area.

As an example of where this could add confusion, if you have a huge economy in the US and a huge economy in China, you basically are canceling out the values with the current axes, if you shifted them, it may change the plots dramatically with the purely 2d representation.

This is essentially what The Economist did. It results in the 'center of mass' being in the northern hemisphere, somewhere near Greenland, because the two biggest economies are both in the hemisphere but at opposite longitudes.

Interesting, do you have any ideas how to apply this to finding the average of multiple points on a sphere?

This is equivalent to what The Economist did. Average of the cartesian points, extend along the radius to the surface.

One way would be to search for points on the sphere which have the closest (weighted) average distance to all the points. Or some similar metric like that.

So instead of finding a point in the middle of the Earth, it would find a point that is geographically closest to the US and China.

Maybe project the x/y coordinates onto a unit sphere, then find the center of mass in spherical coordinates, then convert those spherical coordinates to the relevant 2d coordinate system you're using, discarding the radius (but keeping it for the information it provides).

See also: https://en.wikipedia.org/wiki/Directional_statistics

That's exactly how it's done, for what it worth (though you use 3D cartesian coordinates instead of spherical coordinates).

Treat each point on the sphere as a 3D unit vector, weight as appropriate, and the center of mass is the sum of all of the vectors.

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