How do I determine whether my data is spherically separable? Is there a simple statistical test that I can use to determine whether my data is spherically separable? I am planning to use Kmeans++ to divide 48 dimensional vectors into clusters but I just read that this depends on the assumption that my data is spherically separable…
 A: The two main approaches are:


*

*Visualize (yes, there are methods)

*try clustering and evaluate carefully on your data


Do not rely on any automatic method or statistic.
A: I think the best and easiest thing you can do when you have data is to just implement your model (k-means), train your model, and then validate your model on unseen data.  The validation error tells you how good your model is.  You can safely compare any number of models this way.
Visualization might work for small models, but it's really hard to project the 48-dimensional vectors you have to 2 dimensions and expect to see class separations.  Essentially, your k-means is doing a projection already.
Other answers are pointing out that k-means makes assumptions.  All models make assumptions.  If they make the wrong assumptions, then that will be revealed when you validate.
A: Using this blog post as a reference it appears that it's possible to do better than 'try clustering' and 'visualize': 
1) all variables should have the same variance so I can use Bartlett's test on all variables. 
2) the prior probability for all k clusters are the same (i.e. each cluster has roughly equal number of observations) and this is something I can check as well. 
3) k-means assume the variance of the distribution of each variable is spherical
Now, I'm not sure how to test point 3 which is my question. But, at least these three conditions must hold. So I am not limited to checking whether the variance of the distribution of each variable is spherical. 
