I know that this question is not well defined, but some clusters tend to be elliptical or lie in lower dimensional space whilst the other have nonlinear shapes (in 2D or 3D examples).
Is there any measure of nonlinearity (or "shape") of clusters?
Note that in 2D and 3D space, it is not a problem to see the shape of any cluster, but in higher dimensional spaces it is problem to say something about shape. In particular, are there any measures of how convex cluster is?
I was inspired for this question by many other clustering questions where people talk about clusters but nobody is able to see them (in higher dimensional spaces). Moreover, I know that there are some measures of nonlinearity for 2D curves.
Best Answer
I like Gaussian Mixture models (GMM's).
One of their features is that, in probit domain, they act like piecewise interpolators. One implication of this is that they can act like a replacement basis, a universal approximator. This means that for non-gaussian distributions, like lognormal, weibull, or crazier non-analytic ones, as long as some criteria are met - the GMM can approximate the distribution.
So if you know the parameters of the AICc or BIC optimal approximation using GMM then you can project that to smaller dimensions. You can rotate it, and look at the principal axes of the components of the approximating GMM.
The consequence would be an informative and visually accessible way to look at the most important parts of higher dimensional data using our 3d-viewing visual perception.
EDIT: (sure thing, whuber)
There are several ways to look at the shape.
EDIT:
What does shape mean? They say specificity is the soul of all good communication. What do you mean about "measure"?
Ideas about what it can mean:
Most of the "several ways" are some variation on these.