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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?

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    $\begingroup$ To my knowledge, there are two families of distributions related to your description: 1. Elliptical distributions and 2. Spherical distributions. $\endgroup$ – user10525 Nov 20 '12 at 13:29
  • $\begingroup$ 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. $\endgroup$ – A. Donda Nov 20 '12 at 13:41
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The answer to the first question was given by Procrastinator in a comment: The family is called Elliptical Distributions. The standard textbook reference seems to be

Fang, K., Kotz, S., Ng, K.W., 1990. Symmetric Multivariate and Related Distributions. Chapman and Hall.

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:

Hartikainen, A., Oja, H., 2006. On some parametric, nonparametric and semiparametric discrimination rules, in: Data Depth: Robust Multivariate Analysis, Computational Geometry, and Applications. American Mathematical Society, pp. 61–70.

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