# Generalization of multivariate normal distribution and classification

I am interested in a family of multivariate distributions that can be seen as a generalization of the multivariate normal distribution, insofar as they are defined by an expectation value $\vec \mu$ and a covariance matrix $\Sigma$, plus a monotonously decreasing function $g(d)$ such that the density is $$p(\vec x) \propto g \left ( \Delta(\vec x, \vec \mu) \right )$$ where $$\Delta(\vec a, \vec b) = \sqrt { (\vec a - \vec b)^T \Sigma^{-1} (\vec a - \vec b) }$$ is the Mahalanobis distance. The multivariate normal is of course recovered by $g(d) = \exp (- \frac12 d^2 )$.

My first question is: What is the name of this family of distributions?

It is simple to show that for classification of a given data point to one of two or more classes, each of which is described by such a density with different $\mu$ but identical $\Sigma$ and $g(d)$, optimal classification boundaries are piecewise linear (hyperplanar).

My second question is: Is this a standard result, and if yes, what is the standard literature (textbook) reference for it?

• To my knowledge, there are two families of distributions related to your description: 1. Elliptical distributions and 2. Spherical distributions.
– user10525
Nov 20, 2012 at 13:29
• Hi @Procrastinator, feels weird to have one's post edited by someone else, but I get your point. – As to your comment, I believe Elliptical distributions is exactly what I mean, and Spherical distributions are a special case. Therefore I think what you wrote is not a comment but an answer. Thanks a lot! – Using the newfound terminology, a search for its use in classification still turned up nothing, so my second question is still open. Nov 20, 2012 at 13:41

Regarding the second question, it appears that most literature on classification either considers multivariate normal distributions or completely nonparametric procedures. I did find one publication though that compares classification algorithms based on different estimators of $\vec \mu$ and $\Sigma$, and does so in the context of elliptical distributions: